Showing papers on "Binomial options pricing model published in 1995"
TL;DR: How a very simple and extremely efficient trinomial lattice procedure can be used to price and hedge most types of exotic barriers is explained.
Abstract: B oyle and Lau (hereafter BL) [1994] have illustrated how a naive application of the binomial option pricing algorithm can lead to significantly biased estimates in the prices of a variety of barrier, capped, and vulnerable options, even when the number of time steps is large. The source of the problem arises from the location of the barrier with respect to adjacent layers of nodes in the lattice. BL show that if the layers of the lattice are set up so that the barrier falls between layers of the lattice, the errors may be quite significant. To avoid these errors, they constrain the time partition so that the resulting lattice has layers that are as close as possible to the barrier. While this procedure reduces the size of errors, refining the partition size may not necessarily produce more precise results. Moreover, the BL procedure may be difficult to implement if the barriers are time-varying or if there are multiple barriers. This article provides a simple and highly efficient algorithm that can be used to price and hedge options that have single barriers that are either at constant levels or time-varying as well as contracts that are subject to multiple barriers. First, a lattice is constructed to pass through the barrier points exactly. Second, the stock price partition and the
188 citations
TL;DR: Rubinstein et al. as mentioned in this paper developed a binomial valuation model which simultaneously takes into consideration the most significant differences between standard call options and employee stock options: longer maturity, delayed vesting, forfeiture, non-transferability, dilution, and taxes.
Abstract: In its Exposure Draft, "Accounting for Stock-based Compensation," FASB proposes that either the Black-Scholes or binomial option pricing model be used to expense employee stock options, and that the value of these options be measured on their grant date with typically modest ex-post adjustment. This brings the accounting profession squarely up against the Scylla of imposing too narrow a set rules that will force many firms to misstate considerably the value of their stock options and the Charybdis of granting considerable latitude which will increase non-comparability across financial statements of otherwise similar firms. This, of course, is a common tradeoff afflicting many rules for external financial accounting. It is not my intention to take a position on this issue, but merely to point out the inherent dangers in navigating between these twin perils. To examine this question, this paper develops a binomial valuation model which simultaneously takes into consideration the most significant differences between standard call options and employee stock options: longer maturity, delayed vesting, forfeiture, non-transferability, dilution, and taxes. The final model requires 16 input variables: stock price on grant date, stock volatility, stock payout rate, stock expected return, interest rate, option striking price, option years-toexpiration, option years-to-vesting, expected employee forfeiture rate, minimum and maximum forfeiture rate multipliers, employee's non-option wealth per owned option, employee's risk aversion, employee's tax rate, percentage dilution, and number of steps in the binomial tree. Many of these variables are difficult to estimate. Indeed, a firm seeking to overvalue its options might report values almost double those reported by an otherwise similar firm seeking to undervalue its options. The alternatives of expensing minimum (zero-volatility) option values, whether at grant or vesting date, can easily be gamed by slightly redefining employee stock option contracts, and therefore would not accomplish FASB's goals. As an alternative, FASB could give more careful consideration to exercise date accounting, under which an expense is recognized at the time of exercise equal to the exercise value of the option. This would achieve the long sought external accounting goal of realizing stock options as compensation, while at the same time minimizing the potential for the revised accounting rules to motivate gaming behavior or non-comparable statements. * Mark Rubinstein is a professor of finance at the University of California at Berkeley. This paper arose out of a consulting project for Intel Corporation. The author thanks Robert Sprouse for his accounting courses at Stanford, Jim Ohlson for instructive conversations on accounting over many years, and Stephen Penman for assistance with employee stock options.
159 citations
TL;DR: This work introduces techniques for the sensitivity analysis of option pricing which can be efficiently carried out in the Monte Carlo simulation using an iterative stochastic approximation algorithm.
Abstract: Monte Carlo simulation is one alternative for analyzing options markets when the assumptions of simpler analytical models are violated. We introduce techniques for the sensitivity analysis of option pricing, which can be efficiently carried out in the simulation. In particular, using these techniques, a single run of the simulation would often provide not only an estimate of the option value but also estimates of the sensitivities of the option value to various parameters of the model. Both European and American options are considered, starting with simple analytically tractable models to present the idea and proceeding to more complicated examples. We then propose an approach for the pricing of options with early exercise features by incorporating the gradient estimates in an iterative stochastic approximation algorithm. The procedure is illustrated in a simple example estimating the option value of an American call. Numerical results indicate that the additional computational effort required over that required to estimate a European option is relatively small.
102 citations
TL;DR: The authors developed an arbitrage-free discrete time model to price American-style claims for which domestic term structure risk, foreign term structure, and currency risk are important, which combines a discrete version of the Heath, Jarrow, and Morton (1992) term structure model with the binomial model of Cox, Ross, and Rubinstein (1979) and converges (weakly) to the continuous time models in Amin and Jarrow (1991, 1992).
