Frequently Asked Questions In Quantitative Finance

Chapter 2: FAQs 101

1.Whenever you getdX^2 in a Taylor series expansion
of a stochastic variable you must replace it withdt.
2.Terms that areO(dt^3 /^2 ) or smaller must be ignored.
This means thatdt^2 ,dX^3 ,dt dX, etc. are too small to
keep.

It is difficult to overstate the importance of Ito’s lemmaˆ

in quantitative finance. It is used in many of the deriva-

tions of the Black–Scholes option pricing model and

the equivalent models in the fixed-income and credit

worlds. If we have a random walk model for a stock

priceSand an option on that stock, with valueV(S,t),

then Ito’s lemma tells us how the option price changesˆ

with changes in the stock price. From this follows the

idea of hedging, by matching random fluctuations inS

with those inV. This is important both in the theory of

derivatives pricing and in the practical management of

market risk.

Even if you don’t know how to prove Itˆo’s lemma you

must be able to quote it and use the result.

Sometimes we have a function of more than one

stochastic quantity. Suppose that we have a function

f(y 1 ,y 2 ,...,yn,t)ofnstochastic quantities and time such

that

dyi=ai(y 1 ,y 2 ,...,yn,t)dt+bi(y 1 ,y 2 ,...,yn,t)dXi,

where thenWiener processesdXihave correlationsρij

then

df=



∂f

∂t

+

∑n

i= 1

ai

∂f

∂yi

+^12

∑n

i= 1

∑n

j= 1

ρijbibj

∂^2 f

∂yi∂yj



dt

+

∑n

i= 1

bi

∂f

∂yi

dXi.