Pricing Exotic Options and American Options: A Multidimensional Asymptotic Expansion Approach

This paper introduces a new method for pricing exotic options whose payoff functions depend on several stochastic indices and American options in multidimensional models. This method is based on two ideas. One is an application of the asymptotic expansion method for the law of a multidimensional diffusion process. The other is the combination of the asymptotic expansion method and the method called backward induction. The author applies the method to the problems of pricing call options on the maximum of two assets in the CEV model, average strike options in the Black–Scholes model and American options in the Heston model. Numerical examples show practical effectiveness of the proposed method.     

The Pricing of Options with Default Risk

This paper considers the pricing of options with default risk. The comparative statics of such options can differ from those of ordinary options, and early exercise of such American call options can be optimal. Several examples of options with default risk are considered.     

Valuing and Pricing Retail Leases with Renewal and Overage Options

We consider retail leases with landlord overages options, with tenant renewal options, with both and with neither. We illustrate how the ratio of initial expected sales to the sales threshold can be manipulated to equate the value of the landlord overage options to that of the tenant renewal option at the same initial rent. Not only are the values equal, but the cumulative distributions of potential IRRs on the two leases are nearly identical, suggesting that these leases are equally attractive to risk-averse investors and thus that the same risky discount rate can be used in valuing the leases. In contrast, the appropriate risky discount rate for the overage lease is calculated to be 75–160 basis points greater than that for the renewal lease.     

Pricing European Options by Numerical Replication: Quadratic Programming with Constraints

The paper considers a regression approach to pricing European options in an incomplete market. The algorithm replicates an option by a portfolio consisting of the underlying security and a risk-free bond. We apply linear regression framework and quadratic programming with linear constraints (input = sample paths of underlying security; output = table of option prices as a function of time and price of the underlying security). We populate the model with historical prices of the underlying security (possibly massaged to the present volatility) or with Monte Carlo simulated prices. Risk neutral processes or probabilities are not needed in this framework.     

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.     

The Pricing of Options on Assets with Stochastic Volatilities

One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity.     

Hedging Strategies of Financial Intermediaries: Pricing Options with a Bid-Ask Spread

This paper uses a model similar to the Boyle-Vorst and Ritchken-Kuo arbitrage-free models for the valuation of options with transactions costs to determine the maximum price to be charged by the financial intermediary writing an option in a non-auction market. Earlier models are extended by recognizing that, in the presence of transactions costs, the price-taking intermediary devising a hedging portfolio faces a tradeoff: to choose a short trading interval with small hedging errors and high transactions costs, or a long trading interval with large hedging errors and low transactions costs. The model presented here also recognizes that when transactions costs induce less frequent portfolio adjustments, investors are faced with a multinomial distribution of asset returns rather than a binomial one. The price upper bound is determined by selecting the trading frequency that will equalize the marginal gain from decreasing hedging errors and the marginal cost of transactions.     

Analytic Pricing of Employee Stock Options

We introduce a model that captures the main properties thatcharacterize employee stock options (ESO). We discuss the likelihoodof early voluntary ESO exercise, and the obligation to exerciseimmediately if the employee leaves the firm, except if thishappens before options are vested, in which case the optionsare forfeited. We derive an analytic formula for the price ofthe ESO and in a case study compare it to alternative methods.     

Employee Reload Options: Pricing, Hedging, and Optimal Exercise

Reload options, call options granting new options on exercise,are popularly used in compensation. Although the compound optionfeature may seem complicated, there is a distribution-free dominantpolicy of exercising reload options whenever they are in themoney. The optimal policy implies general formulas for numericalvaluation. Simpler formulas for valuation and hedging followfrom Black–Scholes assumptions with or without continuousdividends. Time vesting affects the optimal policy, but numericalresults indicate that it is nearly optimal to exercise in themoney whenever feasible. The results suggest that reload optionsproduce similar incentives as employee stock options and sharegrants.     

The Credit Risk and Pricing of OTC Options

In the over-the-counter (OTC) markets, the options traded are always subject to credit risk. Therefore the counterparty’s credit risk is a striking factor when pricing options, whereas it is not considered in the classic Black-Scholes models. Based on the first passage time models, this paper develops the credit risk and valuation model for the European options in the OTC markets, incorporating a practical default trigger mechanism. The default probability and the pricing formulae of the OTC options are obtained by using partial differential equation (PDE) techniques, especially Green’s function.     

