Variance swaps, volatility swaps, hedging and bounds under multi-factor Heston stochastic volatility model

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.

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.

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.

Efficient 'Black-Scholes' Approximations of the Heston and Multi-Factor Heston Option Prices for a Correct Spot Price

This paper considers the case of pricing discretely-sampled variance swaps under the class of equity-interest rate hybridization. Our modeling framework consists of the equity which follows the dynamics of the Heston stochastic volatility model, and the stochastic interest rate is driven by the Cox-Ingersoll-Ross (CIR) process with full correlation structure imposed among the state variables. This full correlation structure possesses the limitation to have fully analytical pricing formula for hybrid models of variance swaps, due to the non-affinity property embedded in the model itself. We address this issue by obtaining an efficient semi-closed form pricing formula of variance swaps for an approximation of the hybrid model via the derivation of characteristic functions. Subsequently, we implement numerical experiments to evaluate the accuracy of our pricing formula. Our findings confirm that the impact of the correlation between the underlying and the interest rate is significant for pricing discretely-sampled variance swaps.

In this paper, we will explain how to perfectly hedge under Heston's stochastic volatility model with jump-to-default, which is in itself a generalization of the Merton jump-to-default model and a special case of the Heston model with jumps. The hedging instruments we use to build the hedge will be as usual the stock and the bond, but also the Variance Swap (VS) and a Credit Default Swap (CDS). These instruments are very natural choices in this setting as the VS hedges against changes in the instantaneous variance rate, while the CDS protects against the occurrence of the default event. First, we explain how to perfectly hedge a power payoff under the Heston model with jump-to-default. These theoretical payoffs play an important role later on in the hedging of payoffs which are more liquid in practice such as vanilla options. After showing how to hedge the power payoffs, we show how to hedge newly introduced Gamma payoffs and Dirac payoffs, before turning to the hedge for the vanillas. The approach is inspired by the Post–Widder formula for real inversion of Laplace transforms. Finally, we will also show how power payoffs can readily be used to approximate any payoff only depending on the value of the underlier at maturity. Here, the theory of orthogonal polynomials comes into play and the technique is illustrated by replicating the payoff of a vanilla call option.

A smooth estimator for MC/QMC methods in finance

This paper builds and implements multifactor stochastic volatility models where the main objective is step ahead volatility prediction and to describe its relevance for the equity markets. The paper outlines stylised facts from the volatility literature showing density tails, persistence, mean reversion, asymmetry and long memory, all contributing to systematic data dependencies. As a by-product of the multifactor stochastic volatility model estimation, a long-simulated realization of the state vectors is available. The realization establishes a functional form of the conditional distribution, which is evaluated on observed data convenient for step ahead predictions. The paper uses European equity for relevance arguments and illustrational prediction purposes. Multifactor SV models empower volatility visibility and predictability enriching the amount of information available for equity market participants.

Pricing forward-start variance swaps with stochastic volatility

In this article, we develop an identifiable multi-factor stochastic volatility model for international portfolio choice problems in complete and incomplete markets. Allowing for stochastic covariance between financial asset returns and foreign exchange rates, optimal investment strategies are derived in closed form and welfare losses arising from suboptimal investment strategies are analysed. Moreover, we provide a two-step procedure for estimating as well as calibrating the model parameters and use this ansatz to illustrate optimal investment decisions for the S&P 500, the German blue chip index DAX, and the USD/EUR foreign exchange rate. We find, both theoretically and empirically, that the model satisfies various well-known stylized facts of equity and foreign exchange rate markets and that investors who invest myopically or ignore derivative assets can incur substantial welfare losses implying strong evidence for significant welfare benefits from international diversification across different asset classes.

We introduce a discrete Barndorff-Nielsen and Shephard (BNS) stochastic volatility model, where the non-Gaussian Ornstein-Uhlenbeck process governs the instantaneous variance of an underlying asset. A closed-form solution for pricing variance swaps in financial markets driven by a BN-S type Levy process is derived. Additionally, we propose a discrete-time framework for arbitrage-free pricing of variance swaps, supported by simulations that provide numerical validation of our results. The model fitting and parameter estimation processes are also discussed in detail.

Variance and volatility swaps valuations with the stochastic liquidity risk

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.

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

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.