- Financial Engineering 197
As we can see, in some circumstances delta hedging may be far from satis-
factory. We need to improve the stability of hedging when the underlying asset
price changes considerably and/or some other variables change simultaneously.
In what follows, after introducing some theoretical tools we shall return again
to the current example.
Exercise 9.4
Find the value of the delta neutral portfolio in Exercise 9.3 if the risk-free
rate of interest decreases to 3% on day one.9.1.2 Greek Parameters ..................................
We shall define so-calledGreek parametersdescribing the sensitivity of a port-
folio with respect to the various variables determining the option price. The
strike priceXand expiry dateTare fixed once the option is written, so we
have to analyse the four remaining variablesS,t,r,σ.
Let us write the value of a general portfolio containing stock and some
contingent claims based on this stock as a functionV(S, t, σ, r) of these variables
and denote
deltaV=∂V
∂S,
gammaV=∂^2 V
∂S^2
,
thetaV=∂V
∂t,
vegaV=∂V
∂σ,
rhoV=∂V
∂r.For small changes∆S, ∆t, ∆σ, ∆rof the variables we have the following
approximate equality (by the Taylor formula):
∆V ∼=deltaV×∆S+thetaV×∆t+vegaV×∆σ+rhoV×∆r+1
2
gammaV×(∆S)^2.Hence, a way to immunise a portfolio against small changes of a particular vari-
able is to ensure that the corresponding Greek parameter is equal to zero. For
instance, to hedge against volatility movements we should construct avega neu-
tralportfolio, with vega equal to zero. To retain the benefits of delta hedging,