# Pricing defaultable asset with finite maturity

Assume a stochastic process $X_0 = 0$ and $X_t = \nu t + \sigma W_t$ where $W_t$ is standard Brownian motion and $\nu$ is a drift (can have $\nu \leq 0$ if necessary, but prefer it to be general), and $\sigma > 0$. Furthermore,

• Let $\underline{X} < X_0$ be an exogenously given default threshold.
• Define the first passage time as the random stopping time: $T = \inf\{ 0 < t \mid X_t=\underline{X} \}$.
• Define $0 < \bar{T} < \infty$ as a terminal time.

An asset pays out as follows:

• If default has not happened by the terminal date (i.e., $T > \bar{T}$), the asset pays $p(X)$
• If default does occur (i.e. $T \leq \bar{T}$), the asset pays $0$
• Everything is discounted at rate $r> 0$

QUESTION: Given a $\nu,\sigma^2,\bar{T}, X(0),\underline{X}$, how do we price this asset?

Is this standard? More abstractly (and forgive my poor knowledge of terminology), is this linear combination of standard debt and equity instruments? Looks a lot like a some weird combination of a stock market futures contract and a credit default swap, but I can't figure it out? If it is standard, then we may be able to just add up the solution from some standard formulas.

Attempts at solution: My gut tells me this can be written recursively with a Bellman equation such as: $$r V(t,X) = \nu V_X(t,X) + \frac{1}{2}\sigma^2 V_{XX}(t,X) + V_t(t,X)$$ With boundary values, $$V(t,\underline{X}) = 0, \text{for all t < \bar{T}}$$ and $$V(\bar{T},X) = p(X), \text{for all X \geq \underline{X}}$$

And if we could solve PDE, then the price is: $V(0,X(0))$. Is this correct, or are the boundary values incorrect because they let the $X$ dip below $\bar{X}$ and then go above to gain potentially be higher at $\bar{T}$?

If this is the correct PDF and boundary values, I can attempt a direct solution.

• Might this be some sort of defaultable forward contract? – jlperla Jan 9 '17 at 20:09
• Seems to me as a continuously monitored down-and-out barrier (with payout $p()$) + one-touch put option (with coupon $d$). You could indeed price it with the BS PDE along with appropriate boundary conditions or, simply Monte Carlo. I guess there should even exist a closed form expression under your simple ABM modelling assumption. – Quantuple Jan 10 '17 at 8:13
• Thank you so much for your comments. This is for some theory, so if I need numerical techniques to solve it, then I have written down the problem wrong (or, maybe, the theory is wrong!) – jlperla Jan 10 '17 at 16:56
• @Quantuple I made a small change to remove the $d$, and now it pays nothing if $X$ hits the boundary. Does this change things? Is this now isomorphic to pricing a future on with one-sided default? I looked up "barrier options" but this looks like a future, rather than an option, to me? – jlperla Jan 10 '17 at 16:59
• Have a read through section 0.5 here: google.be/url?sa=t&source=web&rct=j&url=https://…. Does it help you? Although pricing is conducted under GBM modelling assumptions you should have no problem translating what's done to your ABM dynamics – Quantuple Jan 10 '17 at 21:35