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In this paper, we introduce a unifying approach to option pricing under continuous-time stochastic volatility models with jumps. For European style options, a new semi-closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro-differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way. In particular, we focus on a log-normal and a log-uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out-of-the money contracts.Appl. Stochastic Models Bus. Ind. 2017, 33 422-442 F. BAUSTIAN ET AL. allows to have correlated increments of the asset price and the volatility process (as opposed to Stein and Stein [3]), which can mimic a volatility leverage effect observed on many financial markets. However, the model lacks the ability to fit reasonably well-complex option price surfaces [5,6], especially the ones that involve both short-dated and long-dated contracts.To deal with the drawbacks of the first SV models, many modifications have been introduced since, including a dynamic Heston model that involves time-dependent parameters. The case of piece-wise constant parameters in time is studied in Mikhailov and Ngel [7], a linear time dependence in Elices [8] and a more general case is analysed in Benhamou et al. [9]. The latter result introduces only an approximation of the option price. However, Bayer et al. [5] suggest that the general overall shape of the volatility surface does not change in time, at least to a first approximation given by stochastic volatility inspired (SVI) models. Hence, it is desirable to model volatility as a time-homogeneous process. Other generalizations of the Heston model with time-constant parameters include jump processes in asset price, in volatility or in both (e.g. Duffie et al. [10]). As Gatheral [11] notes (and supports by empirical analyses of several authors), a model with jumps in both underlying and volatility, although having more parameter and degrees of freedom, might not provide significantly better market fit than its counterpart with jumps in underlying only. The first SVJD model introduced in [12] adds a log-normally distributed jumps to the diffusion dynamics of the Heston model. Several different jump-diffusion settings were proposed, for example, models postulated by Scott [13] and Yan and Hanson [14] among others.Another possibility to modify standard diffusion SV models is to use a Lévy subordinator as a driving noise of the volatility process. This idea was firstly developed by where both volatility and as...

Abstract:We calibrate Heston stochastic volatility model to real market data using several optimization techniques. We compare both global and local optimizers for different weights showing remarkable differences even for data (DAX options) from two consecutive days. We provide a novel calibration procedure that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods.We test and compare several simulation schemes using the parameters obtained by calibration to real market data. Next to the known schemes (log-Euler, Milstein, QE, Exact scheme, IJK) we introduce also a new method combining the Exact approach and Milstein (E+M) scheme. Test is carried out by pricing European call options by Monte Carlo method. Presented comparisons give an empirical evidence and recommendations what methods should and should not be used and why. We further improve the QE scheme by adapting the antithetic variates technique for variance reduction.