A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.As a concrete example, we specialize to a variant of the Hull-White or Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q.JEL Classification: D52, G13     

Pricing Discrete Barrier Options Under Stochastic Volatility

This paper proposes a new approximation method for pricing barrier options with discrete monitoring under stochastic volatility environment. In particular, the integration-by-parts formula and the duality formula in Malliavin calculus are effectively applied in pricing barrier options with discrete monitoring. To the best of our knowledge, this paper is the first one that shows an analytical approximation for pricing discrete barrier options with stochastic volatility models. Furthermore, it provides numerical examples for pricing double barrier call options with discrete monitoring under Heston and λ-SABR models.     

Index Option Pricing Models with Stochastic Volatility and Stochastic Interest Rates

This paper specifies a multivariate stochasticvolatility (SV) model for the S & P500 index and spot interest rateprocesses. We first estimate the multivariate SV model via theefficient method of moments (EMM) technique based on observations ofunderlying state variables, and then investigate the respective effects of stochastic interest rates, stochastic volatility, and asymmetric S & P500 index returns on option prices. We compute option prices using both reprojected underlying historical volatilities and the implied risk premiumof stochastic volatility to gauge each model's performance through direct comparison with observed market option prices on the index. Our major empirical findings are summarized as follows. First, while allowing for stochastic volatility can reduce the pricing errors and allowing for asymmetric volatility or leverage effect does help to explain the skewness of the volatility smile, allowing for stochastic interest rates has minimal impact on option prices in our case. Second, similar to Melino and Turnbull (1990), our empirical findings strongly suggest the existence of a non-zero risk premium for stochastic volatility of asset returns. Based on the implied volatility risk premium, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information from the options market in pricing options. Finally, both the model diagnostics and option pricing errors in our study suggest that the Gaussian SV model is not sufficientin modeling short-term kurtosis of asset returns, an SV model withfatter-tailed noise or jump component may have better explanatory power.     

Capital Asset Pricing Model and Stochastic Volatility: A Case Study of India

The existing literature demonstrates that under a general equilibrium model, the performance of the Capital Asset Pricing Model (CAPM) can be improved significantly by using conditional consumption and market return volatilities as factors. This article tests the validity of these factors explaining stock return differences using a less developed country (India) as a case study. While the earlier studies used panel data to test CAPM, we use portfolios sorted by size and book-to-market equity (BE/ME) ratio. We found that conditional volatility has a limited effect on firms with large capitalization but a significant impact on small-growth and small-value firms.     

Analysis of the Nonlinear Option Pricing Model Under Variable Transaction Costs

In this paper we analyze a nonlinear Black–Scholes model for option pricing under variable transaction costs. The diffusion coefficient of the nonlinear parabolic equation for the price V is assumed to be a function of the underlying asset price and the Gamma of the option. We show that the generalizations of the classical Black–Scholes model can be analyzed by means of transformation of the fully nonlinear parabolic equation into a quasilinear parabolic equation for the second derivative of the option price. We show existence of a classical smooth solution and prove useful bounds on the option prices. Furthermore, we construct an effective numerical scheme for approximation of the solution. The solutions are obtained by means of the efficient numerical discretization scheme of the Gamma equation. Several computational examples are presented.     

Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing

This paper discusses extensions of the implied diffusion approach of Dupire (1994) to asset processes with Poisson jumps. We show that this extension yields important model improvements, particularly in the dynamics of the implied volatility surface. The paper derives a forward PIDE (PartialIntegro-Differential Equation) and demonstrates how this equationcan be used to fit the model to European option prices. For numerical pricing of general contingent claims, we develop an ADI finite difference method that is shown to be unconditionally stable and, if combined with Fast Fourier Transform methods, computationally efficient. The paper contains several detailed examples fromthe S&P500 market. This revised version was published online in November 2006 with corrections to the Cover Date.     

A Note on Option Pricing for the Constant Elasticity of Variance Model

We study the arbitrage free optionpricing problem for the constant elasticity of variance (CEV) model. To treatthestochastic aspect of the CEV model, we direct attention to the relationship between the CEV modeland squared Bessel processes. Then we show the existence of a unique equivalentmartingale measure and derive the Cox's arbitrage free option pricing formulathrough the properties of squared Bessel processes. Finally we show that the CEVmodel admits arbitrage opportunities when it is conditioned to be strictlypositive.     

Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: An Arrow-Debreu Pricing Approach

Simple analytical pricing formulae have been derived, by different authors and for several derivatives, under the Gaussian Langetieg (1980) model. The purpose of this paper is to use such exact Gaussian solutions in order to obtain approximate analytical pricing formulas under the most general stochastic volatility specification of the Duffie and Kan (1996) model. Using Gaussian Arrow-Debreu state prices, first order stochastic volatility approximate pricing solutions will be derived only involving one integral with respect to the time-to-maturity of the contingent claim under valuation. Such approximations will be shown to be much faster than the existing exact numerical solutions, as well as accurate.     

基于随机波动率假设的权证定价理论评述

随机波动率(SV)模型在衍生品定价和风险管理中的应用开始发挥越来越重要的作用,但是,由于很难得到似然函数的闭型表达式,SV模型的参数估计问题严重限制了它在金融实践领域的普及应用.不过,近年来,学者们提出了许多旨在解决SV模型参数估计问题的有效且可行的新方法,大大推进了SV模型的应用化进程.本文将在权证定价分析的框架内,重点评述SV模型的参数估计方法,并从理论和实证的角度对它们的优点和不足进行简要评介和比较.     

