Discrete Hedging: What Happens When You Can't Hedge Continuously
Black-Scholes says you should hedge continuously. Rebalance your portfolio every instant, forever, and all risk disappears like magic. Beautiful theory. Completely impossible in practice.
Chapter 47 is where Wilmott gets real about this problem. And honestly, this is one of those chapters that separates textbook quants from people who actually trade options for a living.
The Trinomial Problem
Wilmott starts with a dead simple example to make the point. In the binomial model, the stock goes up or down. Two states, one unknown (the hedge ratio), one equation. Perfect hedge. Done.
Now add a third state. Stock goes up, stays flat, or goes down. You still have one hedge ratio to choose, but now you need it to work for three outcomes. Two equations, one unknown. Can’t be done.
That is the whole story of discrete hedging in one paragraph. The moment you move beyond the perfect theoretical world, you cannot eliminate all risk. Period.
What Happens in Practice
Even if we stick with the lognormal model that Black-Scholes loves so much, discrete hedging introduces error. You rebalance once a day instead of every nanosecond, and suddenly there is a gap between what your portfolio should be worth and what it actually is.
Wilmott calls this the hedging error, and it is worth understanding in detail.
The setup: you sell an option at the Black-Scholes price, then delta hedge at regular intervals. In a simulation, the running total hedging error bounces around. Sometimes you are up, sometimes down. By expiration, you might have lost money even though you did everything “right.”
The key observation from the simulations is fascinating. The changes in P&L look very asymmetrical. Lots of small losses, a few large gains. This is not random noise. There is a specific mathematical reason for it.
The Chi-Squared Distribution
Here is the punchline. When you delta hedge, the leading-order random term in your portfolio is proportional to the option’s gamma multiplied by a chi-squared random variable. Not a normal random variable. A chi-squared one.
What does this mean in practice? The chi-squared distribution (with one degree of freedom) has a mean of 1, but 68% of the time the variable is below the mean and only 32% of the time it is above. So if you are long gamma, you lose money on the hedge most of the time (small moves in the underlying), but you gain 32% of the time, and those gains are large enough to offset the losses on average.
The net position over one hedging interval averages out to zero. But it definitely does not feel like zero when you are sitting at your desk watching the P&L.
This explains what traders experience every day. Long gamma positions bleed money slowly and then make it back in bursts. Short gamma positions earn steady small amounts and then give it all back in sudden moves. The math confirms the folklore.
The Size of the Error
The total hedging error by expiration depends on how often you rehedge. The standard deviation of the total error scales with the square root of the hedging interval. So if you hedge daily instead of continuously, and there are 252 trading days in a year, the error is proportional to sqrt(1/252), which is about 6% of a year’s time. Not trivial.
The error is also highly path dependent. If the stock moves far from the strike early on, gamma becomes small and the hedging error settles down. But if the stock hovers around the strike, gamma stays large and the P&L keeps fluctuating until the very end.
This is something every options trader knows intuitively. At-the-money options near expiration are a pain to hedge. Now you know the math behind the pain.
A Better Hedge Than Black-Scholes
This is where it gets really interesting. Wilmott derives a higher-order approximation and shows that the standard Black-Scholes delta is not actually the best hedge when you are hedging discretely.
The approach is straightforward in concept: choose the delta that minimizes the variance of your portfolio over the next time step. The result is a corrected delta:
Optimal delta = Black-Scholes delta + correction term
The correction term depends on higher derivatives of the option value and, critically, on the drift of the underlying asset. This is a big deal. In the standard Black-Scholes world, the drift (mu) never appears. The whole point of risk-neutral pricing is that mu cancels out. But when you hedge discretely, mu comes back.
This means that ideas about “risk-neutral valuation” need to be used with care. There is no perfect hedging in the real world, and the investor is necessarily exposed to the direction the stock is moving. Wilmott calls this “hedging with a view.”
The Adjusted Option Value
The option should not be valued at exactly the Black-Scholes price either, because that assumes perfect hedging. The fair value to a discrete hedger is Black-Scholes plus a correction.
The correction can be expressed as a modified volatility:
sigma_star = sigma * sqrt(1 + correction involving mu, sigma, and delta_t)
With typical parameters and daily rehedging, the volatility correction is one or two percent. Usually not a big deal. But in trending markets where the drift is large, this correction can reach five or ten percent. That is real money.
Real Returns vs Normal Returns
The chapter finishes with an important reality check. All the above analysis assumes normal returns. But real returns are not normal. They have higher peaks and fatter tails.
Wilmott shows data from the Dow Jones index from 1977 to 1996. The empirical distribution has a higher peak and fatter tails than the normal. When you look at the square of returns (which is what matters for hedging error), the situation is even more dramatic.
With the normal distribution, 68% of moves are smaller than the standard deviation. With real data, that number is 78%. So for a long gamma position, you are losing money on nearly four out of five days instead of roughly two out of three. The rare big moves still compensate on average, but the experience is even more asymmetric than theory predicts.
Wilmott provides a table of “hedging error factors” for various assets. Most stocks have factors around 1.3 to 2.3, meaning the actual hedging error is 30% to 130% worse than theory. One asset, Nepool (the price of electricity), has a factor of 7.3. Electricity is a very strange beast, as we learn later in the book.
Which Models Allow Perfect Hedging?
Wilmott provides a useful summary table. Perfect hedging is possible in exactly two worlds: binomial and continuous Black-Scholes. Everything else (trinomial, discrete time, stochastic volatility, jump diffusion, fat tails) does not allow perfect hedging. Since we live in the “everything else” world, understanding hedging error is not optional.
Key Takeaways
The expected P&L from a self-financed, delta-hedged option position is zero. Not the time decay, zero. The common belief that “time decay is the expected P&L” is wrong. Wilmott makes this point explicitly and it is important enough to highlight.
Hedging error is proportional to gamma, proportional to the time between rehedges, and drawn from a chi-squared distribution. It averages to zero but can be very large on any given path. And with real fat-tailed returns, it is even worse than the theory suggests.
The Black-Scholes equation is valid only in the limit of continuous hedging. Everything else is an approximation, and this chapter tells you exactly how good or bad that approximation is.
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