A new approach for option pricing under stochastic volatility

We develop a new approach for pricing European-style contingent claims written on the time T spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time T variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a path-independent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying’s futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitrage-free value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the risk-neutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the risk-neutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, well-behaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives.     

American option valuation under stochastic interest rates

By applying Ho, Stapleton and Subrahmanyam's (1997, hereafter HSS) generalised Geske–Johnson (1984, hereafter GJ) method, this paper provides analytic solutions for the valuation and hedging of American options in a stochastic interest rate economy. The proposed method simplifies HSS's three-dimensional solution to a one-dimensional solution. The simulations verify that the proposed method is more efficient and accurate than the HSS (1997) method. We illustrate how the price, the delta, and the rho of an American option vary between the stochastic and non-stochastic interest rate models. The magnitude of this effect depends on the moneyness of the option, interest rates, volatilities of the underlying asset price and the bond price, as well as the correlation between them. This revised version was published online in June 2006 with corrections to the Cover Date.     

An option pricing approach to the valuation of real estate contaminated with hazardous materials

This paper uses option pricing to examine how the presence of hazardous materials affects real estate value. The property owner has two options. The first option is to remove the hazardous materials at the best time. The second option, embedded in the first one, is to redevelop the property at the best opportunity. The owner has three possible timing strategies with respect to the exercise of these two options: remove the hazardous materials first and retain the option to redevelop the property later, remove and redevelop at the same time, or do nothing. Conditions under which the presence of the hazardous materials may either expedite or postpone the decision to redevelop are also derived. If the regulatory environment does not allow the property owner to make optimal timing decisions with respect to the exercise of these options, then our results provide an indication of the cost of regulation as measured by the additional loss in property value.     

Option pricing: A general equilibrium approach

This article examines option valuation in a general equilibrium framework. We focus on the general equilibrium implications of price dynamics for option valuation. The general equilibrium considerations allow us to derive an alternative option valuation formula that is as simple as the Black and Scholes formula, and that exhibits different behavior with respect to the exercise price and time to expiration. They also help us clarify comparative-statics properties of option valuation formulas in general and of the Black and Scholes model in particular.     

Extending the capital asset pricing model: the reward beta approach

This paper offers an alternative method for estimating expected returns. The proposed reward beta approach performs well empirically and is based on asset pricing theory. The empirical section compares this approach with the capital asset pricing model (CAPM) and the Fama–French three‐factor model. In out‐of‐sample testing, both the CAPM and the three‐factor model are rejected. In contrast, the reward beta approach easily passes the same test. In robustness checks, the reward beta approach consistently outperforms both the CAPM and the three‐factor model.     

Stochastic duration and fast coupon bond option pricing in multi-factor models

Generalizing Cox, Ingersoll, and Ross (1979), this paper defines the stochastic duration of a bond in a general multi-factor diffusion model as the time to maturity of the zero-coupon bond with the same relative volatility as the bond. Important general properties of the stochastic duration measure are derived analytically, and the stochastic duration is studied in detail in various well-known models. It is also demonstrated by analytical arguments and numerical examples that the price of a European option on a coupon bond (and, hence, of a European swaption) can be approximated very accurately by a multiple of the price of a European option on a zero-coupon bond with a time to maturity equal to the stochastic duration of the coupon bond. This revised version was published online in June 2006 with corrections to the Cover Date.     

General closed-form basket option pricing bounds

This article presents lower and upper bounds on the prices of basket options for a general class of continuous-time financial models. The techniques we propose are applicable whenever the joint characteristic function of the vector of log-returns is known. Moreover, the basket value is not required to be positive. We test our new price approximations on different multivariate models, allowing for jumps and stochastic volatility. Numerical examples are discussed and benchmarked against Monte Carlo simulations. All bounds are general and do not require any additional assumption on the characteristic function, so our methods may be employed also to non-affine models. All bounds involve the computation of one-dimensional Fourier transforms; hence, they do not suffer from the curse of dimensionality and can be applied also to high-dimensional problems where most existing methods fail. In particular, we study two kinds of price approximations: an accurate lower bound based on an approximating set and a fast bounded approximation based on the arithmetic-geometric mean inequality. We also show how to improve Monte Carlo accuracy by using one of our bounds as a control variate.     

