Uncertain Parameters: What If You Don't Know the Volatility?

Let us start with an uncomfortable truth. The Black-Scholes equation has three main parameters: volatility, interest rate, and dividend yield. Of these three, not a single one is known with certainty. Sure, you know today’s stock price. You know the expiry date. But the stuff that actually matters for pricing? You are guessing. Chapter 52 of Wilmott’s book takes this discomfort and turns it into a pricing framework.

What Can You Actually Measure?

Wilmott walks through each input to the Black-Scholes equation and grades it on how well you can measure it.

Stock price gets a decent grade. It is quoted in real time, though there is always a bid-ask spread, and different market makers show different numbers. Time to expiry is basically perfect. Today’s date and the expiration date are the easiest things to pin down.

After that, the grades drop fast.

Volatility is the big one. You can measure implied volatility from option prices or historical volatility from past returns. Neither tells you what volatility will actually be over the remaining life of the option. Historical volatility for a typical stock might bounce between 20% and 60% over a few years. That is a huge range. You simply cannot know the true volatility until the option has expired and you can look back at what happened.

Interest rate is also tricky. Black-Scholes technically needs the future path of the instantaneous spot rate, not the yield on a six-month bond. These are different things. Dividends are declared only a few months in advance. Before that, you guess based on history.

The honest answer is: you are building your pricing on a foundation of estimates.

Uncertainty Is Not Randomness

Here is where Chapter 52 gets philosophically interesting. Wilmott makes a sharp distinction between “random” and “uncertain.” In a random model (like stochastic volatility), you do not know what will happen, but you have a probability distribution describing the possibilities. You can compute expected values.

With uncertainty, you cannot even do that. All you know is that a parameter lies somewhere within a range. You have no probability density function, no expected value. You just know the boundaries. Think of it this way: with randomness, you know the shape of the dice. With uncertainty, you only know the dice has between one and six faces, but you have no idea which faces are more likely.

Since there are no probabilities, you cannot talk about expectations. Instead, you think in terms of worst case and best case.

The Worst-Case Approach

The framework from Avellaneda, Levy, Paras, and Lyons works like this. Assume volatility lies in some band, say between 20% and 30%. You do not know what it will be at any given moment, but you know it stays inside that range.

Now follow the standard Black-Scholes hedging argument. Build a portfolio with one option and a delta hedge. Normally, the hedge eliminates risk and you set the portfolio return equal to the risk-free rate. But now you do not know the volatility, so you cannot determine the portfolio return exactly.

The clever move: be pessimistic. Assume the volatility at every moment is whichever value hurts you the most. If you are long gamma (positive gamma), the worst case is low volatility because you need the stock to move to profit from your position. If you are short gamma (negative gamma), the worst case is high volatility because big stock moves hurt you.

This gives you a modified Black-Scholes equation where the volatility switches between its upper and lower bounds depending on the sign of gamma. The equation is nonlinear, which is a big deal. Linear equations are nice because the value of a portfolio equals the sum of its parts. Nonlinear equations break that rule.

Why Nonlinearity Matters

Here is the practical consequence that catches people off guard. In the uncertain volatility model, the value of a portfolio is not the sum of the values of its individual options. A long call valued alone is worth less than a short call valued alone (both in worst-case terms). This makes intuitive sense: if you own a call, the worst case says it is cheap; if you owe a call, the worst case says it is expensive.

But it goes further. The value of any contract depends on what else is in your portfolio. Putting two options together can give a portfolio value that exceeds the sum of the individual worst-case values. Wilmott captures this with a nice formula: the value of the whole is greater than the sum of the parts. He even calls it the “Beatles effect.”

This is fundamentally different from the linear Black-Scholes world, where you can price each option independently and add them up. In the uncertain volatility world, your book structure matters.

The Barrier Option Example

Chapter 52 includes a beautiful example with an up-and-out call. The option has a strike of 100, a barrier at 120, one year to expiry, and a 5% interest rate. Volatility is uncertain, somewhere between 17% and 23%.

The first thing Wilmott does is show what happens if you just price the option twice: once with 17% volatility and once with 23%. You get two curves. Around a stock price of 80, these two curves cross, and the option looks insensitive to volatility.

But that is completely misleading. The uncertain volatility model (which allows volatility to take any value in the range at any time, not just a single constant) produces a worst-case value that is dramatically different from either of the constant-volatility prices. At the stock price where the two curves cross, the uncertain-volatility worst case is much lower.

Why? Because the gamma of a barrier option changes sign. Near the barrier, gamma is negative (the option value curves downward). Far from the barrier, gamma is positive. In the uncertain volatility model, the worst case picks high volatility in the negative-gamma region and low volatility in the positive-gamma region. This is much worse than any single constant volatility.

This is why vega can be extremely dangerous for contracts with gamma that changes sign. The vega tells you sensitivity to a uniform shift in volatility. But the real risk comes from volatility being high precisely where it hurts you and low where it helps.

Uncertain Interest Rate and Dividends

The same framework applies to uncertain interest rates and dividends. For interest rates, if your portfolio has positive value, the worst case is high rates. For dividends, you get jump conditions at each dividend date. You can combine all three uncertainties at once, and the resulting equation is nonlinear in all of them.

The Spread Problem

There is a serious practical issue with this model. The worst-case and best-case prices can be very far apart. Wilmott gives an example: a European call with strike 100, stock at 100, six months to expiry, interest rate between 5% and 6%, and volatility between 20% and 30%. The worst-case long price is $6.85 and the worst-case short price is $9.85. That spread of $3 is much larger than typical market bid-ask spreads.

This means the model, as presented in this chapter, produces ranges too wide to be practically useful. You cannot go to your risk manager with a $3 uncertainty on an option worth about $8.

Fortunately, there is a fix. Static hedging can tighten these spreads dramatically. If you add traded options to your portfolio, the nonlinearity works in your favor. The extra options can offset the regions where gamma changes sign, reducing the gap between best and worst cases. Wilmott promises a full treatment of this in Chapter 60, and it is one of the most useful applications of the uncertain volatility framework.

Why Wilmott Loves This Model

Out of the three main approaches to modeling volatility beyond constant Black-Scholes (deterministic calibrated smile, stochastic volatility, and uncertain volatility), Wilmott says uncertain volatility is “easily my favorite.” That is a strong statement from someone who literally wrote the book.

His reasoning is practical. Greeks like vega and rho differentiate with respect to constants, which can be misleading. The uncertain volatility model produces ranges instead of single numbers, which is more honest about what we actually know. And with static hedging, those ranges can be made tight enough to trade on.

The key lesson: when you do not know a parameter, do not pretend you do. Price for the worst case and let the math handle the uncertainty.

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