Georgios Foufas and Mats G. Larson The main objective of this paper is to develop an adaptive nite element method for computation of the values and dierent sensitivity measures of ordinary European options, barrier options, and lookback options. The options are priced using the Black-Scholes PDE-model, and the resulting PDE:s are of parabolic type in one spatial dimension with different boundary conditions and jump conditions at monitoring dates. The adaptive nite element method is based on a posteriori estimates of the error in desired quantities, which we derive using duality techniques. The suggested adaptive nite element method is stable and gives fast and accurate results.
Gunter Winkler Thomas Apel Uwe Wystup Quoted:
Introduction Due to the smile observed in options markets numerous authors have suggested different models such as generalized Levy processes, fractional Brownian motion, entropy based models , jump diffusions and stochastic volatility models. For vanilla options (put and call options) the dependence of the price on the volatility is monotone, whence using the Black-Scholes formula along with a volatility smile matrix is sufficient. Values of exotic options, however, do not always depend on the volatility in a monotone fashion, whence pricing consistently with the smile requires a more sophisticated model. Therefore, it is important to find efficient ways to calculate exotic option values in exotic models.
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