No abstract

No 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...

In this paper we perform robustness and sensitivity analysis of several continuous-time stochastic volatility (SV) models with respect to the process of market calibration. The analyses should validate the hypothesis on importance of the jump part in the underlying model dynamics. Also an impact of the long memory parameter is measured for the approximative fractional SV model. For the first time, the robustness of calibrated models is measured using bootstrapping methods on market data and Monte-Carlo filtering techniques. In contrast to several other sensitivity analysis approaches for SV models, the newly proposed methodology does not require independence of calibrated parameters -an assumption that is typically not satisfied in practice. Empirical study is performed on data sets of Apple Inc. equity options traded in April and May 2015.

In this paper we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alòs (2012) for Heston (1993) SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models -models utilizing a variance process postulated by Heston (1993). In particular, we inspect in detail the approximation formula for the Bates (1996) model with log-normal jump sizes and we provide a numerical comparison with the industry standard -Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behaviour under a specific SVJ model. Remark 4.5. In particular, we have a closed formula for a log-normal jump diffusion model (e.g. Bates (1996) SVJ model): G n (0,X 0 , v 0 ) = BS 0,X 0 , v 2 0 + n σ 2 J T where we modified the risk-free rate used in the Black-Scholes formula to r * = r − λ e µJ + 1 2 σ 2 J − 1 + n µ J + 1 2 σ 2 J T . (n) 0 = v 2 0 + n σ 2 J T as the new volatility andr n = r − λ e µJ + 1 2 σ 2 J − 1 + n µ J + 1 2 σ 2 J T as the new drift. The parameter n is the number of realized jumps, µ J and σ J are the jump-size parameters and λ is the jump intensity. For simplicity, we denote: c n := −λ e µJ + 1 2 σ 2 J − 1 + n µ J + 1 2 σ 2 J T .As a consequence, we have that d ± x,r n ,ṽ (n) 0