Delta Hedging in Practice: Implied vs Actual Volatility

This chapter is one of the most practically important in the entire book. Wilmott starts with a bold statement: there is money to be made from options because they may be mispriced by the market. He knows the efficient market crowd hates this idea. But volatility arbitrage hedge funds clearly believe it, so let us look at the math.

The core question of Chapter 12: if you think actual volatility will be different from implied volatility, how do you exploit that difference? You delta hedge. But which volatility do you use for your delta?

Actual vs. Implied Volatility

Let me make sure we are on the same page with the terminology.

Actual volatility (also called realized volatility) is the real amount of randomness in the stock price. It is what actually happens day to day. Implied volatility is the number the market plugs into Black-Scholes to get the current option price. These two numbers can and do differ.

Wilmott shows a chart comparing the VIX (an implied volatility measure for the S&P 500) with the rolling realized volatility of the same index. The VIX tends to be higher, averaging around 20% while realized volatility averages around 15%. Both are roughly lognormally distributed, but they are clearly not the same thing.

If you believe actual volatility will be higher than implied, you buy options and delta hedge. If you think it will be lower, you sell options and hedge. But here comes the million-dollar question.

The Big Question: Which Delta?

Suppose implied volatility is 20% but you forecast actual volatility at 30%. You buy the option and plan to delta hedge until expiration. The delta formula requires a volatility input. Do you use 20% or 30%?

This is “one of those questions that people always ask, and one that no one seems to know the full answer to,” as Wilmott puts it.

It turns out both choices are valid, but they produce completely different risk and return profiles.

Case 1: Hedge with Actual Volatility

When you use the actual volatility (30% in our example) to compute your delta, you are replicating a short position in a “correctly priced” option. At expiration, the payoffs cancel out and you pocket exactly the difference between the two Black-Scholes prices:

Guaranteed profit = V(S, t; actual) - V(S, t; implied)

This profit is guaranteed in the sense that by expiration, you will have made exactly this amount. No more, no less (assuming continuous hedging and Black-Scholes assumptions hold).

But there is a catch. On a mark-to-market basis, the path to that final profit is random. Some days you make money, some days you lose. The daily P&L includes a random term that depends on the actual stock movements and even on the real drift of the stock. You know the destination but the journey is bumpy.

Wilmott compares this to owning a bond. You know what you get at maturity, but the price fluctuates in the meantime.

Case 2: Hedge with Implied Volatility

Now let us try using implied volatility (20%) for the delta instead. Something beautiful happens. The daily mark-to-market profit becomes deterministic:

Daily profit = 0.5 * (actual^2 - implied^2) * S^2 * Gamma * dt

There is no random dX term. Every day you make a predictable, positive amount (assuming actual > implied). The profit each day depends on where the stock is (through Gamma), but there is no randomness.

However, the total profit by expiration is now path dependent. If the stock drifts far from the strike, Gamma becomes small and daily profits shrink. The best outcome is when the stock ends up near the strike at expiration, where Gamma is largest.

Wilmott compares this to putting money in the bank. Your account grows steadily, but the final amount depends on the path the interest rate took.

There is another advantage to hedging with implied volatility: you do not even need to know exactly what the actual volatility is. You just need to know that actual is always greater than implied (if you are long) or always less (if you are short). This takes pressure off having a precise volatility forecast.

Comparing the Two Approaches

Here is the tradeoff:

In practice, the choice often depends on your constraints. If you can mark to model (meaning you do not report daily P&L to anyone), you might prefer hedging with actual volatility for the guaranteed outcome. If you must report daily P&L to risk managers, investors, or brokers, hedging with implied volatility gives smoother results day to day.

The General Formula

Peter Carr and Henrard showed that if you hedge using any volatility sigma_h, the total profit is:

PV of profit = [V(S, t; sigma_h) - V(S, t; implied)] + integral of 0.5 * (actual^2 - sigma_h^2) * S^2 * Gamma^h * discount factor

This is a beautiful result. Actual and implied volatility are the two special cases (sigma_h = actual or sigma_h = implied), but you can hedge with any volatility in between or even outside that range. Each choice gives a different expected profit and standard deviation.

Portfolio Optimization

Since the profit formula depends on Gamma, and calls and puts with the same strike and expiration have the same Gamma, you can treat them as equivalent for hedging purposes.

For a portfolio of options with different strikes and expirations, the total profit is just the sum of individual profits (for the expected value). But the variance involves cross-terms because all the options share the same underlying stock.

This opens the door to portfolio optimization. Wilmott gives an example where you maximize expected return while constraining the standard deviation to one. The optimizer finds an ideal risk reversal trade: sell overpriced out-of-the-money puts and buy underpriced out-of-the-money calls. The portfolio costs negative money to set up (you collect premium) and has a positive expected profit.

Stochastic Implied Volatility

Reality is even more interesting. Implied volatility itself moves around over time. Wilmott examines what happens when implied volatility follows its own random process.

If you hedge with actual volatility, the final profit is still guaranteed. Stochastic implied vol does not change this, though it adds extra randomness to the mark-to-market path.

If you hedge with implied volatility when it is stochastic, you need to adjust your delta by the vega to minimize daily P&L variance. The optimal hedge becomes:

Delta_optimal = Delta_implied + (something involving vega and correlation)

This adjustment accounts for the fact that when the stock moves, implied volatility moves too, changing the option value through a channel other than delta.

How Does Implied Volatility Actually Behave?

Wilmott describes two common patterns:

Sticky strike: Each option keeps its own constant implied volatility regardless of where the stock moves. This is more common in equity markets. Your individual option lives in its own little Black-Scholes world.

Sticky delta (sticky moneyness): Implied volatility depends on moneyness (S/E). As the stock moves, the implied volatility of a given option changes because its moneyness changes. This is more common in FX markets where options are quoted by delta rather than strike.

There is even a day-of-the-week effect in implied volatility. The VIX tends to behave differently on different days, partly because weekends create “potential” volatility from news that accumulates while markets are closed.

Key Takeaways

This chapter is essentially a manual for volatility arbitrage. The central insight is that you can profit whenever implied and actual volatility differ, but your choice of hedging volatility affects your risk profile dramatically. Hedging with actual volatility gives a guaranteed but bumpy outcome. Hedging with implied volatility gives a smooth but uncertain outcome.

For most practitioners, the practical takeaway is: hedge with implied volatility if you have to report P&L regularly, hedge with actual volatility if you can hold to maturity without anyone looking over your shoulder.

And the deeper message: the Black-Scholes model is not just a pricing formula. It is a tool for making money when you disagree with the market about volatility. That is what makes it useful in practice, not its ability to produce “correct” prices.

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