FM 5001/5002
Preparation for Financial Mathematics
INSTRUCTOR: SCOT ADAMS
(topics summaries)
Topic 0001
(Foundational Material: Logic and Set Theory)
Link to Topics website
- 0001(1): title slide
- 0001(2): notation: for all, there exists, such that, implies, iff
- 0001(3): transition slide
- 0001(4): limit definition using notation
- 0001(5): QED, box marks end of problem, e.g., i.e.
- 0001(6): def'n: size, cardinality, the empty set, eg size, size of empty set,
- 0001(6): eg: union, intersection, complelement, notation (is an element of)
- 0001(6): notation: \Z:={integers}, \R:={reals}, \Q:={rationals}, \C:={complex numbers}
- 0001(7): eg: disjoint union
- 0001(8): notation: subset, superset
- 0001(9-21): Venn diagrams
- 0001(9-10): union
- 0001(11-12): intersection
- 0001(13-14): if (A \subseteq B), then (A \cap B = A)
- 0001(15): complement
- 0001(16): set-theoretic difference
- 0001(17-21): complement of the union = the intersection of the complements
- 0001(22): def'n: partition
- 0001(22-24): eg/non-eg: partition
- 0001(25): eg: partition
- 0001(26): def'n/notation: ordered pair, contrast with unordered pair
- 0001(26): def'n/notation: Cartesian product, A^n
- 0001(26-27): \R^2 and \R x {3} visualizations
- 0001(28): def'n: Euclidean n-space := \R^n
Topic 0002
(Numbers and sets of numbers)
Link to Topics website
- 0002(1): title slide
- 0002(2): factorial notation and eg
- 0002(3-21): eg/visualization: intervals
- 0002(4-20): SKILL: graph interval
- 0002(5): eg: open, bounded
- 0002(7 and 9): eg: open, unbounded
- 0002(11): eg: compact
- 0002(11): def'n: compact := closed and bounded
- 0002(13 and 15): eg: closed, unbounded, noncompact
- 0002(17): eg: half-open, bounded
- 0002(19): eg: half-open, bounded
- 0002(20): def'n: degenerate
- 0002(20): eg: closed, bounded, compact, degenerate
- 0002(21): eg: neither open nor closed
- 0002(21): eg: both open and closed
- 0002(21): the empty set is not an interval
- 0002(21): SKILL: identify interval (open, closed, half-open, bounded, unbounded, degenerate, nondegenerate)
- 0002(22-24): visualization: \Z
- 0002(24): \Z meets some open intervals, but not others
- 0002(25-26): visualization: \Q
- 0002(26): \Q meets all open intervals
- 0002(26): def'n: dense := meets all open intervals
- 0002(27): def'n: extended real number
- 0002(27): notation: \bar\R := {extended real numbers}
- 0002(27): ordering the extended real numbers
- 0002(28): def'n/eg: extended interval
- 0002(28): def'n/eg: interval
- 0002(29): def'n/eg: lower bound, infimum/inf/greatest lower bound/glb
- 0002(30): def'n/eg: min/minimum
- 0002(31): some subsets of \bar\R have no min
- 0002(32): def'n/eg: upper bound, supremum/sup/least upper bound/lub, max
- 0002(32): some subsets of \bar\R have no max
- 0002(33): eg: sup/max/inf/min
- 0002(34): every subset of \bar\R has an inf and a sup
- 0002(35): def'n/eg: finite union of intervals/fUofI
- 0002(36): def'n/notation/eg: length of an interval
- 0002(37): bounded = finite length, for intervals
- 0002(38): def'n/notation/eg: measure of a fUofI
- 0002(39): Warning: must rewrite the fUofI as a
finite disjoint union of intervals
- 0002(39): SKILL: express a fUofI as a finite disjoint union of intervals
- 0002(39): SKILL: measure of a fUofI
- 0002(40): def'n: partition by intervals
- 0002(41): eg/non-eg: partition by intervals
- 0002(42): def'n: fUofIs = finite unions of intervals
- 0002(42): def'n: partition by fUofIs
- 0002(43): eg: partition by fUofIs
Topic 0003
(Absolute value and distance)
Link to Topics website
- 0003(1): title slide
- 0003(2): def'n/notation: |x| := the absolute value of x
- 0003(3-5): eg: absolute value
- 0003(3-5): SKILL: compute absolute value
- 0003(5): \sqrt{x^2} is |x|, not x
- 0003(6): eg: distance on \R
- 0003(6): distance from a to b is |b-a|, not b-a
- 0003(6): SKILL: compute distance on \R
- 0003(7): (|x-a| \le r) iff (a-r \le x \le a+r)
- 0003(8): (|x-a| < r) iff (a-r < x < a+r)
- 0003(9-12): SKILL: graph an absolute value inequality
- 0003(9-10): graph |x-a| < r
- 0003(11-12): graph |x-a| \le r
- 0003(13-18): SKILL: graph a neighborhood
- 0003(13): graph the 0.5-neighborhood of 1
- 0003(13): graph |x-1| < 0.5
- 0003(14): graph the punctured 0.5-neighborhood of 1
- 0003(14): graph 0 < |x-1| < 0.5
- 0003(15-18): more graphing of neighborhoods and punctured neighborhoods
- 0003(19-20): distance in \R, in \R^2 and in \R^3
- 0003(21): triangle inequality in \R^2
- 0003(22-27): triangle inequality in \R
- 0003(27-30): additivity of error
- 0003(27-30): [ (|x-s| < \sigma) and (|y-t| < \tau) ] implies [(x+y)-(s+t) < \sigma+\tau]
- 0003(31): def'n/notation/eg: x_+ := the positive part of x
- 0003(31): SKILL: compute x_+
- 0003(32): x_+ = (|x|+x)/2
- 0003(32): def'n/notation: x_- := the negative part of x
- 0003(32): x_- = (|x|-x)/2
- 0003(33-34): formula for x_-
- 0003(35): transition slide
- 0003(36): eg: negative part
- 0003(36): SKILL: compute x_-
- 0003(36): x_- \ge 0
- 0003(37): the reproducing equation (x_+ minus x_- is equal to x)
- 0003(37): the absolute value equation (x_+ plus x_- is equal to |x|)
Topic 0004
(Functions and expressions)
Link to Topics website
- 0004(1): title slide
- 0004(2): def'n/eg: function; notation: f:A-->B, f(a)
- 0004(2): def'n: domain, target
- 0004(3): elements of the domain used exactly once
- 0004(4): elements of the target may not be used, or used more than once
- 0004(5-6): eg: the squaring function
- 0004(7-8): G(2,3,4) preferred to G((2,3,4))
- 0004(9): def'n: image, onto, surjective, one-to-one, 1-1, injective
- 0004(9): eg: onto, not 1-1
- 0004(10): eg: 1-1, not onto
- 0004(11): eg: neither 1-1 nor onto
- 0004(12): def'n/eg: bijective = 1-1 and onto
- 0004(13-14): eg: inverses
- 0004(15): def'n: composite
- 0004(16): def'n: identity; redefine inverses
- 0004(17): composition is associative
- 0004(18): composition is not commutative
- 0004(19): x^2, t^2 and the squaring function; notation: evaluation, change in output
- 0004(20): recall inf/infimum/glb, min, sup/supremum/lub, max
- 0004(21): notation: sup_A f = sup_{x\in A} f(x) = etc.
- 0004(22): eg: sup_A f = sup_{x\in A} f(x) = etc.
- 0004(23): notation: inf_A f = inf_{x\in A} f(x) = etc.
- 0004(23): eg: sup_A f = sup_{x\in A} f(x) = etc.
- 0004(24-25): notation: max_A f = etc.; max may not exist
- 0004(26): notation: min_A f = etc.
- 0004(27): The Extreme Value Theorem
- 0004(28-29): def'n: caglad, caglad in x, caglad in t
- 0004(30): def'n/visualization: increasing
- 0004(31): non-visualization: increasing
- 0004(32-33): maximal interval of increase
- 0004(34): def'n/visualization: decreasing, semi-increasing/nondecreasing, semi-decreasing/nonincreasing
- 0004(35): notation: rescaling function
- 0004(36): def'n: scalar := number
- 0004(37): constant = scalar
- 0004(38): notation: addition of functions
- 0004(39): def'n: linear operations, linear combination of functions/expressions, coefficient
- 0004(40): eg: linear combination of expressions of x; domains intersected
- 0004(41): eg: linear combination of expressions of t; domains intersected
- 0004(42): def'n: polynomial in x, degree
- 0004(43): def'n: polynomial in t, degree
- 0004(44): def'n: polynomial in r, degree
- 0004(45): def'n: polynomial, degree
- 0004(46): def'n: constant, linear, quadratic, cubic, quartic, quintic
- 0004(46): SKILL: find the degree of a polynomial
- 0004(47): def'n/eg: constant term, linear term, quadratic term, cubic term, quartic term, quintic term
- 0004(47): SKILL: identify the terms of a polynomial (by degree)
- 0004(48): def'n/eg: constant coefficient, linear coefficient, quadratic coefficient,
cubic coefficient, quartic coefficient, quintic coefficient
- 0004(48): SKILL: identify the coefficients of a polynomial (by degree)
- 0004(49): def'n/eg: leading coefficient
- 0004(49): SKILL: identify the leading coefficient of a polynomial
- 0004(50): def'n/eg: leading term
- 0004(50): SKILL: identify the leading term of a polynomial
- 0004(51): def'n: exponentially bounded / exp-bdd
- 0004(52-53): visualization: exponentially bounded
- 0004(54): eg/non-eg: exp-bdd expressions in x
Topic 0005
(Completing the square)
Link to Topics website
- 0005(1): title slide
- 0005(2): eg/SKILL: collecting like terms; decreasing degree; increasing degree
- 0005(3-6): eg: horizontal translations and replacing x by x-a
- 0005(7): eg: vertical translations and replacing y by y-a
- 0005(8): eg: horizontal dilations and replacing x by x/a
- 0005(9): eg: vertical dilations and replacing y by y/a
- 0005(10-18): eg: eliminate the linear term and graph a quadratic
- 0005(19): eg/SKILL: find the translation that elminates the linear term (in a quadratic)
- 0005(20): transition slide
- 0005(21): eg/SKILL: eliminate the linear term (in a quadratic)
- 0005(22-23): find the translation that elminates the linear term in ax^2 + bx + c
- 0005(24-25): elminate the linear term in ax^2 + bx + c
- 0005(26-30): special case: -(x^2/2) + bx + c
- 0005(26-28): eg: -(x^2/2) + 19x + 5
- 0005(26): replace x by x + (the linear coefficient)
- 0005(29): in general, -(x^2/2) + bx + c
- 0005(29): replace the linear term by (1/2)(linear coefficient)^2
- 0005(30): eg: -(x^2/2) + 8x - 3
Topic 0006
(Combinatorics)
Link to Topics website
- 0006(1): title slide
- 0006(2-6): counting reorderings
- 0006(7-12): counting ways to choose k objects from a set of n
- 0006(7-8): motivation in finance
- 0006(9): notation/eg: "n choose k"
- 0006(12): (5 choose 2) = (5 choose 3); (n choose k) = (n choose n-k)
- 0006(13-14): reordering a partition respecting the comma; counting these reorderings
- 0006(15): each reordering of abcde is (up to removal of a comma) equal to
a reordering of one of the partitions of abcde into: two letters, three letters
- 0006(16-20): listing all the reorderings of abcde by listing partitions, reordering, removing commas
- 0006(21): key point: 5! = (5 choose 2) x (2!) x (3!) = (5 choose 3) x (2!) x (3!)
- 0006(22): (5 choose 2) = [5!] / [(2!)(3!)]
- 0006(23): (n choose k) = [n!] / [(k!)((n-k)!)]
- 0006(24): eg: computation of (100 choose 9)
- 0006(24): SKILL: compute binomial coefficients
- 0006(25): (n choose 1) is a polynomial in n of degree 1
- 0006(26): (n choose 2) is a polynomial in n of degree 2
- 0006(27-28): (n choose 3) is a polynomial in n of degree 3
- 0006(28): SKILL: for fixed d, write (n choose d) as a degree d polynomial in n
Topic 0007
(The binomial formula)
Link to Topics website
- 0007(1): title slide
- 0007(2-3): forumlas for (x+y)^n, with 2^n terms, with duplication
- 0007(4-12): formulas for (x+y)^n, with n+1 terms, without duplication
- 0007(6-8): following the coefficients from n=2 to n=3
- 0007(9): following the monomials in x and y (from x^4 to y^4)
- 0007(9): reorganization of coefficient computation to avoid duplicate writing
- 0007(10-12): rerorganization of coefficients into a triangle
- 0007(12): Pascal's triangle
- 0007(13-19): counting terms of the form x^2 y^3 in the 2^5-term expansion of (x+y)^n, with duplication
- 0007(18): answer is (5 choose 2)
- 0007(19): answer is (5 choose 3)
- 0007(20-22): formula for (x+y)^5 without duplication, using (5 choose k)'s as coefficients
- 0007(21): the binomial formula for (x+y)^5 with (5 choose k) coefficients and k decreasing from 5 to 0
- 0007(22): the binomial formula for (x+y)^5 with (5 choose k) coefficients and k increasing from 0 to 5
- 0007(22): def'n: binomial coefficients
Topic 0008
(Counting monomials)
Link to Topics website
- 0008(1): title slide
- 0008(2): def'n/eg: monomial; eg: degree of a monomial
- 0008(3-9): counting monomials of degree \le d, in n variables
- 0008(3): eg: d=3,n=2; answer is 10; (3+2 choose 2) = (3+2 choose 3) = 10
- 0008(4): eg: d=3,n=4; answer is 35; (3+4 choose 3) = (3+3 choose 4) = 35
- 0008(5-9): eg: d=8,n=4
- 0008(5): answer is difficult; (8+4 choose 4) = 495
- 0008(6-9): counting without listing; 1-1 correspondence; answer is 495
- 0008(10-16): counting monomials of degree = d, in n variables
- 0008(11-15): d=6, n=4; counting without listing; 1-1 correspondence
- 0008(16): answer is (d+n-1 choose d) = (d+n-1 choose n-1)
- 0008(17): transition slide
- 0008(18): a relation between monomial counts
- 0008(19-22): explanation for the relation via Pascal's triangle
Topic 0009
(One variable diff calc)
Link to Topics website
- 0009(1): title slide
- 0009(2): squaring and its derivative,
- as an expression in x, as an expression in t, as a function
- 0009(2): introducing d/dx, d/dt and prime
- 0009(3-4): allowable notation, not allowed, and usually avoided
- 0009(5): cosine and its derivative
- as an expression in x, as an expression in t, as a function
- 0009(6): logarithm and its derivative
- as an expression in x, as an expression in t, as a function
- 0009(7): (d/dx)[f(x)]=f'(x)
- 0009(8-11): the product rule
- 0009(12-13): the quotient rule
- 0009(14-17): the chain rule
- 0009(18-20): practice problem in differentiation
- 0009(21): def'n/eg: logarithmic derivative
- 0009(22): principle of logarithmic differentiation
- 0009(23-25): eg: logarithmic differentiation
Topic 0010
(The IVT and the MVT)
Link to Topics website
- 0010(1): title slide
- 0010(2): def'n: value; Intermediate Value Theorem (IVT)
- 0010(3-7): illustration of the Mean Value Theorem (MVT) by a trip to Chicago
- 0010(8): statement and visualization of MVT
- 0010(9): non-uniqueness of the solution to the MVT equation
- 0010(10-17): applications of the MVT
- 0010(11): the increasing test
- 0010(12): the decreasing test
- 0010(13): the 1-1 test
- 0010(14): the constant test
- 0010(15): equality of derivatives
- 0010(16): the nonincreasing and nondecreasing tests
- 0010(17):stop slide
Topic 0011
(One variable integral calculus review)
Link to Topics website
- 0011(1): title slide
- 0011(2-3): def'n/eg: antiderivative
- 0011(4): def'n/eg: antiderviative wrt (with respect to) x
- 0011(5): d/dx is not 1-1
- 0011(6): equality of derivatives controls the noninjectivity of d/dx
- 0011(6): the set of all antiderivatives of x^2 wrt x
- 0011(7): the (indefinite) integral of x^2 wrt x, notation: \int x^2 dx
- 0011(8-10): (1/3)x^3 + C = (1/3)x^3 + 6C
- 0011(11): indefinite integration and antidifferentiation wrt v
- 0011(12): indefinite integration and antidifferentiation wrt t
- 0011(13): indefinite integration and antidifferentiation wrt s
- 0011(14): indefinite integration and antidifferentiation
- 0011(15-32): area under a curve
- 0011(15): the goal
- 0011(16): three subintervals (3rd partition)
- 0011(17): left endpoint, midpoint, right endpoint for one subinterval
- 0011(18): left endpoints, midpoints, right endpoints
- 0011(19): transition from 3rd to 10th partition
- 0011(20): the 10th partition
