CrashMetrics: Preparing Your Portfolio for the Worst

Value at Risk tells you what to expect on a normal day. But what about the days that are not normal? What about crashes? Chapter 43 introduces CrashMetrics, which is Wilmott’s own creation. If VaR is about routine market conditions, CrashMetrics is the opposite side of the coin. It is about fire sales, panic, and the far-from-orderly liquidation of assets.

Why Banks Go Broke

Wilmott says there are two main reasons banks get into serious trouble.

First: lack of control over traders. Through bad luck, negligence, or straight up dishonesty, someone builds an unmanageable position. Then it is fifty-fifty. Either the trader is a hero and the bank makes a fortune, or the bank loses a fortune and the trader runs. Those are not great odds for the bank.

Second: extreme, unexpected, unhedgeable market moves. Crashes.

CrashMetrics is about the second reason. You cannot prevent crashes, but you can figure out the worst-case damage and protect yourself.

What Makes a Crash Special

A crash is not just a rise in volatility. It has two defining features.

One, it is a sudden fall in prices. Too fast to liquidate your portfolio in any orderly way.

Two, during a crash, all assets fall together. There is no such thing as a crash where half the stocks fall and the rest stay put. Technically, all assets become perfectly correlated. On normal days, correlations between stocks are moderate. That moderate correlation is what makes diversification work. An insurance company can insure your car because individual car accidents are uncorrelated. But insuring against an earthquake is a different matter. When everything moves together, diversification is useless.

This is exactly when VaR fails you. VaR relies on those moderate correlations being stable. During a crash, they all go to one. Data from the 1987 crash clearly shows this: correlations between S&P 500 constituents jumped to nearly one on crash day, then slowly returned to normal over the following days.

The CrashMetrics Approach

CrashMetrics does not care about the probability of a crash. It does not try to predict when a crash will happen. Instead, it asks: given that a crash of some limited size could happen, what is the worst that could happen to my portfolio?

No assumptions about probability distributions. No assumptions about timing. Just a constraint on how big the crash can be, and then find the worst outcome.

For a portfolio of options on a single stock, the change in portfolio value is some function $F(\delta S)$ of the change in the stock price. We want to find the minimum of this function over some range $-\delta S^- \le \delta S \le \delta S^+$.

If we can use a Taylor series approximation (delta-gamma), the portfolio change is:

$$\delta \Pi \approx \Delta \cdot \delta S + \frac{1}{2} \Gamma \cdot (\delta S)^2$$

When gamma is positive, this has a minimum at $\delta S = -\Delta / \Gamma$. Plug that back in and you get the worst-case loss.

But Wilmott warns that in practice, a Taylor series approximation often is not good enough. If you have knock-out options or other exotics in your portfolio, you need to use the actual formulas. A simple delta-gamma is not going to capture the sudden value drop at a barrier.

Platinum Hedging

Knowing the worst case is step one. Step two: make the worst case less bad. This is done by optimal static hedging, which Wilmott calls the Platinum Hedge (named after the plastic card that comes after green and gold).

The idea is straightforward. You buy or sell some hedging contracts (options, for example). Each hedging contract has a delta, a gamma, and a bid-offer spread. Adding these contracts changes your portfolio’s crash exposure. But you also pay a guaranteed cost in the form of the bid-offer spread.

So you are trading a guaranteed small loss (the spread) for a reduced worst-case loss. This is insurance. The Platinum Hedge is the optimal choice of how many hedging contracts to buy to make the worst-case outcome as painless as possible.

After the optimal hedge is in place, the portfolio value as a function of the crash size no longer passes through zero at $\delta S = 0$. You have shifted the whole curve upward at the worst point, at the cost of a small downward shift elsewhere.

The Multi-Asset Model: Crash Coefficients

A real bank portfolio has many stocks, not just one. How do you handle that?

CrashMetrics relates each individual stock’s crash behavior to a benchmark index (like the S&P 500) using crash coefficients. If the index drops by $x$%, then stock $i$ drops by $\kappa_i \cdot x$%. The crash coefficient $\kappa_i$ is not the same as the stock’s beta. It is measured from the extreme tails of return distributions, not from normal-day regressions.

This is an important distinction. Wilmott calls it the “rings of Saturn effect.” On normal days, individual stock returns and market returns show weak correlation and lots of scatter. During a crash, all the points line up tightly on a single line through the origin. The crash coefficient measures the slope of that line.

Preliminary research suggests crash coefficients are more stable over time than betas. And they do not need frequent updating because extreme market events are rare. You can download them from crashmetrics.com.

With crash coefficients, a portfolio of $N$ stocks and their options reduces to a simple two-dimensional problem: portfolio value versus index change. Instead of needing an $N+1$-dimensional graph, you need just two dimensions. The entire CrashMetrics methodology (worst-case finding, Platinum Hedging) carries over using $x$ instead of $\delta S$.

The portfolio change becomes:

$$\delta \Pi \approx D \cdot x + \frac{1}{2} G \cdot x^2$$

Where $D$ is the crash delta and $G$ is the crash gamma of the portfolio.

Time Decay and Multi-Index Models

CrashMetrics can incorporate time effects. Over a longer horizon (not just overnight), your portfolio also loses value from theta decay. This is added as a separate, decoupled term. The worst-case asset move and worst-case time decay can be calculated independently.

For portfolios exposed to multiple markets (US stocks and European stocks, for instance), CrashMetrics extends to a multi-index model. You fit extreme returns to multiple indices and look for worst cases over a rectangle (or some constrained region) in the space of index returns. No correlation between indices is needed, which is nice because crash correlations are nearly impossible to measure anyway.

Margin Hedging

Here is something clever. The thing that actually kills banks during crashes is usually not the paper loss on their portfolio. It is the margin call. When you write options on an exchange, you post margin. When the market crashes, you suddenly need a lot of cash to cover the margin. If you do not have it, you are done.

CrashMetrics handles this by redefining the Greeks. Instead of regular delta and gamma, you work with “margin delta” and “margin gamma” that incorporate the margin percentage. The methodology carries over directly. You find the worst-case margin call, then use Platinum Margin Hedging to minimize it.

The key insight: you can be delta hedged but not margin hedged. Metallgesellschaft learned this the hard way with their oil contracts. They had offsetting positions on paper, but the margin calls on the short-term futures drained their cash.

Counterparty Risk

OTC contracts do not have margin calls. But they have counterparty risk. If your counterparty goes broke during a crash, your “hedged” position is suddenly unhedged. CrashMetrics suggests dividing your portfolio by counterparty and running worst-case analysis on each slice separately.

The CrashMetrics Index

Wilmott also created a CrashMetrics Index (CMI), a kind of Richter scale for financial markets. It measures the magnitude of market moves and signals whether we are in crash territory. Unlike volatility indices (like the VIX), the CMI is not just volatility in disguise. It exploits the correlation spike effect and uses a logarithmic scale. The exact formula is proprietary, but it is based on one timescale, not two, and relies on the fact that during crashes all assets move together.

Why This Matters

CrashMetrics fills a gap that VaR cannot. VaR works in normal conditions. CrashMetrics works in the exact conditions when you need risk management the most. It uses no probability assumptions, works with any portfolio structure, and gives concrete advice on how to protect yourself.

As Wilmott puts it, this analysis is fundamental to the well-being of financial institutions. And for that reason, he takes a non-probabilistic approach. You do not need to know the probability of a crash to protect yourself against one.

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