Unifying pricing formula for several stochastic volatility models with jumps

Abstract

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...

Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure

“…The set of parameters Θ F := {v 0 , κ, θ, σ, ρ, λ, µ J , σ J , H} also includes the Hurst exponent H. As was shown by Lewis (2000) and Baustian, Mrázek, Pospíšil, and Sobotka (2017) respectively, all three models attain a semi-closed form solution not only for plain European options, but also for more complex structured derivatives -this is crucial for our experiments, a single trial will involve 200 calibrations of each model to different data sets. We also did not perform analyses of models with time dependent parameters which were studied by Mikhailov and Nögel (2003), Osajima (2007), Elices (2008), Benhamou, Gobet, and Miri (2010) etc.…”

“…Due to more degrees of freedom, the model should provide a better market fit and as was shown in Duffie, Pan, and Singleton (2000) adding a second jump process to (1) might not improve the fit any more. Instead of considering a stochastic volatility model with jumps in both underlying and variance dynamics, we use an approximative fractional process as in Baustian, Mrázek, Pospíšil, and Sobotka (2017); Pospíšil and Sobotka (2016). Under the approximative fractional model one assumes the same type of jumps as in the previous case, but p…”

Isogeometric analysis in option pricing

“…Following Baustian, Mrázek, Pospíšil, and Sobotka (2017), we consider a general SVJD model which covers several kinds of stochastic volatility processes and also different types of jumps To avoid confusion, the knot at ξ = 3 is not plotted. Fitted NURBS weights are in picture on the right.…”

“…In particular we show, how B-Splines and NURBS can be fitted to smooth, non-smooth or even discontinuous functions easily. We also introduce the considered option pricing models, in particular, the constant volatility jump diffusion model by Merton (1976) (andthe Black andScholes (1973) model as a special case), stochastic volatility model by Bates (1996) (and the Heston (1993) model as a special case) and last but not least a recently proposed approximative fractional stochastic volatility jump diffusion model (Pospíšil and Sobotka 2016;Baustian, Mrázek, Pospíšil, and Sobotka 2017).…”

Calibration and simulation of Heston model

“…al. [23], where authors also extended the Lewis's approach to models with jumps. Yet another formula was later obtained by Attari [24], who uses the Euler's identity to represent separately the real and imaginary part of the integrand that in Attari's representation decays faster than in the original Heston's (or Albrecher's) case.…”