Crash Modeling: Preparing for Market Meltdowns
The jump diffusion models from the previous chapter have a fundamental problem. You have to estimate the probability of a crash, and that is incredibly hard to do. How often does a 15% market drop happen? Once every 5 years? 10 years? 50 years? Nobody really knows. Chapter 58 of Wilmott’s book takes a completely different approach. Instead of guessing crash probabilities, it asks: what if the worst happens?
The Problem with Jump Diffusion
Wilmott is blunt about why jump diffusion is unsatisfactory. Two main weaknesses. First, you cannot hedge through a crash. When the market drops 15% in one day, your delta hedge is useless during that move. Second, the parameters (crash probability and size distribution) are nearly impossible to estimate reliably.
The crash model in this chapter takes a radically different philosophy. We do not make any assumptions about when the crash will happen or how likely it is. We do not even need a probability distribution for the crash size. We just say: the crash can be at most this big, and it will happen at the worst possible time for our portfolio. Whatever comes out of that analysis gives us a lower bound on our portfolio value and tells us how to hedge.
If the crash turns out to be smaller, or happens at a better time, we make more money than expected. This is a conservative approach, and Wilmott argues it is a more honest one than pretending crashes do not exist.
A Simple Example That Breaks Intuition
Before getting into the math, Wilmott gives a beautiful example that catches most people off guard. You hold a long call position, delta hedged in the Black-Scholes fashion. What is the worst that can happen if there is a crash?
Most people’s first instinct: a crash is bad, because the call loses value when the stock drops. Wrong. Remember, you are delta hedged. Your portfolio consists of the call plus a short position in the stock. Look at the portfolio value as a function of the stock price. Because of gamma, the portfolio value is at its minimum right at the current stock price. A move in either direction, up or down, actually increases the portfolio value.
So for a long, delta-hedged call, the worst case is that there is no crash at all. If you flip the signs and consider a short call position, then a crash is bad. And the key question becomes: when is the worst time for a crash to happen?
This is why crash modeling is not trivial. The worst-case timing depends on the specific portfolio, the size of the crash, and the dynamic evolution of the hedge. Positive gamma positions benefit from crashes. Negative gamma positions suffer. The model needs to account for all of this.
The Mathematical Framework
The crash model uses a modified binomial tree. At each time step, the stock can go up (to uS), down (to vS), or crash (to (1-k)S where k is the percentage crash size). So it is really a trinomial tree, but with the third branch going to a much more distant value.
The key idea is to introduce two value functions. V0 is the option value after the crash has happened (since only one crash is allowed in the simplest version). This is just the regular Black-Scholes value. V1 is the option value before the crash, meaning there is still one crash “in the tank.” We need to find V1.
At each node, we set up a hedged portfolio with the option and negative delta of the underlying. When the stock moves, the portfolio value changes differently depending on whether we get a normal up/down move or a crash. We want to choose delta to make the best of the worst outcome.
Case I happens when the crash would actually help our portfolio. In this case the worst outcome is the normal diffusive movement, and we hedge it in the usual Black-Scholes way. The resulting equation is exactly the standard Black-Scholes equation with no crash effects.
Case II happens when the crash would hurt us. Now we choose delta to optimize against the crash scenario. The hedge ratio is different from the usual Black-Scholes delta. It accounts for the jump from S to (1-k)S. The resulting equation is not the standard Black-Scholes equation and leads to different option values.
Which case applies depends on the specific stock price, time, and portfolio composition. For a short call with negative gamma near the strike, you are in Case II. For parts of the portfolio with positive gamma, Case I applies. The model automatically switches between the two cases as it works backward through the tree.
Value at Risk Under the Worst Case
Wilmott works through a concrete example. A portfolio of call options, 17.5% volatility, 6% interest rate, and a potential 15% crash. The Black-Scholes value of the portfolio at a stock price of 100 is 30.5. The worst-case value (assuming the crash happens at the worst possible time) is 21.2. The difference, 9.3, is the Value at Risk under this crash scenario.
This is a much more concrete measure of risk than traditional VaR, which depends on probability distributions. Here we say: if a 15% crash happens, the most we can lose is 9.3, and we have already accounted for optimal hedging during the normal market movements.
The Platinum Hedge: Making the Best of the Worst
The 9.3 VaR comes from the negative gamma in the portfolio around the current stock price. The natural question is: can we reduce it by buying some positive gamma?
This is where optimal static hedging comes in. We look at available traded options and figure out the best quantity to buy or sell to minimize the worst-case loss. In Wilmott’s example, there is a 90-strike call available with a bid of 11.2 and an ask of 12. The question is: how many of these should we buy?
The optimal static hedge (called the Platinum Hedge) turns out to be buying 3.5 of these 90 calls. After this hedge, the Black-Scholes portfolio value is 70.7, and the worst-case value is 65.0. The VaR has been reduced from 9.3 to 5.7. We paid a premium for the static hedge (buying at the ask price of 12), but it was worth it because it dramatically reduced our crash exposure.
Note that the optimal quantity depends on the bid-ask spread of the hedging instruments. If the spread were narrower, the optimal hedge might be different. This is a nice touch of realism that many models ignore.
Extensions: Multiple Crashes and Ranges
The basic model assumes one crash of fixed size k. Wilmott extends it in two directions. First, the crash size can cover a range instead of being fixed. The worst case is then chosen over both timing and size. Second, the model allows multiple crashes. You can either cap the total number of crashes (needing N+1 coupled value functions) or limit the frequency (no crash for at least omega time units after the last one, using a “time since last crash” variable).
Both extensions are tractable. The philosophy stays the same: assume the worst, hedge as best you can, and anything better is profit.
The Key Takeaway
The crash model is different from jump diffusion in a fundamental way. Jump diffusion says “jumps happen with probability lambda and have size distribution P(J).” The crash model says “a jump of at most this size will happen at the worst possible time.” No probability statements, no parameter estimation for the rare event.
This means the model is more conservative but also more robust. You do not need to estimate the probability of a crash, which is the hardest thing to estimate in all of finance. You just need a reasonable upper bound on the crash size. And the Platinum Hedge gives you a concrete, optimal strategy for minimizing the damage.
One final thought from Wilmott. The model does not account for the rise in volatility that typically accompanies a crash. But this is easy to add: just let the post-crash volatility be higher than the pre-crash volatility. If you use the frequency-limited model, you can even have the post-crash volatility decay exponentially back to normal levels. All of this adds realism without fundamentally changing the framework.
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