66 4 Bachelier and Black-Scholes
or the “Gamma” of the option price (as a function ofS)attimet, normalised
by σ
2
2 S(^2) (in the case of the Bachelier model the normalisation was simply
σ^2
2 ). This has a nice economic interpretation and today’s option traders think
in these terms. They speak about “selling or buying convexity” or rather
“going gamma-short or gamma-long” which amounts to the same thing. The
interpretation of (4.19) is that, for the buyer of an option, the convexity of the
functionC(S, T−t)inthevariableScorresponds to a kind of insurance with
respect to price movements ofS. As there is no such thing as a free lunch,
this insurance costs (proportional to the second derivative) and a positive
σ^2
2 S
2 ∂^2 C
∂S^2 is reflected by a negative partial derivative
∂C
∂t ofC(S, T−t)with
respect to timet.
Let us illustrate this fact by reasoning once more heuristically with in-
finitesimal movements of Brownian motion: we want to explain the infinitesi-
mal change of the option price when “time increases by an infinitesimal while
the stock priceSremains constant”. To do so we apply the heuristic ana-
logue of the Brownian bridge: consider the infinitesimal interval [t, t+2dt]
and assume that the drivingQ-Brownian motion ̃W moves in the first half
[t, t+dt]from ̃Wt toW ̃t+εt
√
dt,whereεt is a random variable with
Q[εt=1]=Q[εt=−1] =^12 , while in the second half [t+dt, t+2dt]itmoves
back toW ̃t. What should happen during this time interval to a “hedger”
who proceeds according to the Black-Scholes trading strategyH described
above, which replicates the option? At timetshe holds∂C∂S(St,T−t) units
of the stock. Following first the scenarioεt= +1, the stock has a price of
St+σSt
√
dtat timet+dt. Apart from being happy about this up move-
ment, the hedger now (i.e., at timet+dt) adjusts the portfolio to hold
∂C
∂S
(
St+σSt√
dt, T−(t+dt))
units of stock, which results in a net buy of
∂^2 C
∂S^2 (St,T−t)σSt√
dtunits of stock, where we neglect terms of smaller orderthan
√
dt. In the next half [t+dt, t+2dt] of the interval the stock priceSdrops
again to the valueSt+2dt=Stand the hedger readjusts at timet+2dtthe
portfolio by selling again the∂
(^2) C
∂S^2 (St,T−t)σSt
√
dtunits of stock (neglectingagain terms of smaller order than
√
dt). It seems at first glance that the gains
made in the first half are precisely compensated by the losses in the second
half, but a closer inspection shows that the hedger did “buy high” and “sell
low”: the quantity∂
(^2) C
∂S^2 (St,T−t)σSt
√
dtwas bought at priceSt+σSt√
dtat
timet+dt, and sold at priceStat timet+2dt, resulting in a total loss of
(
∂^2 C
∂S^2(St,T−t)σSt√
dt)(
σSt√
dt)
=σ^2 St^2∂^2 C
∂S^2
(St,T−t)dt. (4.20)Going through the scenarioεt=−1, one finds that the hedger did first
“sell low” and then “buy high” resulting in the same loss (where again we
neglect infinitesimals resulting in effects (with respect to the final result) of
smaller order thandt).