What are the important observations about hedging error

Question

What are the important observations about hedging error?

Answer 1

We can make several significant observations about hedging error.

• It is large: it is O(δt), that is the same order of magnitude as all other terms in the Black–Scholes model. This is typically much bigger than interest received upon the hedged option portfolio.

• On average this is zero: hedging errors are balance out.

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

• The entire hedging error has standard deviation as of √δt: sum of total hedging error is your last error when you get to expiration. When you want to halve the error you will have to hedge four times as frequently.

• Hedging error is drawn by a chi-square distribution: it’s what φ2 is

• When you are long gamma you will lose money around 68% of the time: it is chi-square distribution in action.

But while you make money this will be from the tails, and big sufficient to provide a mean of zero. But short gamma you lose only 32% of the time, if they will be large losses

• In practice φ is not commonly distributed: the fat tails, high peaks we notice in practice, will make the above observation still more extreme, perhaps a long gamma position will lose 80 percent of the time and win only 20percent. But the mean will be zero.