19:09:23 PM    from John Dodson to Everyone:
	Hi everyone. How is the assignment going?
19:09:38 PM    from Will Elliott to Everyone:
	Hey John, I believe I solved the SDE, drift term and M from equations in the book and the tau equation in the problem definition. I am having a hard time taking these results and moving forward with problem 2. I was wondering if you could maybe provide some ideas on progressing from problem #1 to #2. Thanks -- sorry no mic
19:15:17 PM    from John Dodson to Everyone:
	Once you have the parameters for the default stopping time and the contingent payout, you move on valuing the two legs of the swap. This means you will need to calculate some expectations, in particular the expected value of the indicator that default occurs prior to expiration $\chi_{\tau<T}$, and the expected value of  e^{-r t}\chi_{\tau<T}.
19:16:01 PM    from John Dodson to Everyone:
	both of these can be done using the CDF of $\tau$, which I provided recently.
19:17:36 PM    from John Dodson to Everyone:
	the second expected value requires a re-parameterization, basically you need to incorporate the $e^{-rt}$ into the density.
19:18:26 PM    from John Dodson to Everyone:
	Will E, do you mean the contingent payout?
19:19:11 PM    from John Dodson to Everyone:
	The trick here is to recognize that the value of the debt at default $B_\tau$ is $K-p_\tau$, and $p_\tau$ depends on $V_\tau$, which you know.
19:20:13 PM    from John Dodson to Everyone:
	of course the discounting, $e^{-r\tau}$ is random, but the payout itself is known.
19:20:49 PM    from John Dodson to Everyone:
	Does this make sense?
19:21:02 PM    from Doug to Everyone:
	I'm unclear what you mean by re-parameterization. How does that differ from integrating the product of $e^-{rt}$ and the density?
19:21:32 PM    from John Dodson to Everyone:
	It's not, but that's a really hard integral from first principles.
19:22:27 PM    from John Dodson to Everyone:
	it's easier to recognize that E[e^{-r\tau}] is proportional to the CDF of a slightly different $tau$.
19:23:36 PM    from Will Elliott to Everyone:
	John - thanks for the input on #2 I think I understand now
19:30:20 PM    from Doug to Everyone:
	Now I really don't understand what you mean by re-parameterization.  :)  Could you elaborate on "slightly different $tau$", please?
19:30:25 PM    from naili to Everyone:
	Hello John,  in #2, to determine p_tau, the L in the expression should be L or L'? 
19:31:31 PM    from John Dodson to Everyone:
	Nai, $L$ is the boundary for liquidation and $L'$ is the boundary for default, and we know that $L'>L$ as long as $r>0$ (which we can assume). The CDS will payout at default.
19:33:01 PM    from John Dodson to Everyone:
	We can also assume $V_0>L'$, or else the firm would already be in default.
19:35:14 PM    from Boyuan to Everyone:
	John, for part 2, are we evaluating the expectation of the term or just the expression for the term using F_0 information?
19:36:49 PM    from John Dodson to Everyone:
	The payout is $F_0$-measurable. You don't need to evaluate any expectations.
19:37:58 PM    from Boyuan to Everyone:
	OK, so it will be something involving the Brownian motion and tau as a random variable?
19:39:27 PM    from John Dodson to Everyone:
	Boyuan, since we know (under any filtration) what $V_\tau$ is, we know $B_\tau$. Does this make sense?
19:39:37 PM    from Boyuan to Everyone:
	Yes
19:43:09 PM    from Boyuan to Everyone:
	I'm trying to figure out what parameters should be in the final expression and what should not?
19:44:01 PM    from John Dodson to Everyone:
	Boyuan, it does get messy. Don't worry about optimizing too much. $M$ and $\gamma$ will be there, and probably also $L'$ and $\sigma$.
19:45:05 PM    from Boyuan to Everyone:
	OK. Thanks
19:46:01 PM    from Boyuan to Everyone:
	Also, for part 3, you mentioned the integration will be difficult. Any techniques that you think will be required?
19:49:41 PM    from John Dodson to Everyone:
	Boyuan, I recommend incorporating the $e^{-rt}$ into the definition of the density of $\tau$ and finding a $\tau'$ such that $E[e^{-rt}\chi_{t<T}]=const\times P{\tau'<T}$ 
19:52:39 PM    from Boyuan to Everyone:
	Sounds like an affine transformation of $\tau$?
19:53:34 PM    from John Dodson to Everyone:
	no, inverse Gaussians are not invariant under affine transformations. It is another inverse Gaussian.
19:54:23 PM    from Boyuan to Everyone:
	I see... Thanks John!
20:01:20 PM    from Doug to Everyone:
	I'm still not understanding what you're proposing to solve the integral.
20:06:32 PM    from John Dodson to Everyone:
	Doug, since $\int_0^T e^{-rt}f(t;\theta,M)dt$ is hard, convert it to $k\times\int_0^T f(t;\theta',M')dt=k\times F(T;theta',M')$ which you already have a formula for for some $k,\theta',M'$.
20:13:16 PM    from Doug to Everyone:
	I think I understand, now. Thanks, John.
20:35:11 PM    from John Dodson to Everyone:
	Did I miss anyone's question?
20:36:51 PM    from John Dodson to Everyone:
	Sorry about next Sunday. Feel free to contact me through next Saturday, and I will try to respond promptly.
20:37:47 PM    from Doug to Everyone:
	No apology necessary. Thank you for giving us the extra week.
20:38:14 PM    from John Dodson to Everyone:
	Sure. I hope you have found the assignments to be meaningful.
21:02:10 PM    from John Dodson to Everyone:
	Good night