Jump Diffusion: When Markets Jump Instead of Walk
Here is a thing that bothers every honest quant at some point. The lognormal random walk, the thing Black-Scholes is built on, assumes that stock prices move smoothly. Small steps. Continuous paths. Nice and clean. But if you have ever watched a market during a crisis, you know that prices do not always walk. Sometimes they jump. Chapter 57 of Wilmott’s book tackles this head on and introduces jump diffusion models.
The Evidence Is Right There in the Data
Wilmott starts with actual data. He takes Xerox returns from 1986 to 1997, normalizes them, and compares the distribution to a Normal distribution. What does he find? The real distribution has a much higher peak in the center and much fatter tails. In other words, small moves happen more often than the Normal distribution predicts, and large moves also happen more often. The middle ground is where the Normal distribution overestimates.
This is not unique to Xerox. Every market shows this pattern. Equities, currencies, interest rates. The tails are fatter than what a Normal distribution would give you. And when you are hedged against small moves (as most option traders are), those fat tails become a real problem.
Wilmott uses a Q-Q plot to drive this point home. If the returns were truly Normal, the Q-Q plot would be a straight line. Instead it curves away from the straight line in both tails. There is extra weight where it matters most: in the extreme moves that can blow up your portfolio.
Three explanations are commonly given for this non-Normality. Volatility could be stochastic (covered in Chapter 51). Returns might follow some other distribution entirely, like a Pareto-Levy distribution. Or assets can jump in value. This chapter focuses on the third explanation.
The Poisson Process: A Building Block for Jumps
To model sudden jumps, we need a new mathematical tool beyond Brownian motion. Enter the Poisson process. The idea is simple. At any tiny time interval dt, there is a small probability of a jump happening. That probability is proportional to dt and controlled by a parameter called the intensity, usually written as lambda.
So if lambda is 2, you expect roughly 2 jumps per year. But you do not know when they will happen. The jump could come tomorrow or six months from now. And between jumps, everything is quiet.
Wilmott combines the Poisson process with the usual Brownian motion to create the jump-diffusion model. The asset follows its normal random walk most of the time, but every now and then, boom, the price suddenly multiplies by some factor J. If J is 0.9, that is a sudden 10% drop. If J is random itself, drawn from some distribution, the model becomes even more flexible.
This is Merton’s 1976 model, and it remains one of the most important extensions of Black-Scholes in the literature.
The Hedging Dilemma
Here is where things get uncomfortable. In the Black-Scholes world, you can delta hedge continuously and eliminate risk. With jump diffusion, you cannot. When a jump happens, the stock price moves instantly by a large amount. You cannot adjust your hedge during the jump. It is already over before you can react.
So you face a choice. Do you hedge the continuous diffusive part (the normal small moves) or do you try to hedge the jumps? You cannot do both perfectly with one hedge.
Strategy 1: Hedge the diffusion. You set delta equal to the usual partial derivative of the option value with respect to S. This works fine between jumps. But when a jump hits, your portfolio takes a hit that is not hedged. Merton argued that if the jump risk is uncorrelated with the overall market, you should not have to pay a premium for it. Diversifiable risk should not be rewarded. Under this assumption, you take expectations over the jump sizes and arrive at a pricing equation.
The result is a partial integro-differential equation. Unlike Black-Scholes, it links option values at distant stock prices, not just local derivatives. This makes sense: the value of your option here depends on the prices it could instantly jump to.
When the logarithm of J is Normally distributed, there is a nice closed-form solution. The option price becomes a weighted sum of Black-Scholes values, where each term assumes a certain number of jumps have occurred, and the weights come from the Poisson distribution.
Strategy 2: Hedge the jumps too. Instead of just hedging the diffusion, you could choose your delta to minimize the overall variance of the hedged portfolio. This accounts for both the continuous moves and the occasional jumps. The formula for this optimal delta is more complex but the idea is straightforward: find the hedge ratio that makes the portfolio as stable as possible across all outcomes.
The resulting pricing equation is different from Strategy 1. Both reduce to Black-Scholes when the jump intensity lambda equals zero. But for nonzero lambda, they give different option prices. This is one of those places in quant finance where the “right” answer depends on your hedging philosophy.
Perfect Hedging with Multiple Options
There is a third approach. If you know the exact size of the jump but not its timing, you can set up a portfolio of two options and the underlying that hedges both the diffusion and the jump risk. But you end up with one equation and two unknowns, which means a market price of jump risk shows up in your model. Wilmott is not a fan of this, and honestly I am with him on that. Market prices of risk are basically free parameters that you have to estimate, which defeats the purpose of having a model.
If the jump can take two possible values, you need three options plus the underlying. If the jump has n possible states, you need n+1 options. If the jump size has a continuous distribution, you need a continuum of options. So in theory you can hedge perfectly, but in practice the number of instruments required makes it impractical.
Jump Volatility: A Smarter Use of Poisson
The last part of Chapter 57 takes the Poisson process in a different direction. Instead of the stock price jumping, what if volatility jumps?
Think about it. Volatility is often roughly constant for a while, then something happens and it spikes, then it gradually settles back down. This looks less like a diffusion process and more like a jump process with some decay.
Wilmott models volatility as being in one of two states: a low volatility state and a high volatility state. The transitions between states are governed by Poisson processes with their own intensities. When volatility is in the low state, there is some probability per unit time of jumping to the high state, and vice versa.
This gives you two coupled equations, one for the option value in each volatility state. If you hedge the asset price risk with the underlying but take real expectations over the volatility jumps, you get a system that is tractable and captures something real about how markets behave.
The more sophisticated version adds deterministic decay after each jump. Volatility spikes to some high level, then exponentially decays back toward the low level, until the next spike. This matches the empirical observation that volatility clusters and mean-reverts. The option value now depends on the time since the last volatility jump, adding another variable to the problem but keeping the structure clean.
The Honest Downsides
Wilmott ends with a refreshingly honest assessment. Jump diffusion models have three main problems.
First, parameter estimation. The basic lognormal model has one parameter to estimate: volatility. That is the sweet spot. Jump diffusion needs at least the jump intensity and the jump size distribution. More parameters means more estimation error.
Second, the governing equation is harder to solve. It is a partial integro-differential equation instead of a simple diffusion equation. The nonlocal integral term makes numerical solutions more expensive.
Third, and most fundamentally, perfect hedging is impossible. Any hedge you set up will be exposed to jump risk. Wilmott makes a sharp observation here: the standard market practice is to stick with Black-Scholes and pretend your hedge is perfect. He calls this the “eyes wide shut” approach. It is wrong, but it has become market standard because the alternative (admitting your hedge is imperfect) makes people uncomfortable.
Jump diffusion captures something real and important. But using it requires accepting that your position carries unhedgeable risk. For some people, that honesty is a feature. For others, it is a deal-breaker.
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