The Mathematics of Arbitrage

58 4 Bachelier and Black-Scholes

elementary considerations on the binomial model above (compare Corollary
2.2.12).

Theorem 4.2.1.(see, e.g., [RY 91] or Sect. 7.3). Let(Wt)t∈[0,T]be a stan-
dard Brownian motion modelled on(Ω,(Ft)t∈[0,T],P),where(Ft)t∈[0,T]is the
natural (right continuous saturated) filtration generated byW.
ThenPistheuniquemeasureonFTwhich is absolutely continuous with
respect to itself, and under whichW is a martingale.
Correspondingly, for every functionf∈L^1 (Ω,FT,P)there is a unique
predictable processH=(Ht)t∈[0,T]such that

f=E[f]+(H·W)T,

and
E[f|Ft]=E[f]+(H·W)t, 0 ≤t≤T, (4.1)

which implies in particular that(H·W)is a uniformly integrable martingale.

4.3 Bachelier’s Model

We formulate Bachelier’s model in the framework of the formalism developed
above. LetBt≡1andSt=S 0 +σWt,0≤t≤T,whereS 0 is the current
stock price,σ>0 is a fixed constant, andW is standard Brownian motion
on its natural base (Ω,(Ft)t∈[0,T],P).
Fixing the strike priceK, we want to price and hedge the contingent claim

f(ω)=(ST(ω)−K)+∈L^1 (Ω,FT,P). (4.2)

Using the martingale representation theorem we may find a trading strategy
Hs.t.

f=E[f]+(H·W)T (4.3)

=E[f]+(H·S)T,

whereH=Hσ.
Interpreting Theorem 2.4.1 in a liberal way, i.e., transferring its message
to the case where Ω is no longer finite and using Theorem 4.2.1 above, we
conclude thatC(S 0 ,T)=E[f] is the unique arbitrage free price for the call
option defined in (4.2).
Note thatST is normally distributed with meanS 0 and varianceσ^2 T.
Hence

C(S 0 ,T)=

∫∞

K−S 0

(x−(K−S 0 ))g(x)dx, (4.4)

where

g(x)=

1

σ

√

2 πT

e−

x^2

2 σ^2 T. (4.5)