Pricing Options in an Extended Black Scholes Economy with Illiquidity: Theory and Empirical Evidence

This article studies the pricing of options in an extended BlackScholes economy in which the underlying asset is not perfectlyliquid. The resulting liquidity risk is modeled as a stochasticsupply curve, with the transaction price being a function ofthe trade size. Consistent with the market microstructure literature,the supply curve is upward sloping with purchases executed athigher prices and sales at lower prices. Optimal discrete timehedging strategies are then derived. Empirical evidence revealsa significant liquidity cost intrinsic to every option.     

Pricing Options under Stochastic Interest Rates: A New Approach

We will generalize the Black-Scholes option pricing formula by incorporating stochastic interest rates. Although the existing literature has obtained some formulae for stock options under stochastic interest rates, the closed-form solutions have been known only under the Gaussian (Merton type) interest rate processes. We will show that an explicit solution, which is an extended Black-Scholes formula under stochastic interest rates in certain asymptotic sense, can be obtained by extending the asymptotic expansion approach when the interest rate volatility is small. This method, called the small-disturbance asymptotics for Itô processes, has recently been developed by Kunitomo and Takahashi (1995, 1998) and Takahashi (1997). We found that the extended Black-Scholes formula is decomposed into the original Black-Scholes formula under the deterministic interest rates and the adjustment term driven by the volatility of interest rates. We will illustrate the numerical accuracy of our new formula by using the Cox–Ingersoll–Ross model for the interest rates.     

Pricing American Put Options on Defaultable Bonds

In the last two decades, the market of credit derivativeshas expanded rapidly, and the importance of pricing problemsfor credit derivatives has been recognized especially in the last decade.Among these securities, the pricing problems of credit derivativeswith an early exercise, such as American put options,have not received enough attention. In view of this need, this paper develops a continuous stochastic modelof American put options on defaultable bonds.The method of obtaining a solution is based on a new result of the optimalstopping problem for a diffusion process with a jump.Some characterizations of American put options are providedusing partial differential equations.     

Pricing Discrete Barrier Options Under Stochastic Volatility

This paper proposes a new approximation method for pricing barrier options with discrete monitoring under stochastic volatility environment. In particular, the integration-by-parts formula and the duality formula in Malliavin calculus are effectively applied in pricing barrier options with discrete monitoring. To the best of our knowledge, this paper is the first one that shows an analytical approximation for pricing discrete barrier options with stochastic volatility models. Furthermore, it provides numerical examples for pricing double barrier call options with discrete monitoring under Heston and λ-SABR models.     

Pricing Options under Generalized GARCH and Stochastic Volatility Processes

In this paper, we develop an efficient lattice algorithm to price European and American options under discrete time GARCH processes. We show that this algorithm is easily extended to price options under generalized GARCH processes, with many of the existing stochastic volatility bivariate diffusion models appearing as limiting cases. We establish one unifying algorithm that can price options under almost all existing GARCH specifications as well as under a large family of bivariate diffusions in which volatility follows its own, perhaps correlated, process.     

A Stochastic Correlation Model with Mean Reversion for Pricing Multi-Asset Options

We set up a new kind of model to price the multi-asset options. A square root process fluctuating around its mean value is introduced to describe the random evolution of correlation between two assets. In this stochastic correlation model with mean reversion term, the correlation is a random walk within the region from −1 to 1, and it is centered around its equilibrium value. The trading strategy to hedge the correlation risk is discussed. Since a solution of high-dimensional partial differential equation may be impossible, the Quasi-Monte Carlo and Monte Carlo methods are introduced to compute the multi-asset option price as well. Taking a better-of two asset rainbow as an example, we compare our results with the price obtained by the Black–Scholes model with constant correlation.     

Pricing CIR Yield Options by Conditional Moment Matching

We propose an approximation scheme for the pricing of yield options in the CIR model using conditional moment matching based on the gamma and lognormal distributions. This method is fast and simple to implement, and it shows a high degree of accuracy without being subject to the numerical instabilities that can be encountered with more sophisticated approaches.