The Mathematics of Financial Modelingand Investment Management

23-RiskManagement Page 742 Wednesday, February 4, 2004 1:13 PM

742 The Mathematics of Financial Modeling and Investment Management

( ̃

σ

dσt = a ̃ (St, σt )dt + b St, σt )dBt

The above stochastic volatility model can be completed^3 by adding

an asset Yt = C(t,σt,St) that follows the following process:

dYt = rYtdt + F t Y( , tSt )dBˆt

where Bˆ

t is another Brownian motion eventually correlated with the

other two. Note that mathematically there is an infinite family of these

models.

The question of what model applies to a new asset introduced for

completing the market is an empirical one. Note that this new asset is

contractually defined as a function of the stock price. In practice it is an

option. The market will price the new asset according to some economic

pricing principle which is not, however, a principle of absence of arbi-

trage. In this completed market, the underwriter of an option can com-

pletely hedge his/her position. However, the hedging will not be the

same as in the case of constant volatility.

Similar considerations can be repeated for the jump-diffusion mod-

els. Suppose that a lognormal diffusion is given. Consider a Poisson

point process and add a finite jump to the diffusion at every occurrence

of the Poisson process. The resulting model is generally incomplete.

However, it can be completed by adding appropriate contracts. What

type of contracts must be added in each case is not a trivial question.

The Economics of Market Completeness

In discussing market completeness it should be kept in mind that market

completeness means that any risk can be completely hedged. In modern

markets, hedging is typically achieved by taking positions in appropri-

ate contracts such as options or other derivative instruments. In this

way risk is transferred to other entities and hedged. The key question is:

why should there be other entities willing to take the opposite side of a

risky position?

Beside the mathematical details, this is the essence of market com-

pleteness. It means that there is always someone willing to trade, at a

market price, any contingent claim. It is important to reconcile this

notion with that of mathematical completeness. Let’s use the simple

example of European stock options in a market with a risk-free asset

(^3) M.H.A. Davis, Complete-Market Models of Stochastic Volatility, forthcoming in
Proc. Royal Society London (A).