- 0011(21): rectangles (left endpoints, 10th partition)
- 0011(22): 3rd partition, right endpoints
- 0011(23): rectangles (right endpoints, 3rd partition); total area
- 0011(23): right 3rd Riemann sum from a to b of f; R_n S_a^b f
- 0011(24): 3rd partition, midpoints
- 0011(25): rectangles (midpoints, 3rd partition); total area
- 0011(25): midpoint 3rd Riemann sum from a to b of f; M_n S_a^b f
- 0011(26): 3rd partition, left endpoints
- 0011(27): rectangles (left endpoints, 3rd partition); total area
- 0011(27): left 3rd Riemann sum from a to b of f; L_n S_a^b f
- 0011(28): rectangles (left endpoints, 10th partition)
- 0011(29): rectangles (left endpoints, 60th partition)
- 0011(30): In limit, as n-->\infty, all three Riemann sums converge to the same value
- 0011(31): def'n/notation: the definite integral from a to b of f(x) wrt x; \int_a^b f(x) dx
- 0011(32): the definite integral represents the area under the curve
- 0011(33): Riemann sums with varying points
- 0011(34-36): Riemann sums with intervals of varying lengths, mesh -->0
Topic 0012
(Example of a definite integral)
Link to Topics website
- 0012(1): title slide
- 0012(2): visualization: area under y=x^2 from x=0 to x=1; f(x):=x^2
- 0012(3): visualization: R_8 S_0^1 f
- 0012(4): computation of R_8 S_0^1 f as a sum
- 0012(5): computation of R_n S_0^1 f as a sum
- 0012(5): computation of \int_0^1 f as a limit of sums
- 0012(6-7): simplification of the sums (via an IOU); computation of the limit
- 0012(8-21): proof of the IOU
- 0012(8): transition slide; def'n: sequence
- 0012(9): def'n/notation/eg: difference operator on a sequence, \triangle a_n
- 0012(10): the difference of n^4; use of Pascal's triangle
- 0012(11): the coefficients come from Pascal's triangle
- 0012(12): the difference of the quartic n^4 is a cubic (in n)
- 0012(13): the difference of s_n := 1+...+n^2
- 0012(14): the difference of s_n, \triangle s_n, is a quadratic in n
- 0012(14): expect s_n to be a cubic in n
- 0012(15): the difference of n^3
- 0012(16): the difference of n^2 and of n
- 0012(17): matching the first two terms in \triangle s_n
- 0012(18): matching the last term in \triangle s_n
- 0012(19): a cubic in n with the same difference as s_n
- 0012(20): the cubic and s_n differ by a constant; the constant is 0
- 0012(21): bringing the cubic to a common denominator
Topic 0013
(The Fundamental Theorem of Calculus)
Link to Topics website
- 0013(1): title slide
- 0013(2): connecting antidifferentiation to area:
The Fundamental Theorem of Calculus (FTC)
- 0013(2): we'll connect change in position (antideriv. of velocity)
to area under the graph of velocity
- 0013(3): problem of going from velocity to position, esp. change in position
specific problem: given velocity v(t)=t^2, compute [p(11)]-[p(5)]
- 0013(3): start by splitting [5,11] into three subintervals
- 0013(4): transition slide
- 0013(5): estimate of [p(11)]-[p(5)] using midpoint velocities
- 0013(6): transition slide
- 0013(7-9): computation of M_3 S_5^{11} v
- 0013(10): [p(11)]-[p(5)] is approximately equal to M_3 S_5^{11} v
- 0013(11): [p(11)]-[p(5)] is approximately equal to M_n S_5^{11} v, and
error --> 0 as n --> \infty, so:
[p(11)]-[p(5)] is equal to \int_5^{11} t^2 dt
- 0013(11): this connects change in position (antideriv. of velocity)
to area under the graph of velocity
- 0013(12): \int_5^{11} t^2 dt is hard to calculate from the definition, but, as
the change in an antiderivative, it's easy: [11^3/3] - [5^3/3]
- 0013(13): transition slide
- 0013(14): key idea: to compute a definite integral, find an antiderivative,
then evaluate at the limits of integration, then subtract
- 0013(15): the FTC, definite integrals version
- 0013(16-17): (d/dx)\int_5^x t^2 dt = [t^2]_{t:\to x}
- 0013(18): the FTC, antiderivatives version
- 0013(19): change of notation; warning: \int_a^x f(x) dx is not acceptable
- 0013(20): recomputation of \int_0^1 x^2 dx via the FTC
Topic 0014
(Techniques of one variable integration)
Link to Topics website
- 0014(1): title slide
- 0014(2-17): integration by substitution
- 0014(2-3): the general rule for indefinite integrals
- 0014(4): an example
- 0014(5): the general rule for definite integrals
- 0014(5-9): replacing x by x + 5 in a definite integral
- 0014(9): dx unchanged, limits of integration decreased by 5
- 0014(10-17): a definite integral of the exponential of a quadratic
- 0014(17): SKILL: integrate exp(quadratic)
- 0014(18-22):
- 0014(18): the general rule (up up minus the integral of new new)
- 0014(19-20): an example
- 0014(21): another example
- 0014(22): redoing the second example by tabular integration
Topic 0015
(Sequences and series)
Link to Topics website
- 0015(1): title slide
- 0015(2): (recall def'n)/eg: (real) sequence
- 0015(3): tails of a sequence
- 0015(4): def'n/eg: liminf of a sequence
- 0015(5): def'n/eg: limsup of a sequence
- 0015(6): limit exists iff liminf = limsup
- 0015(7-8): eg: liminf and limsup
- 0015(8): limsup = sup of limits of subsequences
liminf = inf of limits of subsequences
- 0015(9): def'n/eg: (real) series
- 0015(10): (-1)+(-1)+(-1)+... often written -1-1-1-...
- 0015(11-12): eg: sequence of partial sums
- 0015(13): recall def'n: extended real number
- 0015(13): def'n: converges to an extended real number
- 0015(13): eg: series = extended real number means
the limit of the partial sums is the extended real number
- 0015(14): eg: sums of series; a series may converge or diverge
- 0015(15): it may diverge to \infty or -\infty
- 0015(16): it may have no sum at all, eg: 1-1+1-1+1-1+... has no sum
- 0015(17): 1+(1/2)+(1/3)+... deferred to later
- 0015(18): recall: linear operations
- 0015(18): def'n: infinite linear combination of functions
- 0015(19): def'n: infinite linear combination of expressions of q
- 0015(20): def'n: infinite linear combination of expressions of s
- 0015(21): recall def'n: polynomial in x; def'n: power series in x
- 0015(22): transition slide
- 0015(23): eg: power series in x
- 0015(24): def'n/eg; power series in u
Topic 0016
(Polynomial approximation)
Link to Topics website
- 0016(1): title slide
- 0016(2): def'n/notation/eg: n-jet of f(x) at a, (J^nf)(a) or J^n_af
- 0016(2): (J^4(sin))(\pi/6)
- 0016(3): \tilde f(x)=f(-x) implies (J^n\tilde f)(a)=...
- 0016(4): if f and g agree to order n at 0, then so do \tilde f and \tilde g
- 0016(5): def'n: second order Maclaurin approximation
- 0016(6): formula for second order Maclaurin approximation
- 0016(7): def'n: third order Maclaurin approximation
- 0016(8): formula for third order Maclaurin approximation
- 0016(9): def'n: nth order Maclaurin approximation
- 0016(10): formula for nth order Maclaurin approximation
- 0016(11): Maclaurin expansion
- 0016(12): def'n: Maclaurin expansion
- 0016(13): eg: Maclaurin expansion
- 0016(14): when is a function equal to its Maclaurin expansion?
- 0016(15): recall: decreasing test, nonincreasing test
- 0016(15-16): antidifferentiation of inequalities
- 0016(17-22): eg: antidifferentiation of inequalities
- 0016(20-22): traveling particle problems
- 0016(23-25): error estimate in third order Maclaurin approxmation at 5
- 0016(26): error estimate in nth order Maclaurin approxmation at a \ge 0
- 0016(27): condition under which a function is equal to the sum of its Maclaurin expansion
- 0016(28): proof of the result
- 0016(29): eg: e^9 is the sum of (the Maclaurin expansion of e^x) evaluated at x:-->9
- 0016(30): nothing special about 9
- 0016(31): transition slide
- 0016(32): various expressions are equal to the sum of their Maclaurin expansions
- 0016(e^x, sin x, cos x
- ln(1+x), for x \in (-1,1]
- 0016(33): the error in the 2nd-order Maclaurin approximation is o(x^2),
i.e., tends to zero faster than x^2
- 0016(34): nothing special about 2
- 0016(35): the error in the nth-order Maclaurin approximation is o(x^n)
i.e., tends to zero faster than x^n
Topic 0017
(Conditional convergence of series)
Link to Topics website
- 0017(1): title slide
- 0017(2): recall some sums of series; 1+(1/2)+(1/3)+... still unknown
- 0017(3): the harmonic series is 1+(1/2)+(1/3)+...; goal: lower bound on 31st partial sum
- 0017(4): transition slide
- 0017(5): grouping of terms
- 0017(6): comparision with another sum
- 0017(7-8): the lower sum is 5/2
- 0017(9): 5/2 is a lower bound on the 31st partial sum of the harmonic series
- 0017(10): transition slide
- 0017(11): partial sums are unbounded; harmonic series diverges to \infty
- 0017(12): alternating harmonic series converges to ln 2
- 0017(13): see more terms
- 0017(14-17): a rearrangement (of the alternating harmonic series) converges to (3/2)(ln 2)
- 0017(18-19): the sum of the positive terms is \infty; the sum of the negative terms is -\infty
- 0017(20): transition slide
- 0017(21-24): there's a rearrangement (of the alternating harmonic series) converges to 1000
- 0017(24): there's a rearrangement that converges to any real number
- 0017(25): transition slide
- 0017(26): there's a rearrangement that sums to \infty; there's a rearrangement that sums to -\infty;
there's a rearrangement that has no sum
- 0017(26): this happens for any series whose positive terms sum to \infty
and whose negative terms sum to -\infty
- 0017(27): def'n: nonnegative series
- 0017(28): def'n/eg: generalized partial sum
- 0017(29): the sum of a nonnegative series is the sup of its generalized partial sums
- 0017(30): every nonnegative series has a sum (possibly \infty) and rearrangement doesn't affect it
- 0017(31): recall the positive/negative parts of x; recall the reproducing/absolute value eq'ns
- 0017(32): if the sums of positive and negative parts are both \infty, then rearrangements converge to any number
- 0017(32): if the sums of positive and negative parts are not both \infty,
then the reproducing equation gives the sum of all rearrangements
- 0017(33): summary
- 0017(34-44): similar issues in integration
- 0017(34): \int_{-\infty}^\infty as a limit of \int_{-K}^K
- 0017(35-37): problem with change of variables
- 0017(38-43): problem persists even assuming smoothness
- 0017(44): problem persists even on a bounded interval
Topic 0018
(Some important indeterminate forms)
Link to Topics website
- 0018(1): title slide
- 0018(2): limits of some rational expressions of n
- 0018(3): continuous compounding motivates limit of [(1+(r/n)]^n
- 0018(4-5): expansion of [(1+(r/n)]^n
- 0018(6-7): computation of the limit
- 0018(7): limit of [(1+(r/n)]^n is e^r
- 0018(7): e^r is the risk-free factor
- 0018(8): [1+(7/n)+o(1/n)]^n --> e^7
- 0018(9-15): proof
- 0018(15): [1+(x/n)+o(1/n)]^n --> e^x
- 0018(16): the renormalized powers of 1-x^2 converge to e^{-x^2}
- 0018(17-22): visualization of these renormalized powers,
and of e^{-x^2}
- 0018(23): x:-->3
- 0018(24): if the 2nd order Macl (Maclaurin) approx of f is 1-7x^2,
then the renormalized powers of f(x) converge to a bell curve,
for x=3
- 0018(25-32): proof
- 0018(33-34): another proof using the "o" notation
- 0018(35): if the 2nd order Macl (Maclaurin) approx of f is 1-ax^2,
then the renormalized powers of f(x) converge to a bell curve,
- 0018(35): eg: the renormalized powers of cos x
- 0018(36-40): visualization of the renormalized powers of cos x,
and of their limit
Topic 0019
(Complex numbers)
Link to Topics website
- 0019(1): title slide
- 0019(2): scalar typically means real number, but not in this topic
- 0019(2): def'n/eg: complex (cx) number
def'n/notation/eg: real part, imaginary part
- 0019(2): SKILL: Finding real and imaginary parts of cx numbers
- 0019(3): SKILL: addition of cx numbers; eg: cx addition
- 0019(4-6): geometry of cx addition
- 0019(7): SKILL: absolute value of a cx number; modulus=absolute value
- 0019(7): notation: absolute value; eg: absolute value
- 0019(8): geometry of absolute value
- 0019(9-12): dist(u,v)=|u-v|
- 0019(12): eg: distance
- 0019(13): SKILL: distance between two cx numbers
- 0019(14): triangle inequality for cx numbers
- 0019(15): SKILL: multiplication of cx numbers; eg: cx multiplication
- 0019(16): addition/multiplication is commutative/associative
- 0019(16): multiplication distributes over addition
- 0019(17): def'n/eg/notation: complex conjugate
- 0019(17): SKILL: complex conjugate
- 0019(17): cx conjugation distributes over addition and multiplication
- 0019(18): proof that cx conjugation distributes over multiplication
- 0019(19): |z|=\sqrt{z \bar{z}}
- 0019(20): transition slide
- 0019(21): absolute value distributes over multiplication
- 0019(22): def'n: e^z; (e^z)(e^w)=e^{z+w}
- 0019(23-25): e^{ix} = (cos x) + i (sin x)
- 0019(26): transition slide
- 0019(27): visualization of re^{i\theta}; argument of a complex number
- 0019(28): geometry of cx multiplication
- 0019(29): SKILL: exponentiating cx numbers; eg: exp of a cx number
- 0019(30-34): exponentiation commutes with cx conjugation
Topic 0020
(Topology)
Link to Topics website
- 0020(1): title slide
- 0020(2): def'n/eg: boundary point
- 0020(3-5): eg: boundary of a set
- 0020(5): boundary of product \ne product of boundaries
- 0020(6-7): eg: boundary
- 0020(8-11): def'n/eg/non-eg: open
- 0020(12-17):def'n/eg/non-eg: closed
- 0020(15): most sets are neither open nor closed
- 0020(18): def'n: clopen; the two clopen sets in \R^n;
- 0020(19-22): def'n/eg/non-eg: compact
- 0020(20): intution of compact sets: all homeomorphic images bounded
- 0020(23): SKILLs
- 0020(identify from picture if a set is open, closed, both or neither
- identify from picture if a set is compact
- identify from description if a set is open, closed, both or neither
- identify from description if a set is compact
- 23): are any sets both open and compact?
- 0020(24): def'n: f is supported on C
- 0020(24): def'n: f is compactly suported/has compact support
- 0020(24): visualization when the domain of f is \R
Topic 0021
(Basics of vector spaces)
Link to Topics website
- 0021(1-2): title slides
- 0021(3): def'n/eg: dot product;def'n: vector; SKILL: vector dot product
- 0021(4-5): a game: determining a vector from its dot products
- 0021(6): def'n: linear operations; SKILLS: vector addition, scalar multiplication
- 0021(7): def'n: linear combination, coefficients: SKILL: linear combinations (l.c.)
- 0021(8): def'n/eg: subspace/vector subspace/linear subspace
- 0021(9): def'n/notation: span/linear span
- 0021(10): transition slide
- 0021(11): more notation for span of a finite set of vectors
- 0021(12): eg of a subspace of \R^4 described as a span of three vectors
- 0021(13): one of the vectors is a l.c. of the others
- 0021(14): transition slide
- 0021(15): that vector can be dropped without affecting the span
- 0021(16): def'n: linearly dependent (l.d.), linearly independent (l.i.)
