7.3 Option Pricing
Consider the value function of a general contingent claim
paying
at time
.
We want to replicate it with a self-financing portfolio. Due to the fact that in Heston's model we have two sources of uncertainty (the Wiener processes
and
)
the portfolio must include the possibility to trade in the money market, the underlying and another derivative security with value function
.
We start with an initial wealth
which evolves according to:
 |
(7.5) |
where
is the number of units of the underlying held at time
and
is the number of derivative securities
held at time
. Since we are operating in a foreign exchange setup, we let
and
denote the domestic and foreign interest rates, respectively.
The goal is to find
and
such that
for all
.
The standard approach to achieve this is to compare the differentials of
and
obtained via Itô's formula.
After some algebra we arrive at the partial differential equation which
must satisfy:
For details on the derivation in the foreign exchange setting see Hakala and Wystup (2002).
The term
is called the market price of volatility risk. Without loss of generality its functional form can be reduced to
, Heston (1993). We obtain a solution to (7.6) by specifying appropriate boundary conditions. For a European vanilla option these are:
where
is a binary variable taking value
for call options and
for put options,
is the strike in units of the domestic currency,
,
is the expiration time in years, and
is the current time.
In this case, PDE (7.6) can be solved analytically using the method of characteristic functions (Heston; 1993). The price of a
European vanilla option is hence given by:
where
,
,
,
,
,
,
,
, and
 |
 |
 |
(7.13) |
 |
 |
 |
(7.14) |
|
|
 |
|
 |
 |
 |
(7.15) |
 |
 |
 |
(7.16) |
 |
 |
 |
(7.17) |
The functions
are the cumulative distribution functions (in the variable
) of the log-spot price after time
starting at
for some drift
. The functions
are the respective densities. The integration in (7.17) can be done with the Gauss-Legendre algorithm using
for
and
abscissas.
The best is to let the Gauss-Legendre algorithm compute the abscissas and weights once and reuse them as constants for all integrations. Finally:
Apart from the above closed-form solution for vanilla options, alternative approaches can be utilized. These include finite difference and finite element methods. The former must be used with care since high precision is required to invert scarce matrices. The Crank-Nicholson, ADI (Alternate Direction Implicit), and Hopscotch schemes can be used, however, ADI is not suitable to handle nonzero correlation. Boundary conditions must be also set appropriately. For details see Kluge (2002). Finite element methods can be applied to price both the vanillas and exotics, as explained for example in Apel, Winkler, and Wystup (2002).
7.3.1 Greeks
The Greeks can be evaluated by taking the appropriate derivatives or by exploiting homogeneity properties of financial markets (Reiss and Wystup; 2001). In
Heston's model the spot delta and the so-called dual delta are given by:
 |
(7.20) |
respectively. Gamma, which measures the sensitivity of delta to the underlying has the form:
 |
(7.21) |
can be computed from (7.6). The formulas for rho are the following:
Note that in a foreign exchange setting there are two rho's - one is a derivative of the option price with respect to the domestic interest rate and the other is a derivative with respect to the foreign interest rate.
The notions of vega and volga usually refer to the first and second derivative with respect to volatility. In Heston's model we use them for the first and second derivative with respect to the initial variance:
where