TUG‐OF‐WAR,MARKET MANIPULATION,AND OPTION PRICING

We develop an option pricing model based on a tug‐of‐war game. This two‐player zero‐sum stochastic differential game is formulated in the context of a multidimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the nonlinear and completely degenerate infinity Laplace operator.     

American options on assets with dividends near expiry

Explicit expressions valid near expiry are derived for the values and the optimal exercise boundaries of American put and call options on assets with dividends. The results depend sensitively on the ratio of the dividend yield rate D to the interest rate r . For D > r the put boundary near expiry tends parabolically to the value rK / D where K is the strike price, while for D ≤ r the boundary tends to K in the parabolic-logarithmic form found for the case D =0 by Barles et al. (1995) and by Kuske and Keller (1998) . For the call, these two behaviors are interchanged: parabolic and tending to rK / D for D < r , as was shown by Wilmott, Dewynne, and Howison (1993) , and parabolic-logarithmic and tending to K for D ≥ r . The results are derived twice: once by solving an integral equation, and again by constructing matched asymptotic expansions.     

A NEW LOOK AT SHORT‐TERM IMPLIED VOLATILITY IN ASSET PRICE MODELS WITH JUMPS

We analyze the behavior of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalization of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (Blumenthal–Getoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short‐end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant model‐independent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on short‐maturity option prices.     

Step‐reset options: Design and valuation

This study proposes a new design of reset options in which the option's exercise price adjusts gradually, based on the amount of time the underlying spent beyond prespecified reset levels. Relative to standard reset options, a step‐reset design offers several desirable properties. First of all, it demands a lower option premium but preserves the same desirable reset attribute that appeals to market investors. Second, it overcomes the disturbing problem of delta jump as exhibited in standard reset option, and thus greatly reduces the difficulties in risk management for reset option sellers who hedge dynamically. Moreover, the step‐reset feature makes the option more robust against short‐term price movements of the underlying and removes the pressure of price manipulation often associated with standard reset options. To value this innovative option product, we develop a tree‐based valuation algorithm in this study. Specifically, we parameterize the trinomial tree model to correctly account for the discrete nature of reset monitoring. The use of lattice model gives us the flexibility to price step‐reset options with American exercise right. Finally, to accommodate the path‐dependent exercise price, we introduce a state‐to‐state recursive pricing procedure to properly capture the path‐dependent step‐reset effect and enhance computational efficiency. © 2002 John Wiley & Sons, Inc. Jrl Fut Mark 22:155–171, 2002     

股票期权:基于人力资本产权的分析

本文认为,企业是一个由人力资本和非人力资本共同订立的特别市场合约,企业家根据其人力资本分享企业剩余,人力资本在产权上永远具有独特性,其产权要求是保证其使用过程中的排他性。文章提出,提高人力资本开发和使用经济效率的唯一有效的途径是满足人力资本的报酬要求,实行充分的激励制度,保证人力资本的全部价值实现。股票期权是公司给股票期权受权人按约定价格和数量在约定时间购买股票的权利,既可避免直接定价的市场效果,又能实现企业家的利益回报,有利于人力资本正态效应的发挥,尽可能地将企业家所创造的收益与其所得保持比较高的正相关关系,在一定程度上是对人力资本参与分配的一种肯定,是对企业家索取剩余权利的承认,是一种人力资本的有效激励方式。     

ON THE TIMING OPTION IN A FUTURES CONTRACT

The timing option embedded in a futures contract allows the short position to decide when to deliver the underlying asset during the last month of the contract period. In this paper we derive, within a very general incomplete market framework, an explicit model independent formula for the futures price process in the presence of a timing option. We also provide a characterization of the optimal delivery strategy, and we analyze some concrete examples.     

PRICING CHAINED OPTIONS WITH CURVED BARRIERS

This paper studies barrier options which are chained together, each with payoff contingent on curved barriers. When the underlying asset price hits a primary curved barrier, a secondary barrier option is given to a primary barrier option holder. Then if the asset price hits another curved barrier, a third barrier option is given, and so on. We provide explicit price formulas for these options when two or more barrier options with exponential barriers are chained together. We then extend the results to the options with general curved barriers.     

PERPETUAL CANCELLABLE AMERICAN CALL OPTION

This paper examines the valuation of a generalized American‐style option known as a game‐style call option in an infinite time horizon setting. The specifications of this contract allow the writer to terminate the call option at any point in time for a fixed penalty amount paid directly to the holder. Valuation of a perpetual game‐style put option was addressed by Kyprianou (2004) in a Black‐Scholes setting on a nondividend paying asset. Here, we undertake a similar analysis for the perpetual call option in the presence of dividends and find qualitatively different explicit representations for the value function depending on the relationship between the interest rate and dividend yield. Specifically, we find that the value function is not convex when r > d . Numerical results show the impact this phenomenon has upon the vega of the option.     

Step Options

Motivated by risk management problems with barrier options, we propose a flexible modification of the standard knock‐out and knock‐in provisions and introduce a family of path‐dependent options: step options . They are parametrized by a finite knock‐out (knock‐in) rate , ρ. For a down‐and‐out step option, its payoff at expiration is defined as the payoff of an otherwise identical vanilla option discounted by the knock‐out factor exp(-ρτ_B^‐) or max(1‐ρτ^-B,0), where &\tau;_B^‐ is the total time during the contract life that the underlying price was lower than a prespecified barrier level ( occupation time ). We derive closed‐form pricing formulas for step options with any knock‐out rate in the range $[0,∞). For any finite knock‐out rate both the step option's value and delta are continuous functions of the underlying price at the barrier. As a result, they can be continuously hedged by trading the underlying asset and borrowing. Their risk management properties make step options attractive "no‐regrets" alternatives to standard barrier options. As a by‐product, we derive a dynamic almost‐replicating trading strategy for standard barrier options by considering a replicating strategy for a step option with high but finite knock‐out rate. Finally, a general class of derivatives contingent on occupation times is considered and closed‐form pricing formulas are derived.     

CONVEXITY OF THE EXERCISE BOUNDARY OF THE AMERICAN PUT OPTION ON A ZERO DIVIDEND ASSET

We show that the optimal exercise boundary for the American put option with non-dividend-paying asset is convex. With this convexity result, we then give a simple rigorous argument providing an accurate asymptotic behavior for the exercise boundary near expiry.     

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 valuation of reset options with multiple strike resets and reset dates

This article makes two contributions to the literature. The first contribution is to provide the closed‐form pricing formulas of reset options with strike resets and predecided reset dates. The exact closed‐form pricing formulas of reset options with strike resets and continuous reset period are also derived. The second contribution is the finding that the reset options not only have the phenomena of Delta jump and Gamma jump across reset dates, but also have the properties of Delta waviness and Gamma waviness, especially near the time before reset dates. Furthermore, Delta and Gamma can be negative when the stock price is near the strike resets at times close to the reset dates. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:87–107,2003     

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.     

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     

A generalization of Rubinstein's “Pay now,choose later”

This article provides quasi‐analytic pricing formulae for forward‐start options under stochastic volatility, double jumps, and stochastic interest rates. Our methodology is a generalization of the Rubinstein approach and can be applied to several existing option models. Properties of a forward‐start option may be very different from those of a plain vanilla option because the entire uncertainty of evolution of its price is cut off by the strike price at the time of determination. For instance, in contrast to the plain vanilla option, the value of a forward‐start option may not always increase as the maturity increases. It depends on the current term structure of interest rates. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28:488–515, 2008     

Valuation and management of money-back guarantee options

In this article, we model money-back guarantees (MBGs) as put options. This use of option theory provides retailers with a framework to optimize the price and the return option independently and under various market conditions. This separation of product price and option value enables retailers to offer an unbundled MBG policy, that is, to allow the customer to choose whether to purchase an MBG option with the product or to buy the product without the MBG but at a lower price. The option value of having an MBG is negatively correlated with the likelihood of product fit and with the opportunity to test the product before purchase, and positively correlated with price and contract duration. Simulation of our model reveals that when customers are highly heterogeneous in their product valuation and probability of need-fit, and if return costs are low, an unbundled MBG policy is optimal. When customers have high likelihood of fit or return costs are excessive, no MBG is the best policy. When customers have small variance in product valuation, but vary greatly in likelihood of product fit, the retailer may prefer to offer a bundled MBG contract, extracting consumer surplus by charging a price close to the valuation level.     

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.     

A Discrete Time Equivalent Martingale Measure

An equivalent martingale measure selection strategy for discrete time, continuous state, asset price evolution models is proposed. The minimal martingale law is shown to generally fail to produce a probability law in this context. The proposed strategy, termed the extended Girsanov principle, performs a multiplicative decomposition of asset price movements into a predictable and martingale component with the measure change identifying the discounted asset price process to the martingale component. However, unlike the minimal martingale law, the resulting martingale law of the extended Girsanov principle leads to weak form efficient price processes. It is shown that the proposed measure change is relevant for economies in which investors adopt hedging strategies that minimize the variance of a risk adjusted discounted cost of hedging that uses risk adjusted asset prices in calculating hedging returns. Risk adjusted prices deflate asset prices by the asset's excess return. The explicit form of the change of measure density leads to tractable econometric strategies for testing the validity of the extended Girsanov principle. A number of interesting applications of the extended Girsanov principle are also developed.     

Semistatic hedging and pricing American floating strike lookback options

We price an American floating strike lookback option under the Black–Scholes model with a hypothetic static hedging portfolio (HSHP) composed of nontradable European options. Our approach is more efficient than the tree methods because recalculating the option prices is much quicker. Applying put–call duality to an HSHP yields a tradable semistatic hedging portfolio (SSHP). Numerical results indicate that an SSHP has better hedging performance than a delta-hedged portfolio. Finally, we investigate the model risk for SSHP under a stochastic volatility assumption and find that the model risk is related to the correlation between asset price and volatility.     

LOCAL RISK MINIMIZATION FOR DEFAULTABLE MARKETS

We study the local risk minimization approach for defaultable markets in a general setting where the asset price dynamics and the default time may influence each other. We find the Föllmer-Schweizer decomposition in this general setting and compute it explicitly in two particular cases, when default time depends on the risky asset's behavior and when only a dependence of discounted asset price on default time is occurring.