A generalization of the Hull and White formula with applications to option pricing approximation

By means of Malliavin calculus we see that the classical Hull and White formula for option pricing can be extended to the case where the volatility and the noise driving the stock prices are correlated. This extension will allow us to describe the effect of correlation on option prices and to derive approximate option pricing formulas.A previous version of this paper has benefited from helpful comments by two anonymous referees.     

Binomial Option Pricing Biases and Inconsistent Implied Volatilities

We evaluate the binomial option pricing methodology (OPM) by examining simulated portfolio strategies. A key aspect of our study involves sampling from the empirical distribution of observed equity returns. Using a Monte Carlo simulation, we generate equity prices under known volatility and return parameters. We price American–style put options on the equity and evaluate the risk–adjusted performance of various strategies that require writing put options with different maturities and moneyness characteristics. The performance of these strategies is compared to an alternative strategy of investing in the underlying equity. The relative performance of the strategies allows us to identify biases in the binomial OPM leading to the well–known volatility smile . By adjusting option prices so as to rule out dominated option strategies in a mean–variance context, we are able to reduce the pricing errors of the OPM with respect to option prices obtained from the LIFFE. Our results suggest that a simple recalibration of inputs may improve binomial OPM performance.     

American stochastic volatility call option pricing: A lattice based approach

This study presents a new method of pricing options on assets with stochastic volatility that is lattice based, and can easily accommodate early exercise for American options. Unlike traditional lattice methods, recombination is not a problem in the new model, and it is easily adapted to alternative volatility processes. Approximations are developed for European C.E.V. calls and American stochastic volatility calls. The application of the pricing model to exchange traded calls is also illustrated using a sample of market prices. Modifying the model to price American puts is straightforward, and the approach can easily be extended to other non-recombining lattices.     

On the duality principle in option pricing: semimartingale setting

The purpose of this paper is to describe the appropriate mathematical framework for the study of the duality principle in option pricing. We consider models where prices evolve as general exponential semimartingales and provide a complete characterization of the dual process under the dual measure. Particular cases of these models are the ones driven by Brownian motions and by Lévy processes, which have been considered in several papers. Generally speaking, the duality principle states that the calculation of the price of a call option for a model with price process S=e ^H (with respect to the measure P) is equivalent to the calculation of the price of a put option for a suitable dual model S′=e ^H′ (with respect to the dual measure P′). More sophisticated duality results are derived for a broad spectrum of exotic options. The second named author acknowledges the financial support from the Deutsche Forschungsgemeinschaft (DFG, Eb 66/9-2). This research was carried out while the third named author was supported by the Alexander von Humboldt foundation.     

Asset pricing under information with stochastic volatility

Based on a general specification of the asset specific pricing kernel, we develop a pricing model using an information process with stochastic volatility. We derive analytical asset and option pricing formulas. The asset prices in this rational expectations model exhibit crash-like, strong downward movements. The resulting option pricing formula is consistent with the strong negative skewness and high levels of kurtosis observed in empirical studies. Furthermore, we determine credit spreads in a simple structural model.      

