Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching

A model is developed for pricing volatility derivatives, such as variance swaps and volatility swaps under a continuous‐time Markov‐modulated version of the stochastic volatility (SV) model developed by Heston. In particular, it is supposed that the parameters of this version of Heston's SV model depend on the states of a continuous‐time observable Markov chain process, which can be interpreted as the states of an observable macroeconomic factor. The market considered is incomplete in general, and hence, there is more than one equivalent martingale pricing measure. The regime switching Esscher transform used by Elliott et al. is adopted to determine a martingale pricing measure for the valuation of variance and volatility swaps in this incomplete market. Both probabilistic and partial differential equation (PDE) approaches are considered for the valuation of volatility derivatives.     

Prices and Asymptotics for Discrete Variance Swaps

Abstract

We study the fair strike of a discrete variance swap for a general time-homogeneous stochastic volatility model. In the special cases of Heston, Hull–White and Schöbel–Zhu stochastic volatility models, we give simple explicit expressions (improving Broadie and Jain (2008a). The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8), 761–797) in the case of the Heston model). We give conditions on parameters under which the fair strike of a discrete variance swap is higher or lower than that of the continuous variance swap. The interest rate and the correlation between the underlying price and its volatility are key elements in this analysis. We derive asymptotics for the discrete variance swaps and compare our results with those of Broadie and Jain (2008a. The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8), 761–797), Jarrow et al. (2013. Discretely sampled variance and volatility swaps versus their continuous approximations. Finance and Stochastics, 17(2), 305–324) and Keller-Ressel and Griessler (2012. Convex order of discrete, continuous and predictable quadratic variation and applications to options on variance. Working paper. Retrieved from http://arxiv.org/abs/1103.2310).     

Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching

In this paper, we consider the problem of pricing discretely-sampled variance swaps based on a hybrid model of stochastic volatility and stochastic interest rate with regime-switching. Our modeling framework extends the Heston stochastic volatility model by including the Cox-Ingersoll-Ross (CIR) stochastic interest rate model. In addition, certain model parameters in our model switch according to a continuous-time observable Markov chain process. This enables our model to capture several macroeconomic issues such as alternating business cycles. A semi-closed form pricing formula for variance swaps is derived. The pricing formula is assessed through numerical implementation, where we validate our pricing formula against the Monte Carlo simulation. The impact of incorporating regime-switching for pricing variance swaps is also discussed, where variance swaps prices with and without regime-switching effects are examined in our model. We also explore the economic consequence for the prices of variance swaps by allowing the Heston-CIR model to switch across three different regimes.     

New approach and analysis of the generalized constant elasticity of variance model

Generally, it is well known that the constant elasticity of variance (CEV) model fails to capture the empirical results verifying that the implied volatility of equity options displays smile and skew curves at the same time. In this study, to overcome the limitation of the CEV model, we introduce a new model, which is a generalization of the CEV model, and show that it can capture the smile and skew effects of implied volatility. Using an asymptotic analysis for two small parameters that determine the volatility shape, we obtain approximated solutions for option prices in the extended model. In addition, we demonstrate the stability of the solution for the expansion of the option price. Furthermore, we show the convergence rate of the solutions in Monte-Carlo simulation and compare our model with the CEV, Heston, and other extended stochastic volatility models to verify its flexibility and efficiency compared with these other models when fitting option data from the S&P 500 index.     

Heston模型下的方差互换定价研究

通过实证分析论证了波动率具有均值回复性质的合理性.在Heston模型下,利用Ito积分推导出了方差互换在其存续期内任意时刻的价格与公平执行价格的定价公式.得到公平执行价格是波动率的平方的初始水平与长期均值水平的线性组合的性质,并利用该性质对Heston模型参数的敏感性进行了分析.     

The Heston model with stochastic elasticity of variance

Empirical evidence suggests that single factor models would not capture the full dynamics of stochastic volatility such that a marked discrepancy between their predicted prices and market prices exists for certain ranges (deep in‐the‐money and out‐of‐the‐money) of time‐to‐maturities of options. On the other hand, there is an empirical reason to believe that volatility skew fluctuates randomly. Based upon the idea of combining stochastic volatility and stochastic skew, this paper incorporates stochastic elasticity of variance running on a fast timescale into the Heston stochastic volatility model. This multiscale and multifactor hybrid model keeps analytic tractability of the Heston model as much as possible, while it enhances capturing the complex nature of volatility and skew dynamics. Asymptotic analysis based on ergodic theory yields a closed form analytic formula for the approximate price of European vanilla options. Subsequently, the effect of adding the stochastic elasticity factor on top of the Heston model is demonstrated in terms of implied volatility surface. Copyright © 2016 John Wiley & Sons, Ltd.     

