Analytically pricing variance and volatility swaps with stochastic volatility, stochastic equilibrium level and regime switching

Pricing variance and volatility swaps with stochastic volatility, stochastic interest rate and regime switching

In this chapter, we give an overview on applications of change of time methods considered in this book in Chapters 4–8 These applications include yet another (among many) derivation of the Black-Scholes formula; the derivation of option pricing formula for a mean-reverting asset in energy finance; pricing of variance, volatility, covariance, and correlation swaps for the classical Heston model; pricing of variance and volatility swaps in energy markets; pricing of financial and energy derivatives with multifactor Levy models; and pricing of variance and volatility swaps and hedging of volatility swaps for the delayed Heston model. This chapter not only describes the applications of the change of time method but also constitutes the ultimate difference between Barndorff-Nielsen-Shiryaev’s book (2010) and present book.

In this paper, the pricing problem of variance and volatility swaps is discussed under a two-factor stochastic volatility model. This model can be treated as a two-factor Heston model with one factor following the CIR process and another characterized by a Markov chain, with the motivation originating from the popularity of the Heston model and the strong evidence of the existence of regime switching in real markets. Based on the derived forward characteristic function of the underlying price, analytical pricing formulae for variance and volatility swaps are presented, and numerical experiments are also conducted to compare swap prices calculated through our formulae and those obtained under the Heston model to show whether the introduction of the regime switching factor would lead to any significant difference.

Variance and Volatility Swaps in Energy Markets

Moment Methods for Exotic Volatility Derivatives

ABSTRACTWe construct a stochastic volatility model considering stochastic liquidity risks when valuing variance and volatility swaps with discrete sampling. We base our model on Heston stochastic volatility, which is adopted for the modeling of stock prices when the market is perfectly liquid. Stock dynamics are further revised by discounting their prices through the employment of mean reverting market liquidity. We convert the stock dynamics under the physical measure into the one under a risk‐neutral measure via measure transform, with which the analytical valuation of variance and volatility swaps is realized. By taking the limit of sampling frequency, we further consider how both swaps with continuous sampling can be priced. Numerical implementation is finally carried out, with which the capability of the constructed model in capturing the influence of the two common types of financial risks can be clear.

Analytically pricing volatility options and capped/floored volatility swaps with nonlinear payoffs in discrete observation case under the Merton jump-diffusion model driven by a nonhomogeneous Poisson process

Long memory and regime switching

We consider a general option pricing framework incorporating the double Heston stochastic volatility, stochastic interest rate, jumps and Markov regime switching. Under the proposed framework, we derive the analytical pricing formulae for European options using Fourier transform technique. Numerical examples illustrate that the option prices and the implied volatility curves under different regimes vary clearly, and the effects of regime-switching and jumps on the option price differ.

Forecasting stock market volatility is an important and challenging task for both academic researchers and business practitioners. The recent trend to improve the prediction accuracy is to combine individual forecasts using a simple average or weighted average where the weight reflects the inverse of the prediction error. However, a problem in the existing forecast combination methods is that the weights remain fixed over time. This may prove inadequate, especially in economics data, where changes in policy regimes may induce structural change in the pattern of forecast errors of the different models, thereby altering the relative effectiveness of each model over time. In this paper, we present a new forecast combination approach where the forecasting results of the Generalized Autoregressive Conditional Heteroskedastic (GARCH), the Exponential GARCH (EGARCH), stochastic volatility(SV), and Moving average(MAV) models are combined based on time-varying weights that can be driven by regime switching in a latent state variable. The results of an empirical study indicate that the proposed method has a better accuracy than the GARCH, EGARCH, SV and MAV models, and also combining forecast methods with constant weights.KeywordsCombining forecastsGARCHStochastic volatilityRegime switchingStock market volatility

This paper uses an extensive set of market data of forward swap rates and swaptions covering 3 July 2002 to 21 May 2009 to identify a two-dimensional stochastic volatility process for the level of rates. The process is identified step by step by increasing the requirement of the model and introducing appropriate adjustments. The first part of the paper investigates the smile dynamics of forward swap rates at their setting dates. Comparing the SABR (with different ) and Heston stochastic volatility models informs us what different specifications of the driving SDEs have to offer in terms of reflecting the dynamics of the smile across dates. The outcome of the analysis is that a normal SABR model () satisfactorily passes all tests and seems to provide a good match to the market. In contrast, we find that the Heston model does not. The next step is to seek a model of the forward swap rates (in their own swaption measure) based on only two Brownian motions that enables a specification with common parameters. It turns out that this can be done by extending the SABR model with a time-dependent volatility function and a mean-reverting volatility process. The performance of the extended (SABR with mean-reversion) model is analysed over several historical dates and is shown to be a stable and flexible choice that allows for good calibration across expiries and strikes. Finally, a time-homogeneous candidate stochastic volatility process that can be used as a driver for all swap rates is identified. This candidate process may in future work be used as a building block for a separable stochastic volatility LIBOR market model or a stochastic volatility Markov-functional model.

An exhaustive survey of research on pricing foreign exchange options with stochastic volatility and stochastic interest rates is presented in this article. The article also includes contributions to pricing of currency options with fixed volatility and deterministic interest rates, domestic and foreign, and in a systematic way extend to results obtained for cases which include stochastic volatility and stochastic interest rates. Since the literature in pricing foreign exchange currency option is quite extensive, the article considers some of the more important contributions.

Variance and volatility swap valuation with stochastic liquidity and regime switching stochastic volatility

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.