- 0021(17): v not in the span of F implies
linear independence of F unaffected by adding v
- 0021(18-19): l.i. iff (the only l.c. equal to 0 is the trivial one)
- 0021(20): def'n: spanning set, basis
- 0021(21): standard basis of \R^n
Topic 0022
(Basics of linear transformations)
Link to Topics website
- 0022(1): title slide
- 0022(2): linear relationships between variables
- 0022(3): transition slide
- 0022(4): exact vs. approximate linear relationships
- 0022(5): a linear relationship as a function L_0 : \R^4 --> \R^2
- 0022(6): transition slide
- 0022(7): def'n/eg: matrix; dimensions of a matrix; def'n/eg: L_M
- 0022(8-10): dotting the rows of the matrix by the vector input
- 0022(10): SKILL: compute L_M(p)
- 0022(11): def'n: linear map between subspaces
- 0022(12): def'n/notation: linear function/linear transformation corresponding to M
- 0022(12-13): a game: determining the matrix M from outputs of L_M
- 0022(14): the jth column of M is L_M(e_j)
- 0022(14): notation: the matrix [A] of a linear transformation A
- 0022(14): L_{[A]}=A and [L_M]=M
- 0022(15): every linear transformation comes from a matix
- 0022(15): no two matrices give the same linear transformation
- 0022(15): def'n: isomorphism/vector space isomorphism
- 0022(15): isomorphic means "the same to a linear algebraist"
- 0022(16): def'n/notation: kernel and image
- 0022(16): kernels and images are subspaces
- 0022(16): onto iff (image = target)
- 0022(17): transition slide
- 0022(18): 1-1 iff (ker = {0})
- 0022(19): def'n/eg: ordered basis
- 0022(20-24): ordered bases of a subspace V in 1-1 correspondence with
isomorphisms between V and Euclidean space
Topic 0023
(Matrix operations)
Link to Topics website
- 0023(1): title slide
- 0023(2-10): eg: given A and B find C s.t. L_C = L_A \circ L_B
- 0023(11): def'n/notation: product of matrices
- 0023(12): to define AB, need that
the number of columns in A is equal to
the number of rows in B
- 0023(12): matrix multiplication is not commutative
- 0023(13): matrix multiplication is associative
- 0023(14): eg: matrix multiplication
- 0023(15): SKILL: matrix multiplication; eg for discussion: matrix multiplication
- 0023(16): associations between vectors, column vectors and row vectors
- 0023(17): L_M(v) "is" Mv; more precisely:
- 0023(17): the column vector of L_M(v) is
M times the column vector of v
- 0023(18): def'n/eg/notation/SKILL: matrix addition
- 0023(19): def'n/eg/notation/SKILL: direct sum of matrices
- 0023(20): def'n/notation: tensor product of matrices
- 0023(21-23): eg: tensor product of matrices
- 0023(24): the standard basis of R^{2x2}
- 0023(24-30): the reproducing equation
- 0023(31): def'n: the multiplication map \scrM
- 0023(31-37): \scrM( A \otimes B ) = AB
- 0023(38): def'n/eg: zero matrix
- 0023(39): zero matrix is an additive identity
- 0023(40): def'n: the diagonal of a square matrix
- 0023(41): the diagonal of a non-square matrix
- 0023(42): def'n: identity matrix
- 0023(43): identity matrix is a left identity
- 0023(44): identity matrix is a right identity
- 0023(45): def'n: inverse; invertible matricies are square
- 0023(45): a matrix cannot have two inverses
- 0023(46): def'n: left conjugate, right conjugate, conjugate
- 0023(46): SKILL: Matrix conjugation
- 0023(47): def'n: transpose
- 0023(47-50): as you pull M across the dot product, it gets transposed
Topic 0024
(Matrix types)
Link to Topics website
- 0024(1): title slide
- 0024(2): def'n: diagonal matrix
- 0024(3): def'ns:
- upper triangular matrix
- strictly upper triangular matrix
- lower triangular matrix
- strictly lower triangular matrix
- 0024(4): def'n: symmetric matrix
- 0024(5): def'n: anti-symmetric matrix
- 0024(6): def'n: nilpotent matrix
- 0024(6-8): any strictly upper triangular matrix is nilpotent
- 0024(8): any conjugate of a strictly upper triangular matrix is nilpotent
- 0024(9): def'n: scalar matrix
- 0024(10): def'n: standard nilpotent
- 0024(11): def'n: Jordan block
- 0024(11): every 1x1 is a Jordan block
- 0024(12): def'n/notation: exponential (exp) of a matrix
- 0024(12): AB=BA implies e^{A+B} = e^A e^B
- 0024(13): multiplication of diagonal matrices
- 0024(14-17): eg: exp of a diagonal matrix
- 0024(17): diagonal matrices are easier to studay than generic matrices
- 0024(18): reflection in the y-axis in \R^2
- 0024(19): counterclockwise rotation in \R^2
- 0024(20): def'n: orthogonal = distance-preserving
- 0024(21): orthogonal = length-preserving
- 0024(22): orthogonal = dot-product-preserving
- 0024(23): M orthogonal iff M^t M = I
Topic 0025
(Intro to row and col ops)
Link to Topics website
- 0025(1): title slide
- 0025(2-7): solving a system of linear equations
- 0025(8-10): solving the same system using row operations in matrices
- 0025(11): the primary elementary row operations, with examples
- 0025(12): elementary row operations are left multiplications
- 0025(13-15): the secondary elementary row operations, with examples
- 0025(16-20): interchanging two rows via primary row operations
- 0025(21): the primary elementary column operations
- 0025(22): the secondary elementary column operations
- 0025(23-24): elementary column operations are right multiplications
- 0025(25-26): elementary matrices are invertible
- 0025(27): def'n/eg: column-padded identity
- 0025(28): SKILL: row magic
- 0025(29): eg of row magic
- 0025(30): SKILL: row canonical form
- 0025(31-33): eg of row canonical form
- 0025(34): column-padded identity, column magic and column canonical form
- 0025(35): how to go from row canonical form to fully canonical form
- 0025(36): SKILL: fully canonical form
- 0025(37): eg of fully canonical form
- 0025(38): decomposition theorem: any matrix = (prod of elem)(fully canonical)(prod of elem)
- 0025(39-43): eg decomposition theorem
- 0025(44): SKILL: decomposition theorem
- 0025(44): SKILL: list n by k fully canonical matrices; eg: list 4 x 3 fully canonical matrices
Topic 0026
(Row and col ops and linear algebra)
Link to Topics website
- 0026(1): title slide
- 0026(2): onto map {1,2,3} --> {1,2,3,4}? No
- 0026(2): onto map \R^3 --> \R^4? Yes, even continuous
- 0026(2): onto linear map \R^3 --> \R^4? No; start of proof
- 0026(2-4): proof that there's no onto linear map \R^3 --> \R^4
- 0026(5): if there's an onto linear \R^n --> \R^k, then n\ge k
- 0026(6): 1-1 map {1,2,3,4} --> {1,2,3}? No
- 0026(6): 1-1 map \R^4 --> \R^3? Yes, but not continuous
- 0026(6): 1-1 linear map \R^4 --> \R^3? No; similar proof
- 0026(7): if there's an 1-1 linear \R^n --> \R^k, then n\le k
- 0026(7): recall: if there's an onto linear \R^n --> \R^k, then n\ge k
- 0026(7): if there's an isomorphism \R^n --> \R^k, then n = k
- 0026(8): transition slide
- 0026(9): any two bases of a subspace have the same size
- 0026(10): any set of k+1 vectors in \R^k is linearly dependent
- 0026(11): transition slide
- 0026(12): any subspace of \R^k has a basis; start of proof
- 0026(13): end of proof
- 0026(14): def'n/eg: dimension of a subspace of Euclidean space
- 0026(15): SKILL: determine if a vector is in the span of some others
- 0026(15): elementary row operations don't change the row span
- 0026(15-17): eg: determine if a vector is in the span of some others
- 0026(18): SKILL: extract a basis from a spanning set
- 0026(18): algorithm for that skill
- 0026(19): SKILL: determine if a set of vectors is linearly independent, and, if not,
express one of them as a linear combination of the others
- 0026(19): algorithm for that skill
- 0026(20): SKILL: find the kernel and image of a linear transformation
- 0026(20): elementary row operations don't change the row span
- 0026(20): elementary column operations don't change the column span
- 0026(21): eg: find the kernel and image of linear transformations
- 0026(22-23): eg: find the kernel of a linear transformation
- 0026(24): eg: find the image of a linear transformation
- 0026(25): elementary row and column operations change
neither the dimension of the kernel nor the dimension of the image
- 0026(26): dimensions of kernel and image of a fully canonical matrix
- 0026(27): theorem: dim kernel + dim image = dim domain; intuition for this theorem
- 0026(28-30): proof of this theorem
- 0026(31): for square matrices, 11 properties that are equvalent to invertible
- 0026(32): if a matrix has a left and right inverse, then they're equal
- 0026(32): for square matrices, XY = I implies YX = I
- 0026(33): for non-square matrices:
- a short wide matrix times a tall thin matrix can sometimes equal the identity
- a tall thin matrix times a short wide matrix can never equal the identity
- a non-square matrix may have a left inverse
- a non-square matrix may have a right inverse
- a non-square matrix cannot have both at once
- 0026(34-40): Inversion of (square) matrices
- 0026(35): recall that elementary matrices are invertible
- 0026(36-37): eg: finding a left inverse (of a square matrix)
- 0026(38): algorithm for finding left inverses of square matrices
- 0026(39): transition slide
- 0026(40): a left inverse of a square matrix is a two-sided inverse
- 0026(41): SKILL: determine if a matrix is invertible
- 0026(41): SKILL: given an invertible matrix, find its inverse
- 0026(41): SKILL: given a collection of linear systems of equations with constant left side, but varying right side,
find an efficient algorithm for solving them
- 0026(42-44): convert the collection of systems to a collection of matrix equations
- 0026(45): efficient algorithm for solving them
Topic 0027
(Determinants exist)
Link to Topics website
- 0027(1): title slide
- 0027(2): def'n/eg: oriented parallelogram
- 0027(3): transition slide
- 0027(4): visualization of oriented parallelogram
- 0027(5): def'n;eg: oriented 3-parallelpiped
- 0027(5): visualization discussion
- 0027(6): def'n: standard oriented parallelogram
- 0027(6): def'n: standard oriented 3-parallelpiped
- 0027(7): def'n: oriented n-parallelpiped
- 0027(7): def'n: standard oriented n-parallelpiped
- 0027(8): def'n: degenerate oriented n-parallelpiped
- 0027(8): def'n/eg: degenerate oriented parallelograms
- 0027(9): def'n/eg: degenerate oriented 3-parallelpiped
- 0027(10): def'n/non-eg: positive oriented n-parallelpiped
- 0027(11): transition slide
- 0027(12): explanation, via visualization, of why ((1,3),(2,4)) is not positive
- 0027(13): def'n: negative oriented n-parallelpiped
- 0027(13): def'n: signed n-volume of an oriented n-parallelpiped
- 0027(13): def'n/eg: signed area of an oriented parallelogram
- 0027(14): eg: signed area of an oriented parallelogram
- 0027(15): transition slide
- 0027(16): IOU: signed area of ((1,3),(4,2))
- 0027(17): def'n/eg: AP, where A is an nxn matrix and P is an oriented n-parallelpiped
- 0027(18): def'n/eg/notation: "has a determinant"
- 0027(19): eg: determinant; det(AB)=[det(A)][det(B)]
- 0027(19): det(diagonal) = product of diagonal entries
- 0027(20): geometry of shearing
- 0027(21): shearing preserves area
- 0027(22): shearing preserves signed area
- 0027(23): det(shear) = 1; any elementary matrix has a determinant
- 0027(24): any matrix has a determinant
- 0027(25): sv(AP) = [det(A)][sv(P)], for *all* A and P
- 0027(25): vol(AP) = [|det(A)|][vol(P)], for all A and P
- 0027(26-28): calculating the volume of L_A ( [-a,b] x [-c,d] )
- 0027(29): eg: deteterminant of the matrix [ (1,3) (2,4) ] is -10
- 0027(30-31): sv ( (1,3) , (2,4) ) = -10
Topic 0028
(Properties of determinants)
Link to Topics website
- 0028(1): title slide
- 0028(2): transpose does not affect determinant
- 0028(3-8): analysis of the determinant of the direct sum
- 0028(8): the determinant of the direct sum is the product of the determinants
- 0028(9-11): determinant is alternating in columns
- 0028(12-13): determinant is alternating in rows
- 0028(13): if two columns are equal, determinant = 0
- 0028(13): if two rows are equal, determinant = 0
- 0028(14): determinant is additive in the second column (for 2x2 matrices)
- 0028(15-16): determinant is additive in columns
- 0028(17-21): determinant is multilinear in columns
- 0028(22): determinant is additive in rows
- 0028(22): determinant is multilinear in rows
- 0028(23-25): if a row is zero, then the determinant is zero
- 0028(26-28): if a column is zero, then the determinant is zero
- 0028(29-32): formula for the determinant in the 2x2 case
- 0028(33-39): expanding along the first column in the 3x3 case
- 0028(40): formula for the determinant in the 3x3 case
- 0028(41): expanding along the first column in the 4x4 case
- 0028(42): expanding along the first row in the 4x4 case
- 0028(42): description of the formula for the determinant in the 4x4 case
- 0028(43): determiant(upper triangular) = product of diagonal entries
- 0028(43): determiant(lower triangular) = product of diagonal entries
- 0028(44): invertible iff (determinant is nonzero)
- 0028(45-51): a square matrix multiplied by (its transposed cofactor matrix)
is equal to (its determinant times the identity)
- 0028(51): matrix of minors, cofactor matrix, transposed cofactor matrix
- 0028(52): the inverse of a square matrix is
its transposed cofactor matrix divided by its determinant
- 0028(53-54): Cramer's rule for finding one unknown in a system of linear equations
Topic 0029
(Bilinear forms and quadratic forms)
Link to Topics website
- 0029(1): title slide
- 0029(2): def'n/eg: polynomial in x
- 0029(2): def'n: polynomial in x,y
- 0029(3): def'n: polynomial in x,y,z
- 0029(4): def'n: homogeneous polynomial in x,y,z
- 0029(5): eg: homogeneous polynomial in x,y,z
- 0029(5): def'n: homogeneous polynomial \R^3 --> \R^5
- 0029(6): names of variables don't matter
- 0029(7): single-variable polynomial approximation by Maclaurin approximations
- 0029(7): Black-Scholes as a function \R^4 --> \R^2
- 0029(8): describing the Black-Scholes function F : \R^4 --> \R^2
- 0029(9): Approximating G(w,x,y,z) = F(100+w,97+x,0.01+y,0.2+z), for small w,x,y,z
- 0029(10): description of second-order approximation by polynomial in w,x,y,z
- 0029(11): this approximation is called the second order Maclaurin approximation of G
- 0029(11-12): some synonyms:
- constant = degree 0
- linear = degree 1
- quadratic = deggree 2
- cubic = deggree 3
- quartic = deggree 4
- 0029(12): do we need "quadratic algebra", "cubic algebra", "quartic algebra", etc
- 0029(13): no because tensor algebra reduces all of these to linear algebra
- 0029(13): we only need this reduction for quadratic approximations, where it's relatively easy
- 0029(14): start of naïve formulation of quadratic tensor algebra
- 0029(14): reduction to scalar-valued functions
- 0029(14): def'n/eg: quadratic form
- 0029(15): def'n: bilinear form
- 0029(16-17): def'n/notation: the matrix [B] of a bilinear form B
- 0029(18-19): a bilinear form is determined by its matrix
- 0029(20): def'n/eg: symmetric bilinar form
- 0029(20): a bilinear form is symmetric iff its matrix is symmetric
- 0029(21): from bilinear forms to quadratic forms: "restrict to the diagonal"
- 0029(22-25): more than one bilinear form gives rise to the same quadratic form
- 0029(26): square matrices are not in 1-1 correspondence with quadratic forms
- 0029(27): symmetric matrices are in bijective correspondence with quadratic forms
- 0029(27): description of the map in one direction (matrices to forms)
- 0029(28): start of search for the inverse (forms to matrices)
- 0029(29): goal: a procedure for going from quadratic forms to symmetric bilinear forms
- 0029(30): eg: going from a quadratic form to a symmetric matrix
- 0029(31-33): going from that symmetric matrix to a symmetric bilinear form
- 0029(34): recall going directly from a symmetric bilinear form to a quadratic form (restricting to the diagonal
- 0029(34-36): going directly from a quadratic form to a symmetric bilinear form
- 0029(36): def'n: polarization; the polarization formula
- 0029(36): the polarization map (from quadratic forms to symmetric biliear forms)
- 0029(37): def'n/notation: the matrix, Q_M, of a quadratic form
- 0029(37): recall notation for the matrix, [B], of a symmetric bilinear form
- 0029(37): recall notation for the bilinear form, B_M, of a matrix
- 0029(37): recall formula for B_M in terms of L_M
- 0029(37): formula for Q_M in terms of L_M
- 0029(38): multiplication is the polarization of squaring
- 0029(39): dot product is the polarization of length squared
- 0029(40-42): BIG IDEAs:
- polynomial algebra reduces to linear algebra
- everything reduces to polynomial algebra
- ergo: everyting boils down to linear algebra
Topic 0030
(Cauchy-Schwarz)
Link to Topics website
- 0030(1): title slide
- 0030(2-7): how big can v \cdot w be when |v| \le 3 and |w| \le 5
- 0030(4): upper bounds of 17, 377/9, 25
- 0030(4): general upper bounds of [9t+(25/t)]/2, for all t
- 0030(4): want to minimize [9t+(25/t)]/2
- 0030(6): the geometric mean is the minimum of the modified arithmetic means
- 0030(7): minimum upper bound of 15
- 0030(7): Cauchy-Schwarz for dot product
- 0030(8): transition slide
- 0030(9-10): Cauchy-Schwarz for a general positive semidefinite quadratic form
- 0030(10): def'n: positive semidefinite; def'n: positive definite
- 0030(11-12): buzz phrase: the absolute polarization at v,w of a positive semidefinite quadratic form
is less than or equal to
the geometric mean of its values at v and w.
