7.6. SENSITIVITY, HEDGING AND THE “GREEKS” 261
The concept of Gamma is important when the hedged portfolio cannot
be adjusted continuously in time according to ∆(S(t)). If Gamma is small
then Delta changes very little withS. This means the portfolio requires only
infrequent adjustments in the hedge-ratio. However, if Gamma is large, then
the hedge-ratio Delta is sensitive to changes in the price of the underlying
security.
According to the Black-Scholes formula, we have
Γ =1
S
√
2 πσ√
T−texp(−d^21 /2)Notice that Γ>0, so the call option value is always concave-up with respect
toS. See this in Figure 7.7.
Theta: The time decay factor
TheTheta(Θ) of a European claim with value functionV(S,t) is defined
as
Θ =∂V
∂t.
Note that this definition is the rate of change with respect to the real (or
calendar) time, some other authors define the rate of change with respect to
the time-to-expirationT−t, so be careful when reading.
The Theta of a claim is sometimes refereed to as the time decay of the
claim. For a European call option on a non-dividend-paying stock,
Θ =−
S·σ
2√
T−t·
exp(−d^21 /2)
√
2 π−rKexp(−r(T−t))Φ(d 2 ).Note that Θ for a European call option is negative, so the value of a European
call option is decreasing as a function of time, confirming what we intuitively
deduced before. See this in Figure 7.7.
Theta does not act like a hedging parameter as do Delta and Gamma.
Although there is uncertainty about the future stock price, there is no un-
certainty about the passage of time. It does not make sense to hedge against
the passage of time on an option.
Note that the Black-Scholes partial differential equation can now be writ-
ten as
Θ +rS∆ +1
2
σ^2 S^2 Γ =rV.