Modeling Default Risk in Stock Price Distributions and Option Values

Summary

The document considers how default changes the probability distribution of a stock price at a future date. It proposes representing the probability mass associated with survival through a maturity-dependent factor multiplying a standard lognormal price density. In a simplified setting with zero rates and dividends, the resulting call value is the usual Black–Scholes value scaled by that factor.

The author then describes calibrating the factor to an implied-volatility term structure while assuming constant volatility and no volatility skew or smile. The proposed setup is illustrative and leaves key modeling questions open: the stock price after default is undefined in the framing, and the suggested extension to stochastic or state-dependent default effects is not worked out. It is a conceptual calibration proposal, not a validated pricing model or empirical study.

Key ideas

• The author asks how to represent the probability that a stock remains defined and positive when default is possible.
• A deterministic maturity factor scales a lognormal stock-price density in the simplified example.
• The factor can be calibrated to match a term structure of implied volatilities under restrictive assumptions.
• Stochastic and local default effects are proposed as extensions but are not developed.

Tags

# Default risk and stock price probability distributions


# Default risk and stock price probability distributions












First of all, I realise this question might border on `meta-finance', so I'd be totally OK if it gets closed.

Having said that, the question itself:

Given a stock $S$, in the absence of default it is quite clear that the probability the stock price at time $T$ takes values in $(0,\infty)$ is $1$ for all $T$.

In the presence of default, however, the price at $T$ isn't even defined if the company defaulted at $t<T$. Can we express this mathematically as $P(0<S_T < \infty) = f(T) \leq 1$?

To illustrate what I mean with a simple example, consider the probability density of the stock price (zero interest rate and dividend yield for now) given by $$ q(S_T,T | S_t,t) = e^{-c(T,t)} p(S_T,T | S_t,t) $$ where $p$ is the usual lognormal density with constant volatility $\sigma$ and $c(t,T)$ is a positive deterministic function of $t$ and $T$. In this simple example let's assume that we only observe a term structure of IVs and no skew/smile.

Then, $$ E\left[(S_T-K)_+ \right] = e^{-c(T,t)} C^{BS} (S_t,K,\sigma,T) $$ We can calibrate $c(t,T)$ to the IV TS through $$ C^{BS} (S_t,K,IV(T),T) = e^{-c(T,t)} C^{BS} (S_t,K,\sigma,T) $$

I am working on generalising this to stochastic $c$ and/or a local $c$ to get skew/smile.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.