Topic 0031
(Rotations, reflections, orthogonal transformations)
Link to Topics website
- 0031(1): title slide
- 0031(2): recall definition/notation of the length of a vector
- 0031(2): v \cdot v is the length squared of v
- 0031(2): def'n: normal; def'n/notation: orthogonal
- 0031(2): def'n: orthonormal
- 0031(3): def'n/notation: Kronecker delta
- 0031(3): rephrase definition of orthonormal, using the Kronecker delta
- 0031(4): fact: a matrix is orthogonal iff its columns "are" orthonormal
- 0031(5): transition slide
- 0031(6): fact: a matrix is orthogonal iff its rows "are" orthonormal
- 0031(7-13): Gram-Schmidt Orthonormalization
- 0031(7): description of the algorithm
- 0031(8-13): a worked example with three vectors in \R^4
- 0031(14): SKILLS:
- find the length of a vector
- determine if two vectors are orthogonal
- Gram-Schmidt orthonormalization
- 0031(15): fact: [R orthogonal] implies [(det R = 1) or (det R = -1)]
- 0031(15): def'n: rotation; def'n: reflection
- 0031(15): def'n: orthogonal linear transformation
- 0031(15): def'n: rotation linear transformation
- 0031(15): def'n: reflection linear transformation
- 0031(16): any normal vector "is" the first column of some rotation matrix
- 0031(16-17): eg: a certain normal vector in \R^4 "is" the first column of some rotation matrix
- 0031(18): proof that: any normal vector "is" the first column of some rotation matrix
- 0031(19): def'n: unit vector (same as normal vector)
- 0031(19): SKILL: Given a unit vector v, make a rotation matrix whose first column "is" v
- 0031(20): fact: for any two unit vectors v and w, there is a rotation R s.t. L_R(v)=w
- 0031(21): fact: for any two vectors v and w of the same length, there is a rotation R s.t. L_R(v)=w
- 0031(22): def'n: angle between two vectors
- 0031(23): if a = e_1 and b is in the span of e_1,e_2,
then a \cdot b is the cosine of the angle between a and b
- 0031(24): for any two unit vectors a and b, a \cdot b is the cosine of the angle between a and b
- 0031(25): for any two vectors a and b, a \cdot b is the product of three quantities:
- the length of a
- the length of b
- the cosine of the angle between a and b
- 0031(26): summary of all the equivalent definitions of orthogonal
- 0031(26): orthogonal = rotation or reflection; formula for a \cdot b
- 0031(27): restate Cauchy-Schwarz
- 0031(27): def'n/eg: perfectly correlated
- 0031(28): def'n/eg: perfectly anti-correlated
- 0031(29): def'n/eg: positively correlated
- 0031(30): def'n/eg: negatively correlated
- 0031(31): def'n/eg: uncorrelated
- 0031(32-33): motivation for Spectral Theory
Topic 0032
(Eigenvalues and eigenvectors)
Link to Topics website
- 0032(1): title slide
- 0032(2): description of Ramagrog, with Ramatins and Grogali
- 0032(3): transition slide
- 0032(4-6): population after one year
- 0032(6): population development as two scalar equations
- 0032(7-8): population development as one matrix equation, matrix B
- 0032(9): transition slide
- 0032(10): a way of finding the population after 10 years
- 0032(11): big herds and little herds; initial population in herds
- 0032(12): population development in herds
- 0032(13): transition slide
- 0032(14): population after 10 years in herds
- 0032(15): transition slide
- 0032(16): eigenvector/eigenvalue terminology
- 0032(16): initial population as a linear combination of eigenvectors
- 0032(17): the big idea of eigenvectors and eigenvalues: simplifying computations
- 0032(18): transition from herd count to Ramatin Grogali count
- 0032(19): transition from Ramatin Grogali count to herd count
- 0032(20): ease of computation of diagonal matrices
- 0032(21): the population development matrix (called B) is a conjugate of a diagonal matrix D; that is, B = CDC^{-1}
- 0032(22-24): going from eigenvectors and eigenvalues to B = CDC^{-1}
- 0032(24-26): double check B = CDC^{-1}
- 0032(27-28): computation of B^2 and B^{10}, in terms of C and D
- 0032(29): some questions
- given B, how to find C and D, i.e., how to diagonalize B, i.e., how to find eigenvectors and eigenvalues
- are all square matrices diagonalizable?
- if some are not, how to ideentify and diagonalize those that are?
- if some are not, how can we find a computationally good form for them?
- 0032(30-31): nonzero nilpotent matrices are not diagonalizable
- 0032(32): strictly upper triangular matrices are nilpotent
- 0032(33): how to find eigenvalues
- 0032(34): transition slide
- 0032(35-44): how to find eigenvectors for each eigenvalue
- 0032(45): transition slide
- 0032(46-49): how to diagonalize, given eigenvectors and eigenvalues
- 0032(49): how to take 10th power, given diagonalization
- 0032(50): def'n: characteristic polynomial
- 0032(51): def'n: eigenvalue = characteristic root
- 0032(52): transition slide
- 0032(53): def'n: eigenvector; def'n: eigenspace
- 0032(54): transition slide
- 0032(55): 0 is never and eigenvalue, but is in every eigenspace
- 0032(56): SKILL: given a square matrix, find its eigenvalues and,
for each eigenvalue, find a basis for its eigenspace
Topic 0033
(Diagonalization of matrices)
Link to Topics website
- 0033(1): title slide
- 0033(2-19): how to identify and diagonalize matrices that are diagonalizable?
- 0033(2-10): real diagonalizable and complex diagonalizable are different
- 0033(11-12): the diagonalization algorithm
- 0033(13-17): where the diagonalization algorithm can fail
- 0033(18): criterion for diagonalizability:
the multiplicity of each eigenvalue is the dimension of its eigenspace
- 0033(19): SKILL: Given a square matrix, determine if it's diagonalizable.
If it is, diagonalize it.
- 0033(20): Jordan blocks are not diagonalizable
- 0033(21-25): recall: scalar, standard nilpotent, Jordan block, direct sum
- 0033(26): direct sum is diagonalizable iff the summands are
- 0033(26): Jordan canonical form:
every matrix is conjugate to a direct sum of Jordan blocks
- 0033(26): a direct sum of Jordan blocks is diagonalizable iff all the blocks are 1 x 1
- 0033(27-49): Is Jordan form computationally good?
- 0033(28-36): tenth power of a Jordan block
- 0033(37-48): exponential of a Jordan block
- 0033(49): powers and exponential distribute over direct sum
- 0033(49): Jordan form is computationally good.
- 0033(50-55): the exponential of an antisymmetric 2 x 2 matrix is a rotation matrix
Topic 0034
(The Spectral Theorem)
Link to Topics website
- 0034(1): title slide
- 0034(2): recall orthogonal matrix
- 0034(3): recall rotation matrix, reflection matrix
- 0034(3): recall orthogonal linear transformation, rotation linear transformation, reflection linear transformation
- 0034(4): recall the quadratic form of a symmetric matrix; recall polarization
- 0034(4): def'n: equivalent quadratic forms
- 0034(5): formula for Q_S \circ L_X
- 0034(6): def'n: t-equivalent
- 0034(6): two quadratic forms are equivalent iff their matrices are t-equivalent
- 0034(7): proof of formula for Q_S \circ L_X
- 0034(8): SKILL: Given S and X, produce a matrix N such that Q_S \circ L_X = Q_N
- 0034(9): def'n: diagonal quadratic form
- 0034(10): eg/non-eg: diagonal quadratic form
- 0034(11): SKILL: recognize whether a given quadratic form is diagonal
- 0034(12): def'n: rotationally diagonalizable
- 0034(12): goal: rotationally diagonalizable = symmetric
- 0034(13-17): preliminaries
- 0034(13): fact: the eigenvalues of a symmetric (real) matrix are real
- 0034(14): fact: if two matrices are conjugate,
then they have the same characteristic polynomial,
and the same eigenvalues
- 0034(15): fact: the eigenvalues of an upper triangular matrix are its diagonal entries
- 0034(16): fact: conjugation of a symmetric matrix by an orthogonal matrix yields a symmetric matrix
- 0034(17): recall the definition of rotationally diagonalizable
- 0034(17): fact: any rotationally diagonalizable matrix is symmetric
- 0034(17): statement of the Spectral Theorem
- 0034(18): transition slide
- 0034(19): statement of the Spectral Corollary
- 0034(19): proof of the Spectral Corollary, given the Spectral Theorem
- 0034(20): the idea of the Spectral Corollary: any quadratic form can be easily studied by diagonalizing it
- 0034(22): transition slide
- 0034(22-36): application of the Spectral Corollary to graphing the level set of a quadratic form, Q
- 0034(22): goal is to get rid of the mixed term
- 0034(23-25): how to proceed if we have a rotation R such that Q \circ R is diagonal
(and we know the coeffients of that diagonal form)
- 0034(25): the graph ends up being an hyperbola
- 0034(26): how to find the rotation and the coefficients of the diagonalized form?
- 0034(27-28): setup
- 0034(29): the eigenvalues, a and b, of the matrix, P, of the quadratic form
- 0034(30): recall that these eigenvalues must be real
- 0034(31): transition slide
- 0034(32): exercise to find an a-eigenvalue, normalize it,
and make it the first column of a rotation matrix R_0
- 0034(32): the first column of R_0^{-1} P R_0 has entries a_0 and 0
- 0034(33-34): full computation of R_0^{-1} P R_0
- 0034(35): transition slide
- 0034(36): proof that, if R is the linear transformation corresponding to R_0, then Q \circ R is diagonal
- 0034(37): a direct sum of rotationally diagonalizable matrices is again rotationally diagonalizable
- 0034(38): recall that a rotationally diagonalizable matrix is symmetric
- 0034(38): recall the statement of the Spectral Theorem (now proved in the 2x2 case, as part of diagonliazing Q)
- 0034(39): transition slide
- 0034(40-42): proof of the Spectral Theorem in the 3x3 case, ginen the 2x2 case
Topic 0035
(Principal component analysis)
Link to Topics website
- 0035(1): title slide
- 0035(2): a pairwise-orthogonal collection of nonzero vectors must be linearly independent
- 0035(3): start of "Principal Component Analysis" (PCA) or "Singular Value Decomposition" (SVD)
- 0035(4): the PCA/SVD theorem
- 0035(5-6): motivation: any matrix, after some post-processing (multiplication on the left by an orthogonal matrix),
has pairwise-orthogonal (i.e., uncorrelated) rows, and so is easier to study
- 0035(7): motivation in terms of tracking measurements from general measuring devices
- 0035(8): motivation in terms of tracking financial assets
- 0035(9): a 2x5 example with the two rows almost equal, general description
- 0035(10): transition slide
- 0035(11): the 2x5 example, worked out in detail
- 0035(12-14): show that the data in the original matrix is driven by one factor, distorted by noise
- 0035(15): easier version of PCA/SVD; proof of this easier version
- 0035(16-22): preliminaries to proving the PCA/SVD theorem
- 0035(23): proof of the PCA/SVD theorem
- 0035(24): transition slide
- 0035(25-26): motiviation in terms of approximating a matrix by a nearby matrix of whose rows span a smaller subspace
- 0035(27-34): application to estimating movie preferences
- 0035(27): setup and notation
- 0035(27): panel of composite peope, a collection of composite movies
- 0035(28): change to the composites
- 0035(29): use of PCA to get simple ratings
- 0035(30): formula for the rating person 0 gives to composite movies 1,...,q
- 0035(31): proof of that formula
- 0035(32): recall the formula, and that it works for composite movies 1,...,q
- 0035(33): we approximate that the formula will work for movie 0
- 0035(34): the approximation for person 0's rating of movie 0
- 0035(34): difficulty if the singular values are 0, or close to 0
Topic 0036
(Cayley's Theorem)
Link to Topics website
- 0036(1): title slide
- 0036(2): def'n: matrix extension
- 0036(2): statement of Cayley's Theorem
- 0036(3-4): matrix extension for polynomials of two or more variables
- 0036(5): notation: matrix is T, E^{ij} is the standard basis of \R^{nxn}
- 0036(5): notation: I:=identity, P is {polynomials in T},
- 0036(5): notation: det : \R^{nxn} --> \R is determinant
- 0036(5): notation: DET : P^{nxn} --> P is its matrix extension
- 0036(6): DET in the 3x3 case
- 0036(7): recall A\otimes B
- 0036(7): (A \otimes B)(C \otimes D) = (AC) \otimes (BD)
- 0036(8-9): the reproducing equation
- 0036(10): T \otimes I and I \otimes \Lambda
- 0036(11): transition slide
- 0036(12): the matrix extension formula of the characteristic polynomial of T
- 0036(13-15): proof of that formula in the 3x3 case
- 0036(16): recall the statement of Cayley's Theorem
- 0036(17): U := (T \otimes I) - (I \otimes T); Want: DET(U)=0
- 0036(18): T \otimes I , I \otimes T both in P^{nxn}
- 0036(19): transition slide
- 0036(20): recall transposed cofactor matrix, notation: tr-cof
- 0036(21): TR-COF, it's matrix extension
- 0036(22): [tr-cof(M)]M = M[tr-cof(M)] = (det M)I
- 0036(22): [TR-COF(M)]M = M[TR-COF(M)] = I \otimes (DET M)
- 0036(23): mult is linear and mult ( A\otimes B) = AB
- 0036(24): C := TR-COF(U);
- 0036(24): Want: mult ( C (T \otimes I) ) = mult ( C (I \otimes T) )
- 0036(25): entries of C all are in P, and so commute with T
- 0036(26): computation of mult ( C (T \otimes I) ) and mult ( C (I \otimes T) )
- 0036(27): conclusion of proof
Topic 0037
(Multivariable polynomial approximation)
Link to Topics website
- 0037(1): title slide
- 0037(2): single variable linear approximation
- 0037(3-4): visualization of single variable linear approximation
- 0037(5-15): a two variable linear approximation problem
- 0037(16): the general formula for two variable linear approximation
- 0037(17-18): def'n: partial derivatives for two variables
- 0037(19-22): the general formula for two variable linear approximation
in the language of partial derivatives
- 0037(23): transition slide
- 0037(24-25): simplifying the notation, using p=(x,y), \triangle p = (h,k)
- 0037(26): def'n: gradient for two variables; notation: \nabla g, g'
- 0037(27): simplifying notation using gradient
- 0037(28): comparison with single variable linear approximation
- 0037(29): def'n: graph; a the mountain climber's problem
- 0037(30-32): solving the mountain climbers problem
- 0037(33): practice problems in partial derivatives
- 0037(34): SKILL: compute partial derivatives, compute the gradient
- 0037(35): SKILL: compute second partials; partials commute
- 0037(36-37): def'n: n variable partial derivatives
- 0037(38-42): alternate notations for partial deriviatives
- 0037(43): def'n: gradient for n variables; notation: \nabla g, g'
- 0037(44): SKILL: comptute partial derivatives, compute the gradient
- 0037(45): notations for higher order partial derivatives
- 0037(46): SKILL: compute the second order partial derivatives
- 0037(47): recall single variable second order Maclaurin approximation
- 0037(47): def'n: two variable second order Maclaurin approximation
- 0037(48): transition slide
- 0037(49-50): compute coefficients for the two variable second order Maclaurin approximation
- 0037(51-52): the formula for the two variable second order Maclaurin approximation
- 0037(53): transition slide
- 0037(54): exercise: third order Maclaurin approximation to f(x,y)
- 0037(54): exercise: second order Maclaurin approximation to f(x,y,z)
- 0037(55): transition slide
- 0037(56-58): rewriting the second order Maclaurin approximation to f(x,y)
in terms of the gradient and Hessian
- 0037(56): def'n: gradient; def'n: Hessian
- 0037(59): gradient and Hessian approximations for vector-valued functions
- 0037(60): notations for the gradient, gradient matrix and Hessian
- 0037(61): def'n: k-jet at 0 of a function of n-variables with values in R^q
- 0037(62): def'n: Maclaurin approximation for a function of n-variables with values in R^q
- 0037(63): SKILLS:
- gradient and Hessian of a function of n variables
- k-jet of a function of n variables
- k'th order Maclaurin approximation of a function of n variables
- number of terms in the k'th order Maclaurin approximation of a function of n variables
- number of entries in the k-jet at 0 of a function of n variables
- 0037(64): the second order Maclaurin approximation of the Black-Scholes function
- 0037(65): transition slide
- 0037(66-75): explaining the meaning of "agrees to order two"
Topic 0038
(Vector fields and ODEs)
Link to Topics website
- 0038(1): title slide
- 0038(2): def'n: vector field; def'n: linear vector field
- 0038(3): e.g. of vector field: the tornado
- 0038(3): def'n: flowline
- 0038(4-5): visualization of a a flowline of the tornado
- 0038(5): guess ( cos t , sin t ) based on the visualization
- 0038(6): checking that ( cos t , sin t ) is right
- 0038(7): existence of short-time flowlines
- 0038(7): a specific example of a short-time flowline in one dimension
that traverses \R in finite time
- 0038(8): def'n: footed (footpoint), extends, maximal
- 0038(9): transition slide
- 0038(10): existence and uniques of maximal flowlines
- 0038(10): explanation of why the specific example above
does not have a long-time flowline
- 0038(11): an ODE (Ordinary Differential Equation) and an equivalent flowline problem
- 0038(11): the continuous compounding ODE
- 0038(12): an ODE system with initial value conditions and an equivalent flowline problem
- 0038(13): the Euler method
- 0038(14): the Euler method on the tornado flowline
- 0038(15-24): the Euler method for a continuous compounding ODE
- 0038(25): transition slide
- 0038(26): checking that the Euler method gave the correct solution
- 0038(27): an alternate solution to the continuous compounding ODE, via logarithmic change of variables
- 0038(28-45): a third approach via fixpoint methods in infinite dimensional function spaces
- 0038(28): reformulation of the ODE problem as a fixpoint problem for an integral mapping
- 0038(29): transition slide
- 0038(30): def'n: contraction; def'n: contraction factor
- 0038(30): fact: in a certain distance, the integral mapping is a contraction with contraction factor 1/2
- 0038(31-41): proof of the fact
- 0038(31): the distance used in the uniform distance
- 0038(42): why a contraction mapping leads to a fixpoint in complete metric spaces
- 0038(43): the uniform distance is complete
- 0038(44-45): finding the short-time flowline for the continuous compounding ODE
- 0038(46): a coupled linear system of ODES and the corresponding flowline problem
- 0038(47): rewriting the ODE system as a matrix ODE
- 0038(48): transition slide
- 0038(49): solving the matrix ODE using the matrix exponential map
- 0038(50): comparision of the matrix ODE solution with the continuous compounding ODE solution
- 0038(51): finding the constant from the initial condition
- 0038(52-54): writing the solution with the correct constant
- 0038(54): exercise: diagonalize the matrix to compute the exponentials that solve the matrix ODE
- 0038(55): SKILLS:
- NOTE: "integrate" means find the flowline at an arbitrary footpoint, i.e., find all the flowlines
- SKILL: integrate a constant vector field
- SKILL: integrate a homogeneous linear vector field
- SKILL: integrate an inhomogeneous linear vector field
- 0038(56): finding flowlines for a constant vector field
- 0038(56): finding flowlines for an inhomogeneous linear vector field
- 0038(57): def'n: functional
- 0038(57): def'n: local max; def'n: local min
- 0038(57): goal: maximize/minimize a smooth functional
- 0038(57): related goal: find local max/min for a smooth functional
- 0038(58): def'n: critical point; local max at a implies critical point at a
- 0038(58): second derivative test
- 0038(59): reverse gradient flow to minimize a functional
- 0038(60): gradient flow to maximize a functional
Topic 0039
(The multivariable chain rule)
Link to Topics website
- 0039(1): title slide
- 0039(2): the chain rule for composites \R^k --> \R^m --> \R^n
- 0039(2): discussion of a formula for the Hessian of a composite \R^k --> \R^m --> \R
- 0039(3-9): the chain rule for composites \R --> \R^m --> \R
- 0039(9): discussion of a formula for the Hessian of a composite \R --> \R^m --> \R
- 0039(10-20): using the chain rule to show:
If q(x,y) is the 2nd order Maclaurin approximation to g(x,y),
then g(x,y) = q(x,y) + o(x^2+y^2) for (x,y) --> (0,0)
Topic 0040
(Lagrange mult, constrained optimization)
Link to Topics website
- 0040(1): title slide
- 0040(2): problem: find the major and minor axes of 3x^2+2xy+3y^2=8
- 0040(2-11): solution of the problem via Spectral Theory, i.e., via diagonalization of the matrix of 3x^2+2xy+3y^2
- 0040(12-30): solution of the problem via constrained optimization
- 0040(12): reprhasing the minor axis problem as a constrainted minimization problem
- 0040(13): transition slide
- 0040(14): level sets of f(x,y) = x^2+y^2 compared to level sets of Q(x,y) = 3x^2+2xy+3y^2
- 0040(15): comparing the gradients of f and Q at a point
- 0040(15): key point: the gradient is perpendicular to the level set
- 0040(16): a starting point one E = Q^{-1}(8)
- 0040(17-18): which way to go on E to make f decrease
- 0040(19-20): why the decrease ceases at the endpoint of a minor axis
- 0040(21): def'n: critical point for a constrained optimization problem
- 0040(22-30): finding the critical points for minimizing f constrained to E
- 0040(31): the general format of a constrained minimization problem, with one scalar constraint
- 0040(31): def'n: objective; def'n: constraint; def'n: Lagrange multiplier
- 0040(31): counting equations and unknowns
- 0040(32): def'n: non-smooth points for one constraint
- 0040(33-35): example that shows we must check non-smooth points as well as critical points
- 0040(36): the general format of a constrained minimization problem, with k scalar constraints
- 0040(36): counting equations and unknowns
- 0040(37): def'n: non-smooth points for k constraints
Topic 0041
(Multivariable change of variables)
Link to Topics website
- 0041(1): title slide
- 0041(2): start of multivariable integral calculus
- 0041(3): single variable change of variables formula
- 0041(4-5): some special cases, as they're presented in a calc course
- 0041(6): goal is to find an analog for two variable functions
- 0041(7): setting up the definition of the Riemann integral for functions of two variables
- 0041(8-20): review of the definition of the Riemann integral for functions of one variable
- 0041(8): the setup
- 0041(9): a partition
- 0041(10): picking points
- 0041(11): focus on one subinterval
- 0041(12): name it
- 0041(13): look at the graph
- 0041(14): form the rectangle
- 0041(15): estimate the integral over that subinterval as the area of the rectangle
- 0041(16): add over all the boxes to estimate the integral over the full interval as a Riemann sum
- 0041(17): make a sequence of partitions with mesh --> 0
- 0041(18): transition slide
- 0041(19-20): define the integral as a limit of Riemann sums
- 0041(21-28): repeate this for functions of two variables
- 0041(21): the setup
- 0041(22): a partition
- 0041(23): picking points
- 0041(24): focus on one subset
- 0041(25): estimate the integral over a subset as the volume of a box
- 0041(26): sum to estimate the integral
- 0041(27): make a sequence of partitions with mesh --> 0
- 0041(28): define the integral as a limit of Riemann sums
- 0041(29): Fubini's Theorem on rectangles
- 0041(30-43): the two-variable change of variables formula
- 0041(30): the setup and goal
- 0041(31): the answer
- 0041(32): estimating the LHS and RHS of the answer via Riemann sums
- 0041(33): reducing the problem to an estimation of area of the image of a subset
- 0041(34): zoom in on a subset, and parametrize the box by h and k
- 0041(34): estimate the change of variables by linear approximation
- 0041(34): estimate the image of the subset by linear approximation
- 0041(35): estimate the area of the image of the subset by linear approximation
- 0041(36): the absolute value of the determinant relates area before a linear map to area afterward
- 0041(37): the area of the box is the base time the height
- 0041(38-39): complete the required estimation of the area of the image of a subset
- 0041(40): complete the proof that the LHS and RHS are close
- 0041(41): transition slide
- 0041(42-43): complete the proof that the LHS and RHS are equal
- 0041(44-49): polar coordinates
- 0041(49): replacements:
- x by r \cos \theta
- y by r \sin \theta
- x^y + y^2 by r^2
- dx dy by r dr d\theta
- 0041(50-53): proof that \int_{-\infty}^\infty e^{-x^2/2} dx is equal to \sqrt{2\pi}
- 0041(54): spherical coordinates
Topic 0042
(Variations on Stokes' Theorem)
Link to Topics website
- 0042(1): title slide
- 0042(2): start of Green's Theorem and Cauchy's Theorem
- 0042(3): def'n: directed line segment; def'n: starting point; def'n: ending point
- 0042(4): def'n: standard parametrization; def'n: constant velocity
- 0042(5): def'n: simple chain
- 0042(6): def'n: rectangle
- 0042(7): e.g.: open rectangle
- 0042(8): descriptive def'n and notation: counterclockwise boundary
- 0042(9): e.g.: counterclockwise boundary
- 0042(10): formal def'n and notation: counterclockwise boundary
- 0042(11): def'n and notation: line integral in the plane over a directed line segment
- 0042(12): def'n and notation: line integral in the plane over a chain
- 0042(13): statement of Green's Theorem
- 0042(14): def'n: zero-form in two variables; def'n: one-form in two variables
- 0042(14): def'n: exterior derivative of a zero-form in two variables
- 0042(15): SKILL: compute exterior derivative of a zero-form
- 0042(16): the conventions for computing wedge products
- 0042(17): zero-forms move through wedge products
- 0042(18): an example computation of a wedge product in two variables
- 0042(19): an example computation of an exterior derivative of a one-form in two variables
- 0042(20): def'n: two-form in two variables
- 0042(20): def'n: exterior derivative of a one-form in two variables
- 0042(21): transition slide
- 0042(22): an example computation of an exterior product in two variables
- 0042(22): def'n: integral of a two-form in $x$ and $y$ over a rectangle in $\R^2$.
- 0042(23): restatement of Green's Theorem in the language of forms
- 0042(24-26): example computations of wedge products of three one-forms in three variables
- 0042(27-29): example computations of exterior derivatives of one-forms in three variables
- 0042(30): an example computation of a line integral of a one-form in two variables
- 0042(31-32): an example computation of a line integral of a one-form in three variables
- 0042(33): recall the definition of line integral over a directed line segment in \R^2
- 0042(34): def'n: line integral over a directed line segment in \C
- 0042(35): recall the definition of line integral over a chain in \R^2
- 0042(36): def'n: line integral over a chain in \C
- 0042(37): recall Green's Theorem on rectangles in \R^2
- 0042(38): Green's Theorem on rectangles in \C
- 0042(39-41): exterior derivative of 1-form in one variable is always zero (over \R)
- 0042(42): expecting 0 in Green's Theorem on rectangles in \C
- 0042(43): def'n: complex differentiable
- 0042(43): exercise: show that f(z)=e^{3z} is complex differentiable
- 0042(44): transition slide
- 0042(45): example of a function (namely, f(z)=|z|^2) that is smooth, but not complex differentiable
- 0042(46-57): proof that f(z)=|z|^2 is not complex differentiable
- 0042(57): necessary condition for complex differentiability
- 0042(58-60): going from that necessary condition to the Cauch-Riemann equations
- 0042(60): the Cauchy-Riemann equations
- 0042(61): finding the real and imaginary parts of e^{z^2/2}
- 0042(61): exercise to verify the Cauchy-Riemann equations for the real and imaginary parts of e^{z^2/2}
- 0042(62): recall Green's Theorem for rectangles in \C
- 0042(63): in the analytic case, the exterior derivative vanishes
- 0042(64): Cauchy's Theorem for rectangles in \C
- 0042(65-75): proof of Green's Theorem for rectangles in \R^2
Topic 0043
(Planimeters)
Link to Topics website
- 0043(1): title slide
- 0043(2-3): description of a simple design of a planimeter; the curve C
- 0043(4-7): modeling the rate of turning of the wheel
- 0043(8): goal is to show that the total turning of the wheel is proportional to the area enclosed inside C
- 0043(9): the vector fields R and W and V
- 0043(9): computation of R
- 0043(10): computation of W
- 0043(11): the scalar quantities c and s
- 0043(11-12): computation of V in terms of c and s
- 0043(13): transition slide
- 0043(14-15): computation of c and s
- 0043(16): transition slide
- 0043(17): recall the goal (proportionalty of area enclosed to total turning of the wheel)
- 0043(18): \gamma parametrizes the curve C
- 0043(19): \gamma = ( \alpha , \beta )
- 0043(19): p and q are the components of V
- 0043(20): \omega = p(x,y) dx + q(x,y) dy
- 0043(20-25): the total turning of the wheel as a line integral of \omega over C via the parametrization \gamma
- 0043(26): \omega in terms of s, x, c and y
- 0043(27): Stokes' Theorem for the curve C in \R^2
- 0043(28): transition slide
- 0043(29): R := the region enclosed inside C, A := the area enclosed as an integral over R of dx \wedge dy
- 0043(30-37): computation of d\omega, the exterior derivative of \omega
- 0043(37): d\omega is proportional to dx \wedge dy
- 0043(38): the total turning of the wheel is propotonal to the area enclosed inside C
Topic 0044
(Problems in integration)
Link to Topics website
- 0044(1): title slide
- 0044(2): def'n: \Phi as an antiderivative of H := e^{-x^2/2} / \sqrt{2\pi}
- 0044(2): the definite integral of H from p to \infty
- 0044(3): the definite integral of e^{ax} H dx from p to \infty
- 0044(4): the definite integral of ( e^{ax} - b ) H dx from p to \infty
- 0044(5-6): the definite integral of ( e^{ax} - b )_+ H dx from -\infty to \infty
- 0044(7-16): indefinite integral of x^k H dx
- 0044(7): k=0 case
- 0044(8): k=1 case
- 0044(9): simplest antiderivative in the k=1 case
- 0044(10-11): from k=2 to k=0
- 0044(12): conclusion of the k=2 case
- 0044(13): from k=3 to k=1
- 0044(14): conclusion of the k=3 case
- 0044(15): from k to k-2
- 0044(16): exercise: the k=4 case
- 0044(17-21): the definite integral of x^k H dx from -\infty to \infty
- 0044(17): setting up the problem
- 0044(18): solution when k is even
- 0044(19): solution when k is odd
- 0044(20): transition slide
- 0044(21): the two cases (k even and k odd) on one slide
Topic 0045
(Basics of PCRVs)
Link to Topics website
- 0045(1): title slide
- 0045(2): def'n: piecewise constant random variable (PCRV)
- 0045(2): finitely many pieces
- 0045(2): \Omega := [0,1]
- 0045(2): e.g. of a PCRV (called X)
- 0045(3): transition slide
- 0045(4): intuition behind PCRVs (Tyche chooses elements of \Omega)
- 0045(5): some probability computations
- 0045(6): def'n and e.g.s: deterministic
- 0045(6): pieces can have zero length
- 0045(7): another e.g. of a PCRV (called Y)
- 0045(7): use of "almost surely" and "surely"
- 0045(8): modeling a coin-flip: C_1
- 0045(8): modeling two coin-flips: C_1 and C_2 (bad choice of C_2)
- 0045(9): good choice of C_2
- 0045(10): def'n: distribution of a PCRV
- 0045(11): e.g.: distribution of X
- 0045(12): e.g.: distribution of Y
- 0045(13): e.g.: distribution of C_1 and C_2
- 0045(13): C_1 and C_2 are identically distributed, but are not equal
- 0045(14): def'n: joint distribution
- 0045(14): to get distribution of X + Y, it is enough to know the joint distribution of (X,Y)
- 0045(15): e.g. to show that the distributions of X and Y do not determine the distribution of X + Y
- 0045(16): def'n: f(A), where f is a function and A is a PCRV
- 0045(17): def'n: mean (same as expectation) of a PCRV; notation: E[T] = mean (or expectation) of T
- 0045(17): mean is linear
- 0045(17): def'n: T^\circ, where T is a PCRV
- 0045(17): variance of a PCRV; notation: Var[T] = variance of T
- 0045(17): variance is always nonnegative
- 0045(17): a PCRV is deterministic iff its variance is zero
- 0045(17): variance = second raw moment minus the square of the first, i.e., Var[T] = (E[T^2]) - (E[T])^2
- 0045(18-22): proof that variance = (second raw moment) minus (the square of the first)
- 0045(23): recall X, compute E[X]
- 0045(23): intuition: mean is a measure of (average) size
- 0045(24): transition slide
- 0045(25): compute Var[X]
- 0045(25): intuition: variance is a measure of risk
- 0045(26): return-loving vs. risk-averse
- 0045(27): variance is not linear; it's quadratic
- 0045(28): def'n: covariance is the polarization of variance; notation: Cov[S,T] is the covariance of S and T
- 0045(28): Cauchy-Schwarz; def'n: correlation; notation: Corr[S,T] is the correlation of S and T
- 0045(28): correlation is between -1 and 1; correlation is not defined if either PCRV is deterministic
- 0045(28): def'n: uncorrelated
- 0045(29): transition slide
- 0045(30): covariance is (the expectation of the product) minus (the product of the expectations)
- 0045(30): uncorrelated iff (covariance zero) iff (variance of sum = sum of variances) iff (expectation product = product of expectations)
- 0045(31-32): S-T deterministic implies Var[S]=Var[T]
- 0045(33-34): calculation of variance of a binary PCRV
- 0045(35): transition slide
- 0045(36): def'n: standard PCRV
- 0045(36): coin-flipping variables are standard
- 0045(37): def'n: standard deviation; notation: SD[S] is the standard deviation of S
- 0045(37): def'n: U_\circ, for a non-deterministic PCRV U; the "standardization of U"
- 0045(38): any non-deterministic PCRV is "almost" standard
- 0045(39): SD[cS] = |c| SD[S]
- 0045(39-40): variance measures risk, but standard deviation measures it better
- 0045(41): identically distributed implies same mean, variance, standard deviation
- 0045(41): need identical joint distribution for same covariance or for same standard deviation
- 0045(42): standard deviation of a binary PCRV; def'n: arithmetic mean and geometric mean
- 0045(43): def'n: indicator function
- 0045(43-45): problems of calculating probabilities can be converted into problems of calculating expected values
- 0045(46): applications to finance: reduction of risk
- 0045(47): comparing additivity and subadditivity of standard deviation, for perfectly correlated and uncorrelated portfolios, respectively
- 0045(48): description of modern portfolio theory, as a constrained optimization problem
- 0045(49-52): given standard deviations and correlations for two assets, and a requirement to hold a certain amount of one of them,
find the amount of the other to short so as to minimize the standard deviation of the portfolio -- example problem
Topic 0046
(Cholesky decomposition)
Link to Topics website
- 0046(1): title slide
- 0046(2): given Z_1,...,Z_n uncorrelated, with variance 1, goal is to
find linear combinations of the Z_j that have prescribed variance-covariance matrix
- 0046(2): necessary conditions on a variance-covariance matrix: symmetric, positive semidefinite
- 0046(3): careful mathematical phrasing of the problem
- 0046(4-9): converting the problem into a linear algebra problem:
given C, solve C=AA^t, for A
- 0046(10-11): another way to convert the problem to C=AA^t
- 0046(12): using the Spectral Theorem to solve C=AA^t for A
- 0046(12): Spectral Theorem is hard to implement because it requires finding zeroes of a polynomial
- 0046(13-24): an easier way to solve C=AA^t, using row and column operations -- an example
- 0046(25): first statement of the Cholesky decomposition (positive definite C, lower triangular A s.t. C=AA^t)
- 0046(26-29): example of using row and column operations to get a lower triangular A
- 0046(30-34): example with a positive semidefinite C
- 0046(35): improved Cholesky decomposition (positive semidefinite C, lower triangular A s.t. C=AA^t)
- 0046(36-37): why the row/column algorithm works for positive semidefinite C:
a zero on the diagonal implies zeroes all through that row and column
- 0046(38): a worked example, done by solving equations (as opposed to row/column operations)
- 0046(39-41): another worked example, done by solving equations
- 0046(42): for semideifnite, a zero on the diagonal implies zeroes all through that row and column
- 0046(43): second improved Cholesky decomposition (positive semidefinite C, upper triangular A s.t. C=AA^t)
- 0046(44): third improved Cholesky decomposition (positive semidefinite C, lower triangular B s.t. C=BB^t)
- 0046(45): fourth improved Cholesky decomposition (positive semidefinite C, upper triangular B s.t. C=BB^t)
- 0046(46): SKILL: given a positive semidefinite, symmetric matrix, find all four Cholesky decompositions of it
- 0046(47): end of linear algebra
Topic 0047
(Cond prob, indep and the CLT)
Link to Topics website
- 0047(1): title slide
- 0047(2-6): conditional probability of one event given another -- motivating example
- 0047(7): def'n: conditional probability of one event given another
- 0047(8): coin-flip example (C_1 and C_2) to motivate independence of events
- 0047(9): def'n: two indpendent events when second one has positive probability
- 0047(10): toward a better definition
- 0047(11): def'n: two independent events (the probability of both is the product of the probabilities)
- 0047(12): def'n: two independent PCRVs
- 0047(13): e.g.: C_1 and C_2 are independent PCRVs
- 0047(14): def'n: independence of three or more events
- 0047(14): def'n: independence of three or more PCRVs
- 0047(15): e.g.: an independent sequence of PCRVs -- C_1,C_2,C_3,...
