Frequently Asked Questions In Quantitative Finance

172 Frequently Asked Questions In Quantitative Finance

smooth, monotonically increasing P&L but at the cost of

not knowing how much money you will make.

The profit each time step is

1

2

(

σ^2 − ̃σ^2

)

S^2 idt,

whereiis 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 or your posi-

tion 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 useφto denote a normally distributed random

variable with mean of zero and variance one, then the

profit you make over a time stepδtdue to the gamma

is simply

1

2 σ

(^2) S (^2) δtφ (^2).
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 inter-
vals.
The Black–Scholes models prices in theexpectedvalue
of this expression. You will recognize the^12 σ^2 S^2 from
the Black–Scholes equation. So thehedging erroris