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.     

Implementing Option Pricing Models When Asset Returns Are Predictable

The predictability of an asset's returns will affect the prices of options on that asset, even though predictability is typically induced by the drift, which does not enter the option pricing formula. For discretely-sampled data, predictability is linked to the parameters that do enter the option pricing formula. We construct an adjustment for predictability to the Black-Scholes formula and show that this adjustment can be important even for small levels of predictability, especially for longer maturity options. We propose several continuous-time linear diffusion processes that can capture broader forms of predictability, and provide numerical examples that illustrate their importance for pricing options.     

Properties of Multinomial Lattices with Cumulants for Option Pricing and Hedging

In this paper, we analyze properties of multinomial lattices that model general stochastic dynamics of the underlying stock by taking into account any given cumulants (or moments). First, we provide a parameterization of multinomial lattices, and demonstrate that mean, variance, skewness, and kurtosis of the underlying may be matched using five branches. Then, we investigate the convergence of the multinomial lattice when the basic time period approaches zero, and prove that the limiting process of the multinomial lattice that matches annualized mean, variance, skewness and kurtosis is given by a compound Poisson process. Finally, we illustrate the effect of higher order moments in the underlying asset process on the price of derivative securities through numerical experiments using the multinomial lattice, and provide a comparison with jump-diffusion models.     

Dynamic Spanning in the Consumption-Based Capital Asset Pricing Model

Under the assumptions of the Consumption-based Capital AssetPricing Model (CCAPM), Pareto optimal consumption allocationsare characterized by each agent's consumption process beingadapted to the filtration generated by the aggregate consumptionprocess of the economy. The wealth processes of the agents,however, are adapted to the finer filtration generated by aggregateconsumption and the conditional distribution of future aggregateconsumption. Therefore, in order to achieve pareto optimal consumptionallocations, a sufficiently varied set of assets must existsuch that any wealth process adapted to this finer filtrationcan be implemented by dynamically trading in that set of assets.We provide sufficient conditions for the existence of such aset of assets based on dynamically trading contingent claimson aggregate consumption. In addition, we give sufficient conditionsfor the existence of equilibria in a dynamically effectivelycomplete market in which agents are only able to trade in contingentclaims on aggregate consumption, the market portfolio of firms,and a (numeraire) zero-coupon bond. We demonstrate the roleof short- and long-term contingent claims on aggregate consumptionfor the implementation of Pareto optimal allocations inthe presenceof short- and long-term risks. In addition, in the presenceof personal risks, we demonstrate the role of insurance contracts.JEL Classification: G13.     

Dynamic Spanning in the Consumption-Based Capital Asset Pricing Model

Under the assumptions of the Consumption-based Capital Asset Pricing Model (CCAPM), Pareto optimal consumption allocations are characterized by each agent's consumption process being adapted to the filtration generated by the aggregate consumption process of the economy. The wealth processes of the agents, however, are adapted to the finer filtration generated by aggregate consumption and the conditional distribution of future aggregate consumption. Therefore, in order to achieve Pareto optimal consumption allocations, a sufficiently varied set of assets must exist such that any wealth process adapted to this finer filtration can be implemented by dynamically trading in that set of assets. We provide sufficient conditions for the existence of such a set of assets based on dynamically trading contingent claims on aggregate consumption. In addition, we give sufficient conditions for the existence of equilibria in a dynamically effectively complete market in which agents are only able to trade in contingent claims on aggregate consumption, the market portfolio of firms, and a (numeraire) zero-coupon bond. We demonstrate the role of short- and long-term contingent claims on aggregate consumption for the implementation of Pareto optimal allocations in the presence of short- andlong-term risks. In addition, in the presence of personal risks, we demonstrate the role of insurance contracts.     

An Event Option Pricing Model with Scheduled and Unscheduled Announcement Effects

There is considerable evidence supporting the time-varying distribution of asset returns. There is also ample evidence that scheduled announcement events such as money supply announcements (in the case of foreign exchange), earnings announcements (in the case of stocks), and crop reports (in the case of commodities), as well as random unscheduled events, can affect the level and volatility of asset returns. This study provides an Event Model for European call options which explicitly addresses effects of these two classes of events. This specification requires estimation of more parameters, but it could provide a more accurate basis for pricing options than previous Poisson jump-diffusion models. Parametric analysis shows that the standard models under price the options relative to the Event Model. The Event Model may be particularly useful in pricing short-term deep out-of-the-money options when scheduled events are present in the market.     

Option Pricing with Heterogeneous Expectations

This paper re-derives the finite mixture option pricing model of Ritchey (1990), based on the assumption that the option investors hold heterogeneous expectations about the parameters of the lognormal process of the underlying asset price. By proving that the model admits no riskless arbitrage, this paper justifies that the entire family of finite mixture of lognormal distributions is a desirable candidate set for recovering the risk-neutral probability distributions from contemporaneous options quotes. The parametric method derived from the model is significantly simpler than the nonparametric method of Rubinstein (1994) for recovering the risk-neutral probability distributions from contemporaneous option prices.     

Pricing dynamic binary variables and their derivatives

Many important assets or business ventures have cash flows that are not derivatives of a market security but are nevertheless dependent on some variable that is correlated with market prices. This includes many real option projects. This paper presents a methodology using a binary framework for pricing such assets by projection onto the market space. Under certain conditions, the result has the property that, given this price process, no risk-averse investor would choose to invest in this asset either long or short.     

The Asset Pricing Palette: Cash Flows, Returns and Trading Behavior

Asset pricing is the topic of the 2001 Eastern Finance Association Symposium and the five papers selected for this collection, which are summarized below, span a broad range of subjects that fall under the umbrella of the determinants of market prices. For example, the Schwartz and Moon article that introduces the symposium uses real options methodology to value firms whose cash flows are subject to multiple sources of uncertainty while the Luders and Peisl and Mixon analytical models that close the selections incorporate dual stochastic processes to derive relationships between information flow, trading volume and price volatility that are consistent with empirical evidence. In between, Mishra and O'Brien present new evidence on the important of index and factor selection when estimating the required return on equity and Spahr and Schwebach revisit the issue of time diversification by reintroducing a statistical construct from earlier times. Each of the works included here makes an important contribution to our understanding of the asset pricing process in a distinct area and opens new doors onto avenues for future research.     

Options and efficiency: Some experimental evidence

This paper examines the information aggregation role of options when agents possess diverse information about possible asset returns. We construct two identical experimental markets: one with and one without options. We find that options speed the information aggregation process. Asset markets that have parallel option markets aggregate traders' diverse information faster than markets where options trading is not available. Implied ranges were calculated from asset and option prices and compared to the actual ranges. These comparisons suggest that options may provide a means for agents to coordinate beliefs about asset values.     

A Refined Binomial Lattice for Pricing American Asian Options

We present simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model. We introduce a new refined version of the Cox-Ross-Rubinstein (1979) binomial lattice of stock prices. Each node in the lattice is partitioned into nodelets, each of which represents all paths arriving at the node with a specific geometric stock price average. The upper bound uses an interpolation idea similar to the Hull-White (1993) method. From the backward-recursive upper-bound computation, we estimate a good exercise rule that is consistent with the refined lattice. This exercise rule is used to obtain a lower bound on the option price using a modification of a conditional-expectation based idea from Rogers-Shi (1995) and Chalasani-Jha-Varikooty (1998). Our algorithms run in time proportional to the number of nodelets in the refined lattice, which is smaller than n^4/20 for n > 14 periods.     

Some Tests of the Risk-Return Relationship Using Alternative Asset Pricing Models and Observed Expected Returns

This study examines the performance of three asset pricing models: the CAPM, the APT and the UAPT using observed expected returns from a three-phase dividend discount model with Value Line analyst estimates of future company-level earnings, dividends and growth rates. Our study is the first we know of to test the three major asset pricing models using observed expected returns. Our results are similar to prior research using ex post (realized) returns in that we find that the UAPT using macroeconomic factors is the best performing model, followed by the APT and the CAPM. However, our results also suggest that the importance of macroeconomic factors is much greater to expected returns than to realized returns, and the corresponding R2 values for models using expected returns are much higher than for models using realized returns. Combining our results for the UAPT with those of Marston and Harris (1993) for the CAPM suggests that these models are more successful in tests using observed expected returns than in tests using realized returns as proxies for expected returns. Unit root tests suggest that monthly observed expected returns follow the classic random walk without drift model while monthly realized returns do not.     

Modeling Conditional Skewness in Stock Returns

Abstract

In this paper, we propose a new GARCH-in-Mean (GARCH-M) model allowing for conditional skewness. The model is based on the so-called z distribution capable of modeling skewness and kurtosis of the size typically encountered in stock return series. The need to allow for skewness can also be readily tested. The model is consistent with the volatility feedback effect in that conditional skewness is dependent on conditional variance. Compared to previously presented GARCH models allowing for conditional skewness, the model is analytically tractable, parsimonious and facilitates straightforward interpretation.Our empirical results indicate the presence of conditional skewness in the monthly postwar US stock returns. Small positive news is also found to have a smaller impact on conditional variance than no news at all. Moreover, the symmetric GARCH-M model not allowing for conditional skewness is found to systematically overpredict conditional variance and average excess returns.     

Portfolio Construction for Tests of Asset Pricing Models

Portfolios are commonly used in finance literature to study asset‐pricing models. In business practice portfolios are used to detect abnormal performance in certain asset groups or to construct reference assets. However, analyses on practical issues related to portfolio construction are surprisingly few. This paper presents and discusses issues related to portfolio return calculation from theoretical and practical perspectives. Special attention is given both to smaller and emerging stock markets. These stock markets often share common features like low liquidity, multiple stock series, and changes in foreign ownership restrictions that greatly affect portfolio construction.     

Pricing Options on an Asset with Bernoulli Jump-Diffusion Returns

The price movements of certain assets can be modeled by stochastic processes that combine continuous diffusion with discrete jumps. This paper compares values of options on assets with no jumps, jumps of fixed size, and jumps drawn from a lognormal distribution. It is shown that not only the magnitude but also the direction of the mispricing of the Black-Scholes model relative to jump models can vary with the distribution family of the jump component. This paper also discusses a methodology for the numerical valuation, via a backward induction algorithm, of American options on a jump-diffusion asset whose early exercise may be profitable. These cannot, in general, be accurately priced using analytic models. The procedure has the further advantage of being easily adaptable to nonanalytic, empirical distributions of period returns and to nonstationarity in the underlying diffusion process.     

Option Pricing Bounds and the Elasticity of the Pricing Kernel

In this paper we use power functions as pricing kernels to derive option-pricing bounds. We derive option pricing bounds given the bounds of the elasticity of the true pricing kernel. The bounds of the elasticity of the true pricing kernel are closely related to the bounds of the representative investor's coefficient of relative risk aversion. This methodology produces a tighter upper call option bound than traditional approaches. As a special case we show how to use the Black–Scholes formula to obtain option pricing bounds under the assumption of lognormality.     

VIX期权定价与校正

VIX期权作为波动率衍生品能为金融机构提供有效的市场风险对冲工具。文献中对VIX期权定价的实证分析误差都很大,原因在于模型的选取误差以及校正方法和样本选取不妥。通过在VIX模型中加入均值回复因素和跳因素,可以使VIX过程更加合理,也可以使VIX期权定价精度更高。通过对VIX期权市场中间报价进行校正,得到了4个文献模型的参数估计,并比较4个模型的定价精度和正向隐含波动率偏斜拟合效果。     

Option Prices Under Generalized Pricing Kernels

In this paper analytical solutions for European option prices are derived for a class of rather general asset specific pricing kernels (ASPKs) and distributions of the underlying asset. Special cases include underlying assets that are lognormally or log-gamma distributed at expiration date T. These special cases are generalizations of the Black and Scholes (1973) option pricing formula and the Heston (1993) option pricing formula for non-constant elasticity of the ASPK. Analytical solutions for a normally distributed and a uniformly distributed underlying are also derived for the class of general ASPKs. The shape of the implied volatility is analyzed to provide further understanding of the relationship between the shape of the ASPK, the underlying subjective distribution and option prices. The properties of this class of ASPKs are also compared to approaches used in previous empirical studies. JEL Classification: G12, G13, C65 Erik Lüders is an assistant professor at Laval University and a visiting scholar at the Stern School of Business, New York University.     

Pricing exotic options using MSL-MC

Today, better numerical approximations are required for multi-dimensional SDEs to improve on the poor performance of the standard Monte Carlo pricing method. With this aim in mind, this paper presents a method (MSL-MC) to price exotic options using multi-dimensional SDEs (e.g. stochastic volatility models). Usually, it is the weak convergence property of numerical discretizations that is most important, because, in financial applications, one is mostly concerned with the accurate estimation of expected payoffs. However, in the recently developed Multilevel Monte Carlo path simulation method (ML-MC), the strong convergence property plays a crucial role. We present a modification to the ML-MC algorithm that can be used to achieve better savings. To illustrate these, various examples of exotic options are given using a wide variety of payoffs, stochastic volatility models and the new Multischeme Multilevel Monte Carlo method (MSL-MC). For standard payoffs, both European and Digital options are presented. Examples are also given for complex payoffs, such as combinations of European options (Butterfly Spread, Strip and Strap options). Finally, for path-dependent payoffs, both Asian and Variance Swap options are demonstrated. This research shows how the use of stochastic volatility models and the θ scheme can improve the convergence of the MSL-MC so that the computational cost to achieve an accuracy of O(ε) is reduced from O(ε^?3) to O(ε^?2) for a payoff under global and non-global Lipschitz conditions.     

资产定价:理论演进及应用研究

资产定价理论是现代金融理论的核心.本文通过对资产定价理论的综述,揭示了从传统资产定价理论到行为资产定价理论的演进脉络,并对各理论及相应模型的内涵和应用进行了描述,最后对传统资产定价理论和行为资产定价理论进行了比较,以期对我国金融理论和实践的发展有所帮助.