On calibration of stochastic and fractional stochastic volatility models. (English) Zbl 1346.91238

Mrázek, Milan; Pospíšil, Jan; Sobotka, Tomáš

Eur. J. Oper. Res. 254, No. 3, 1036-1046 (2016).

Summary: In this paper we study optimization techniques for calibration of stochastic volatility models to real market data. Several optimization techniques are compared and used in order to solve the nonlinear least squares problem arising in the minimization of the difference between the observed market prices and the model prices. To compare several approaches we use a popular stochastic volatility model firstly introduced by S. L. Heston [“A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Rev. Financ. Stud. 6, No. 2, 327–343 (1993; doi:10.1093/rfs/6.2.327)] and a more complex model with jumps in the underlying and approximative fractional volatility. Calibration procedures are performed on two main data sets that involve traded DAX index options. We show how well both models can be fitted to a given option price surface. The routines alongside models are also compared in terms of out-of-sample errors. For the calibration tasks without having a good knowledge of the market (e.g. a suitable initial model parameters) we suggest an approach of combining local and global optimizers. This way we are able to retrieve superior error measures for all considered tasks and models.

MSC:

91G20	Derivative securities (option pricing, hedging, etc.)
60H30	Applications of stochastic analysis (to PDEs, etc.)
35R60	PDEs with randomness, stochastic partial differential equations
35R11	Fractional partial differential equations
62P05	Applications of statistics to actuarial sciences and financial mathematics
91B70	Stochastic models in economics
91G80	Financial applications of other theories

Keywords:

fractional stochastic volatility model; Heston model; option pricing; calibration; optimization

Software:

Mathematica

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