Frequently Asked Questions In Quantitative Finance

Chapter 5: Models and Equations 277

e−r(T−t)

σ

√

2 π(T−t)

∫∞

0

e

−

(

ln(S/S′)+

(

r−D−^12 σ^2

)

(T−t)

) 2

/ 2 σ^2 (T−t)

Payoff(S′)

dS′

S′

,

where

σ=

√

1

T−t

∫T

t

σ(τ)^2 dτ,

D=

1

T−t

∫T

t

D(τ)dτ

and

r=

1

T−t

∫T

t

r(τ)dτ.

The.parameters represent the ‘average’ of the para-
meters from the current time to expiration. For the
volatility parameter the relevant average is the root-
mean-square average, since variances can be summed
but standard deviations (volatilities) cannot.

The above is a very general formula which can be
greatly simplified for European calls, puts and
binaries.

Multi-dimensional lognormal random

walks

There is a formula for the value of a European non-path-
dependent option with payoff of Payoff(S 1 ,...,Sd)at
timeT: