A MULTINOMIAL APPROXIMATION FOR AMERICAN OPTION PRICES IN LÉVY PROCESS MODELS

This paper gives a tree-based method for pricing American options in models where the stock price follows a general exponential Lévy process. A multinomial model for approximating the stock price process, which can be viewed as generalizing the binomial model of Cox, Ross, and Rubinstein (1979) for geometric Brownian motion, is developed. Under mild conditions, it is proved that the stock price process and the prices of American-type options on the stock, calculated from the multinomial model, converge to the corresponding prices under the continuous time Lévy process model. Explicit illustrations are given for the variance gamma model and the normal inverse Gaussian process when the option is an American put, but the procedure is applicable to a much wider class of derivatives including some path-dependent options. Our approach overcomes some practical difficulties that have previously been encountered when the Lévy process has infinite activity.     

A new simple square root option pricing model

This study derives a simple square root option pricing model using a general equilibrium approach in an economy where the representative agent has a generalized logarithmic utility function. Our option pricing formulae, like the Black–Scholes model, do not depend on the preference parameters of the utility function of the representative agent. Although the Black–Scholes model introduces limited liability in asset prices by assuming that the logarithm of the stock price has a normal distribution, our basic square root option pricing model introduces limited liability by assuming that the square root of the stock price has a normal distribution. The empirical tests on the S&P 500 index options market show that our model has smaller fitting errors than the Black–Scholes model, and that it generates volatility skews with similar shapes to those observed in the marketplace. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark     

Simultaneous option prices and an implied risk-free rate of interest: A test of the Black-Scholes models

Two parameters in the Black-Scholes model, the risk-free rate of interest and standard deviation of stock returns, cannot be directly observed. Nevertheless, it is possible to simultaneously solve for the two parameters by using the prices of two different options written on the same security. If the Black-Scholes model is valid, then the implied interest rate from one repair of options should equal the implied interest rate from another pair of options for a given trading day. The analysis reexamines simultaneous option price data from a previous study using the implied interest rate test, and the results support the validity of the Black-Scholes model if we consider the bid/ask spread of option prices and that options are traded over discrete intervals.     

Hedging under the influence of transaction costs: An empirical investigation on FTSE 100 index options

The Black–Scholes (BS; F. Black & M. Scholes, 1973) option pricing model, and modern parametric option pricing models in general, assume that a single unique price for the underlying instrument exists, and that it is the mid‐ (the average of the ask and the bid) price. In this article the authors consider the Financial Times and London Stock Exchange (FTSE) 100 Index Options for the time period 1992–1997. They estimate the ask and bid prices for the index, and show that, when substituted for the mid‐price in the BS formula, they provide superior option price predictors, for call and put options, respectively. This result is reinforced further when they .t a non‐parametric neural network model to market prices of liquid options. The empirical .ndings in this article suggest that the ask and bid prices of the underlying asset provide a superior fit to the mid/closing price because they include market maker's, compensation for providing liquidity in the market for constituent stocks of the FTSE 100 index. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:471–494, 2007     

Pricing of moving‐average‐type options with applications

Moving‐average‐type options are complex path‐dependent derivatives whose payoff depends on the moving average of stock prices. This article concentrates on two such options traded in practice: the moving‐average‐lookback option and the moving‐average‐reset option. Both options were issued in Taiwan in 1999, for example. The moving‐average‐lookback option is an option struck at the minimum moving average of the underlying asset's prices. This article presents efficient algorithms for pricing geometric and arithmetic moving‐average‐lookback options. Monte Carlo simulation confirmed that our algorithms converge quickly to the option value. The price difference between geometric averaging and arithmetic averaging is small. Because it takes much less time to price the geometric‐moving‐average version, it serves as a practical approximation to the arithmetic moving‐average version. When applied to the moving‐average‐lookback options traded on Taiwan's stock exchange, our algorithm gave almost the exact issue prices. The numerical delta and gamma of the options revealed subtle behavior and had implications for hedging. The moving‐average‐reset option was struck at a series of decreasing contract‐specified prices on the basis of moving averages. Similar results were obtained for such options with the same methodology. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:415–440, 2003     

The Pricing of Catastrophe Equity Put Options with Default Risk

In this paper, we present a pricing model for catastrophe equity put options with default risk by assuming that the default of the option issuer may occur at any time prior to maturity of the option. Catastrophic events are assumed to occur according to a doubly stochastic Poisson process, and stock price is affected by the catastrophe losses, which follow the compound doubly stochastic Poisson process. As for default risk, we adopt typical structural approaches, and we also allow the correlation between the underlying stock and the assets of the option issuer. Under this framework, we derive a pricing formula for catastrophe equity put options with default risk. Finally, numerical analysis is presented to illustrate effects of default risk on catastrophe equity put option prices.     

