Spanning and completeness with options

The role of ordinary options in facilitating the completionof securities markets is examined in the context of a modelof contigent claims sufficiently general to accommodate thecontinuous distributions of asset pricing theory and optionpricing theory. In this context, it is shown that call optionswritten on a single security approximately span all contingentclaims written on this security and that call options writtenon portfolios of call options on individual primitive securitiesapproximately span all contingent claims that can be writtenon these primitive securities. In the case of simple options,explicit formulas are given for the approximating options andportfolios of options. These results are applied to the pricingof contingent claims by arbitrage and to irrelevance propositionsin corporate finance.     

Hedging with small uncertainty aversion

We study the pricing and hedging of derivative securities with uncertainty about the volatility of the underlying asset. Rather than taking all models from a prespecified class equally seriously, we penalise less plausible ones based on their “distance” to a reference local volatility model. In the limit for small uncertainty aversion, this leads to explicit formulas for prices and hedging strategies in terms of the security’s cash gamma.     

An alternative approach to the valuation of American options and applications

In this paper we examine the structure of American option valuation problems and derive the analytic valuation formulas under general underlying security price processes by an alternative but intuitive method. For alternative diffusion processes, we derive closed-form analytic valuation formulas and analyze the implications of asset price dynamics on the early exercise premiums of American options. In this regard, we introduce useful and interesting diffusion processes into American option-pricing literature, thus providing a wide range of choices of pricing models for various American-type derivative assets. This work offers a useful analytic framework for empirical testing and practical applications such as the valuation of corporate securities and examining the impact of options trading on market micro-structure.     

Financial Modeling in a Fast Mean-Reverting Stochastic Volatility Environment

We present a derivative pricing and estimation methodology for a class of stochastic volatility models that exploits the observed 'bursty' or persistent nature of stock price volatility. Empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by- tick fluctuations of the index value, but it is fast mean- reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter provides a simple procedure to 'fit the skew' from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log- moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk are not needed for the asymptotic pricing formulas for European derivatives, and we derive the formula for a knock-out barrier option as an example. The extension to American and path-dependent contingent claims is the subject of future work. This revised version was published online in August 2006 with corrections to the Cover Date.     

Risk measures for derivatives with Markov-modulated pure jump processes

We consider a regime-switching HJB approach to evaluate risk measures for derivative securities when the price process of the underlying risky asset is governed by the exponential of a pure jump process with drift and a Markov switching compensator. The pure jump process is flexible enough to incorporate both the infinite, (small), jump activity and the finite, (large), jump activity. The drift and the compensator of the pure jump process switch over time according to the state of a continuous-time hidden Markov chain representing the state of an economy. The market described by our model is incomplete. Hence, there is more than one pricing kernel and there is no perfect hedging strategy for a derivative security. We derive the regime-switching HJB equations for coherent risk measures for the unhedged position of derivative securities, including standard European options and barrier options. For measuring risk inherent in the unhedged option position, we first need to mark the position into the market by valuing the option. We employ a well-known tool in actuarial science, namely, the Esscher transform to select a pricing kernel for valuation of an option and to generate a family of real-world probabilities for risk measurement. We also derive the regime-switching HJB-variational inequalities for coherent risk measures for American-style options.     

Changes in Expected Security Returns,Risk, and the Level of Interest Rates

Regressions of security returns on treasury bill rates provide insight about the behavior of risk in rational asset pricing models. The information in one-month bill rates implies time variation in the conditional covariances of portfolios of stocks and fixed-income securities with benchmark pricing variables, over extended samples and within five-year subperiods. There is evidence of changes in conditional “betas” associated with interest rates. Consumption and stock market data are examined as proxies for marginal utility, in a general framework for asset pricing with time-varying conditional covariances.     

Financial innovation, values and volatilities when markets are incomplete

The traditional pricing methodology in finance values derivative securities as redundant assets that have no impact on equilibrium prices and allocations. This paper considers a model with incomplete markets in which the valuation of derivative securities cannot be treated independently from the valuation of the primary securities. The model constitutes a framework for the analysis of the consequences of financial innovation (creation of new contracts or modification of existing contracts). We provide a numerical counterexample to the popular belief that financial innovations that increase the volatilities of traded securities are bad. In this example the introduction of an option increases the volatility of the rate of return on the underlying stock, yet the creation of this asset is unanimously supported by investors.I am grateful to H. Polemarchakis and to an anonymous referee for insightful suggestions.     

