Option Pricing Bounds in Discrete Time

Upper and lower bounds are derived for call options traded at discrete intervals. These bounds are independent of assumptions on the stock price distribution other than a restriction satisfied by the stock being “non-negative beta.” The development of the bounds relies on the single-price law and arbitrage arguments. Both single-period and multiperiod results are produced, and put option bounds follow by extension. The bounds exist as equilibrium values given a consensus on stock price distribution; they are also valid for empirical studies, being adjustable for dividends and commissions.     

Option Replication in Discrete Time with Transaction Costs

Option replication is discussed in a discrete-time framework with transaction costs. The model represents an extension of the Cox-Ross-Rubinstein binomial option pricing model to cover the case of proportional transaction costs. The method proceeds by constructing the appropriate replicating portfolio at each trading interval. Numerical values of these prices are presented for a range of parameter values. The paper derives a simple Black-Scholes type approximation for the option prices with transaction costs and demonstrates numerically that it is quite accurate for plausible parameter values.     

The Valuation of Multivariate Contingent Claims in Discrete Time Models

There are several examples in the literature of contingent claims whose payoffs depend on the outcomes of two or more stochastic variables. Familiar cases of such claims include options on a portfolio of options, options whose exercise price is stochastic, and options to exchange one asset for another. This paper derives risk neutral valuation relationships (RNVRs) in a discrete time setting that facilitate the pricing of such complex contingent claims in two specific cases: joint lognormally distributed underlying variables and constant proportional risk aversion on the part of investors, and joint normally distributed underlying variables and constant absolute risk aversion preferences, respectively. This methodology is then applied to the valuation of several interesting complex contingent claims such as multiperiod bonds, multicurrency option bonds, and investment options.     

The Equilibrium Valuation of Risky Discrete Cash Flows in Continuous Time

This paper values a contingent claim to discrete stochastic cash flows generated by a Poisson arrival process with a randomly varying intensity parameter. In the most general case, both the size and the arrival intensity of cash flows may correlate wih state variables in a continuous time economy. Assuming the conditions of an intertemporal capital aset pricing model, solutions for the value of the contingent claim can be found using various techniques. The paper suggests immediate applications to the valuation of insurance contracts, the decision to build a firm with unknown future investment opportunities, and the pricing of mortgage-backed securities.     

Dividend Forecast Biases in Index Option Valuation

Since the early days of option pricing theory,the assumption that the dividends on the underlying stock or index over the life of the contract are known has not been challenged. We examine the sensitivity of index option prices to the assumption of dividend uncertainty. We consider a number of issues related to the forecasting of dividends and build a dividend forecasting model that passes several rigorous tests for unbiasedness. We then generate option prices using contemporary market levels and interest rates. We find that prices generated with the actual dividends are unbiased with respect to those generated using the forecasted dividends. The magnitudes of the forecast errors, however, are sufficiently large to suggest a concern, but the percentage errors are consistently small, typically amounting to less than two percent of the option price. We conclude that the convenient assumption that the stream of future dividendsis known is probably innocuous. This revised version was published online in November 2006 with corrections to the Cover Date.     

Displaced Diffusion Option Pricing

This paper develops a new option pricing formula that pushes the underlying source of risk back to the risk of individual assets of the firm. The formula simultaneously encompasses differential riskiness of the assets of the firm, their relative weights in determining the value of the firm, the effects of firm debt, and the effects of a dividend policy with both constant and random components. Although this setting considerably generalizes the Black-Scholes [1] analysis, it nonetheless produces a formula via riskless arbitrage arguments that, given estimated inputs, is as easy to use as the Black-Scholes formula.     

Option Valuation with Systematic Stochastic Volatility

We use an extension of the equilibrium framework of Rubinstein ( 1976 ) and Brennan ( 1979 ) to derive an option valuation formula when the stock return volatility is both stochastic and systematic. Our formula incorporates a stochastic volatility process as well as a stochastic interest rate process in the valuation of options. If the “mean,” volatility, and “covariance” processes for the stock return and the consumption growth are predictable, our option valuation formula can be written in “preference-free” form. Further, many popular option valuation formulae in the literature can be written as special cases of our general formula.     

Valuation of a Repriceable Executive Stock Option

We introduce a path-dependent executive stock option. The exercise price might be reduced when both the firm’s stock price and a stock market index fall greatly. The repriceable executive stock option has a simple payoff that may be used for realistic executive rewards. We show the valuation formula, and compute the probability of the repriceable executive stock option expiring in-the-money. Both price and probability are important pieces of quantitative information when choosing an executive compensation package.     

Conditions for Myopic Valuation and Serial Independence of the Market Excess Return in Discrete Time Models

In a multiperiod pure exchange world with investors displaying HARA-preferences, conditions for period-by-period application of one-period asset pricing models are derived first. The future investment opportunity set may be uncertain, provided that in every period a specific market portfolio variable depending on preferences is known as of the preceding date. This variable need not be completely deterministic. Second, conditions for a serially independent market excess return are derived. These conditions render serial independence very unlikely. Hence estimation methods assuming serial independence are likely to yield biased results.     

Option Pricing with Threshold Diffusion Processes

The threshold diffusion (TD) model assumes a piecewise linear drift term and piecewise smooth diffusion term, which can capture many nonlinear features and volatility clustering often observed in financial time series data. We solve the problem of option pricing with a TD asset pricing process by deriving the minimum entropy martingale measure, which is the risk-neutral measure closest to the underlying TD probability measure in terms of Kullback-Leibler divergence, given the historical regime-switching pattern. The proposed valuation model is illustrated with a numerical example.     