- 0047(16): for indpendent PCRVs, the joint distribution is determined by the individual distributions
- 0047(17): independent implies uncorrelated
- 0047(18): independence is preserved under postcomposition with functions
- 0047(19): proof that independence is preserved under postcomposition with functions
- 0047(20): A,B independent implies that, for all function f,g,
f(A),g(B) are uncorrelated
- 0047(20): converse also true, but not proved
- 0047(21): D_n = C_1 + ... + C_n
- 0047(21): graph and distribution of D_2
- 0047(22): standardization of D_n is D_n/\sqrt{n}
- 0047(23): start of preview of Central Limit Theorem (CLT)
- 0047(24): statement of the CLT in terms of convergence in distribution to a standard normal Z
- 0047(24): explanation of convergence in distribution to Z via "test function"s
- 0047(25): transition slide
- 0047(26): two possible meanings of "test function"
- 0047(27): transition slide
- 0047(28): example problem using the CLT
- 0047(29): def'n: augmented expectation; notation: AE[X]
- 0047(29-31): expectation almost asymptotically commutes with exponentiation
- 0047(32-37): from the mean and standard deviation of an iid sum to the mean and standard deviation of the summands
- 0047(37): an uncorrelated portfolio is better because the standard deviations don't add; they subadd
- 0047(38): the formula for Var[A+B] shows that it's good to have A and B uncorrelated if they represent future asset values
- 0047(38): even better is if A and B have negative correlation
- 0047(39): remember Cauchy-Schwarz
- 0047(39): def'n: perfectly correlated; def'n: perfectly anti-correlated;
- 0047(40): remember the definition of correlation, for non-deterministic PCRVs
- 0047(40): perfectly correlated, uncorrelated and perfectly anti-correlated, in terms of correlation
- 0047(41): def'n: positively correlated; def'n: negatively correlated; recall the def'n of uncorrelated
- 0047(41): positively correlated, uncorrelated and negatively correlated, in terms of correlation
- 0047(42): recall the definition of standard deviation; motivation behind standard deviation; formula for SD[cX]
- 0047(43-45): for perfectly correlated PCRVs, standard deviations add
- 0047(46): for uncorrelated PCRVs, variances add
Topic 0048
(Bayes' Law)
Link to Topics website
- 0048(1): title slide
- 0048(2): recall the definition of conditional probability
- 0048(3): an incorrect cancellation formula for Pr[C|A]
- 0048(3): a correct cancellation formula for Pr[(B and C)|A]
- 0048(4): long form vs. short form
- 0048(5): suggestion: move to the long form before doing cancellation
- 0048(6-7): s cancellation formula for Pr[B]
- 0048(8): def'n/notation: {E}_A^x; def'n/notation: {E}_A^o; examples
- 0048(9): more examples of {E}_A^o; factors without A disappear
- 0048(10): def'n/notation: Odds[A]
- 0048(11): def'n/notation: Odds[A|B], conditional odds of A given B
- 0048(11): example of odds in horse-racing
- 0048(12): [probability of (sickness) given (positive medical test)] vs. [probability of (positive medical test) given (sickness)]
- 0048(13): Bayes' Law
- 0048(14-15): application of Bayes' Law to computing [probability of (sickness) given (positive medical test)]
- 0048(16-18): computing [probability of (positive medical test) given (not sick)]
- 0048(19): applying {\bullet}_A^o to Bayes' Law for Pr[A|B]
- 0048(19): result is the odds form of Bayes' Law
- 0048(20): transition slide
- 0048(21): def'n/notation: LQ^A[B], the likelihood quotient of B with respect to A
- 0048(21): general philosophy of the odds form of Bayes' Law:
to update the odds based on new information, multiply by a carefully chosen "likelihood quotient"
- 0048(22): checking the odds form of Bayes' Law for sickness and positive medical test
- 0048(23-26): second level of Bayes' Law
- 0048(23): second level of Bayes' Law in terms of probabilities
- 0048(24): apply {\bullet}_S^o
- 0048(25): def'n/notation: LQ^A[C|B], the conditional likelihood quotient of C given B with respect to A
- 0048(26): the odds form of the second level of Bayes' Law
- 0048(27): application of the second level of Bayes' Law, via high-risk group
- 0048(28): explicit formula for LQ^A[C|B]
- 0048(29): general philosophy of the odds form of Bayes' Law:
to update the odds based on new information, multiply by a carefully chosen "likelihood quotient"
- 0048(30): the third level of Bayes' Law
- 0048(30): general philosophy of the odds form of Bayes' Law:
to update the odds based on new information, multiply by a carefully chosen "likelihood quotient"
- 0048(31-33): cancellation formulas for likelihood quotients
- 0048(34-36): proving the second and third level of Bayes' Law using the first level,
together with cancellation formulas from likelihood quotients
- 0048(36): general philosophy of the odds form of Bayes' Law:
to update the odds based on new information, multiply by a carefully chosen "likelihood quotient"
Topic 0049
(Conditional expectation)
Link to Topics website
- 0049(1): title slide
- 0049(2-3): an example of calculating the conditional expectation of a PCRV given an event
- 0049(4): notation: E[X|W], the conditional expectation of X, given W
- 0049(4): a formula for E[X|W] in terms of integration
- 0049(5): def'n: conditional expectation of a PCRV given an event
- 0049(6-14): the conditional expectation of one PCRV given another
- 0049(7-10): a worked example based on two PCRVs, X and Y
- 0049(7): calculation of E[Y|X=a] for two values of a
- 0049(8): transition slide
- 0049(9): def'n of the PCRV E[Y|X] and the graph of of E[Y|X]
- 0049(10): general philosophy: average Y over the level sets of X
- 0049(11-14): a worked example based on two PCRVs, X' and Y
- 0049(11): calculation of E[Y|X'=a] for two values of a
- 0049(12): transition slide
- 0049(13): def'n of the PCRV E[Y|X'] and the graph of of E[Y|X']
- 0049(14): general philosophy: average Y over the level sets of X'
- 0049(15-23): conditional expectation of a PCRV given a partition
- 0049(16): calculation of the level sets of X, where X is defined above
- 0049(16): def'n: partition of a PCRV; notation: \scrP_V
- 0049(16): calculation \scrP := \scrP_X
- 0049(17): calculation of the level sets of X', where X' is defined above
- 0049(17-18): calculation \scrP := \scrP_X = \scrP_{X'}
- 0049(19): calculation of E[Y|\scrP], where Y is defined above
- 0049(20): def'n of E[V|I], where I is an interval
- 0049(20): def'n of E[V|\scrQ], where \scrQ is a partition of \Omega by intervals; \Omega:=[0,1]
- 0049(21): def'n: E[V|W], where V and W are PCRVs
- 0049(21): difficulty: sets in the partition of W may not be intervals, but, rather, fUofIs (finite unions of intervals)
- 0049(22): def'n of E[V|I], where I is a fUofI
- 0049(22): def'n of E[V|\scrQ], where \scrQ is a partition of \Omega by fUofIs; \Omega:=[0,1]
- 0049(23): new difficulty
- 0049(23): fix that difficulty by only considering PCRVs W whose level sets all have positive size
- 0049(23): more on that difficulty later
- 0049(24-73): more on conditional expectations given PCRVs and partitions
- 0049(24): def'n: a.e. constant
- 0049(24): def'n: measurability of a PCRV w.r.t. a partition
- 0049(24): philosophy: a partition is a measurement of information
- 0049(24): philosophy: a PCRV is measurable w.r.t. a partition if the information of the partition is sufficient to calculate the PCRV a.s.
- 0049(25): the idea of starting with Y, which is *not* \calP-measurable and perturbing as little as possible into a PCRV that is
- 0049(26): one way of interpreting "as little as possible": keep the same conditional expectations on sets in \calP
- 0049(27-28): another interpretation: develop a distance between PCRVs and minimize the distance
- 0049(27): recall distance in the plane
- 0049(28): L^2 distance between PCRVs and L^2 norm of a PCRV
- 0049(29): a worked example with two PCRVs W and V
- 0049(29): averaging V over the level sets of W gives U, i.e., U = E[V|W]
- 0049(29): goal: interpret U as a the L^2-closest PCRV to V that is (\scrP_W)-measurable
- 0049(29): let X be a generic (\scrP_W)-measurable PCRV (with values x and y)
- 0049(30): write out X and V
- 0049(31): compute X-V
- 0049(32): compute (X-V)^2 and then E[(X-V)^2] which is the squared L^2 norm of X-V
- 0049(33): finish off the minimization problem
- 0049(34): difficulty: some level sets may have size 0, and so you can't average over them
- 0049(35): make a change to W so that one level set has size 0
- 0049(35): average over those level sets that do have positive size
- 0049(36): make arbitrary choices elsewhere
- 0049(37): then E[V|W] is not well-defined, and, instead of (U=E[V|W]), we write (U=E[V|W] a.s.)
- 0049(38): another possibility for E[V|W]
- 0049(39): the Tower Law
- 0049(39): def'n: finer
- 0049(39): P finer than Q implies any Q-measurable PCRV is P-measurable
- 0049(40): transition slide
- 0049(41): the Tower Law (not proved)
- 0049(42-47): recall the coin-flipping variables C_1,C_2,...
- 0049(48): recall the Central Limit Theorem
- 0049(48): def'n: PCRV approximation
- 0049(49-51): key goal of probability theory:
define random variables and expectation so that every PCRV approximation has a limit in distribution
- 0049(52-53): for example: we seek a random variable Z such that (C_1 + ... + C_n) / \sqrt{n} ---> Z in distribution
- 0049(54-70): SKIPPED
- 0049(71): independent implies (deterministic conditional expectation) implies uncorrelated
- 0049(72): uncorrelated does not imply (deterministic conditional expectation) does not imply independent
- 0049(73): SKIPPED
- 0049(74-77): linearity of conditional expectation (or: taking out what you know)
- 0049(74): linearity of E[\bullet] over \R
- 0049(74): linearity of E[\bullet|P] over P-measurable PCRVs
- 0049(74): special case when P is as coarse as possible
- 0049(74): special case: E[\bullet|P] commutes with multiplication by P-measurable PCRVs
- 0049(75): why this commutativity is called "taking out what you know"
- 0049(76-77): proof of linearity of E[\bullet|P] over P-measurable PCRVs
Topic 0050
(Stirling's Formula)
Link to Topics website
- 0050(1): title slide
- 0050(2): def'n of asymptotics, notation: a_n \tilde b_n
- 0050(2): some illustrations of asymptotics
- 0050(2): (x_n \tilde 5/n) implies (1+x_n)^n ---> e^5
- 0050(3): the graph of \ln
- 0050(4-5): trapezoidal approximation of \int_1^4 \ln x dx, and notation for the error, as a sum of "slivers" c_1+c_2+c_3
- 0050(6): transition slide
- 0050(7): the slivers, on translation form disjint subsets in a rectangle of area \ln 2
- 0050(8): c_1+c_2+c_3 \le \ln 2
- 0050(9): generalizing to n trapezoids, with n slivers, to estimate \int_1^n \ln x dx, with error
- 0050(10-12): computing the exact value of \int_1^n \ln x dx, by integration by parts
- 0050(13): exponentiation of estimate with error gives a formula for n!
- 0050(14): the error factor K_n is asymptotic to some constant K
- 0050(14-15): from the aysmptotics of K_n to the asymptotics of n!