基于随机波动率假设的权证定价理论评述

随机波动率(SV)模型在衍生品定价和风险管理中的应用开始发挥越来越重要的作用,但是,由于很难得到似然函数的闭型表达式,SV模型的参数估计问题严重限制了它在金融实践领域的普及应用。不过,近年来,学者们提出了许多旨在解决SV模型参数估计问题的有效且可行的新方法,大大推进了SV模型的应用化进程。本文将在权证定价分析的框架内,重点评述SV模型的参数估计方法,并从理论和实证的角度对它们的优点和不足进行简要评介和比较。     

基于分数维随机利率的分数跳-扩散外汇期权定价

假设利率为分数维随机利率,外汇汇率服从分数跳一扩散过程,并且波动率为常数,期望收益率为时间的非随机函数,本文利用保险精算方法,得出了看涨、看跌外汇欧武期权的一般定价公式,并建立了平价公式。     

存款保险定价研究:期权定价法和预期损失定价法的进展

合理的存款保险定价可有效减少道德风险和逆向选择问题。本文梳理了国内外关于存款保险定价的两种主要方法——期权定价法和预期损失定价法及其最新发展情况。期权定价法的核心是将存款保险看作存款保险机构以银行资产为标的发行的一份看跌期权,之后学者从股利发放、监管宽容、系统性风险等多个角度进行拓展。预期损失定价法主要根据边际损失与边际保费收入相等来进行保费厘定,以探寻如何通过更科学的方法更精确地测量银行的预期损失。此外,本文讨论了存款保险定价方法对我国的启示。     

基于B-S期权定价模型的可转换债券定价实证分析

可转换债券是一种混合金融衍生工具,它把相应的股票看涨期权内嵌在传统的公司债券之中,具有债券和股票的双重性质,因而可转债的定价问题逐渐为企业和投资者所关注。本文借助Black—Scholes定价模型研究定价理论,对Black-Scholes定价模型进行修正,体现了红利发放对可转换债券定价的影响。     

A Stochastic Correlation Model with Mean Reversion for Pricing Multi-Asset Options

We set up a new kind of model to price the multi-asset options. A square root process fluctuating around its mean value is introduced to describe the random evolution of correlation between two assets. In this stochastic correlation model with mean reversion term, the correlation is a random walk within the region from −1 to 1, and it is centered around its equilibrium value. The trading strategy to hedge the correlation risk is discussed. Since a solution of high-dimensional partial differential equation may be impossible, the Quasi-Monte Carlo and Monte Carlo methods are introduced to compute the multi-asset option price as well. Taking a better-of two asset rainbow as an example, we compare our results with the price obtained by the Black–Scholes model with constant correlation.     

A Non-Parametric Option Pricing Model: Theory and Empirical Evidence

In this paper, we propose an empirically-based, non-parametric option pricing model to evaluate S&P 500 index options. Given the fact that the model is derived under the real measure, an equilibrium asset pricing model, instead of no-arbitrage, must be assumed. Using the histogram of past S&P 500 index returns, we find that most of the volatility smile documented in the literature disappears.     

A Complete Markovian Stochastic Volatility Model in the HJM Framework

This paper considers a stochastic volatility version of the Heath, Jarrow and and Morton (1992) term structure model. Market completeness is obtained by adapting the Hobson and Rogers (1998) complete stochastic volatility stock market model to the interest rate setting. Numerical simulation for a special case is used to compare the stochastic volatility model against the traditional Vasicek (1977) model.     

Pricing Annuity Guarantees Under a Regime-Switching Model

Abstract

We consider the pricing problem of equity-linked annuities and variable annuities under a regimeswitching model when the dynamic of the market value of a reference asset is driven by a generalized geometric Brownian motion model with regime switching. In particular, we assume that regime switching over time according to a continuous-time Markov chain with a finite number state space representing economy states. We use the Esscher transform to determine an equivalent martingale measure for fair valuation in the incomplete market setting. The paper is complemented with some numerical examples to highlight the implications of our model on pricing these guarantees.     

基于非对称随机波动模型的人民币汇率波动特征研究

本文使用非对称随机波动模型,对2005年7月22日至2012年9月5日期间美元兑人民币汇率的波动特征进行了实证分析。模型拟合检验结论显示,非对称随机波动模型能够很好地拟合美元兑人民币汇率波动过程中存在的时变性、持续性和非对称性特征。来自MCMC估计结果进一步表明:美元兑人民币汇率波动过程存在的非对称特征不同于在股票市场普遍发现的"放大利空,缩小利好"型的"杠杆效应",而是突出表现为"放大利好,缩小利空"。但波动的非对称效应和强度较弱,这意味着央行在采取措施干预和管理汇率波动时,在时机选择和力度把握上不仅要充分考虑到人民币汇率波动的时变性和持续性特征,而且更应注意汇率波动的非对称性及非对称类型。     

The GARCH Option Pricing Model: A Modification of Lattice Approach

Ritchken and Trevor (1999) proposed a lattice approach for pricing American options under discrete time-varying volatility GARCH frameworks. Even though the lattice approach worked well for the pricing of the GARCH options, it was inappropriate when the option price was computed on the lattice using standard backward recursive procedures, even if the concepts of Cakici and Topyan (2000) were incorporated. This paper shows how to correct the deficiency and that with our adjustment, the lattice method performs properly for option pricing under the GARCH process. JEL Classification: C10, C32, C51, F37, G12