The Cliquet Option: A Volatility Case Study

Every few chapters, Wilmott stops talking about theory and shows you a concrete product that exposes why the theory matters so much. Chapter 56 does exactly this. The cliquet option is a structured product that looks innocent on the surface but hides extreme sensitivity to volatility modeling. If you price it with the wrong volatility assumptions, you can be off by a factor of ten in your risk estimate. That is not a rounding error. That is a blowup waiting to happen.

What Is a Cliquet Option?

A cliquet (sometimes called a ratchet option) is a series of forward-starting options linked together. The version Wilmott examines works like this:

You have a five-year contract with annual fixings. At each fixing date, the contract records the percentage return of the underlying over the past year, but caps it at 8% (the local cap). At the end of five years, you add up all these capped annual returns. If the total is above 16% (the global floor), you get the total. If it is below 16%, you still get 16%.

So the holder gets guaranteed downside protection (at least 16% over five years) plus upside participation (capped at 8% per year). This looks like a safe, boring product. It is anything but.

Why Gamma Changes Sign

The subtlety is in the cap and floor structure. For a standard call option, the payoff is convex (curves upward), and gamma is always positive. For the cliquet, the per-period payoff is capped. This cap creates a point where the payoff stops being convex and becomes flat (or even starts curving the other way when you include the global floor).

This means gamma changes sign. Somewhere between the previous fixing price and the cap level, the option value has a point of inflection. Below that point, gamma is positive. Above it, gamma is negative.

And here is the really nasty part: the point where gamma changes sign is not at a fixed stock price. It depends on the relative move from the last fixing. So at each annual reset, the inflection point shifts. A deterministic volatility surface that is calibrated to today’s vanilla options cannot capture this behavior because the relevant skew is a forward skew, and it pivots around a point that moves.

Wilmott says this directly: any deterministic volatility model fitted to vanilla prices is not going to properly model the risk from changing volatility. Even if you let the local volatility surface move up and down and rotate, it still cannot get the forward skew dynamics right for this product.

The PDE Approach

Wilmott prices the cliquet using finite differences rather than Monte Carlo. The uncertain volatility model requires knowing gamma at every point, and Monte Carlo does not naturally produce gamma. By introducing a similarity variable (the ratio of stock price to last fixing price), the four-dimensional problem reduces to three dimensions, making finite differences practical.

Between fixings, you solve the standard Black-Scholes equation. At each fixing date, you apply a jump condition to update the accumulated return. With constant 25% volatility, the plots show the non-convex shape but hide the real story. The cliquet looks tame with constant volatility.

The Uncertain Volatility Bombshell

Now Wilmott runs five pricing calculations with the same 20% to 30% volatility range:

1. Constant 25% volatility (the midpoint)
2. Constant 20% volatility (the lower bound)
3. Constant 30% volatility (the upper bound)
4. Uncertain volatility, best case (volatility ranges freely from 20% to 30%, chosen to maximize value)
5. Uncertain volatility, worst case (volatility ranges freely from 20% to 30%, chosen to minimize value)

The three constant-volatility curves (1, 2, and 3) are almost on top of each other. The contract value at 25% volatility is about 0.173. At 20% it is slightly higher. At 30% it is slightly lower. The difference between the extreme constant volatilities is about 0.0013, or roughly 0.75% of the mid price.

Looking at this, you might conclude the cliquet is not very sensitive to volatility. Vega would tell you the same thing. And you would be completely wrong.

The best-case and worst-case curves (4 and 5) are far from the constant-volatility cluster. The best case is much higher. The worst case is much lower. The spread between best and worst is about 0.0183, or roughly 10.5% of the mid price.

The true sensitivity to volatility is 14 times larger than what vega suggests.

Read that again. Fourteen times. This is not a minor discrepancy. If you are relying on vega to manage the risk of this product, you are underestimating your exposure by more than an order of magnitude.

Why the Gap Is So Large

The reason is the changing sign of gamma. When you use a constant volatility, the positive-gamma region and the negative-gamma region partially cancel each other out. Increasing volatility helps in the positive-gamma area but hurts in the negative-gamma area. The net effect is small because the two effects roughly offset.

But in the uncertain volatility model, volatility does not have to be the same everywhere. It is high where gamma is negative and low where gamma is positive (worst case). There is no cancellation. The worst-case scenario picks the volatility that hurts you everywhere simultaneously.

This is the fundamental difference between sensitivity to a parameter (vega) and sensitivity to a process (actual realized volatility path). Vega measures what happens when you shift a single number uniformly. The uncertain volatility model measures what happens when the number can vary freely within a range.

For any product with gamma that changes sign, these are radically different questions.

The Volatility Range Table

Wilmott includes a table showing cliquet values for different volatility ranges at five years before expiry. On the diagonal (where the upper and lower bounds are equal, meaning constant volatility), you see the constant-volatility values. They are all close together, ranging from about 0.173 to 0.175 as you go from 18% to 30%.

Off the diagonal, the story changes. For a range of 22% to 27%, the worst-case value is 0.165 and the best-case value is 0.183. For a range of 18% to 30%, the spread is even wider.

The diagonal values barely move. The off-diagonal values spread apart dramatically. This table is probably the single most powerful illustration in the entire book of why vega is inadequate for exotic options.

The Code

Wilmott includes Visual Basic code for pricing the cliquet with uncertain volatility using finite differences. The code is not production quality (he admits it “leaves much to be desired”), but it is transparent and shows exactly how the uncertain volatility model works in practice.

The key line is where the volatility is chosen based on the sign of gamma:

If Gamma > 0 Then Vol = VolMin

Otherwise, Vol defaults to VolMax. That is the entire uncertain volatility logic. At every grid point and every time step, check the sign of gamma and pick the volatility that gives the worst outcome. The rest is standard explicit finite-difference Black-Scholes.

This simplicity is part of the appeal. The uncertain volatility model does not require fancy calibration or stochastic simulation. It requires a finite-difference solver, a range for volatility, and one if-then statement.

The Lesson

Sensitivity to volatility is not the same as vega. For any product where gamma changes sign, vega dramatically underestimates the true exposure. The uncertain volatility model answers the question that actually matters: given that volatility could be anything within a range, how bad can things get?

The cliquet’s sensitivity is not to the level of volatility or to the current skew. It is to the rotation of forward volatility skew around a point near the risk-neutral expected stock price. No deterministic volatility surface can capture this.

Wilmott closes by calling the cliquet “pure genius, albeit an evil genius.” It looks safe but embeds enormous hidden volatility risk. Understanding why requires the full toolkit of chapters 52 through 56, starting with one simple admission: we do not know what volatility will be.

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