192 Mathematics for Finance
9.1 Hedging Option Positions..................................
The writer of a European call option is exposed to risk, as the option may end
up in the money. The risk profile for a call writer isCEerT−(S(T)−X)+,where
CEerTis the value at the exercise timeTof the premiumCEreceived for the
option and invested without risk. Theoretically, the loss to the writer may be
unlimited. For a put option the risk profile has the formPEerT−(X−S(T))+,
with limited loss, though still possibly very large compared to the premiumPE
received. We shall see how to eliminate or at least reduce this risk over a
short time horizon by taking a suitable position in the underlying asset and, if
necessary, also in other derivative securities written on the same asset.
In practice it is impossible to hedge in a perfect way by designing a single
portfolio to be held for the whole period up to the exercise timeT. The hedging
portfolio will need to be modified whenever the variables affecting the option
change with time. In a realistic case of non-zero transaction costs these mod-
ifications cannot be performed too frequently and some compromise strategy
may be required. Nevertheless, here we shall only discuss hedging over a single
short time interval, neglecting transaction costs.
9.1.1 Delta Hedging .....................................
The value of a European call or put option as given by the Black–Scholes
formula clearly depends on the price of the underlying asset. This can be seen
in a slightly broader context.
Consider a portfolio whose value depends on the current stock priceS=
S(0) and is hence denoted byV(S).Its dependence onScan be measured
by the derivative ddSV(S),called the deltaof the portfolio. For small price
variations fromStoS+∆Sthe value of the portfolio will change by
∆V(S)∼=d
dSV(S)×∆S.
The principle ofdelta hedgingis based on embedding derivative securities in a
portfolio, the value of which does not alter too much whenSvaries. This can
be achieved by ensuring that the delta of the portfolio is equal to zero. Such a
portfolio is calleddelta neutral.
We take a portfolio composed of stock, bonds and the hedged derivative
security, its value given by
V(S)=xS+y+zD(S),where the derivative security price is denoted byD(S) and a bond with current
value 1 is used. Specifically, suppose that a single derivative security has been