The Mathematics of Arbitrage

4.4 The Black-Scholes Model 65

The bottom line of these considerations on the role ofris: when we as-
sumed thatr= 0 in the above comparison of the Bachelier and Black-Scholes
option pricing methodology, this assumption did not restrict the generality
of the argument. It also applies tor= 0 as Bachelier denoted the relevant
quantities in terms of “true prices”, i.e., forward prices.

4.)What is the partial differential equation satisfied by the solution (4.17)
of the Black-Scholes formula? Again we specialise to the caser=0inorder
to focus on the heart of the matter, but we note that now wedo restrict the
generalityand refer to any introductory text to Mathematical Finance (e.g.,
[LL 96]) for the Black-Scholes partial differential equation in the case of a
riskless rate of interestr=0.
From the Martingale Representation Theorem 4.2.1 we know that the
Black-Scholes option priceprocess

C(St,T−t)t∈[0,T]

is a martingale under the measureQdefined in (4.14). Hence, denoting by

( ̃Wt)t∈[0,T]a standard Brownian motion underQ,usingdSt=σStdW ̃t,and
working under the measureQ, we deduce from Itˆo’s formula

dCt=dC(St,T−t)=

∂C

∂S

σStd ̃Wt+

(

σ^2

2

S^2 t

∂^2 C

∂S^2

+

∂C

∂t

)

dt.

We first observe, using againσStd ̃Wt = dSt, that — similarly as in
the context of Bachelier — the replicating trading strategyHtis given by
∂C
∂S(St,T−t). In the lingo of finance this quantity is called the “Delta” of
the option (which depends onStandt) and the trading strategyHis called
“delta-hedging”.
Next we pass to the drift term: asC(St,T−t)isaQ-martingale we
infer that it must vanish, which yields the “Black-Scholes partial differential
equation”

∂C

∂t

(S, T−t)=−

σ^2

2

S^2

∂^2 C

∂S^2

(S, T−t), forS≥ 0 ,t≥ 0. (4.19)

This is the multiplicative analogue of the heat equation (4.11) and may, in
fact, easily be reduced to a heat equation (with drift) by passing to logarithmic
coordinatesx=ln(S).
Exactly as in Bachelier’s case we may proceed by solving the partial dif-
ferential equation (4.19) for the boundary conditionC(S, T−T)=C(S,0) =
(S−K)+andC(0,t) = 0 to obtain the Black-Scholes formula.

In the lingo of finance, the quantity−∂C∂t is called the “Theta” and the

quantity∂

(^2) C
∂S^2 the “Gamma” of the option. Hence the p.d.e. (4.19) allows for
the following economic interpretation: when time to maturityT−tdecreases
(andSremains fixed), the loss of value of the option is equal to the “convexity”