Advances in Risk Management

202 MODEL RISK AND FINANCIAL DERIVATIVES

Now, in reality, the trader does not know perfectly equation (10.1),
and may therefore use a mis-specified and/or mis-estimated model. By
mis-specified, we mean a model different from Black-Scholes, such as an
arithmetic Brownian motion with time-varying parameters, or a mean-
reverting diffusion process. By mis-estimated, we mean that the trader uses
the Black and Scholes model, but mis-estimates the parametersμand/orσ.
In either cases, the wrong option pricing model will give a priceCˆ(t) for
the option that differs from the true (market) priceC(t). Moreover, the wrong

option pricing model will also provide an incorrect hedge ratio∂
Cˆ(t)
∂S(t). At time
t, the trader’s replicating portfolio will then be worth:

(t)=−C(t)+

∂Cˆ(t)

∂S(t)

S(t)+

(

Cˆ(t)−∂

Cˆ(t)

∂S(t)

S(t)

)

(10.5)

which is no longer necessarily equal to zero. The portfolio instantaneous
variations are:

d(t)=−dC(t)+

∂Cˆ(t)

∂S(t)

dS(t)+

(

Cˆ(t)−∂

Cˆ(t)

∂S(t)

S(t)

)

rdt (10.6)

Using equations (10.5) and (10.6) and rearranging terms yields:

d(t)=

(

Cˆ(t)−C(t)

)

rdt

+

[

∂Cˆ(t)

∂S(t)

−

∂C(t)

∂S(t)

]

(μ−r)S(t)dt

+

[

∂Cˆ(t)

∂S(t)

−

∂C(t)

∂S(t)

]

σS(t)dW(t) (10.7)

This equation summarizes the consequences of model risk for our trader.

The first term is a pricing error. The trader uses the model price (Cˆ(t)) to

determine the initial investment to set up the hedging portfolio. IfCˆ(t)

differs from the market priceC(t), the initial investment is excessive or

insufficient, and the difference is carried through time at the risk-free rate.

As a consequence, the delta hedge strategy is no longer self-financing. In

particular, at some point, the hedger may need to borrow and infuse

external funds in order to maintain the delta-hedge. Since the amount

borrowed may become larger than the value of his portfolio, this signifies

that delta hedging with model risk can lead to bankruptcy.

The second term results from (i) the difference between the true delta
parameter and the delta given by the model, and (ii) the difference