Investing Under the Threat of a Crash
In the previous chapter, we learned how to allocate money between risky and safe assets in continuous time. The answer was clean: hold a constant fraction in stocks and rebalance. But that result assumed the world follows a smooth lognormal random walk with no sudden jumps. What if you know that a crash could happen at any moment? Not a small dip, but a real crash, like October 1987. Chapter 67 of Wilmott’s book, written with Ralf Korn, tackles exactly this question. The answer is no longer a constant fraction, and the way the optimal allocation changes over time matches our intuition perfectly.
The Problem with Standard Models
The lognormal model that underlies Black-Scholes and Merton’s asset allocation work does not produce crashes. The stock price follows a continuous path. It can go down, slowly or quickly, but it cannot jump overnight from 100 to 70. Yet this is exactly what happens in real crashes.
Jump-diffusion models (like Merton’s from Chapter 57) do include jumps, but they only protect you on average. An investor following a jump-diffusion-based strategy will still suffer large losses during an actual crash. The hedging works in expectation but not for any individual crash event.
Wilmott and Korn take a different approach. They do not try to model the exact probability distribution of crashes. Instead, they assume you know two things: the maximum possible size of the crash (as a percentage drop), and the maximum number of crashes that can occur before your time horizon. Beyond that, you know nothing. You do not know when the crash will happen, how big it will actually be, or even if it will happen at all.
This is a worst-case framework. You are looking for the portfolio that gives you the best outcome even in the worst scenario.
Two Extreme Strategies
To build intuition, consider two extreme approaches.
Strategy 1: Play it safe. Put everything in bonds until a crash happens. After the crash (if it happens), you switch to the optimal stock allocation from the crash-free model. What is the worst case? The crash never happens, and you are stuck earning the bond rate the whole time while stocks are going up without you.
Strategy 2: Ignore the crash. Invest at the optimal Merton level (the fraction that maximizes growth in the crash-free world). What is the worst case? A maximum-size crash hits at the worst possible time, and your portfolio takes a massive hit.
Which strategy is better depends mainly on the time horizon. With a long horizon, the stock investment has time to grow, so Strategy 2 looks better. With a short horizon, the crash risk dominates, so Strategy 1 looks safer.
But neither extreme is actually optimal. Strategy 1 is too conservative. Strategy 2 is too aggressive. The best strategy lives between them, balancing the upside of stock investment against the downside of a crash.
The Balance Point
The optimal strategy has a beautiful structure. It is the portfolio where two worst cases yield the same outcome. Specifically, the expected utility after an immediate crash of maximum size equals the expected utility if no crash ever occurs.
Think about what this means. If you invested more in stocks, your no-crash outcome would improve but your crash outcome would worsen. If you invested less, the opposite. The optimal point is where these two scenarios are perfectly balanced.
This is an equilibrium condition. The crash and no-crash scenarios both give you the same worst-case utility. You cannot improve one without hurting the other.
The Optimal Portfolio Is Not Constant
In the crash-free Merton model, the optimal fraction in stocks is constant over time. But under the threat of a crash, the optimal fraction depends on time, and in a way that matches our intuition perfectly.
When the time horizon is far away, you invest more aggressively in stocks. Even if a crash hits, you have plenty of time to recover. The growth potential of stocks is too valuable to pass up.
As the time horizon approaches, you reduce your stock allocation. If a crash hits close to your retirement date (or whatever your horizon represents), you do not have time to recover. The risk is not worth it.
At the horizon itself, the optimal stock allocation is zero. A crash right at the end would be devastating, so you protect yourself completely.
This is exactly what financial advisors tell you to do: reduce your stock exposure as you approach retirement. The Wilmott-Korn model provides a rigorous mathematical justification for that advice.
Numerical Examples
Wilmott provides numerical examples with realistic parameters: stock drift of 20%, risk-free rate of 5%, volatility of 40%, maximum crash size of 20%.
Short time horizon (T = 1 year): The optimal crash-aware portfolio starts well below the crash-free optimal level (which is about 0.94). Even at the beginning of the year, the crash-aware investor holds less stock. As the year progresses, the allocation drops to zero at expiry. The best constant portfolio in the crash setting actually turns out to be zero, meaning a short-horizon investor worried about crashes should just hold bonds.
Long time horizon (T = 10 years): The picture changes dramatically. The optimal crash-aware portfolio starts much higher and stays high for most of the period. Only as time approaches the horizon does it drop toward zero. The best constant portfolio in the crash setting is now substantial, even exceeding the time-varying optimal portfolio at some points. With ten years to go, the growth opportunity is too valuable to sacrifice entirely.
The contrast between short and long horizons is the main practical takeaway. Your attitude toward crashes should depend on how much time you have to recover.
Multiple Crashes
The base model assumes at most one crash. But the extension to multiple crashes is straightforward through backward induction. If you have solved the optimal strategy for n-1 crashes, you can use that as the starting point for the n-crash problem. The role of the “after crash” value function from the one-crash case is replaced by the value function from the (n-1)-crash case.
As you might expect, more possible crashes mean less stock investment. Wilmott shows the three-crash case with the same parameters as the long-horizon example. The optimal allocation drops compared to the one-crash case but is still significantly positive for most of the time period. Even under the threat of three crashes, investing in stocks is worthwhile as long as the horizon is long enough.
Multiple Stocks
The model extends naturally to n stocks. The key difference from the single-stock case is that there can be more than one equilibrium strategy. With multiple stocks, you can construct equilibrium portfolios using different subsets of stocks. The optimal strategy is the equilibrium portfolio that delivers the highest worst-case utility.
In the multi-stock crash model, all stocks become highly correlated at the crash time. They all fall together, with the size of each stock’s drop being a multiple of some index-level crash. This is realistic. In real crashes, correlations go to one. Everything drops at the same time.
The optimal portfolio process is still deterministic (it depends only on time, not on the stock prices), and it can be found by backward induction from the terminal condition.
Changing Volatility After a Crash
Wilmott notes an important practical refinement. After a crash, volatility typically increases. This means the post-crash value function should be computed with a higher volatility parameter. In the multi-crash case, each “layer” might need a different volatility, reflecting the regime change that crashes tend to trigger.
What the Model Does Not Include
The paper ends by acknowledging several extensions left for future work:
Including the possibility of consumption (not just terminal wealth). Solving the problem for general utility functions (not just log utility). Adding liquidity constraints. Using derivatives for portfolio insurance alongside the stock and bond allocation.
These are all important practical considerations, but the basic framework already gives valuable insights without them.
Key Takeaways
Three big ideas from this chapter. First, under the threat of a crash, constant portfolio allocations are no longer optimal. The optimal strategy reduces stock exposure as your time horizon approaches, exactly matching the common financial advice of getting more conservative near retirement.
Second, the optimal portfolio balances two competing worst cases: the scenario where a crash hits immediately and the scenario where no crash ever occurs. You invest at the level where both scenarios give the same worst-case utility.
Third, the length of your time horizon matters enormously. With a long horizon, the optimal strategy involves significant stock investment even under crash threat, because you have time to recover. With a short horizon, the optimal allocation can drop to zero, because the crash risk dominates the growth opportunity.
Wilmott finishes with a satisfying observation. It is “particularly pleasing when a model produces results that tie in with intuition.” The recommendation to reduce risky exposure as retirement approaches is something every financial planner says. Now we know the math agrees.
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