Pricing a Defaultable Asset with a Brownian First-Passage Barrier
Summary
The document formulates a finite-maturity asset whose payoff depends on whether a drifted Brownian state process reaches a lower default threshold before maturity. If the threshold is hit, the asset pays nothing; otherwise, it pays a function of the terminal state, with both outcomes discounted at a constant rate. The question asks whether the price can be found by solving a backward valuation PDE with an absorbing boundary at default and a terminal payoff condition.
This is a first-passage or barrier-style pricing problem. The proposed PDE uses the process drift and diffusion as its differential operator, and the suggested valuation is the solution at the initial time and state. The document raises a relevant boundary concern: paths that hit the threshold should be treated as defaulted even if they could otherwise later recover. It does not provide a solution or derive the boundary conditions, and leaves the terminal payoff function unspecified. Pricing therefore requires resolving the model’s stopping and boundary details before applying a formula.
Key ideas
- The asset pays a terminal-state-dependent amount only if the process avoids a lower threshold through maturity.
- The threshold is modeled as a first-passage default event, so a hit before maturity triggers zero payoff.
- A backward PDE with drift, diffusion, discounting, an absorbing default boundary, and a terminal condition is proposed.
- The document poses but does not solve the pricing problem or verify its boundary conditions.
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Full text
# Pricing defaultable asset with finite maturity
# 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.