Pricing Derivatives on Financial Securities Subject to Credit Risk

This article provides a new methodology for pricing and hedging derivative securities involving credit risk. Two types of credit risks are considered. The first is where the asset underlying the derivative security may default. The second is where the writer of the derivative security may default. We apply the foreign currency analogy of Jarrow and Turnbull (1991) to decompose the dollar payoff from a risky security into a certain payoff and a “spot exchange rate.” Arbitrage-free valuation techniques are then employed. This methodology can be applied to corporate debt and over the counter derivatives, such as swaps and caps.     

International asset pricing under segmentation and PPP deviations

We analyze the impact of both purchasing power parity (PPP) deviations and market segmentation on asset pricing and investor's portfolio holdings. The freely traded securities command a world market risk premium and an inflation risk premium. The securities that can be held by only a subset of investors command two additional premiums: a conditional market risk premium and a segflation risk premium. Our model is empirically supported with important implications for tests of international asset pricing.     

Asset Pricing Using Trading Volumes in a Hidden Regime-Switching Environment

By utilizing information about prices and trading volumes, we discuss the pricing of European contingent claims in a continuous-time hidden regime-switching environment. Hidden market sentiments described by the states of a continuous-time, finite-state, hidden Markov chain represent a common factor for an asset’s drift and volatility, as well as its trading volumes. Using observations about trading volumes, we present a filtered estimate of the hidden common factor. The asset pricing problem is then considered in a filtered market, where the hidden drift and volatility are replaced by their filtered estimates. We adopt the Esscher transform to select an equivalent martingale measure for pricing and derive a partial-differential integral equation for the option price.     

Constructing Asset Pricing Models With Specific Factor Loadings

We demonstrate how one can build pricing formulae in which factors other than beta may be viewed as determinants of asset returns. This is important conceptually as it demonstrates how the additional factors can compensate for a market portfolio proxy that is mis‐specified, and also shows how such a pricing model can be specified ex ante. The procedure is implemented by first selecting an ‘orthogonal’ portfolio which falls on the mean‐variance efficient frontier computed from the empirical average returns, variances and covariances on the equity securities of a large sample of firms. One then determines the inefficient index portfolio which leads to a vector of betas that when multiplied by the average return on the orthogonal portfolio, and which when subtracted from the vector of average returns for the firms comprising the sample, yields an error vector that is equal to the vector of numerical values for the variables that are to form the basis of the asset pricing formula. There will then be a perfect linear relationship between the vector of average returns for the firms comprising the sample, the vector of betas based on the inefficient index portfolio and such other factors that are deemed to be important in the asset pricing process. We illustrate computational procedures using a numerical example based on the quality of information contained in published corporate financial statements.     

Pricing futures on geometric indexes: A discrete time approach

Several futures contracts are written against an underlying asset that is a geometric, rather than arithmetic, index. These contracts include: the US Dollar Index futures, the CRB-17 futures, and the Value Line geometric index futures. Due to the geometric averaging, the standard cost-of-carry futures pricing formula is improper for pricing these futures contracts. We assume that asset prices are lognormally distributed, and capital markets are complete. Using the concepts of equivalent martingale measure and the risk-neutral valuation relationships in conjunction with discrete time methodology, we derive closed-form pricing formulas for these contracts. Our pricing formulas are consistent with the ones obtained via a continuous time paradigm.

The effects of model errors and market imperfections on financial institutions writing derivative warrants: Simulation evidence from Taiwan

This paper uses the historical simulation approach, similar to that of Green and Figlewski [Journal of Finance 54 (1999) 1465] to investigate the effects of market imperfections and model errors on financial institutions writing derivative warrants. The paper undertakes specific exploration of the effects of particular market imperfections such as transaction costs and the primary market process. An understanding of model risk in the valuation and trading of derivative securities is particularly important for emerging markets, because asset returns are too fat-tailed to be lognormal and errors in volatility forecasts tend to be very significant. Simulation evidence shows that model risk is extremely significant for issuers of derivative warrants in emerging markets such as Taiwan. While exploring possible remedial strategies for market imperfections and model errors, simulation evidence suggests that a search for a better pricing model for the Taiwan derivative warrants market is called for.     

Assessing the Real Estate Pricing Puzzle: A Diagnostic Application of the Stochastic Discounting Factor to the Distribution of REIT Returns

This article applies a general asset-pricing framework and the volatility bounds methodology of Hansen and Jagannathan (1991) to REIT returns. The state of real estate asset pricing remains somewhat of a puzzle relative to the identification of state variables and the structural form of models. This article offers a framework whereby real estate asset-pricing models and data can be diagnosed to answer questions about the shortcomings. In addition, several nominated discount processes are investigated for success in pricing real estate securities. Although the nominated specifications demonstrate some success in satisfying the restrictions on the first and second moments of the real estate returns distribution, they do not successfully price the securities under a no-arbitrage condition. This result calls into question previous real estate performance studies that employ these risk-adjustment processes.     

