The Pricing of Exotic Options by Monte–carlo Simulations in a Lévy Market With Stochastic Volatility

Abstract

Recently, stock price models based on Lévy processes with stochastic volatility were introduced. The resulting vanilla option prices can be calibrated almost perfectly to empirical prices. Under this model, we will price exotic options, like barrier, lookback and cliquet options, by Monte-Carlo simulation. The sampling of paths is based on a compound Poisson approximation of the Lévy process involved. The precise choice of the terms in the approximation is crucial and investigated in detail. In order to reduce the standard error of the Monte-Carlo simulation, we make use of the technique of control variates. It turns out that there are significant differences with the classical Black-Scholes prices.

Recursive Algorithms for Pricing Discrete Variance Options and Volatility Swaps Under Time-Changed Lévy Processes

“…In our numerical calculations, we adopted the set of parameter values of the NIG-CIR and VG-CIR models from Schoutens and Symens (2003), and the Heston model from Zhu and Lian (2011). The values of the model parameters are listed in Table 1.…”

“…Therefore, the simulation time taken for the VG process is significantly longer than that of the NIG process. We followed the approach by Schoutens and Symens (2003) in the Monte Carlo simulation, where 100,000 sample paths for the NIG-CIR model and 10,000 paths for the VG-CIR model were simulated in our calculations. The recursive algorithms were coded in Mathematica to take advantage of its built-in Bessel functions.…”

On European and Asian option pricing in the generalized hyperbolic model

“…Much current research is dusted to the attempt to combine the good static properties of Lévy processes and semi-heavy tail distributions with more sophisticated dynamic modelling techniques [1,12,15,29,114]. Since the generalized hyperbolic model and the variance-gamma model (which is in fact a limitting case of the generalized hyperbolic model, as we will see) are the most common non-Black-Scholes models, we will have a deeper look at them in the following.…”

Efficient Pricing of Barrier Options on High Volatility Assets Using Subset Simulation

“…Although these methods are generally powerful, they depend on being able to accurately model the option with PDEs and cannot be used in all circumstances (other approaches used in the pricing of exotic derivatives include the method of lines (Chiarella et al, 2012), where the Greeks are also estimated, robust optimization techniques (Bandi and Bertsimas, 2014), applicable also to American options, finite-difference based approaches (Wade et al, 2007), where a Crank-Nicolson smoothing strategy to treat discontinuities in barrier options is presented, and regime-switching models (Elliott et al, 2014;Rambeerich and Pantelous, 2016)). As a result, Monte Carlo simulation (MCS) is often used for option pricing (Schoutens and Symens, 2003) and particularly for barrier options (Glasserman and Staum, 2001).…”