Improving lattice schemes through bias reduction

Lattice schemes for option pricing, such as tree or grid/partial differential equation (p.d.e.) methods, are usually designed as a discrete version of an underlying continuous model of stock prices. The parameters of such schemes are chosen so that the discrete version “best” matches the continuous one. Only in the limit does the lattice option price model converge to the continuous one. Otherwise, a discretization bias remains. A simple modification of lattice schemes which reduces the discretization bias is proposed. The modification can, in theory, be applied to any lattice scheme. The main idea is to adjust the lattice parameters in such a way that the option price bias, not the stock price bias, is minimized. European options are used, for which the option price bias can be evaluated precisely, as a template to modify and improve American option methods. A numerical study is provided. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:733–757, 2006     

A two‐mean reverting‐factor model of the term structure of interest rates

This article presents a two‐factor model of the term structure of interest rates. It is assumed that default‐free discount bond prices are determined by the time to maturity and two factors, the long‐term interest rate, and the spread (i.e., the difference) between the short‐term (instantaneous) risk‐free rate of interest and the long‐term rate. Assuming that both factors follow a joint Ornstein‐Uhlenbeck process, a general bond pricing equation is derived. Closed‐form expressions for prices of bonds and interest rate derivatives are obtained. The analytical formula for derivatives is applied to price European options on discount bonds and more complex types of options. Finally, empirical evidence of the model's performance in comparison with an alternative two‐factor (Vasicek‐CIR) model is presented. The findings show that both models exhibit a similar behavior for the shortest maturities. However, importantly, the results demonstrate that modeling the volatility in the long‐term rate process can help to fit the observed data, and can improve the prediction of the future movements in medium‐ and long‐term interest rates. So it is not so clear which is the best model to be used. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23: 1075–1105, 2003     

The performance of VIX option pricing models: Empirical evidence beyond simulation

We examine the pricing performance of VIX option models. Such models possess a wide‐range of underlying characteristics regarding the behavior of both the S&P500 index and the underlying VIX. Our tests employ three representative models for VIX options: Whaley ( 1993 ), Grunbichler and Longstaff ( 1996 ), Carr and Lee ( 2007 ), Lin and Chang ( 2009 ), who test four stochastic volatility models, as well as to previous simulation results of VIX option models. We find that no model has small pricing errors over the entire range of strike prices and times to expiration. In particular, out‐of‐the‐money VIX options are difficult to price, with Grunbichler and Longstaff's mean‐reverting model producing the smallest dollar errors in this category. Whaley's Black‐like option model produces the best results for in‐the‐money VIX options. However, the Whaley model does under/overprice out‐of‐the‐money call/put VIX options, which is opposite the behavior of stock index option pricing models. VIX options exhibit a volatility skew opposite the skew of index options. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark31:251–281, 2011     

Empirical performance of alternative pricing models of currency options

This article examines the out‐of‐sample pricing performance and biases of the Heston’s stochastic volatility and modified Black‐Scholes option pricing models in valuing European currency call options written on British pound. The modified Black‐Scholes model with daily‐revised implied volatilities performs as well as the stochastic volatility model in the aggregate sample. Both models provide close and similar correspondence to actual prices for options trading near‐ or at‐the‐money. The prices generated from the stochastic volatility model are subject to fewer and weaker aggregate pricing biases than are the prices from the modified Black‐Scholes model. Thus, the stochastic volatility model may provide improved estimates of the measures of option price sensitivities to key option parameters that may lead to more effective hedging and speculative strategies using currency options. © 2000 John Wiley & Sons, Inc. Jrl Fut Mark 20:265–291, 2000     

The characteristics of interest rates and stock variances implied in option prices

Using the Black-Scholes option pricing model, this study simultaneously estimates stock return variances and interest rates implied in market option prices. Results show that the shorter term options exhibit greater implied stock variances than do longer term options. Implied interest rates, however, appear to be constant over the different expirations. Overall, the implied interest rates exhibited higher correlations with Treasury bill rates than with other money market rates, but they were consistently about one-fourth higher than the Treasury bill rates.     

