Dynamic hedging, or delta hedging, means the continuous buying or selling of the underlying asset according to some formula or algorithm so that risk is eliminated from an option position. The key point in this is what formula do you use, and, given that in practice you can t hedge continuously, how should you hedge discretely? First, get your delta correct, and this means use the correct formula and estimates for parameters, such as volatility. Second, decide when to hedge based on the conflicting desires of wanting to hedge as often as possible to reduce risk, but as little as possible to reduce any costs associated with hedging.

Example

The implied volatility of a call option is 20% but you think that is cheap and volatility is nearer 40%. Do you put 20% or 40% into the delta calculation? The stock then moves, should you rebalance, incurring some inevitable transactions costs, or wait a bit longer while taking the risks of being unhedged?

Long answer

There are three issues, at least, here. First, what is the correct delta? Second, if I don t hedge very often how big is my risk? Third, when I do rehedge, how big are my transaction costs?

What is the correct delta?

Let's continue with the above example, implied volatility 20% but you believe volatility will be 40%. Does 0.2 or 0.4 go into the Black-Scholes delta calculation, or perhaps something else? First let me reassure you that you won t theoretically lose money in either case (or even if you hedge using a volatility somewhere in the 20 to 40 range) as long as you are right about the 40% and you hedge continuously. There will however be a big impact on your P&L depending on which volatility you input.

If you use the actual volatility of 40% then you are guaranteed to make a profit that is the difference between the Black-Scholes formula using 40% and the Black-Scholes formula using 20%.

where V(S, t; a) is the Black-Scholes formula for the call option and a denotes actual volatility and a is implied volatility.

That profit is realized in a stochastic manner, so that on a marked-to-market basis your profit will be random each day. This is not immediately obvious, nevertheless it is the case that each day you make a random profit or loss, both equally likely, but by expiration your total profit is a guaranteed number that was known at the outset. Most traders dislike the potentially large P&L swings that you get by hedging, using the forecast volatility that they hedge using implied volatility.

When you hedge with implied volatility, even though it is wrong compared with your forecast, you will still make money. But in this case the profit each day is non-negative and smooth, so much nicer than when you hedge using forecast volatility. The downside is that the final profit depends on the path taken by the underlying. If the stock stays close to the strike then you will make a lot of money. If the stocks goes quickly far into or out of the money then your profit will be small. Hedging using implied volatility gives you a nice, smooth, monotonically increasing P&L but at the cost of not knowing how much money you will make.

The profit each time step is

where Г' is the Black-Scholes gamma using implied volatility. You can see from this expression that as long as actual volatility is greater than implied, you will make money from this hedging strategy. This means that you do not have to be all that accurate in your forecast of future actual volatility to make a profit.

How big is my hedging error?

In practice you cannot hedge continuously. The Black-Scholes model, and the above analysis, requires continuous rebalancing of your position in the underlying. The impact of hedging discretely is quite easy to quantify.

When you hedge you eliminate a linear exposure to the movement in the underlying. Your exposure becomes quadratic and depends on the gamma of your position. If we use 0 to denote a normally distributed random variable with mean of zero and variance one, then the profit you make over a time step St due to the gamma is simply

This is in an otherwise perfect Black-Scholes world. The only reason why this is not exactly a Black-Scholes world is because we are hedging at discrete time intervals.

The Black-Scholes model prices in the expected value of this expression. You will recognize the a2S2T from the Black-Scholes equation. So the hedging error is simply

This is how much you make or lose between each rebalancing.

We can make several important observations about hedging error.

• It is large: it is O(St), which is the same order of magnitude as all other terms in the Black-Scholes model. It is usually much bigger than interest received on the hedged option portfolio.

• On average it is zero: hedging errors balance out.

• It is path dependent: the larger gamma, the larger the hedging errors.

• The total hedging error has standard deviation of *J~Si : total hedging error is your final error when you get to expiration. If you want to halve the error you will have to hedge four times as often.

• Hedging error is drawn from a chi-square distribution: that's what 4>2 is

• If you are long gamma you will lose money approximately 68% of the time: this is chi-square distribution in action. But when you make money it will be from the tails, and big enough to give a mean of zero. Short gamma you lose only 32% of the time, but they will be large losses

• In practice 0 is not normally distributed: the fat tails, high peaks we see in practice, will make the above observation even more extreme, perhaps a long gamma position will lose 80% of the time and win only 20%. Still the mean will be zero

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