276 Frequently Asked Questions In Quantitative Finance
Equity, Foreign Exchange and
Commodities
The lognormal random walk
The most common and simplest model is the lognormal
random walk:
dS+μSdt+σSdX.
The Black–Scholes hedging argument leads to the fol-
lowing equation for the value of non-path-dependent
contracts,
∂V
∂t
+^12 σ^2 S^2
∂^2 V
∂S^2
+(r−D)S
∂V
∂S
−rV= 0.
The parameters are volatilityσ, dividend yieldDand
risk-free interest rater. All of these can be functions of
Sand/ort, although it wouldn’t make much sense for
the risk-free rate to beSdependent.
This equation can be interpreted probabilistically. The
option value is
e−
∫T
tr(τ)dτEtQ[Payoff(ST)] ,
whereSTis the stock price at expiry, timeT,and
the expectation is with respect to the risk-neutral ran-
dom walk
dS=r(t)Sdt+σ(S,t)SdX.
Whenσ,Dandrare only time dependent we can write
down an explicit formula for the value of any non-path-
dependent option without early exercise (and without
any decision feature) as
