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Chapter 11
Sensitivities and Greeks
A key task in financial engineering is the fast and accurate calculation of sensitiv-
ities of market models with respect to model parameters. This becomes necessary,
for example, in model calibration, risk analysis and in the pricing and hedging of
certain derivative contracts. Classical examples are variations of option prices with
respect to the spot price or with respect to time-to-maturity, the so-called “Greeks”
of the model. For classical, diffusion type models and plain vanilla type contracts,
the Greeks can be obtained analytically. With the trends to more general market
models of jump-diffusion type and to more complicated contracts, closed form solu-
tions are generally not available for pricing and calibration. Thus, prices and model
sensitivities have to be approximated numerically.
Here, we consider the general class of Markov processes
X
, including stochastic
volatility and Lévy models as described before. We distinguish between two classes
of sensitivities. The sensitivity of the solution
V
to variation of a model parameter,
like the Greek Vega (
∂
σ
V
) and the sensitivity of the solution
V
to a variation of
state spaces such as the Greek Delta (
∂
x
V
). We show that an approximation for
the first class can be obtained as a solution of the pricing PIDE with a right hand
side depending on
V
. For the second class, a finite difference like differentiation
procedure is presented which allows obtaining the sensitivities from the forward
price without additional calls to the forward solver.
11.1 Option Pricing
We consider the process
X
to model the dynamics of a single underlying, a basket or
an underlying and its “background” volatility drivers in case of stochastic volatility
models. For notational simplicity, only we assume that the interest rate is zero, i.e.
r
0. As shown in the previous sections provided some smoothness assumptions,
the fair price of a European style contingent claim with payoff
g
and underlying
X
,
i.e.
=
= E
g(X
T
)
x
,
v(t,x)
|
X
t
=
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