A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Networks

We propose a nonparametric method for estimating the pricing formula of a derivative asset using learning networks. Although not a substitute for the more traditional arbitrage-based pricing formulas, network-pricing formulas may be more accurate and computationally more efficient alternatives when the underlying asset's price dynamics are unknown, or when the pricing equation associated with the no-arbitrage condition cannot be solved analytically. To assess the potential value of network pricing formulas, we simulate Black-Scholes option prices and show that learning networks can recover the Black-Scholes formula from a two-year training set of daily options prices, and that the resulting network formula can be used successfully to both price and delta-hedge options out-of-sample. For comparison, we estimate models using four popular methods: ordinary least squares, radial basis function networks, multilayer perceptron networks, and projection pursuit. To illustrate the practical relevance of our network pricing approach, we apply it to the pricing and delta-hedging of S&P 500 futures options from 1987 to 1991.     

Barriers to international investing and market segmentation: Evidence from Indian GDR market

In this paper we evaluate market segmentation and its effect on the pricing of cross-listed securities using Indian Global Depositary Receipts (GDRs). When international capital markets are segmented, cross-listed securities may trade at different prices. We test this market segmentation hypothesis using a theoretical and empirical model developed along the lines of Hietala [Hietala, P.T., 1989, Asset pricing in partially segmented markets: Evidence from the Finnish market, Journal of Finance 44, 697–718]) and Foerster and Karolyi [Foerster, S.R., Karolyi, A.G., 1999, The effects of market segmentation and investor recognition on asset prices: Evidence from foreign stocks listing in the United States, Journal of Finance 54, 981–1013; Foerster, S.R., Karolyi, A.G., 1999, The long-run performance of global equity offerings, Working Paper, Ohio State University]. Our model looks at a specific type of market segmentation in India, where capital flow barriers are such that domestic investors are allowed to invest only in domestic securities, while the foreign investors can invest in dollar-denominated Indian GDRs as well as other foreign securities. Tests on these GDRs indicate that foreign investors, who hold these depositary receipts, estimate the expected returns at a lower level than the domestic investors do. This leads to the GDRs being priced at a premium over the exchange rate adjusted prices of the underlying Indian securities. GDR index returns are affected by both domestic and international factors, while the underlying Indian securities are affected only by domestic variables.     

Risky asset pricing based on safety first fund management

We study a portfolio selection model based on Kataoka's safety-first criterion (KSF model in short). We assume that the market is complete but without risk-free asset, and that the returns are jointly elliptically distributed. With these assumptions, we provide an explicit analytical optimal solution for the KSF model and obtain some geometrical properties of the efficient frontier in the plane of probability risk degree z _α and target return r _α. We further prove a two-fund separation and tangency portfolio theorem in the spirit of the traditional mean-variance analysis. We also establish a risky asset pricing model based on risky funds that is similar to Black's zero-beta capital asset pricing model (CAPM, for short). Moreover, we simplify our risky asset pricing model using a derivative risky fund as a reference for market evaluation.     

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.      

Liquidity biases in asset pricing tests

Microstructure noise in security prices biases the results of empirical asset pricing specifications, particularly when security-level explanatory variables are cross-sectionally correlated with the amount of noise. We focus on tests of whether measures of illiquidity, which are likely to be correlated with the noise, are priced in the cross-section of stock returns, and show a significant upward bias in estimated return premiums for an array of illiquidity measures in Center for Research in Security Prices (CRSP) monthly return data. The upward bias is larger when illiquid securities are included in the sample, but persists even for NYSE/Amex stocks after decimalization. We introduce a methodological correction to eliminate the biases that simply involves weighted least squares (WLS) rather than ordinary least squares (OLS) estimation, and find evidence of smaller, but still significant, return premiums for illiquidity after implementing the correction.     

Asset Trading Volume with Dynamically Complete Markets and Heterogeneous Agents

Trading volume of infinitely lived securities, such as equity, is generically zero in Lucas asset pricing models with heterogeneous agents. More generally, the end‐of‐period portfolio of all securities is constant over time and states in the generic economy. General equilibrium restrictions rule out trading of equity after an initial period. This result contrasts the prediction of portfolio allocation analyses that portfolio rebalancing motives produce nontrivial trade volume. Therefore, other causes of trade must be present in asset markets with large trading volume.