The Mathematics of Financial Modelingand Investment Management

23-RiskManagement Page 741 Wednesday, February 4, 2004 1:13 PM

Risk Management 741

volatility models are models where prices are diffusion processes but the

volatility term is driven by a separate process. In discrete time, all models

make jumps while stochastic volatility models become the ARCH and

GARCH models. Let’s briefly discuss completeness in relation to stochas-

tic volatility models.

A standard geometric-diffusion model is complete as there is a

unique equivalent martingale measure Q (see Chapter 15) under which

the model can be written as

dSt = rStdt + σStdBt

where r is the risk-free rate, σ is the volatility constant, and B is a stan-

dard Brownian motion. If a stock price follows this model, any contingent

claim can be uniquely replicated. In particular, options can be replicated

as a portfolio formed with the stock and the risk-free asset. Options are

redundant securities. Anyone who has underwritten an option can com-

pletely hedge its risk by constructing an appropriate self-financing repli-

cation strategy.

The same reasoning can be applied in the case of N geometric

Brownian motions. In this case, there is still a unique equivalent martin-

gale measure under which the model can be written as

N

dSi j j

t = rSt

idt +

∑σjStdBt

j = 1

Suppose now that volatility is not constant but that it is a time-

dependent process. The simplest two-factor, stochastic-volatility model

can be written, in the physical probability measure, as

dSt = μStdt + σtStdBt

dσt = aS( t, σt)dt + b S( t, σt)Bt

σ

where Bσ

t is another standard Brownian motion eventually correlated

with Bt. In this case, however, there are infinite equivalent martingale

measures in which the model can be written as

dSt = rStdt + σtStdB ̃ t