The Relationship Between Risk and Maturity In A Stochastic Setting

This paper explores the interest rate sensitivity of the prices of bonds and other securities when the instantaneous interest rate follows a Markov process. We show that whenever the interest rate describes a diffusion process the sensitivity of zero-coupon bonds increases with maturity. More generally, we characterize the risk-maturity relationship for contingent claims. This investigation yields a new property of option prices in the case where the underlying security price is a diffusion.     

Comment on “A new simple square root option pricing model”

Câmara A. and Wang Y.‐H. ( 2010 ) introduce a simple square root option pricing model where the square root of the stock price is governed by a normal distribution. They show that their three‐parameter option pricing model can outperform the Black–Scholes option pricing model. We demonstrate that their assumption possesses an internal inconsistency in that the square root of the stock price can take on negative values. We generalize and revise their assumption so that the internal inconsistency can be avoided, and introduce a new square root option pricing model. The difference in option prices calculated from the two models may not be trivial. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Complete Models with Stochastic Volatility

The paper proposes an original class of models for the continuous-time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of exponentially weighted moments of historic log-price. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that, unlike many other models of nonconstant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preference-independent options prices.
We find a partial differential equation for the price of a European call option. Smiles and skews are found in the resulting plots of implied volatility.     

VIX option pricing

Substantial progress has been made in developing more realistic option pricing models for S&P 500 index (SPX) options. Empirically, however, it is not known whether and by how much each generalization of SPX price dynamics improves VIX option pricing. This article fills this gap by first deriving a VIX option model that reconciles the most general price processes of the SPX in the literature. The relative empirical performance of several models of distinct interest is examined. Our results show that state‐dependent price jumps and volatility jumps are important for pricing VIX options. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:523–543, 2009     

A term structure model for dividends and interest rates

Over the last decade, dividends have become a standalone asset class instead of a mere side product of an equity investment. We introduce a framework based on polynomial jump‐diffusions to jointly price the term structures of dividends and interest rates. Prices for dividend futures, bonds, and the dividend paying stock are given in closed form. We present an efficient moment based approximation method for option pricing. In a calibration exercise we show that a parsimonious model specification has a good fit with Euribor interest rate swaps and swaptions, Euro Stoxx 50 Index dividend futures and dividend options, and Euro Stoxx 50 Index options.     

The Chinese warrant bubble: A fundamental analysis

We investigate the information content in Chinese warrant prices based on an option pricing framework that incorporates short‐selling and margin‐trading constraints in the underlying stock market. We show that Chinese warrant prices can be explained under this pricing framework. On the basis of this new model, we develop a price deviation measure to quantify stock market investors' unobserved demand for short selling or margin trading due to market constraints. We find that warrant‐price deviations are driven by underlying stock valuation to a great extent. Chinese warrant prices, save for the time around expiration dates, are better characterized as derivatives than as pure bubbles.     

Too many options? Theory and evidence on option exchange design

A model of option exchange design is proposed and tested. The model allows investors to choose among several exchange‐traded options based on a trade‐off between standardization costs and liquidity/transaction costs. It employs a spatial economics approach to provide results for the existence of markets for particular option contracts on the exchange, a comparison of exchange design by a social planner and a profit‐maximizing monopolist (corresponding to the idea that most derivatives exchanges centralize the design and creation of option contracts), and comparative statics that can potentially aid decision makers in the design of option exchanges. In the empirical work, open interest is analyzed for Chicago Board Options Exchange (CBOE) options on the stocks in the S&P 100 index. In accordance with the model's predictions, open interest forms a previously undocumented seesaw pattern across strike prices, clustering around certain strike prices, and dropping off for the adjacent strike prices. © 2006 Wiley Periodicals, Inc. Jrl Fut Mark 26:533–570, 2006     

Equilibrium Pricing in the Presence of Cumulative Dividends Following a Diffusion

The paper presents some security market pricing results in the setting of a security market equilibrium in continuous time. The theme of the paper is financial valuation theory when the primitive assets pay out real dividends represented by processes of unbounded variation. In continuous time, when the models are also continuous, this is the most general representation of real dividends, and it can be of practical interest to analyze such models.
Taking as the starting point an extension to continuous time of the Lucas consumption-based model, we derive the equilibrium short-term interest rate, present a new derivation of the consumption-based capital asset pricing model, demonstrate how equilibrium forward and futures prices can be derived, including several examples, and finally we derive the equilibrium price of a European call option in a situation where the underlying asset pays dividends according to an Itô process of unbounded variation. In the latter case we demonstrate how this pricing formula simplifies to known results in special cases, among them the famous Black–Scholes formula and the Merton formula for a special dividend rate process.