期权定价理论及在我国资产评估领域应用的分析

本文首先介绍了布莱克--斯科尔斯期权定价模型(B-S模型)的产生背景、前提假设与期权定价理论的发展简史;然后介绍了B-S模型的一些变型,接着又介绍了二叉树期权定价模型及其假设,以及B-S模型在股指期权、货币期权、期货期权及实物期权等领域中的应用;最后,作者结合会计准则与评估准则,强调了继续研究期权定价理论的必要性和迫切性.     

Telling from Discrete Data Whether the Underlying Continuous–Time Model Is a Diffusion

Can discretely sampled financial data help us decide which continuous-time models are sensible? Diffusion processes are characterized by the continuity of their sample paths. This cannot be verified from the discrete sample path: Even if the underlying path were continuous, data sampled at discrete times will always appear as a succession of jumps. Instead, I rely on the transition density to determine whether the discontinuities observed are the result of the discreteness of sampling, or rather evidence of genuine jump dynamics for the underlying continuous-time process. I then focus on the implications of this approach for option pricing models.     

Fair Valuation of Participating Life Insurance Contracts with Jump Risk

The purpose of this article is to value participating life insurance contracts when the linked portfolio is modeled by a jump-diffusion. More precisely, this process has a Brownian component and a compound Poisson one, where the jump size is driven by a double exponential distribution. Specifically here, the bankruptcy risk of the insurance company is considered. Thus, market and credit risks are taken into account. A quasi-closed-form formula is obtained in fair value for the price of the considered life insurance contract. This allows us to investigate the impact of strategic parameters as well as structural ones, as is shown in the numerical section of this paper. In particular, we study the impact on the contract of the volatility, jump intensity, jump asymmetry, company leverage, guaranteed rate, participation rate and level of the default barrier, and comment on how they are likely to increase the probability of early default of the issuer.     

Option Pricing for Pure Jump Processes with Markov Switching Compensators

This paper proposes a model for asset prices which is the exponential of a pure jump process with an N-state Markov switching compensator. We argue that such a process has a good chance of capturing all the empirical stylized regularities of stock price dynamics and we provide a closed form representation of its characteristic function. We also provide a parsimonious representation of the (not necessarily unique) risk neutral density and show how to price and hedge a large class of options on assets whose prices follow this process.     

Valuation of a Developmental Drug as a Real Option

Valuing a capital investment as a real option (or series of options) has advantages over standard DCF valuation when the investment creates the future flexibility to delay, abandon, or expand an element of the project based on the resolution of a major source of uncertainty. The uncertainty is generally dealt with using a “volatility” term that aims to reflect the variability in the future value of the underlying asset. But there are certain situations in which the uncertainty has a second dimension. For example, drugs in development can be abandoned either because of bad technical outcomes (the drug doesn't work) or unfavorable resolutions of market risk (though the drug works, its market potential turns out to be too limited). In an article published earlier in this journal, the authors illustrated the valuation of an early‐stage pharma R&D investment using a real options approach in which the market and technical risks were folded together into the volatility parameter. In this article, the authors explain why they have concluded that this is an incorrect approach and then show how to handle market and technical risk as two separate dimensions of risk in valuing an R&D program. The potential use of this technique extends beyond pharma and biotech R&D to any situation in which the outcome of an important uncertainty is independent of the resolution of market risk associated with the underlying asset.     

Option Valuation with Information Costs: Theory and Tests

In this paper, the valuation of stock and index options is analyzed in the context of Merton's model of capital market equilibrium with incomplete information. It is possible to derive a partial differential equation for options in such a context. The derivation gives more understanding of the way an option's future payoff is discounted to the present. In order to estimate some of its parameters, the model is calibrated to market prices. It is tested using market prices and the authors' valuation formula. It is found that model prices are not significantly different from market prices, especially when out-of-the-money and deep-in-the-money options are considered. The model gives an explanation to the “strike bias” and the “smile effect.” Simulations of models based respectively on stochastic volatilities and gamma processes, are in accordance with the findings in this paper concerning biases in the Black and Scholes model, especially for pricing deep-in-the-money and out-of-the-money options. Even if the estimation method has its drawbacks, the costs of gathering and processing information regarding the option and its underlying asset play a central role in explaining the biases observed in the Black and Scholes model and help also the understanding of the U-shaped curve known as the smile of volatilities.     

Local Expected Shortfall-Hedging in Discrete Time

This paper proposes a self-financing trading strategy that minimizes theexpected shortfall locally when hedging a European contingentclaim. A positive shortfall occurs if the hedger is not willing to follow a perfect hedging or a superhedging strategy. In contrast to the classicalvariance criterion, the expected shortfall criterion dependsonly on undesirable outcomes where the terminal value of the writtenoption exceeds the terminal value of the hedgeportfolio. Searching a strategy which minimizes the expected shortfallis equivalent to the iterative solution of linear programs whosenumber increases exponentially with respect to the number of tradingdates. Therefore, we partition this complex overall problem intoseveral one-period problems and minimize the expected shortfall onlylocally, i.e., only over the next trading period. This approximation is quite accurate and the number of linear programs to be solved increases only linearly with respect to the number of trading dates.