Advances in Risk Management

356 MODEL SELECTION AND HEDGING OF FINANCIAL DERIVATIVES

a number of tradable (non-redundant) securities, typically liquid vanilla
options. In what follows, we will assume that agents are allowed to trade
an additional option, sayD,onStto risk manage their books. The latter
assumption, together with the no arbitrage condition and the market price
of volatility risk, uniquely determines the risk neutral probability measure
P∗under which discounted asset prices are martingales. For convenience
we will useβtas numéraire. The corresponding Wiener process underP∗
will be denoted byW=(W^1 ,W^2 ). We shall further assume that the covari-
ance process is given by〈W^1 ,W^2 〉=ρtfor a given choice of the constant
correlation coefficientρ.
In order to introduce the concept of hedging error, we define two different
dynamics forStunderP∗, specifically:

dSt=rStdt+γ(t,S,v)dWt^1

dvt=φ(t,v)dt+φ(t,v)dWt^2

(18.1)

and

dSt=rStdt+(t,S,v)dWt^1

dvt=(t,v)dt+!(t,v)dWt^2

(18.2)

We will also assume that the generic functionsγ,φ,θ,,,!satisfy the
usual conditions of existence and unicity to the solutions of the above SDE’s
(see for instance Jacod and Shiryaev (1987)). In our set-up, equation (18.1)
represents the real (unknown to the trader) dynamics ofSt, while equation
(18.2) corresponds to the model used by traders as a proxy of the “true” mar-
ket dynamics. We also assume that the correlation between Wiener noises
in the “wrong” model (18.2) isρ′ =ρ. Our goal is to estimate the hedging
error that traders incur when using model (18.2) to risk-manage a vanilla
European option written onSt.
It should be noted that our modeling setup is fairly generic as many well-
known models can be recovered by appropriately specifying the functions
γ,φ,θ,,,!. For instance, the following special cases are well-known:

1Ifγ(t,S,v)=vSt,φ=θ=0 we recover the Black–Scholes-model.

2Ifγ(t,S,v)=

√

vtSt,φt=a(b−vt) andθ=α

√

vtfor given constantsa,b,α

we recover the Heston (1993) stochastic volatility model (in this casevtis

the square of the volatility, i.e. the instantaneous variance).

3Ifγ(t,S,v)=vtSt,φt=a(b−vt) andθ=αfor given constantsa,b,αwe

recover the Stein–Stein (1991) stochastic volatility model.

It is in principle possible to extend the results presented in section 18.3 to
more general processes (in particular to include the presence of jumps in the
state variables) but the mathematical setup needed is more involved and it
will be addressed in a separate paper (Di Graziano and Galluccio, 2005).