Abstract
Both barrier options and the Heston stochastic volatility model are omnipresent in real-life applications of financial mathematics. In this paper, we apply the Heath–Platen (HP) estimator (as first introduced by Heath and Platen in [12]) to price barrier options in the Heston model setting as an alternative to conventional Monte Carlo methods and PDE based methods. We demonstrate the superior performance of the HP estimator via numerical examples and explain this performance by a detailed look at the underlying theoretical concept of the HP estimator.
Funding statement: This work was supported by the DFG-research training group 1932 “Stochastic models for innovations in the engineering sciences” of which Sema Coskun has been a doctoral researcher and Ralf Korn is the spokesperson. Both gratefully acknowledge the support of the DFG. The contents of this work is based on parts of the first author’s dissertation at the Department of Mathematics of TU Kaiserslautern.
A Appendix
To be able to present one example of derivation of the Greeks of a down-and-out call option, we start with giving the BS pricing formula of the relevant option. The formula is given in [11] as
where
with
Since we have a piecewise formula conditioned on the barrier level against the strike price, we derive the Greeks also in two parts. First, we consider the case
and, finally,
Now, we deal with the derivatives of the second case
Eventually, we get the value for the vega for
The remaining second order Greeks can be obtained in a similar manner, see also [4].
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