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