Lie-algebraic approach for pricing moving barrier options with time-dependent parameters

In this paper we apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters. The value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process. By exploiting the dynamical symmetry of the pricing partial differential equations, the new approach enables us to derive the analytical kernels of the pricing formulae straightforwardly, and thus provides an efficient way for computing the prices of the moving barrier options. The method is also able to provide tight upper and lower bounds for the exact prices of CEV barrier options with fixed barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, our new approach could facilitate more efficient comparative pricing and precise risk management in equity derivatives with barriers by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.     

Variance swaps under the threshold Ornstein–Uhlenbeck model

Variance swap is a typical financial tool for managing volatility risk. In this paper, we evaluate different types of variance swaps under a threshold Ornstein–Uhlenbeck model, which exhibits both mean reversion and regime switching features in the underlying asset price. We derive the analytical solution for the joint moment generating function of log‐asset prices at two distinct time points. This enables us to price various types of variance swaps analytically. Copyright © 2017 John Wiley & Sons, Ltd.     

Asymptotic approach to the pricing of geometric asian options under the CEV model

This paper studies the pricing of Asian options whose payoffs depend on the average value of an underlying asset during the period to a maturity. Since the Asian option is not so sensitive to the value of underlying asset, the possibility of manipulation is relatively small than the other options such as European vanilla and barrier options. We derive the pricing formula of geometric Asian options under the constant elasticity of variance (CEV) model that is one of local volatility models, and investigate the implication of the CEV model for geometric Asian options.     

On the pricing and hedging of volatility derivatives

The paper considers the pricing of a range of volatility derivatives, including volatility and variance swaps and swaptions. Under risk-neutral valuation closed-form formulae for volatility-average and variance swaps for a variety of diffusion and jump-diffusion models for volatility are provided. A general partial differential equation framework for derivatives that have an extra dependence on an average of the volatility is described. Approximate solutions of this equation are given for volatility products written on assets for which the volatility process fluctuates on a timescale that is fast compared with the lifetime of the contracts, analysing both the 'outer' region and, by matched asymptotic expansions, the 'inner' boundary layer near expiry.     

Market calibration under a long memory stochastic volatility model

In this article, we study a long memory stochastic volatility model (LSV), under which stock prices follow a jump-diffusion stochastic process and its stochastic volatility is driven by a continuous-time fractional process that attains a long memory. LSV model should take into account most of the observed market aspects and unlike many other approaches, the volatility clustering phenomenon is captured explicitly by the long memory parameter. Moreover, this property has been reported in realized volatility time-series across different asset classes and time periods. In the first part of the article, we derive an alternative formula for pricing European securities. The formula enables us to effectively price European options and to calibrate the model to a given option market. In the second part of the article, we provide an empirical review of the model calibration. For this purpose, a set of traded FTSE 100 index call options is used and the long memory volatility model is compared to a popular pricing approach – the Heston model. To test stability of calibrated parameters and to verify calibration results from previous data set, we utilize multiple data sets from NYSE option market on Apple Inc. stock.     

Robust Approximations for Pricing Asian Options and Volatility Swaps Under Stochastic Volatility

Abstract

We show that if the discounted Stock price process is a continuous martingale, then there is a simple relationship linking the variance of the terminal Stock price and the variance of its arithmetic average. We use this to establish a model-independent upper bound for the price of a continuously sampled fixed-strike arithmetic Asian call option, in the presence of non-zero time-dependent interest rates (Theorem 1.2). We also propose a model-independent lognormal moment-matching procedure for approximating the price of an Asian call, and we show how to apply these approximations under the Black–Scholes and Heston models (subsection 1.3). We then apply a similar analysis to a time-dependent Heston stochastic volatility model, and we show how to construct a time-dependent mean reversion and volatility-of-variance function, so as to be consistent with the observed variance swap curve and a pre-specified term structure for the variance of the integrated variance (Theorem 2.1). We characterize the small-time asymptotics of the first and second moments of the integrated variance (Proposition 2.2) and derive an approximation for the price of a volatility swap under the time-dependent Heston model ( Equation (52)), using the Brockhaus–Long approximation (Brockhaus, and Long, 2000 Brockhaus, O. and Long, D. 2000. Volatility Swaps made simple. Risk, 13(1) January: 92–96.  [Google Scholar]). We also outline a bootstrapping procedure for calibrating a piecewise-linear mean reversion level and volatility-of-volatility function (Subsection 2.3.2).     

