Variance and Volatility Swaps in Energy Markets

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.

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

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.

In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is the multi-factor Heston stochastic volatility model. We obtain pricing formulas for the weighted variance swap and approximate expressions for the weighted volatility swap. The bounds of the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to compute in real time and can be applied efficiently for practical applications. We study the problem of hedging volatility swap with variance swap. We also determined the optimal amount of the underlying asset that has to be held for minimizing the hedging error by taking positions in options and weighted variance swap. From the numerical analysis, a couple of important features of the usefulness of the multi-factor Heston stochastic volatility model are discussed.

This paper presents a robust methodology for the valuation of options on variance swaps and volatility swaps. While existing literature has often focused on the pricing of the swaps themselves under stochastic volatility models, the valuation of options on these second-order derivative products remains a challenge, particularly within a framework that captures empirical market properties such as volatility clustering and long memory. Our approach addresses this problem by relying on the precise calculation of the optimal exercise prices (or strike rates) for variance and volatility swaps. To achieve this, we introduce and utilize a multifractional extension of the classic SABR model. This enhanced version of the model incorporates a deterministic and time-varying Hurst function, h(t), which allows for the modeling of a richer and more flexible volatility dynamics, capable of replicating the varying persistence phenomena observed in real markets. The proposed methodology is detailed comprehensively. We derive the theoretical formulas for the swap prices, which form the basis for calculating the optimal exercise prices. The valuation of options (call and put) on these swaps is then performed by employing advanced numerical techniques, adapted to the complexity of the multifractional process.

Volatility, Variance and Covariance Swaps for BN-S Type Levy Process Driven Financial Markets

Stochastic interest rate volatility modeling with a continuous-time GARCH(1, 1) model

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.

Moment Methods for Exotic Volatility Derivatives

Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this paper is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled swap values are finite, given that the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values.

This article discusses the valuation and hedging of volatility swaps within the frame of a GARCH(1,1) stochastic volatility model. First we use a general and flexible partial differential equation (PDE) approach to determine the first two moments of the realized variance in a continuous or discrete context. Next, and also the main contribution of the paper, is a closed-form approximate solution for the so-called convexity correction, when the risk-neutral process for the instantaneous variance is a continuous time limit of a GARCH (1,1) model. Following this, we provide a numerical example using S&P 500 data.

Pricing volatility derivatives under the modified constant elasticity of variance model

Some of the newest and most recent financial products are variance and volatility swaps, which are useful for volatility hedging and speculation. The market for variance and volatility swaps has been growing, and many investment banks and other financial institutions are now actively quoting volatility swaps on various assets: stock indexes, currencies, as well as commodities. In this chapter, the authors model financial markets with semi-Markov volatilities and price variance, volatility, covariance and correlation swaps. Numerical evaluations of variance, volatility, covariance and correlations swaps with semi-Markov volatility are also presented. The novelty of the chapter lies in the pricing of volatility swaps in the closed form, and the pricing of covariance and correlation swaps in a market with two risky assets. Volatility swaps are forward contracts on future realized stock volatility, while variance swaps are similar contracts on variance, the square of the future volatility.

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