Stochastic Volatility Meets Mean-Variance Analysis
Wilmott does not like the market price of risk. He says so right at the start of Chapter 54, and his reasoning is solid. The market price of volatility risk is not directly observable. You can only back it out from option prices, and that only works if the people setting those prices are using the same model you are. If you refit the model a few days later and get a different answer, was the market wrong before? Or is it wrong now? You end up chasing your own tail.
This chapter offers an alternative. Instead of trying to build a perfect risk-neutral model that gives you a single “correct” price, Wilmott says: accept that you cannot perfectly hedge when volatility is stochastic. Instead, compute the mean and variance of your hedged portfolio, and use those to make trading decisions.
The Setup
The model is classical. The stock follows geometric Brownian motion with some drift, and its volatility follows a separate stochastic process with drift alpha and diffusion beta. The two Brownian motions driving the stock and the volatility can be correlated.
You build a portfolio: one option, hedged with delta shares of the underlying. In the standard Black-Scholes world with known constant volatility, you can perfectly hedge the option and the hedged portfolio is risk-free. But when volatility is stochastic, you have two sources of randomness (stock and volatility) and only one hedge instrument (the stock). You cannot eliminate both risks with just delta hedging.
The result: your hedged portfolio is not risk-free. Its value at expiry is random. Wilmott proposes studying the statistical properties of this random portfolio value, specifically its mean and its variance.
Computing the Mean
The mean of the portfolio value satisfies a PDE that looks a lot like the risk-neutral pricing equation for stochastic volatility. In fact, it is almost the same, but with a specific “market price of risk” that drops out naturally from the analysis. The effective market price of risk turns out to be proportional to the correlation between the stock and volatility, times the stock’s excess return over the risk-free rate, divided by the stock’s volatility.
This is different from the classical approach where you fit the market price of risk to option prices. Here it comes from the physical (real-world) dynamics. You do not need to calibrate to the option market at all.
Computing the Variance
The variance equation is trickier. Wilmott derives a PDE for the variance that has a “forcing term” proportional to the square of the derivative of the mean with respect to volatility. Think about what this means: wherever the mean value of your portfolio is sensitive to the current volatility level, the variance of your portfolio is large.
This is intuitive. If the option value (the mean) changes a lot when volatility shifts, and you cannot hedge that shift with the stock alone, then you are exposed to that risk. The variance captures exactly this unhedgeable exposure.
Two special cases make the variance zero. First, if volatility is deterministic (no randomness in volatility at all). Second, if the correlation between the stock and volatility is exactly plus or minus one. In the second case, movements in the stock perfectly predict movements in volatility, so delta hedging the stock also hedges the volatility risk.
Choosing the Best Delta
Here is a subtle but important point. The usual Black-Scholes delta is the derivative of the option value with respect to the stock price. But when volatility is stochastic and correlated with the stock, there is a better choice.
Wilmott picks the delta that minimizes the variance of the hedged portfolio. This optimal delta is not the same as the partial derivative of the option value with respect to S. It includes an extra correction term that accounts for the correlation between S and volatility. When the stock moves, volatility tends to move too (think of the leverage effect: stock drops, volatility rises). The optimal delta adjusts for this secondary effect.
The correction is proportional to the correlation and to how sensitive the mean is to volatility. If your option is very sensitive to volatility (like a barrier option near the barrier), this correction can be significant.
From Mean and Variance to Prices
Now you have the mean and variance of your hedged portfolio at any point in time. How do you turn this into a price?
Wilmott assumes the distribution of outcomes is roughly normal (a bold assumption, but a starting point). Then if you want to be, say, 95% confident that you make money, you price the option at:
sell price = mean + 1.645 * sqrt(variance) buy price = mean - 1.645 * sqrt(variance)
The 1.645 comes from the 95th percentile of the normal distribution. If you only need 84% confidence (one standard deviation), use 1.0 instead.
This gives a spread around the mean. The wider the spread, the more confident you are. But wider spreads mean fewer trades, because the market will often be inside your band and you will not trade.
Wilmott shows three charts that illustrate this tradeoff. As you increase the confidence multiplier (he calls it xi): the expected profit per trade goes up, the number of trades goes down, and the total profit potential (profit per trade times number of trades) has a peak somewhere in the middle. Be too greedy and you never trade. Be too generous and you give away your edge.
The Barrier Option Example
To make this concrete, Wilmott prices and hedges a short up-and-out call with strike 100, barrier at 110, and 30 days to expiry. Barrier options are notoriously hard to hedge because near the barrier, the option value has a sharp kink. Small moves in the stock price cause large swings in the option value.
Without any static hedge, the mean of the short position is about -1.11 and the variance is 0.33. Using the formula with one standard deviation (84% confidence), the option should be priced at about $1.68.
But look at that variance. It is large relative to the mean. This tells you the hedge is poor. The expected outcome is modest, but there is a wide range of possible outcomes. Near the barrier, the mean changes rapidly with respect to volatility, which feeds the variance through the forcing term.
Static Hedging to the Rescue
This is where the chapter gets really practical. Suppose you can trade vanilla call options with different strikes. Can you add them to your portfolio to reduce the variance?
Yes. Because the pricing model is nonlinear (the “price” is mean plus something times the square root of variance), putting vanillas into your portfolio changes the overall mean and variance in a way that can dramatically improve your position.
Wilmott sets up six vanilla calls with strikes ranging from 90 to 115 and finds the optimal quantities to trade. The optimization maximizes the “value” of the portfolio (mean minus cost of the static hedge plus adjustment for variance).
The results are striking. After the optimal static hedge:
- The cost of the hedge position is about $1.19
- The mean improves from -1.11 to +0.04
- The variance drops from 0.33 to 0.05
- The option can now be sold for about $1.38 instead of $1.68
You can either sell the option cheaper (and win more trades) or sell it at the original price and pocket more profit. Either way, the static hedge has created real value by reducing model risk.
Different Strategies for Different Players
“Value” means different things to different people. The sell side might minimize variance. The buy side might maximize the Sharpe ratio. The framework accommodates all of these by changing the objective function. The math stays the same.
The Key Insight
Risk-neutral pricing pretends you can perfectly hedge. When volatility is stochastic, you cannot. Rather than fitting an unobservable market price of risk, Wilmott says: measure the imperfection. Compute mean and variance, set your confidence level, and use static hedging to reduce the imperfection. You know your expected outcome, you know the range of possible outcomes, and you have a framework for improving both.
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