Chapter 2: FAQs 173
simply
1
2 σ(^2) S (^2) δt(φ (^2) −1).
This is how much you make or lose between each rebal-
ancing.
We can make several important observations about
hedging error.
- It is large: it isO(δt) 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√
δt: 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 whatφ^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 practiceφis 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.
How much will transaction costs reduce my profit? To reduce
hedging error we must hedge more frequently, but
the downside of this is that any costs associated with