260 CHAPTER 7. THE BLACK-SCHOLES MODEL
we first considered options, see Options. The increase in security value inS
is visible in Figure 7.7.
Delta Hedging
Notice that for any sufficiently differentiable functionF(S)
F(S 1 )−F(S 2 )≈
dF
dS(S 1 −S 2 )
Therefore, for the Black-Scholes formula for a European call option, using
our current notation ∆ =∂V/∂S,
(V(S 1 )−V(S 2 ))−∆(S 1 −S 2 )≈ 0or equivalently for small changes in security price fromS 1 toS 2 ,
V(S 1 )−∆S 1 ≈V(S 2 )−∆S 2.In financial language, we express this as:
“long 1 derivative, short ∆ units of the underlying asset is
market neutral for small changes in the asset value.”We say that the sensitivity ∆ of the financial derivative value with respect
to the asset value gives thehedge-ratio. The hedge-ratio is the number of
short units of the underlying asset which combined with a call option will
offset immediate market risk. After a change in the asset value, ∆(S) will
also change, and so we will need to dynamically adjust the hedge-ratio to
keep pace with the changing asset value. Thus ∆(S) as a function ofS
provides a dynamic strategy for hedging against risk.
We have seen this strategy before. In the derivation of Black-Scholes
equation, we required that the amount of security in our portfolio, namely
φ(t) be chosen so thatφ(t) =VS. See Derivation of the Black-Scholes Equa-
tion The choiceφ(t) =VSgave us a risk-free portfolio which must change in
the same way as a risk-free asset.
Gamma: The convexity factor
TheGamma(Γ) of a derivative is the sensitivity of ∆ with respect toS: