Pricing Variance Swaps Research Articles

Pricing Variance Swaps with Cash Dividends

We derive a simple formula for the price of a variance swap when the underlying has cash dividends.

On the Valuation of Variance Swaps with Stochastic Volatility

This paper is an extension to a recent paper Zhu and Lian (2009), in which a closed-form exact solution was presented for the price of variance swaps with a particular definition of the realized variance. Here, we further demonstrate that our approach is quite versatile and can be used for other definitions of the realized variance as well. In particular, we present a closed-form formula for the price of a variance swap with the realized variance in the payoff function being defined as a logarithmic return of the underlying asset at some pre-specified discretely sampling points. The simple formula presented here is a result of successfully finding an exact solution of the partial differential equation (PDE) system based on the Heston's (1993) two-factor stochastic volatility model. A distinguishable feature of this new solution is that the computational time involved in pricing variance swaps with discretely sampling time has been substantially improved.

Volatility and Dividends - Volatility Modelling with Cash Dividends and Simple Credit Risk

This article shows how to incorporate cash dividends and credit risk into equity derivatives pricing and risk management. In essence, we show that in an arbitrage-free model the stock price process upon default must have the form St = (F ⁄ t i Dt)Xt + Dt where X is a (local) martingale with X0 = 1, the curve F ⁄ is the \risky forward and D is the ∞oor imposed on the stock price process in the form of appropriately discounted future dividends. This has already been shown in [1]. We show that the method presented is the only such method which is consistent with the assumption of cash dividends and simple credit risk. We discuss the implications for implied volatility, no-arbitrage conditions and we derive a version of Dupire’s formula which handles cash dividend and credit risk properly. We discuss pricing and risk management of European options, PDE methods and in quite some detail variance swaps and related derivatives such as gamma swaps, conditional variance swaps and corridor variance swaps. Indeed, to the our best if our knowledge, this is the flrst article which shows the correct handling of cash dividends when pricing variance swaps.

A Closed-Form Exact Solution for Pricing Variance Swaps With Stochastic Volatility

In this paper, we present a highly efficient approach to price variance swaps with discrete sampling times. We have found a closed-form exact solution for the partial differential equation (PDE) system based on the Heston's two-factor stochastic volatility model embedded in the framework proposed by Little and Pant. In comparison with the previous approximation models based on the assumption of continuous sampling time, the current research of working out a closed-form exact solution for variance swaps with discrete sampling times at least serves for two major purposes: (i) to verify the degree of validity of using a continuous-sampling-time approximation for variance swaps of relatively short sampling period; (ii) to demonstrate that significant errors can result from still adopting such an assumption for a variance swap with small sampling frequencies or long tenor. Other key features of our new solution approach include the following: (1) with the newly found analytic solution, all the hedging ratios of a variance swap can also be analytically derived; (2) numerical values can be very efficiently computed from the newly found analytic formula.

Fast Calibration of the Affine and Quadratic Models

Using the recent work of Alos and Ewald on option pricing approximations we extend their approach to some specific jump-diffusion models with stochastic interest rates, compute the Greeks and improve the accuracy of the approximations. Further, we obtain analytical solutions to the price of variance swap and volatility swap. Using these results we derive approximations to the equivalent implied volatility surface, and we relate the at-the-money forward term-structure of the surface when the correlation is set to zero to the volatility swap. To conclude we use in the FFT both a change of variable and the approximated call prices as control variates in the computation of more general jump-diffusion models, reducing the variance, making the call price square integrable and drastically increasing the speed of convergence.

Forecasting the Term Structure of Variance Swaps

Recently, Diebold and Li (2003) obtained good forecasting results for yield curves in a reparametrized Nelson-Siegel framework. We analyze similar modeling approaches for price curves of variance swaps that serve nowadays as hedging instruments for options on realized variance. We consider the popular Heston model, reparametrize its variance swap price formula and model the entire variance swap curves by two exponential factors whose loadings evolve dynamically on a weekly basis. Generalizing this approach we consider a reparametrization of the three-dimensional Nelson-Siegel factor model. We show that these factors can be interpreted as level, slope and curvature and how they can be estimated directly from characteristic points of the curves. Moreover, we analyze a semiparametric factor model. Estimating autoregressive models for the factor loadings we get termstructure forecasts that we compare in addition to the random walk and the static Heston model that is often used in industry. In contrast to the results of Diebold and Li (2003) on yield curves, no model produces better forecasts of variance swap curves than the random walk but forecasting the Heston model improves the popular static Heston model. Moreover, the Heston model is better than the flexible semiparametric approach that outperforms the Nelson-Siegel model.

Apreçamento de Ativos Referenciados em Volatilidade

Um swap de volatilidade é um ativo contingente que negocia a volatilidade realizada no futuro. Um swap de variância é um contrato similar que negocia a variância de um ativo. Diferentemente de uma opção, cuja exposição à volatilidade está contaminada pela dependência ao preço do ativo, um swap de volatilidade fornece uma exposição pura à mesma. Adota-se uma forma funcional para descrever a dinâmica da variância, conforme sugerido por Heston (1993), permitindo a determinação, sob uma medida Q-martingale, do preço livre de arbitragem para os swaps de volatilidade e de variância. Como um resultado adicional, o modelo de Heston é calibrado as opções de dólar negociadas na BM&F e seus parâmetros são utilizados para o apreçamento dos swaps de volatilidade e variância para a taxa de câmbio R$ / USD.

Forecasting the Term Structure of Variance Swaps

Recently, Diebold and Li (2003) obtained good forecasting results for yield curves in a reparametrized Nelson-Siegel framework. We analyze similar modeling approaches for price curves of variance swaps that serve nowadays as hedging instruments for options on realized variance.We consider the popular Heston model, reparametrize its variance swap price formula and model the entire variance swap curves by two exponential factors whose loadings evolve dynamically on a weekly basis. Generalizing this approach we consider a reparametrization of the three-dimensional Nelson-Siegel factor model. We show that these factors can be interpreted as level, slope and curvature and how they can be estimated directly from characteristic points of the curves. Moreover, we analyze a semiparametric factor model.Estimating autoregressive models for the factor loadings we get termstructure forecasts that we compare in addition to the random walk and the static Heston model that is often used in industry. In contrast to the results of Diebold and Li (2003) on yield curves, no model produces better forecasts of variance swap curves than the random walk but forecasting the Heston model improves the popular static Heston model. Moreover, the Heston model is better than the flexible semiparametric approach that outperforms the Nelson-Siegel model.

Approximate Integration of Stochastic Differential Equations

Approximate Integration of Stochastic Differential Equations

Comment on Merton and Samuelson

Affine jump-diffusion models have been the mainstream in options pricing because of their analytical tractability. Popular affine jump-diffusion models, however, are still unsatisfactory in describing the options data and the problem is often attributed to the diffusion term of the unobserved state variables. Using prices of variance-swaps (i.e., squared VIX) implied from options prices, we provide fresh evidence regarding the misspecification of affine jump-diffusion models, as variance-swap prices are affine functions of the state variables in a broader class of models that do not restrict the diffusion term of the state variables. We apply the nonparametric methodology used by Aït-Sahalia (1996b), supplemented with bootstrap tests and other parametric tests, to the S&P 500 index options data from January 1996 to September 2008. We find that, while the affine diffusion term of the state variables may contribute to the misspecification as the literature has suggested, the affine drift of the state variables, jump intensities, and risk premiums are also sources of misspecification.