关于Shibor利率期权波动率定价机制的研究

Shibor自2007年发布以来,已成为人民币利率市场的一个重要定价基准,对金融衍生品、债券的定价起着十分重要的作用,由于人民币利率衍生品市场尚处于发展的初期,与美元Libor利率期权等较为成熟市场相比,目前Shibor利率期权缺少成熟的市场报价。本文通过风险中性的定价方程反解参数的方法,利用Shibor利率掉期曲线对Shibor利率上下限期权的隐含波动率进行计算,从而探讨对Shibor利率期权的定价。     

On option pricing in binomial market with transaction costs

Option replication is studied in a discrete-time framework with proportional transaction costs. The model represents an extension of the Cox-Ross-Rubinstein binomial option-pricing model to cover the case of proportional transaction costs for one risky asset with different interest rates on bank credit and deposit. Contingent claims are supposed to be 2-dimensional random variables. Explicit formulas for self-financing strategies are obtained for this case.Received: March 2004, Mathematics Subject Classification (2000): 62P05JEL Classification: G11, G13The authors are grateful to an anonymous referee for numerous helpful comments and to Yulia Romaniuk for final corrections. The paper was partially supported by grant NSERC 264186.     

Modeling time series information into option prices: An empirical evaluation of statistical projection and GARCH option pricing model

This paper compares the empirical performances of statistical projection models with those of the Black–Scholes (adapted to account for skew) and the GARCH option pricing models. Empirical analysis on S&P500 index options shows that the out-of-sample pricing and projected trading performances of the semi-parametric and nonparametric projection models are substantially better than more traditional models. Results further indicate that econometric models based on nonlinear projections of observable inputs perform better than models based on OLS projections, consistent with the notion that the true unobservable option pricing model is inherently a nonlinear function of its inputs. The econometric option models presented in this paper should prove useful and complement mainstream mathematical modeling methods in both research and practice.     

Conditional beta pricing models: A nonparametric approach

We propose a two-stage procedure to estimate conditional beta pricing models that allows for flexibility in the dynamics of asset betas and market prices of risk (MPR). First, conditional betas are estimated nonparametrically for each asset and period using the time-series of previous data. Then, time-varying MPR are estimated from the cross-section of returns and betas. We prove the consistency and asymptotic normality of the estimators. We also perform Monte Carlo simulations for the conditional version of the three-factor model of Fama and French (1993) and show that nonparametrically estimated betas outperform rolling betas under different specifications of beta dynamics. Using return data on the 25 size and book-to-market sorted portfolios, we find that the nonparametric procedure produces a better fit of the three-factor model to the data, less biased estimates of MPR and lower pricing errors than the Fama–MacBeth procedure with betas estimated under several alternative parametric specifications.     

A generalized approach to optimal hedging with option contracts

In this paper, we develop a theoretical model in which a firm hedges a spot position using options in the presence of both quantity (production) and basis risks. Our optimal hedge ratio is fairly general, in that the dependence structure is modeled through a copula function representing the quantiles of the hedged position, and hence any quantile risk measure can be employed. We study the sensitivity of the exercise price which minimizes the risk of the hedged portfolio to the relevant parameters, and we find that the subjective risk aversion of the firm does not play any role. The only trade-off is between the effectiveness and cost of the hedging strategy.     

Real option pricing with mean-reverting investment and project value

In this work, we are concerned with valuing the option to invest in a project when the project value and the investment cost are both mean-reverting. Previous works on stochastic project and investment cost concentrate on geometric Brownian motions (GBMs) for driving the factors. However, when the project involved is linked to commodities, mean-reverting assumptions are more meaningful. Here, we introduce a model and prove that the optimal exercise strategy is not a function of the ratio of the project value to the investment V/I – contrary to the GBM case. We also demonstrate that the limiting trigger curve as maturity approaches traces out a nonlinear curve in (V, I) space and derive its explicit form. Finally, we numerically investigate the finite-horizon problem, using the Fourier space time-stepping algorithm of Jaimungal and Surkov [2009. Lev´y based cross-commodity models and derivative valuation. SIAM Journal of Financial Mathematics, to appear. http://www.ssrn.com/abstract=972837]. Numerically, the optimal exercise policies are found to be approximately linear in V/I; however, contrary to the GBM case they are not described by a curve of the form V*/I*=c(t). The option price behavior as well as the trigger curve behavior nicely generalize earlier one-factor model results.