CRITICAL PRICE NEAR MATURITY FOR AN AMERICAN OPTION ON A DIVIDEND-PAYING STOCK IN A LOCAL VOLATILITY MODEL

We consider an American put option on a dividend-paying stock whose volatility is a function of the stock value. Near the maturity of this option, an expansion of the critical stock price is given. If the stock dividend rate is greater than the market interest rate, the payoff function is smooth near the limit of the critical price. We deduce an expansion of the critical price near maturity from an expansion of the value function of an optimal stopping problem. It turns out that the behavior of the critical price is parabolic. In the other case, we are in a less regular situation and an extra logarithmic factor appears. To prove this result, we show that the American and European critical prices have the same first-order behavior near maturity. Finally, in order to get an expansion of the European critical price, we use a parity formula for exchanging the strike price and the spot price in the value functions of European puts.     

Knock‐in American options

A knock‐in American option under a trigger clause is an option contract in which the option holder receives an American option conditional on the underlying stock price breaching a certain trigger level (also called barrier level). We present analytic valuation formulas for knock‐in American options under the Black‐Scholes pricing framework. The price formulas possess different analytic representations, depending on the relation between the trigger stock price level and the critical stock price of the underlying American option. We also performed numerical valuation of several knock‐in American options to illustrate the efficacy of the price formulas. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:179–192, 2004     

QUANTO LOOKBACK OPTIONS

The lookback feature in a quanto option refers to the payoff structure where the terminal payoff of the quanto option depends on the realized extreme value of either the stock price or the exchange rate. In this paper, we study the pricing models of European and American lookback options with the quanto feature. The analytic price formulas for two types of European-style quanto lookback options are derived. The success of the analytic tractability of these quanto lookback options depends on the availability of a succinct analytic representation of the joint density function of the extreme value and terminal value of the stock price and exchange rate. We also analyze the early exercise policies and pricing behaviors of the quanto lookback options with the American feature. The early exercise boundaries of these American quanto lookback options exhibit properties that are distinctive from other two-state American option models.     

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.     

The Valuation of American Options on Multiple Assets

In this paper we provide valuation formulas for several types of American options on two or more assets. Our contribution is twofold. First, we characterize the optimal exercise regions and provide valuation formulas for a number of American option contracts on multiple underlying assets with convex payoff functions. Examples include options on the maximum of two assets, dual strike options, spread options, exchange options, options on the product and powers of the product, and options on the arithmetic average of two assets. Second, we derive results for American option contracts with nonconvex payoffs, such as American capped exchange options. For this option we explicitly identify the optimal exercise boundary and provide a decomposition of the price in terms of a capped exchange option with automatic exercise at the cap and an early exercise premium involving the benefits of exercising prior to reaching the cap. Besides generalizing the current literature on American option valuation our analysis has implications for the theory of investment under uncertainty. A specialization of one of our models also provides a new representation formula for an American capped option on a single underlying asset.     

A Forward Monte Carlo Method for American Options Pricing

This study proposes a forward Monte Carlo method for the pricing of American options. The main advantage of this method is that it does not use backward induction as required by other methods. Instead, the proposed approach relies on a wise determination about whether a simulated stock price has entered the exercise region. The validity of the proposed method is supported by the mathematical proofs for the vanilla cases. With some adaption, it is shown that this forward method can be extended to price other American style options such as chooser and exchange options. This study demonstrates the effectiveness of the proposed approach using a series of numerical examples, revealing significant improvements in numerical efficiency and accuracy in contrast with the standard regression‐based method of Longstaff and Schwartz (2001). © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:369‐395, 2013     

Robustness of the Black and Scholes Formula

Consider an option on a stock whose volatility is unknown and stochastic. An agent assumes this volatility to be a specific function of time and the stock price, knowing that this assumption may result in a misspecification of the volatility. However, if the misspecified volatility dominates the true volatility, then the misspecified price of the option dominates its true price. Moreover, the option hedging strategy computed under the assumption of the misspecified volatility provides an almost sure one-sided hedge for the option under the true volatility. Analogous results hold if the true volatility dominates the misspecified volatility. These comparisons can fail, however, if the misspecified volatility is not assumed to be a function of time and the stock price. The positive results, which apply to both European and American options, are used to obtain a bound and hedge for Asian options.     

再装期权定价及其在经理激励中的应用

期权及其定价理论是目前金融管理,金融工程研究的前沿与热点问题。在标的资产的价格服从指数O-U过程模型假设下,运用Girsanov定理获得了该过程的唯一等价鞅测度。借助期权定价的鞅方法,得出了再装期权定价模型的定价公式。同时,将此模型用于经理股票期权激励中并进行了分析。     

Double continuation regions for American and Swing options with negative discount rate in Lévy models

In this paper, we study perpetual American call and put options in an exponential Lévy model. We consider a negative effective discount rate that arises in a number of financial applications including stock loans and real options, where the strike price can potentially grow at a higher rate than the original discount factor. We show that in this case a double continuation region arises and we identify the two critical prices. We also generalize this result to multiple stopping problems of Swing type, that is, when successive exercise opportunities are separated by i.i.d. random refraction times. We conduct an extensive numerical analysis for the Black–Scholes model and the jump‐diffusion model with exponentially distributed jumps.     