Abstract: We develop an arbitrage-free discrete time model to price American-style claims for which domestic term structure risk, foreign term structure risk, and currency risk are important This model combines a discrete version of the Heath, Jarrow, and Morton (1992) term structure model with the binomial model of Cox, Ross, and Rubinstein (1979) It converges (weakly) to the continuous time models in Amin and Jarrow (1991, 1992) The general model is "path dependent" and can be implemented with arbitrary volatility functions to value claims with maturity up to five years The model is illustrated with applications to long-dated American currency warrants and a cross-rate swap from the quanto class Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies
97 citations
TL;DR: In this article, the authors derived pricing formulas for a variety of full and partial lookback options, where monitoring takes place at not necessarily equally-spaced points in time, and showed that monitoring the underlying price discretely instead of continuously may have a significant effect on the prices of lookback option but does not introduce new hedging problems.
Abstract: We show that in the world of Black and Scholes (1973) lookback options where the underlying price is monitored discretely instead of continuously can be priced in semi-closed form. We derive pricing formulas for a variety of full and partial lookback options, where monitoring takes place at not necessarily equally-spaced points in time. Analysis of the results shows that monitoring the underlying price discretely instead of continuously may have a significant effect on the prices of lookback options but does not introduce new hedging problems.
61 citations
TL;DR: In this article, the effects of stochastic volatility on the pricing and hedging of long-term foreign currency options were examined, and it was shown that the traditional method leads to small pricing errors for short-term options, but does a poor job in pricing longterm options.
59 citations
TL;DR: An efficient simulation technique for simulating a group of set-associative caches in a single pass through the address trace, where all caches have the same line size but varying associativities and varying number of sets.
Abstract: Set-associative caches are widely used in CPU memory hierarchies, I/O subsystems, and file systems to reduce average access times. This article proposes an efficient simulation technique for simulating a group of set-associative caches in a single pass through the address trace, where all caches have the same line size but varying associativities and varying number of sets. The article also introduces a generalization of the ordinary binomial tree and presents a representation of caches in this class using the Generalized Binomial Tree (gbt). The tree representation permits efficient search and update of the caches. Theoretically, the new algorithm, GBF_LS, based on the gbt structure, always takes fewer comparisons than the two earlier algorithms for the same class of caches: all-associativity and generalized forest simulation. Experimentally, the new algorithm shows performance gains in the range of 1.2 to 3.8 over the earlier algorithms on address traces of the SPEC benchmarks. A related algorithm for simulating multiple alternative direct-mapped caches with fixed cache size, but varying line size, is also presented.
55 citations
TL;DR: In this paper, the authors characterize all possible stock price models that can be approximated by the binomial models and derive the corresponding approximations for the pricing formulas and introduce two additional randomizations in binomial price models seeking more general and more realistic limiting models.
Abstract: Cox, Ross, and Rubinstein [6] introduced a binomial option price model and derived the seminal Black–Scholes pricing formula. In this paper we characterize all possible stock price models that can be approximated by the binomial models and derive the corresponding approximations for the pricing formulas. We introduce two additional randomizations in the binomial price models seeking more general and more realistic limiting models. The first type of model is based on a randomization of the number of price changes, the second one on a randomization of the ups and downs in the price process.As a result we also obtain price models with fat tails, higher peaks in the center, nonsymmetric etc., which are observed in typical asset return data. Following similar ideas as in [6] we also derive approximating option pricing formulas and discuss several examples.
32 citations
TL;DR: In this article, the authors consider the convergence of the Black-Scholes option pricing theory to the valuation of barrier options and show how various types of barrier option can be priced either by backward induction or by closed binomial formulas.
Abstract: The extension of the Black-Scholes option pricing theory to the valuation of barrier options is reconsidered. Working in the binomial framework of CRR we show how various types of barrier options can be priced either by backward induction or by closed binomial formulas. We also consider analytically and numerically the convergence of the prices in discrete time to their continuous-time limits. The arising numerical problems are solved by quadratic interpolation. Furthermore, the case of American barrier options is analyzed in detail. For American barrier call options, binomial formulae and their limit results are given. Finally, the binomial approach is applied to contracts with local and partial barrier checks.
24 citations
TL;DR: In this article, the authors studied the problem of convergence of discrete-time option values to continuous time option values, and proved that local risk minimization possesses an inherent stability property under discretization.
Abstract: We study the problem of convergence of discrete-time option values to continuous-time option values. While previous papers typically concentrate on the approximation of geometric Brownian motion by a binomial tree, we consider here the case where the model is incomplete in both continuous and discrete time. Option values are defined with respect to the criterion of local risk-minimization and thus computed as expectations under the respective minimal martingale measures. We prove that for a jump-diffusion model with deterministic coefficients, these values converge; this shows that local risk-minimization possesses an inherent stability property under discretization.
