Probability in Finance: Density Functions and First-Exit Times
Most of derivative pricing theory goes out of its way to avoid thinking about probability. The whole point of hedging and no-arbitrage is to eliminate uncertainty. You do not need to know where the stock is going; you just need to build a portfolio that does not care. But Chapter 10 of Wilmott’s book asks us to step back and look at the randomness underneath. Where might the stock actually end up? How long before it hits a certain level? These questions matter for American options, for speculation, and for understanding what the math is really doing.
The Transition Probability Density Function
The fundamental object in this chapter is the transition probability density function p(y, t; y’, t’). In plain terms: given that a random variable y has value y at time t, what is the probability that it takes a value between a and b at some future time t'?
Prob(a < y’ < b) = integral from a to b of p(y, t; y’, t’) dy’
Think of y as a stock price and t as today. The density function tells you the distribution of possible future stock prices. It is a function of four things: where you start (y, t) and where you end up (y’, t’).
This function satisfies two different PDEs, depending on which pair of variables you differentiate with respect to.
The Forward Equation (Fokker-Planck)
The forward Kolmogorov equation (also called the Fokker-Planck equation) involves derivatives with respect to the future state y’ and future time t’. Wilmott derives it using a trinomial tree approximation, which is the most intuitive approach.
The idea: the random variable can go up by some amount delta_y, stay the same, or go down. The probabilities of each move are chosen to match the mean and variance of the continuous-time random walk. By expanding in Taylor series and taking limits, you get:
dp/dt’ = (1/2) * d2(B^2 * p)/dy’^2 - d(A * p)/dy’
Where A(y, t) is the drift and B(y, t) is the volatility of the random walk.
Use this equation when you know where you are now and want to see the distribution of future outcomes. For example, you know the stock is at 100 today and want to know the probability distribution of prices in 6 months.
For the lognormal random walk (dS = muSdt + sigmaSdW), the special solution starting from a known initial point is:
p(S, t; S’, t’) = [1 / (S’ * sigma * sqrt(2pi(t’-t)))] * exp(-(log(S’/S) - (mu - sigma^2/2)(t’-t))^2 / (2sigma^2(t’-t)))
This is the lognormal distribution that forms the backbone of the entire Black-Scholes framework. The stock price at any future time is lognormally distributed, with the mean return controlled by mu and the spread controlled by sigma.
The Backward Equation
The backward Kolmogorov equation involves derivatives with respect to the current state y and current time t:
dp/dt + A * dp/dy + (1/2) * B^2 * d2p/dy2 = 0
Use this equation when you have a specific future state in mind and want to know the probability of reaching it from various starting points. The derivation is more subtle. Wilmott uses a pub crawl example to explain it.
You are in the office at 5pm (current state). At 6pm, you will be at the pub, still at the office, or at Madame Jojo’s (one step forward in time). You want to know the probability of being in bed at midnight (the future target). That probability equals the probability of going to the pub at 6pm times the probability of reaching bed from the pub, plus the probability of staying at the office times the probability of reaching bed from the office, plus the probability of going to Madame Jojo’s times the probability of reaching bed from there.
This reasoning leads directly to the backward equation. And here is the important observation: the backward Kolmogorov equation looks almost identical to the Black-Scholes equation.
First-Exit Times
The first-exit time is when a random variable first reaches a specified boundary. When will the stock double? How long before an interest rate drops to 2%? When should an American option be exercised?
Probability of Exit
To find the probability of leaving a given range before a given time, solve the backward equation with specific boundary and final conditions:
- If y is on the boundary of the region, the probability of exit is 1 (you are already out)
- If y is inside the region at the final time t’, the probability is 0 (no time left to exit)
Expected Exit Time
Once you have the cumulative distribution of exit times, you can compute the expected first-exit time u(y, t). It satisfies:
A * du/dy + (1/2) * B^2 * d2u/dy2 + du/dt = -1
The -1 on the right side is like a clock ticking. For the time-homogeneous lognormal random walk, the expected time for a stock starting at S to leave the range (S_0, S_1) has a closed-form solution involving logarithms.
Optimal Stopping: When to Sell
Here is a practical application. You hold some investment. It goes up, it goes down, it is not really trending. When should you sell?
Wilmott sets this up as an optimal stopping problem. You want to sell at the time that maximizes the expected present value of the sale price. The governing equation for the maximum expected value V(S) is:
muSV’ + (1/2)sigma^2S^2V’’ - rV = 0
with the constraint V(S) >= S (you can always just sell now for S).
For a lognormal asset, the result depends on the relationship between the drift mu and the risk-free rate r:
- If mu > r: Never sell. The asset’s expected growth exceeds the discount rate, so waiting is always better. (There is no finite optimal strategy.)
- If mu < r: Sell immediately. The asset grows slower than the discount rate, so holding it destroys value.
This is a simplified analysis that ignores risk (Wilmott notes this caveat), but it gives clean, intuitive results.
The Connection to Black-Scholes: Risk-Neutral Pricing
Here is where everything comes together. The backward Kolmogorov equation for the present value of an expected payoff looks like:
dV/dt + muSdV/dS + (1/2)sigma^2S^2d2V/dS2 - rV = 0
Compare this to the Black-Scholes equation:
dV/dt + rSdV/dS + (1/2)sigma^2S^2d2V/dS2 - rV = 0
The only difference: mu is replaced by r. If we pretend that the stock grows at the risk-free rate r instead of its actual drift mu, then the option price equals the discounted expected payoff:
V(S, t) = e^(-r(T-t)) * E[Payoff(S_T)]
where the expectation is taken under the risk-neutral random walk:
dS = rSdt + sigmaSdW
This is the core insight of risk-neutral pricing. You do not need to know the actual drift mu of the stock. The market eliminates it through hedging. What remains is a world where everything grows at the risk-free rate on average, and option prices are just discounted expected values in that artificial world.
Wilmott calls this the main contribution of the martingale approach to pricing. He also calls it “a bit of a one-trick pony” and “a bit of a dead end,” noting that the most interesting problems in finance arise precisely when this simple relationship breaks down.
A Common Misconception
Wilmott debunks a popular myth: delta is NOT the probability of an option ending in the money.
The probability of a call ending in the money is:
N(d1’) where d1’ = [log(S/E) + (mu + sigma^2/2)(T-t)] / [sigma*sqrt(T-t)]
The delta of a call involves:
N(d1) where d1 = [log(S/E) + (r - D + sigma^2/2)(T-t)] / [sigma*sqrt(T-t)]
Two differences: mu versus r (real world versus risk-neutral world), and a sign difference. Both matter. The option price has nothing to do with real-world probabilities, and the plus/minus difference means the numbers are different even structurally.
Why Probability Still Matters
Even though hedging eliminates the need for probability in pricing, probability thinking is essential for:
American options: The value depends on sigma through hedging, but the expected exercise time depends on mu. You know the payoff is hedged, but you do not know if you will still hold the option at expiry.
Speculation: If you do not hedge (or cannot hedge perfectly), your P&L is at the mercy of actual price movements. Understanding the real probability distribution is critical.
Risk management: VaR, stress testing, and scenario analysis all require thinking about what might actually happen, not just what the risk-neutral world says.
Wilmott closes by noting that a thorough understanding of financial markets requires acknowledging the underlying randomness, even if the most elegant parts of the theory manage to work around it.
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