Modelling and Simulation of Stochastic Volatility in Finance

Front Cover
Universal-Publishers, 2008 - Business & Economics - 220 pages
0 Reviews
The famous Black-Scholes model was the starting point of a new financial industry and has been a very important pillar of all options trading since. One of its core assumptions is that the volatility of the underlying asset is constant. It was realised early that one has to specify a dynamic on the volatility itself to get closer to market behaviour. There are mainly two aspects making this fact apparent. Considering historical evolution of volatility by analysing time series data one observes erratic behaviour over time. Secondly, backing out implied volatility from daily traded plain vanilla options, the volatility changes with strike. The most common realisations of this phenomenon are the implied volatility smile or skew. The natural question arises how to extend the Black-Scholes model appropriately. Within this book the concept of stochastic volatility is analysed and discussed with special regard to the numerical problems occurring either in calibrating the model to the market implied volatility surface or in the numerical simulation of the two-dimensional system of stochastic differential equations required to price non-vanilla financial derivatives. We introduce a new stochastic volatility model, the so-called Hyp-Hyp model, and use Watanabe's calculus to find an analytical approximation to the model implied volatility. Further, the class of affine diffusion models, such as Heston, is analysed in view of using the characteristic function and Fourier inversion techniques to value European derivatives.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Introduction
1
Stochastic volatility models
5
21 Transformed OrnsteinUhlenbeck models
6
22 Affine diffusion stochastic volatility models
13
Monte Carlo methods
29
32 QuasiMonte Carlo
32
33 Path construction methods
34
34 Fractional Fourier transformation for spectral path construction
46
53 Optimal choice of alpha
97
54 Numerical results
109
Numerical integration schemes for stochastic volatility models
113
61 Numerical integration of meanreverting CEV processes
115
62 Numerical integration of stochastic volatility models
127
63 Multidimensional stochastic volatility models
146
Conclusion
161
Balanced Milstein Methods for ordinary SDEs
163

European option pricing for transformed OrnsteinUhlenbeck stochastic volatility models
51
42 Asymptotic approximation using Watanabes expansion
60
Optimal Fourier inversion for affine diffusion models
91
52 Integration domain change
95
Proofs
179
Bibliography
189
Index
200
Copyright

Common terms and phrases

Bibliographic information