TL;DR: This approach generalizes the traditional methodology by relaxing the assumption of a frictionless spot market (or even the existence of a spot market) and that the underlying commodity is storable, and is consistent with short sale constraints in the spot market for the underlying commodities.
Abstract: Summary This paper provided an analytic synthesis of the option pricing literature, using a term structure of futures prices approach. Postulating a process for the evolution of the term structure of futures prices, it is shown how to price derivative secutities in an arbitrage-free manner. Complete markets are assumed. This approach generalizes the traditional methodology by relaxing the assumption of a frictionless spot market (or even the existence of a spot market) and that the underlying commodity is storable. Thus, this method is consistent with short sale constraints in the spot market for the underlying commodity. When short sale restrictions are removed, the traditional option pricing models are shown to be obtainable as special cases. This includes the binomial model of CRR, as well as its applications to index options, currency options and commodity options. The new interest rate options models of HJM are also shown to be a subset of this framework. A brief discussion of how to empirically implement the model is also provided. References are given to reviews of the empirical literature and historic surveys of the model development.
01 Jan 1995
TL;DR: This work expresses a functional implementation of binomial queues which is both elegant and efficient, and quantifies some of the differences with other functional implementations.
Abstract: Efficient implementations of priority queues can often be clumsy beasts. We express a functional implementation of binomial queues which is both elegant and efficient. We also quantify some of the differences with other functional implementations. The operations decreaseKey and delete always pose a problem without destructive update, we show how our implementation may be extended to express these.
TL;DR: In this paper, a method of accelerating the pricing of American options in binomial lattices in a Black-Scholes environment is described, which can be applied to pricing American options on interest rate derivatives and options involving multiple assets.
Abstract: This article describes a method of accelerating the pricing of American options in binomial lattices in a Black-Scholes environment The standard backward induction method for solving an option valuation problem involves computations at every node of the binomial option price tree We show that many of the intermediate calculations are actually unnecessary, and eliminating them leads to a dramatic increase in computational efficiency Test cases demonstrate that valuing an American put option can be accelerated by at least an order of magnitude, while yielding the identical estimate given by the standard Cox, Ross, and Rubinstein binomial tree In addition, we discuss how similar techniques may be applied to pricing American options on interest rate derivatives and options involving multiple assets
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TL;DR: In this article, the authors provide a unified and easily applicable approach to pricing and hedging Black-Scholes type options on stocks, bonds, forwards, futures, exchange rates and indices.
Abstract: The paper deals with the valuation and hedging of non path- dependent European options on one or several underlyings in a model of an international economy which allows for both interest rate and exchange rate risk. The contingent claims may pay off in arbitrary currencies. Using martingale theory we provide a unified and easily applicable approach to pricing and hedging Black-Scholes type options on stocks, bonds, forwards, futures, exchange rates and indices. We also cover the pricing and hedging of options to exchange two Black-Scholes type options for one another. This class of payoffs includes for instance spread options. We give near explicit solutions for pricing and hedging of these contracts and discuss several numerical techniques for the evaluation of our formulas. In particular, using the theory of large deviation, we are able to give sharp estimates for the quality of Monte Carlo simulations.
TL;DR: In this paper, the authors characterize the relationship between the prices of options and stocks on which the options are written in a general equilibrium model where options are non-redundant assets.
Abstract: The traditional valuation formulas for options were derived in a complete market setting and were based on the no-arbitrage principle. If the asset structure is incomplete, the presence of options affects the linear subspace spanned by the payoffs of the existing assets, and the pricing of options and underlying primary assets becomes a simultaneous valuation problem. We characterize the relationship between the prices of options and the prices of the stocks on which the options are written in a general equilibrium model where options are non-redundant assets. Contrary to the predictions of the Black-Scholes-Merton theory, in our model investor preferences have an impact on the relationship between the prices of primary and derivative assets. Copyright 1995 by The editors of the Scandinavian Journal of Economics.
TL;DR: In this paper, the authors examined the relationship between investors' preferences and the binomial option pricing model of Cox, Ross, and Rubinstein (CRR) and showed that the independence of the CRR pricing model from investors' preference is a result of a special choice of binomial parameters made by CRR.
Abstract: This paper reexamines the relationship between investors' preferences and the binomial option pricing model of Cox, Ross, and Rubinstein (CRR). It is shown that the independence of the binomial option pricing model from investors' preferences is a result of a special choice of binomial parameters made by CRR. For a more general choice of binomial parameters, risk neutrality cannot be obtained in discrete time. This analysis reveals the essential difference between the “risk neutral” valuation approach of Cox and Ross and the equivalent martingale approach of Harrison and Kreps in a discrete time framework.