Development of Computational Algorithms for Evaluating Option Prices Associated with Square-Root Volatility Processes

The stochastic volatility model of Heston (Rev Financ Stud 6:327–343, 1993) has been accepted by many practitioners for pricing various financial derivatives, because of its capability to explain the smile curve of the implied volatility. While analytical results are available for pricing plain Vanilla European options based on the Heston model, there hardly exist any closed form solutions for exotic options. The purpose of this paper is to develop computational algorithms for evaluating the prices of such exotic options based on a bivariate birth-death approximation approach. Given the underlying price process S _t , the logarithmic process U _t  = logS _t is first approximated by a birth-death process B^U_tB^U_t via moment matching. A second birth-death process B^V_tB^V_t is then constructed for approximating the stochastic volatility process V _t through infinitesimal generator matching. Efficient numerical procedures are developed for capturing the dynamic behavior of { B^U_t , B^V_t }\{ B^U_t , B^V_t \} . Consequently, the prices of any exotic options based on the Heston model can be computed as long as such prices are expressed in terms of the joint distribution of { S _t ,V _t } and the associated first passage times. As an example, the prices of down-and-out call options are evaluated explicitly, demonstrating speed and fair accuracy of the proposed algorithms.     

American option pricing under stochastic volatility: an empirical evaluation

Over the past few years, model complexity in quantitative finance has increased substantially in response to earlier approaches that did not capture critical features for risk management. However, given the preponderance of the classical Black–Scholes model, it is still not clear that this increased complexity is matched by additional accuracy in the ultimate result. In particular, the last decade has witnessed a flurry of activity in modeling asset volatility, and studies evaluating different alternatives for option pricing have focused on European-style exercise. In this paper, we extend these empirical evaluations to American options, as their additional opportunity for early exercise may incorporate stochastic volatility in the pricing differently. Specifically, the present work compares the empirical pricing and hedging performance of the commonly adopted stochastic volatility model of Heston (Rev Financial Stud 6:327–343, 1993) against the traditional constant volatility benchmark of Black and Scholes (J Polit Econ 81:637–659, 1973). Using S&P 100 index options data, our study indicates that this particular stochastic volatility model offers enhancements in line with their European-style counterparts for in-the-money options. However, the most striking improvements are for out-of-the-money options, which because of early exercise are more valuable than their European-style counterparts, especially when volatility is stochastic.     

Pricing exotic derivatives exploiting structure

In this paper we introduce a new fast and accurate numerical method for pricing exotic derivatives when discrete monitoring occurs, and the underlying evolves according to a Markov one-dimensional stochastic processes. The approach exploits the structure of the matrix arising from the numerical quadrature of the pricing backward formulas to devise a convenient factorization that helps greatly in the speed-up of the recursion. The algorithm is general and is examined in detail with reference to the CEV (Constant Elasticity of Variance) process for pricing different exotic derivatives, such as Asian, barrier, Bermudan, lookback and step options for which up to date no efficient procedures are available. Extensive numerical experiments confirm the theoretical results. The MATLAB code used to perform the computation is available online at http://www1.mate.polimi.it/∼marazzina/BP.htm.     

基于CEV过程带交易费的脆弱期权定价的数值算法

主要研究基于CEV过程且支付交易费的脆弱期权定价的数值计算问题.首先通过构造无风险投资组合,导出了基于CEV过程且支付交易费用的脆弱期权定价的偏微分方程模型;其次应用有限差分方法将定价模型离散化,并设计数值算法;最后以看跌期权为例进行数值试验,分析各定价参数对看跌期权价值的影响.     

The Heston stochastic volatility model in Hilbert space

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein–Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein–Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modeling of forward curves in energy and commodity markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics.     

On the application of Wishart process to the pricing of equity derivatives: the multi-asset case

Given the inherent complexity of financial markets, a wide area of research in the field of mathematical finance is devoted to develop accurate models for the pricing of contingent claims. Focusing on the stochastic volatility approach (i.e. we assume to describe asset volatility as an additional stochastic process), it appears desirable to introduce reliable dynamics in order to take into account the presence of several assets involved in the definition of multi-asset payoffs. In this article we deal with the multi asset Wishart Affine Stochastic Correlation model, that makes use of Wishart process to describe the stochastic variance covariance matrix of assets return. The resulting parametrization turns out to be a genuine multi-asset extension of the Heston model: each asset is exactly described by a single instance of the Heston dynamics while the joint behaviour is enriched by cross-assets and cross-variances stochastic correlation, all wrapped in an affine modeling. In this framework, we propose a fast and accurate calibration procedure, and two Monte Carlo simulation schemes.