- 0050(16): IOU: K = \sqrt{2\pi}; this yields Stirling's Formula
- 0050(17): Know that n! is asymptotic to a certain formula involving K
- 0050(18): transition slide
- 0050(19): I_n := \int_0^{\pi/2} \sin^n(x) dx
- 0050(19-24): a formula relating I_n to I_{n-2} via integration by parts
- 0050(24): a recursive formula for I_n in terms of I_{n-2}
- 0050(25): transition slide
- 0050(26): recall the definition of asymptotics, some asymptotics involving I_n-2
- 0050(27): I_n is asymptotic to I_{n-2}
- 0050(28): I_{n-1} is between I_n and I_{n-2}, so all three are asymptotic
- 0050(28): I_n is asymptotic to I_{n-1}
- 0050(28): relpacing n by n+1, I_{n+1} is asymptotic to I_n
- 0050(28): replacing n by 2n, I_{2n+1} is asymptotic to I_{2n}
- 0050(29-37): calculation of I_0 through I_9
- 0050(38): two formulas, one for I_{2n}, and another for I_{2n+1}
- 0050(39): the two formulas are asymptotic
- 0050(40-45): rewriting this asymptotic relation in terms of factorials
- 0050(46): transition slide
- 0050(47): cancellation of terms from this rewritten asymptoic formula yields that 2\pi is asymptotic to K^2
- 0050(47): two constants are asymptotic iff they are equal, so 2\pi = K^2
- 0050(47): K is positive, completing the IOU
- 0050(48): Stirling's formula is proved
Topic 0051
(From Stirling to the CLT)
Link to Topics website
- 0051(1): title slide
- 0051(2): recall Stirling's formula, giving asymptotics of n!
- 0051(2-4): recall asymptotics and some examples of asymptotics
- 0051(4): if x_n is asymptotic to y_n and q_n is a sequence of integers that tends to infinity,
then x_{q_n} is asymptotic to y_{q_n}
- 0051(4): from Stirling's formula to asymptotics of, say, (2n)!
- 0051(5-11): applied coin-flipping
- 0051(5): modeling male height
- 0051(6-9): modeling gravitational acceleration as an extremely mildly probabilistic phenomenon, with N in the denominator instead of \sqrt{N}
- 0051(9): maybe all physics should be probabilistics, but sometimes we can't see it because, e.g., N is in the denominator
- 0051(10): applying coin-flipping to finance, an expected payout problem
- 0051(11): a (simplistic) probabilistic model of stock price evolution, with N=10^{10^{100}} time subperiods
- 0051(11): computation of that expected payout, assuming a 68% probability that (H-T)/\sqrt{N} is between -1 and 1
- 0051(11): our goal is now to prove that that probability really is 68%
- 0051(12): changing from [the constant N=10^{10^{100}}] to [2n, with n a variable integer]; we will eventually look at the n=N/2 case
- 0051(13): an example path when n=9, i.e., with 2n=18 time subperiods, based on 18 coin-flips
- 0051(14): probability of a random path (with n=9) ending at 4/\sqrt{18}
- 0051(15): transition slide
- 0051(16-22): for n=9, building a histogram (a.k.a. bar graph) illustrating probabilities of ending at various numbers
- 0051(22): the span of the n=9 histogram
- 0051(23): generalize to arbitrary n, and compute, in the nth histogram:
- the heights of all the bars
- the span of the histogram
- 0051(24): to prove the 68% percent goal from above, we study the n=N/2 histogram
- 0051(25): f_n(x) is defined as the function whose graph is the top of the nth histogram
- 0051(25): goal: f_n(x) ---> e^{-x^2/2}/\sqrt{2\pi}, as n ---> \infty
- 0051(26-28): when n > x^2/2, x is in the span of the nth histogram
- 0051(29): statement of Central Limit Theorem (CLT) in this notation
- 0051(29): for n > x^2/2, x is in the span of the nth histogram, k_n is the number of the bar under which it lies and h_n is the height of that bar
- 0051(29): we wish to show that h_n ---> e^{-x^2/2}/\sqrt{2\pi}
- 0051(30): we focus on x=7
- 0051(31-34): aysmptotics of k_n
- 0051(35): transition slide
- 0051(36-38): asymptotics of n + k_n and of n - k_n
- 0051(39): transition slide
- 0051(40-42): develop the formula for h_n
- 0051(42): recall Stirling's formula
- 0051(43): transition slide
- 0051(44-52): asymptotics of h_n
- 0051(53-54): h_n ---> e^{-x^2/2}/\sqrt{2\pi}, for x=7, except for two IOUs
- 0051(55): a new IOU: general result about 1^\infty indeterminate forms
- 0051(55-60): using that general result to prove the first two IOUs
- 0051(61-62): proof of the general result
Topic 0052
(Piecewise constant processes)
Link to Topics website
- 0052(1): title slide
- 0052(2): def'n: (\triangle t)-piecewise constant process ((\triangle t)-PCP)
- 0052(2): def'n: X_t
- 0052(2): idea of def'n: an evolving PCRV, but can only change at multiples of \triangle t
- 0052(3): a 2-PCP is a 1-PCP
- 0052(4): a (\triangle t)-PCP is a ((1/2)(\triangle t))-PCP
- 0052(5): def'n: piecewise constant process ((\triangle t)-PCP)
- 0052(6): def'n: \triangle X, when X is a PCP
- 0052(7-8): recall coin-flipping sequence C_1,C_2,...
- 0052(9): a PCP X that approximates Brownian motion (BM)
- 0052(10): transition slide
- 0052(11): for that X (that approximates BM), the CLT gives good approximation for E[g(X_1)]
- 0052(12): recall definition of bounded function \R ---> \R
- 0052(12): recall definition of PCRV approximation
- 0052(13): def'n: bounded fuction X ---> R, where X is any set, (e.g., X=\R^k)
- 0052(14): def'n: PCP approximation
- 0052(15): example of a PCP approximation: the standard Brownian motion approximation
- 0052(16): with X^{(N)} = that standard BM approximation, do the following:
- problem: for all t\ge0, compute the limit, as n--->\infty, of E[( e^{X_t^{(N)} - 5 )_+]
- subproblem: compute the limit, as n--->\infty, of: E[( e^{X_3^{(N)} - 5 )_+]
- 0052(16-23): solution to the subproblem, using the CLT
- 0052(24): guess solution to the problem, based on solution to the subproblem
- 0052(24): a second subproblem: compute the limit, as n--->\infty, of: E[( e^{X_\pi^{(N)} - 5 )_+]
- 0052(25-26): solution to the second subproblem, except for an IOU
- 0052(27): transition slide
- 0052(28-32): proof of the IOU
- 0052(33-36): motivation of Stochastic Differential Equations via "the random bank"
Topic 0053
(Functional analysis)
Link to Topics website
Topic 0054
(The heat equation)
Link to Topics website
Topic 0055
(One period pricing and hedging)
Link to Topics website
- 0055(1): title slide
- 0055(2-8): a pricing problem involving Dan and Alice
- 0055(3): description of a foward (or futures) contract
- 0055(3): description of an option contract
- 0055(4): transition slide
- 0055(5): the payoff or (contingent) claim
- 0055(6-8): rewriting the payoff using (\bullet)_+ notation
- 0055(9): transition slide
- 0055(10-11): pricing step 1: model the underlying
- 0055(11): explanation of the 1-subperiod 70-30 CRR model
- 0055(12): transition slide
- 0055(13-17): pricing step 2: calibrate the model
- 0055(13): explanation of drift and volatility
- 0055(14): transition slide
- 0055(15-16): calculation of u and d from the drift and volatility
- 0055(17): calculation of the uptick and downtick factors
- 0055(18-25): pricing step 2: find a perfect hedging strategy
- 0055(19): transitio slide
- 0055(20): description of the hedging portfolio (a.k.a. "the hedge")
- 0055(21): using the payoff to find the ending value of the hedge
- 0055(22): transition slide
- 0055(23): setting up two equations in two unknowns to find the hedging parameters
- 0055(24): a third equation in a third unknown
- 0055(25): solving the equations
Topic 0056
(Risk-neutrality and Delta-hedging)
Link to Topics website
- 0056(1): title slide
- 0056(2): recall the three equations in three unknowns from the preceding topic
- 0056(2): probability is used in the calibration step (step 2), but continues to be useful in the hedging step (step 3)
- 0056(2): trick: imagine another universe with uptick and downtick probabilities that help us in the calculation of the price
- 0056(3): in *our* universe (modeled with 70-30 probabilities) two expected value problems
- 0056(4): transition slide
- 0056(5): the expected value and return of 1 US dollar, invested in the bank (following our 70-30 model)
- 0056(6-7): the expected value and return of 1 US dollar, invested in Euros (following our 70-30 model)
- 0056(8): transition slide
- 0056(9): investors in our universe are "risk-averse"
- 0056(9): imagine a universe where they are "risk-neutral"
- 0056(10): computation of new probabilities (60-40)
- 0056(11): recall Euro return under real world probabilities
- 0056(12): $1 Euro expected value under the new (60-40) probabilities
- 0056(13): $1 bank expected value under the new (60-40) probabilities
- 0056(14): $1 Euro and bank expected return under the new (60-40) probabilities
- 0056(14): the 60-40 world is the risk-neutral world
- 0056(15): $3 bank expected return under the risk-neutral (60-40) probabilities
- 0056(16): $2 Euro expected return under the risk-neutral (60-40) probabilities
- 0056(17): ($3 bank plus $2 Euro) expected return under the risk-neutral (60-40) probabilities
- 0056(17): in the risk-neutral world all portfolios have the same expected return
- 0056(18): using this constancy of expected returns to compute the price of the option that Alice sells to Dan
- 0056(19): coin-flippers got price
- 0056(20): recall the diagram yielding the price and hedging parameters
- 0056(20): difference in the hedging parameters (x-y) is the price (called "?")
- 0056(20): as pricers, we know ?, so, if we can compute x, we can solve x-y=? for y
- 0056(21): naïve volatility of Euro, bank and the hedging portfolio
- 0056(22): 40 Euros would not have enough naïve volatility
- 0056(23): we need 50 Euros; number of Euros (called "x") is the naïve volatility of the option divided by that of the Euro
- 0056(24): pricers got hedge
Topic 0057
(Pricing/hedging in 3 subperiods)
Link to Topics website
- 0057(1): title slide
- 0057(2): a pricing problem involving Harry and Gail involving a three monnth call option on stock
- 0057(3): Gail's model (three subperiod 90-10 CRR)
- 0057(4-8): calibration to get the one month uptick and downtick factors
- 0057(9): computing the one month logarithmic risk-free factor
- 0057(10): transition slide
- 0057(11): computing the one monnth risk-free factor
- 0057(11): comparing the uptick and downtick factors to the risk-free factor
- 0057(12): dropping the 90-10 real-world probabilities
- 0057(13): computing the risk-neutral probabilities
- 0057(14): all portfolios have the same expected return each month
- 0057(15): setting up the template of prices and hedging parameters
- 0057(16-17): putting the share prices in the template
- 0057(18-19): using the payoff function to work out the contingent claim = the ending hedge values
- 0057(20): working out the hedge values after two months, using risk-neutrality
- 0057(21): working out the hedge values after one month, using risk-neutrality
- 0057(22): working out the starting hedge value = the price of the option, using risk-neutrality
- 0057(23-24): going directly from the contingent claim to the option price, using risk-neutrality
- 0057(25): summary
- work out the underlying in forward time
- use the payoff function to find the contingent claim
- use risk-neutrality to work back to the price (either in stages or in a single step)
- 0057(26): transition slide
- 0057(27): plan to use \Delta-hedging, a.k.a. "pricers got hedge"
- 0057(28): working out one hedging parameter (number of shares) as the needed naïve volatility of the option divided by the naïve volatility of a single share
- 0057(29): a comment on rounding error
- 0057(30): filling that hedging parameter into the template
- 0057(31): too few shares gives too little naïve volatility; too many gives too much
- 0057(32-33): working out the other hedging parameter (bank loan) at that same node via the equation of the node
- 0057(34): filling in all the other hedging parameters into the template
- 0057(34): underlying in forward time, the derivative in backward time, hedging parameters in any order
- 0057(35): answers to more decimals
Topic 0058
(Pricing/hedging many subperiods, 1)
Link to Topics website
- 0058(1): title slide
- 0058(2): a pricing problem involving Harry and Gail involving a 30 day call option on stock
- 0058(2): Gail's model (N subperiod 50.001-49.999 CRR, with N = 30 x 24 x 60 x 60 = number of seconds in 30 days)
- 0058(3-7): calibration to get the one second uptick and downtick factors
- 0058(8-10): computing the one second logarithmic risk-free factor
- 0058(10): computing r := the one second risk-free factor
- 0058(10-11): the one-second risk-free factor is the average of the uptick and downtick factors
- 0058(12): transition slide
- 0058(13): recall the one-second uptick and downtick factors and the one-second risk-free factor
- 0058(14): goal: price via hedging; difficulty: many subperiods; savlation: CLT
- 0058(15): the payoff function
- 0058(16): computing the risk-neutral probabilities
- 0058(17): returns are the same in the risk-netural world
- 0058(18): the evolution of the share price in our model, organized by a large recombinant tree
- 0058(18): the ending share prices
- 0058(19): the contingent claim, via the payoff function
- 0058(20): V = price of option = initial value of hedging portfolio
- 0058(21): e^{rN}V = expected contingent claim
- 0058(21): the probability of each outcome in the contingent claim
- 0058(22): an equivalent coin-flipping game
- 0058(23): hard to find the expected value in the coin-flipping game
- 0058(24): first do a probability problem, then move to an expected value problem
- 0058(24): the probability problem: compute Pr[-\sqrt{N}
Topic 0059
(Central Limit Theorem)
Link to Topics website
- 0059(1): title slide
- 0059(2): recall the probability problem: compute Pr[-\sqrt{N}0059(3): rephrasing the problem in terms of the standardization, X := (H-T)/\sqrt{N}, of H-T
- 0059(4-5): notation H_j, T_j, D_j
- 0059(5): the program: D_1, D_1/7, D_2, D_N, X = D_N/\sqrt{N}
- 0059(6-9): understanding D_1
- 0059(6-7): the distribution of D_1
- 0059(8): its generating function and Fourier transform (FT)
- 0059(9): inverse Fourier transform
- 0059(10-11): understanding D_1/7
- 0059(11): (the FT of the distribution of D_1/7) is obtained by replacing t by t/7 in
(the FT of the distribution of D_1)
- 0059(12-13): understanding D_2
- 0059(13): (the FT of the distribution) of D_2 is the square of (the FT of the distribution of D_1)
- 0059(14): understanding D_N
- 0059(15): understanding X
- 0059(16): transition slide
- 0059(17): themes (generating functions, Fourier transforms, Fourier analysis, infinite-dimensional spectral theory)
- 0059(17): recall the probability problem: compute Pr[-10059(17): (the FT of the distribution) of X is approximately equal to e^{-t^2/2}
- 0059(18): transition slide
- 0059(19): ( a random variable whose distribtion has FT equal to e^{-t^2/2} ) should be close to X
- 0059(19): approximate answer to the probability problem
- 0059(20): description of the distribution of a certain random variable Z
- 0059(21): Z has infinite support
- 0059(22): correcting a small mistake
- 0059(23): Pr[Z=7] = 0
- 0059(24): Pr[20059(25): generating function and Fourier transform of Z
- 0059(26): Z is close to X
- 0059(27): approximate answer to the probability problem: : Pr[-10059(27): Berry-Esseen Theorem bounds the error
- 0059(28): next: expected value in the coin-flipping game
Topic 0060
(Pricing/hedging many subperiods, 2)
Link to Topics website
- 0060(1): title slide
- 0060(2): recall the coin-flipping game and the expected value problem
- 0060(2): goal: the approximate expected value, E, of f(e^{Hu+Td})
- 0060(3): the expected value of f(D_2)
- 0060(4): the expected value of g(D_2), for any g
- 0060(5-7): the expected value of f(Z)
- 0060(8): the expected value of g(Z), for any exp-bdd g
- 0060(9-10): highlighting the change from Z to x, and then integrating against e^{-x^2/2}/\sqrt{2\pi} =: h(x)
- 0060(11): recall X = D_N / \sqrt{N}, with N = 30 x 24 x 60 x 60, and D_N = H_N - T_N = H - T = C_1 + ... + C_N
- 0060(12): the approximate expected value of g(X), for any exp-bdd g
- 0060(13): subgoal: choose g so that g(X) = f(e^{Hu+Td})
- 0060(14): make sure that g is exp-bdd
- 0060(15-16): writing H and T in terms of X
- 0060(17-19): writing e^{Hu+Td} as Ce^{kX}
- 0060(20): figuring out g and checking that g is exp-bdd
- 0060(21): figuring out C and k
- 0060(22): setting up an integral that is approximately equal to E, our goal
- 0060(23): transition slide
- 0060(24-30): evaluating the integral
- 0060(30): E is approximately 121.07
- 0060(31): recall the coin-flipping game, recall V = e^{-rN} E is the price of the option
- 0060(31): compute V approximately
- 0060(31): V is approximately 120.76
- 0060(32-36): computing E exactly as a sum involving binomial coefficients
- 0060(36): to two decimals, E is 121.11; the approximation we found was accurate to 4 cents
- 0060(37): to two decimals, V is 120.80; the approximation we found was accurate to 4 cents
- 0060(37-41): computing the (real-world) expected return of the stock after one year, from its drift and volatility
- 0060(41): let S_1 denote the stock price after 1 year
- 0060(41): augmented drift := [drift] + [1/2][(volatility)^2] = [the augmented expectation of (ln S_1]
- 0060(41): key point: [the expected return on the stock] is [the expectation of (the exponential of (ln S_1))], which is:
- the exponential of (the augmented expectation of (ln S_1)), or
- the exponential of the augmented drift
- 0060(41): the expected return is NOT the exponential of the drift
Topic 0061
(Intro to the Black-Scholes Formula)
Link to Topics website
- 0061(1): title slide
- 0061(2): set up a T-year European call option on one share of a stock struck at K
- 0061(2): S_0 := current price of the stock
- 0061(2): more notation for payoff function, drift, volatility, logarithmic risk-free factor
- 0061(2): asterisk subscript is used for annualized figures, unsubscripted figures are over T years
- 0061(2): choose (modeled) real-world uptick probability p, and let q := 1-p
- 0061(2): our sequence of models: the n-subperiod (p,q) CRR model, with n = 1,2,3,...