Tick sizes and relative rates of price discovery in stock,futures, and options markets: Evidence from the Taiwan stock exchange

This study examines the competition in price discovery among stock index, index futures, and index options in Taiwan. The price‐discovery ability of the Taiwan Top 50 Tracker Fund, an exchange‐traded fund based on the Taiwan 50 index is examined. The authors find that, after the minimum tick size in the stock market decreases, the bid–ask spreads of the component stocks of the stock index and the Taiwan Top 50 Tracker Fund get lower, and the contribution of the spot market to price discovery increases. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 29:74–93, 2009     

Stock index futures markets: stochastic volatility models and smiles

This study examined whether the inclusion of an appropriate stochastic volatility that captures key distributional and volatility facets of stock index futures is sufficient to explain implied volatility smiles for options on these markets. I considered two variants of stochastic volatility models related to Heston (1993). These models are differentiated by alternative normal or nonnormal processes driving log‐price increments. For four stock index futures markets examined, models including a negatively correlated stochastic volatility process with nonnormal price innovations performed best within the total sample period and for subperiods. Using these optimal stochastic volatility models, I determined the prices of European options. When comparing simulated and actual options prices for these markets, I found substantial differences. This suggests that the inclusion of a stochastic volatility process consistent with the objective process alone is insufficient to explain the existence of smiles. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:43–78, 2001     

Nonparametric American option pricing

A nonparametric method is introduced to accurately price American-style contingent claims. This method uses only historical stock price data, not option price data, to generate the American option price. The accuracy of this method is tested in a controlled experimental environment under both Black, F and Scholes, M (1973) and Heston, S (1993) assumptions, and an error-metric analysis is performed. These numerical experiments demonstrate that this method is an accurate and precise method of pricing American options under a variety of market conditions. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:717–748, 2008     

BLACK–SCHOLES REPRESENTATION FOR ASIAN OPTIONS

Asian options are securities with a payoff that depends on the average of the underlying stock price over a certain time interval. We identify three natural assets that appear in pricing of the Asian options, namely a stock S, a zero coupon bond B^T with maturity T, and an abstract asset A (an “average asset”) that pays off a weighted average of the stock price number of units of a dollar at time T. It turns out that each of these assets has its own martingale measure, allowing us to obtain Black–Scholes type formulas for the fixed strike and the floating strike Asian options. The model independent formulas are analogous to the Black–Scholes formula for the plain vanilla options; they are expressed in terms of probabilities under the corresponding martingale measures that the Asian option will end up in the money. Computation of these probabilities is relevant for hedging. In contrast to the plain vanilla options, the probabilities for the Asian options do not admit a simple closed form solution. However, we show that it is possible to obtain the numerical values in the geometric Brownian motion model efficiently, either by solving a partial differential equation numerically, or by computing the Laplace transform. Models with stochastic volatility or pure jump models can be also priced within the Black–Scholes framework for the Asian options.     

Index options open interest and stock market returns

This study finds that the growth of index options open interest has a significant relation with future stock market returns. We propose a theoretical model that considers hedgers and informed traders in the options market and suggests that hedgers fully utilize options according to their expectations of future stock returns. The empirical results show that the growth of out-of-the-money call options open interest is significantly related with future stock market returns. These findings provide supporting evidence for our theoretical model.     

An efficient approximation method for American exotic options

The authors suggest a modified quadratic approximation scheme, and apply this scheme to American barrier (knock‐out) and floating‐strike lookback options. This modified scheme introduces an additional parameter into the quadratic approximation method, originally suggested by G. Barone‐Adesi and R. Whaley (1987), to reduce pricing errors. When the barrier is close to the underlying asset's current price, the approximation formula is more accurate than lattice methods because the optimal exercise boundary is independent of the underlying asset's current price. That is, the proposed method overcomes the “near‐barrier” problem that occurs in lattice methods. In addition, the pricing error decreases when the underlying asset's volatility is high. This approximation scheme is more efficient than B. Gao, J. Huang, and M. Subrahmanyam's (2000) method. As a second application of the modified approximation scheme, the authors provide an approximation formula for American floating‐strike lookback options which is the first approximation formula ever suggested in the literature. Compared to S. Babbs' (2000) binomial approach, our approximation method is more efficient after controlling for pricing errors, and is more accurate after controlling for computing time. © 2007 Wiley Periodicals, Inc. Jrl Fut Mark 27:29–59, 2007     

Quantification of sovereign risk: Using the information in equity market prices

This paper proposes a new approach to quantify the sovereign risk. We use the information of the stock price index as a proxy for the equity value of the country, and introduce a size parameter, which is a conversion factor of the stock price index to the equity value of the country, and adopted the extended Black–Scholes–Merton option-pricing model for the calculation of the probability of default. Our model is applied on two countries, Argentina as the case of debt crisis and Thailand as currency crisis. We demonstrated a capability of our model as an early warning indicator for both crises.     

关于股票指数期货定价问题的思考

中国设立股指期货迫在眉睫 ,而定价问题是建立股票指数期货市场的核心问题。作者首先介绍了通过持仓成本定价模型确定股指期货理论价格 ,进而对模型的假设条件是否成立进行了探讨 ,并对建立中国股指期货市场提出了几点政策建议。     

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.     

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     

Price discovery in the German equity index derivatives markets

This article examines the intraday price discovery process among stock index, index futures, and index options in Germany using DAX index securities and intraday transactions data. The three index securities contribute to a common factor, but the spot index and index futures have substantially larger information shares than index options. Moreover, the returns of the three index securities exhibit feedback effects, with futures being dominant. Because the trading costs of the futures appear to be the lowest of the three and those of the options to be the highest, the results are consistent with the transaction cost hypothesis. © 1999 John Wiley & Sons, Inc. Jrl Fut Mark 19: 619–643, 1999