Abstract
This chapter discusses the problem of pricing an American call option when the underlying dynamics follow the Heston’s stochastic volatility and the Cox-Ingersoll-Ross (CIR) stochastic interest rate. We use a partial differential equation (PDE) approach to obtain a numerical solution. The call is formulated as a free boundary PDE problem on a finite computational domain with appropriate boundary conditions. It is solved with the time discrete method of lines which is found to be accurate and efficient in producing option prices, early exercise boundaries and option hedge ratios like delta and gamma. The method of lines results are compared with those from a finite difference approximation of the corresponding linear complementarity formulation which were obtained with PSOR and the Sparse Grid approach.
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Notes
- 1.
Of course, since we are using a numerical technique we could in fact use more general processes for \(S\) and \(v\). The choice of the Heston processes is driven partly by the fact that this has become a very traditional stochastic volatility model and partly because the transform methods do not easily handle the more general variance processes.
- 2.
In fact, if it is assumed that the market prices of risk associated with the uncertainty driving the variance process and the interest rate process have the form \(\lambda _v \sqrt{v}\) and \(\lambda _r \sqrt{r}\), respectively, where \(\lambda _v\) is a constant (this was the assumption in Heston (1993)) and \(\lambda _r\) is a constant. In addition \(\kappa _v^{\mathbb P}, \theta _v^{\mathbb P}\) and \(\kappa _r^{\mathbb P}, \theta _r^{\mathbb P}\) are the corresponding parameters under the physical measure. Then \(\kappa _v=\kappa _v^{\mathbb P}+\lambda _v\sigma _v, \theta _v=\frac{\kappa _v^{\mathbb P}\theta _v^{\mathbb P}}{\kappa _v^{\mathbb P}+\lambda _v\sigma _v}\); \(\kappa _r=\kappa _r^{\mathbb P}+\lambda _r\sigma _r, \theta _r=\frac{\kappa _r^{\mathbb P}\theta _r^{\mathbb P}}{\kappa _r^{\mathbb P}+\lambda _r\sigma _r}\).
- 3.
- 4.
All ODEs have been solved by use of the implicit trapezoidal rule, discussed for example by Shampine (1994).
- 5.
We test Eq. (32) at each grid point and find the grid points at which \(S - K -R_{m,n}(S) - W^l_{m,n}(S)\) changes sign. We then use Newton’s method to search for the value of \(S^*\) by fitting a cubic spline through four points around of this point.
- 6.
We remind the reader that at \(S^*\) the first of the free boundary conditions (7) becomes \(V_{m,n}^l(S^*)=1.\)
- 7.
A thorough error analysis of the multi-linear interpolation operator can be found in Reisinger (2008) who gives a generic derivation for linear difference schemes through an error correction technique employing semi-discretisations and obtains error formulae as well.
- 8.
The combination method requires, theoretically, smoothness of mixed derivatives of the solution. This is obviously not the case here due to the non-smooth payoff. However, if the payoff is aligned with the grid, which is the case in our problem, then good results have been observed for the combination method (see Leentvaar and Oosterlee (2008)). This is probably due to the rapid smoothing of the payoff in the first few time steps.
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We would like to thank one referee for his/her comments which improved this chapter.
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Kang, B., Meyer, G.H. (2014). Pricing an American Call Under Stochastic Volatility and Interest Rates. In: Dieci, R., He, XZ., Hommes, C. (eds) Nonlinear Economic Dynamics and Financial Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-07470-2_17
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