- 0061(2): the models "tend to Black-Scholes"
- 0061(2): goal: limit of (the option price under the nth model), as n ---> \infty
- 0061(3): the evoluation (over one subperiod) of the stock price, and its logarithm
- 0061(4-7): setting up the equations that must be solved to do the calibration
- 0061(8-10): finding the risk-neutral probabilities
- 0061(11): V_n := inital value of the hedge in the nth model = option price in the nth model
- 0061(11): goal is limit of V_n, as n ---> \infty
- 0061(11-14): finding a summation formula for the price of the option in the nth model
- 0061(14): the CRR Option Pricing Formula
- 0061(15): K' := present value of the strike price
- 0061(15): def'n: bogus at the money quotient, at the money quotient, logarithmic at the money quotient
- 0061(16): def'n: Black-Scholes center
- 0061(17): def'n: Black-Scholes interval with endpoints d_- and d_+
- 0061(18): transition slide
- 0061(19): theorem: \lim V_n = [S_0][\Phi(d_+)] - [K'][\Phi(d_-)]
- 0061(20): version 0 of the Black-Scholes Option Pricing Formula (BSOPF) is [S_0][\Phi(d_+)] - [K'][\Phi(d_-)]
- 0061(21): simplicity and centrality of the BSOPF
- 0061(22): inputs, outputs, asymptotic output
- 0061(23-27): change to annualized figures
- 0061(28-29): centrality and simplicity of BSOPF using annualized figures
- 0061(30-32): writing out the BSOPF using annualized figures
- 0061(32): first version of the BSOPF
- 0061(33-34): rewriting K' in terms of K, and adjusting the BSOPF to involve K not K'
- 0061(34): second version of the BSOPF
- 0061(35-36): rewriting the BSOPF in terms of forward prices
- 0061(36): third version of the BSOPF
- 0061(37): transition slide
- 0061(38): first, second and third version of BSOPF are present, neutral and forward formulas
- 0061(39): version zero is a time-normalized present formula
Topic 0062
(Testing the B-S Formula)
Link to Topics website
- 0062(1): title slide
- 0062(2): reall versions zero to four of the BSOPF (Black-Scholes Option Pricing Formula)
- 0062(2): do these formulas approximate the exact CRR price, with a large number of subperiods?
- 0062(3): recall: Kyle and Gail, with N = 30 x 24 x 60 x 60 subperiods
- 0062(4): recall: the exact CRR price
- 0062(5): transition slide
- 0062(6): recall the logarithmic risk-free factor
- 0062(7): transition slide
- 0062(8): recall the volatility and strike price
- 0062(8): recall the initial stock price
- 0062(8): recall the formulas for K', for d_+ and for d_-
- 0062(9): compute K'
- 0062(10): compute the logarithmic at the money quotient
- 0062(11): compute d_+ and d_-
- 0062(12): transition slide
- 0062(13): compute \Phi(d_+) and \Phi(d_-)
- 0062(14): transition slide
- 0062(15): compute the BSOPF price and compare with the exact CRR price
- 0062(16): the approximation is good
- 0062(17): transition slide
- 0062(18): BSOPF as a function, BlSch, of five inputs:
- tenor T
- annual volatility \sigma_*
- annual logarithmic risk-free factor r_*
- initial price of the stock S_0
- strike price K
- 0062(18): for fixed T,r_*,S_0 and K, this function of \sigma_* is increasing
- 0062(19): def'n: implied volatility
- 0062(20): Why teach BS (Black-Scholes)? Comparison with home mortgage interest rates.
- 0062(21): def'n: volatility smile and skew
- 0062(22): def'n: volatility surface
Topic 0063
(Prelim to the TCLT and B-S)
Link to Topics website
- 0063(1): title slide
- 0063(2): recall decay rates on Maclaurin error estimates
- 0063(3): decay rate on error estimate for e^{x_n} \approx 1 + x_n + (x_n)^2/2, with x_n ---> 0 as n ---> \infty
- 0063(3): decay rate on error estimate for e^{3x_n} \approx 1 + 3x_n + 9(x_n)^2/2, with x_n ---> 0 as n ---> \infty
- 0063(4): def'n: h(x) = e^{-x^2}/\sqrt{2\pi}
- 0063(4): def'n: Z_n ---> Z in distribution
- 0063(5): remembering the formula: change every Z_n to x and then integrate against h
- 0063(6): equivalent definition for Z_n ---> Z in distribution
- 0063(6): fact: adding a sequence of scalars tending to zero
- 0063(6): fact: multiplying by a sequence of scalars tending to one
- 0063(7): def'n: Z_n ---> Z in distribution against continuous exp-bdd
- 0063(7): same two facts for convergence in distribution against continuous exp-bdd
- 0063(8-9): recall: dividing the PCRV by 7 causes the Fourier transform to have t replaced by t/7
- 0063(10): recall: the Fourier transform for D_2 is the square of the one for D_1
- 0063(11-12): recasting D_2 as a sum of iid PCRVs whose distributions have Fourier transform cos t
- 0063(12): if you add two independent PCRVs, the Fourier transform is the product of the two Fourier transforms
- 0063(13): the idea of convolution of distributions
- 0063(14): Fourier transform simplifies convolution to multiplication
- 0063(15): if two PCRVs have distributions with nearly equal Fourier transforms, then their distributions are nearly equal
- 0063(16): if a sequence of PCRVs have distributions whose Fourier transforms approach e^{-t^2/2}, then the PCRVs approach Z in distriution
- 0063(17): notation for the Fourier transform of the distribution
- 0063(18): facts about Fourier transforms
- 0063(18): def'n: identically distributed (i.d.) set of PCRVs
- 0063(19): def'n and notation: [mean/expectation, variance, standard deviation (SD), Fourier transform of distribution (FTD)] of an i.d. set, A, of PCRVs
- 0063(19): def'n: c+A, cA
- 0063(19): (A is i.d.) implies (cA is i.d.)
- 0063(19): formulas for mean, variance and SD of cA
- 0063(20): formulas for mean, variance and SD of c+A
- 0063(21): def'n: S := \{ standard PCRVs \}
- 0063(21): S is not i.d.
- 0063(22): an i.d. subset of S has mean 0 and variance 1
- 0063(22): an i.d. set meeting S must be contained in S
- 0063(23): def'n: iid n-fold sum, notation \sum_n
- 0063(24): (A is i.d.) implies (\sum_n A is i.d.)
- 0063(24): formulas for mean, variance, SD and FTD of \sum_n A
- 0063(24): formula for [ \sum_n on ((c/n)+A) ]
- 0063(24): formula for [ \sum_n on (cA) ]
- 0063(24-25): renormalized iid sum preserves and reflects standardness
- 0063(26): def'n: B^{p,u}_{q,d}, specific set of binary variables
- 0063(26): B^{p,u}_{q,d} is i.d.
- 0063(26): multplying a constant by B_{q,d}^{p,u}
- 0063(26): adding a constant to B_{q,d}^{p,u}
- 0063(27): def'n: B^p_q, general set of binary variables
- 0063(27): analysis of \sum^n B^p_q
- 0063(28): the iid sum of general binaries is the union of the iid sums of specific binaries
- 0063(29-30): \sum^n B^p_q is invariant under adding a constant or multiplying by a nonzero constant
- 0063(31): recall the definition of X_\circ, standardization
- 0063(31): \sum^n B^p_q is invariant under standardization
- 0063(32): iid products
- 0063(32): exp of an iid sum is an iid product
- 0063(32): formula for the mean of the an iid product
Topic 0064
(the Triangular CLT)
Link to Topics website
- 0064(1): title slide
- 0064(2): statement of the TCLT (Triangular Central Limit Theorem)
- 0064(2): the sequence p_n of probabilities is bounded away from 0 and 1
- 0064(2): Y_n is an standard iid sum of n binary variables, using probabilities p_n and 1 - p_n
- 0064(2): recall the definition of convergence to Z in distribution
- 0064(2): recall the definition of B^p_q
- 0064(3): the triangle of variables
- 0064(4): recall that the iid sum of general binaries is the union of the iid sums of specific binaries
- 0064(5): q_n := 1 - p_n
- 0064(6): start of proof of the TCLT
- 0064(7): identify Y_n as an iid sum of specific binary variables, using u_n and d_n
- 0064(8): transition slide
- 0064(9-11): find the equations to solve to find u_n and d_n
- 0064(11): solve the equations to find u_n and d_n
- 0064(12): show that u_n and d_n are bounded
- 0064(13): transition slide
- 0064(14-16): compute the Fourier transform [\phi(t/\sqrt{n})]^n of the distribution of Y_n
- 0064(17): goal is to show that [\phi(t/\sqrt{n})]^n convergest to e^{-t^2/2}
- 0064(18): transition slide
- 0064(19): focus on t=5
- 0064(19-21): second order approximation of \phi(5/\sqrt{n}), with error term
- 0064(22): transition slide
- 0064(23): [ 1 + (x/n) + o(1/n) ]^n ---> e^x finishes the proof of the TCLT
- 0064(24): TCLT, version 2
- 0064(25): def'n of convergence in distribution to \sigma Z + \mu
- 0064(26-28): equivalent definitions for convergence in distribution to \sigma Z + \mu
- 0064(29-40): TCLT, version 3
- 0064(29): start of statement; note that PCRVs may be nonstandard
- 0064(30): conclusion of statement
- 0064(31): transition slide
- 0064(32): start of proof: def'n of \mu_n and \sigma_n, standardization of the PCRVs, and use of first TCLT
- 0064(33): transition slide
- 0064(34-36): replacing \sigma_n by its limit
- 0064(37-39): replacing \mu_n by its limit
- 0064(40): recall the definion of convergence in distribution to \sigma Z + \mu
- 0064(40): conclude the proof
Topic 0065
(1st proof of Black-Scholes)
Link to Topics website
- 0065(1): title slide
- 0065(2-3): notation
- f(x) is the payoff function (for a call option)
- T is the tenor
- S_t is the stock price at time T
- X = \ln ( S_T / S_0 )
- r is the logarithmic risk-free rate over time T
- \mu is the drift of the stock over time T
- \sigma is the drift of the stock over time T
- 0065(3): we select as sequence of models (the n-subperiod (p,q) CRR models, with n=1,2,...)
- 0065(3): this models X as a sequence of PCRVs, X_n, in \Sum^n B^p_q
- 0065(4): change X to X_n for the modeled real world
- 0065(5): transition slide
- 0065(6): let u_n and d_n be the (unknown) uptick and downtick amounts
- 0065(6): then X_n \in \Sum^n B^{p,u_n}_{q,d_n}
- 0065(6): start calibration to find u_n, d_n from \mu,\sigma
- 0065(7): find the risk-neutral (R-N) probabilities p_n and q_n
- 0065(7): IOU: for n sufficiently large, the risk-free rate is between the downtick and uptick factors
- 0065(8): transition slide
- 0065(9): in the R-N world, change X_n to \tidle X_n \in \Sum^n B^{p_n,u_n}_{q_n,d_n}
- 0065(9-10): verify that the expeted value of e^{\tilde X_n} is e^r,
i.e., an investment in bank or stock yields the same expected return
- 0065(11-12): comparison of the real world (about which little is known for sure), the modeled real world, and the R-N world
- 0065(12): key point: in the R-N world, the nth forward option price is equal to the expected payout
- 0065(13): transition slide
- 0065(14): \mu_n := the R-N drift; \sigma_n := the R-N volatility
- 0065(15): compare calibration formulas that allow us to go from \mu,\sigma to u_n,d_n
to formulas that take us from u_n,d_n to \mu_n,\sigma_n
- 0065(16): transition slide
- 0065(17): more IOUs: as n ---> \infty, we have
- Girsanov: \sigma_n ---> \sigma
- p_n ---> p, q_n ---> q
- \mu_n ---> r - ( \sigma^2 / 2 )
- key point: the limit of \mu_n does not involve \mu
- as a result, \mu does not appear in the Black-Scholes Option Pricing Formula
- 0065(17): standardize \tilde X_n, getting ( \tilde X_n - \mu_n ) / \sigma_n
- 0065(18-19): apply TCLT version 2 to see that ( \tilde X_n - \mu_n ) / \sigma_n ---> Z in distribution against continuous exp-bdd
- 0065(20): replacing \sigma_n by its limit \sigma
- 0065(21): replacing \mu_n by its limit \nu := r - ( \sigma^2 / 2 )
- 0065(21): then Z_n := ( \tilde X_n - \nu ) / \sigma ---> Z in distribution againts continuous exp-bdd
- 0065(22): finding a test function g so that the forward option price is E[g(Z_n)]
- 0065(23-25): using Z_n ---> Z to reduce calculation of E[g(Z_n)] to an integration problem
- 0065(25): IOU: the solution to the integration problem
- 0065(26): conclude that the nth option price is given by version 0 of the Black-Scholes formula
Topic 0066
(IOUs in proof of B-S)
Link to Topics website
- 0066(1): title slide
- 0066(2-8): IOU: the solution to the integration problem
- 0066(2): recall the setup
- 0066(3): computing g(x) and recalling that h(x) = e^{-x^2/2} / \sqrt{2\pi}
- 0066(3): writing out the integral to be computed
- 0066(4): finding the lower limit of integration that allows us to drop the positive part
- 0066(5): splitting the resulting integral into two integrals, and computing one of them
- 0066(6): changing x to (x plus the linear coefficient) to handle the second integral
- 0066(6): some computations and simplificiations in the second integral
- 0066(7): transition slide
- 0066(8): computing the second integral
- 0066(9-12): IOU: for n sufficiently large, the risk-free rate is between the downtick and uptick factors
- 0066(9): completion of calibration
- 0066(10): limit (as n ---> \infty) of u_n\sqrt{n} is positive
- 0066(10): limit (as n ---> \infty) of d_n\sqrt{n} is negative
- 0066(11): limit (as n ---> \infty) of (r/n)\sqrt{n} is 0
- 0066(12): conclusion of proof
- 0066(13-23): IOU: p_n ---> p, q_n ---> q
- 0066(13): recall the calibration formulas for p_n, q_n
- 0066(13): get at asymptotics of p_n and q_n by studying the asymptotics of the formula
p_n e^{u_n} + q_n e^{d_n} = e^{r/n}
- 0066(13): asymptotics of e^{(c/\sqrt{n})+(k/n)}
- 0066(14): asymptotics of e^{u_n}
- 0066(15): asymptotics of e^{d_n}
- 0066(16): asymptotics of e^{r/n}
- 0066(17): asymptotics of p_n e^{u_n} + q_n e^{d_n}
- 0066(18-19): asymptotics of p_n e^{u_n} + q_n e^{d_n} = e^{r/n}
- 0066(20): asymptotics of p_n \alpha + q_n \beta
- 0066(21): transition slide
- 0066(22-23): asymptotics of p_n and q_n
- 0066(24-25): IOU: \sigma_n ---> \sigma
- 0066(24): (u_n - d_n) \sqrt{n} is constant
- 0066(25): (formula \sigma_n = ...) minus (formula \sigma = ...)
- 0066(25): show that \sigma_n - \sigma tends to 0
- 0066(26-30): IOU: \mu_n ---> r - ( \sigma^2 / 2 )
- 0066(26): recall that the expeted value of e^{\tilde X_n} is e^r,
i.e., an investment in bank or stock yields the same expected return
- 0066(26-27): recall that W_n := (\tilde X_n - \mu_n) / \sigma ---> Z in distribution against exp-bdd
- 0066(27): multiply E[e^{\tilde X_n}] = e^r by e^{-\mu_n} and write left hand side in terms of W_n
- 0066(27-29): use the fact that W_n ---> Z to express the limit of e^{r-\mu_n} as an integral
- 0066(30): computing that integral, show that e^{r-\mu_n} ---> e^{\sigma^2/2}
- 0066(30): conclude that r-\mu_n ---> \sigma^2/2, i.e., that \mu_n ---> r - ( \sigma^2 / 2 )