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The Black–Scholes model is the rare closed-form formula for pricing options, but its shortcomings are well known. Adding stochastic volatility, American or Bermudan early exercise, non-diffusive jumps in the returns process, and/or other “exotic” payoff features quickly takes one into the realm where approximate answers must by computed by numerical methods. Binomial and other lattice models, Monte Carlo simulation, and numerical solution of the valuation partial differential equation are the standard techniques. They are well known to eventually arrive at as close an approximation to the exact model value as one wants but can require a large amount of calculation and execution time to get there. In this article, the authors present what amounts to the culmination of a series of papers on using quadrature methods to solve option problems of increasing complexity. Quadrature entails modeling and approximating the transition densities between critical points in the option’s lifetime. Depending on the specifics of the option’s payoff, looking at only a few points in time can greatly reduce the amount of calculation needed to price it. For example, a Bermudan option that can be exercised at three possible early dates needs quadrature calculations for only those three dates plus expiration, while other numerical methods require calculations for every date and possible asset price. This article summarizes how to apply quadrature to standard option problems, from plain-vanilla Black–Scholes up to Bermudan and American options under a stochastic volatility jump-diffusion returns process. The procedure can be speeded up even more by use of Richardson extrapolation and caching techniques that allow reuse of results calculated in intermediate steps. The range of derivatives models that can be valued extremely efficiently using quadrature is very broad.

American exercise presents difficulties for option valuation. For an in-the-money option, it becomes necessary at each point in time to consider whether to exercise or to hold on, based on expectations about the stock price at option expiration and also about the optionality value of potential exercise on every date when exercise will be possible in the future. An “Asian” option payoff presents its own valuation problems because the payoff is based on the arithmetic average of correlated lognormal prices, which is not lognormal. Adding stochastic volatility makes both of these problems much harder. But in this article, Russo and Staino are able to develop a lattice technique that deals with all three of these difficulties. The stochastic variable that the tree is built around is the variance, while both the current asset price and the running average of past prices to be included in calculating the payoff at expiration are carried along as auxiliary variables, in the form of sets of values at each node that span the range of possible values along all of the paths that lead to that node. Although there are several approximations in the procedure, accuracy is excellent, and execution is relatively fast.

Numerical solution methods for option pricing fall into two broad classes, either the lattice framework or Monte Carlo simulation. Monte Carlo simulation is problematic for American options, while the Binomial and Trinomial approaches have difficulty incorporating stochastic jumps. The probability density at expiration that is generated by a lattice typically has different skewness and kurtosis than is embedded in option market prices. The Binomial can be fitted imposing a terminal density with the right moments, using the Johnson system of distributions for example, but the problem still remains that the number of distinct nodes to calculate increases quadratically with the number of time steps. This article proposes the “Willow Tree” method, which combines a small number of large time steps that connect to many possible next period nodes, with the constraint that the terminal density, constructed from the Johnson system, has the same mean, volatility, skewness, and kurtosis as the risk neutral density extracted from prices in the options market. The result is a much faster and more accurate valuation than may be achieved through existing approaches.

The binomial model is a workhorse for valuing options with early exercise possibilities, including options with American or Bermudan exercise and barrier options. The procedure is straightforward when the underlying asset does not pay dividends, and it can be readily adapted to a payout at a fixed proportional rate. But discrete dividends cause trouble because they make the tree “splinter” rather than recombining at each time step. If the tree recombines, an up move followed by a down move produces the same asset price as a down move followed by an up move. But if the intermediate date corresponds to an ex-dividend date and the same fixed amount is subtracted from the up and down nodes, the tree no longer recombines at the subsequent step. The number of nodes in a non-recombining tree increases at a geometric rate, quickly leading to unreasonably long execution times. A number of alternative procedures have been developed to deal with this problem. In this article, Areal and Rodrigues adopt a new approach, accepting the splintering of the binomial lattice but then applying several techniques to accelerate the calculations. They show that their procedures are faster and more accurate than the current methods that force the tree to recombine despite discrete dividend payouts.

The Binomial model and similar lattice methods are workhorses of practical derivatives valuation. But returns processes more realistic than lognormal diffusions with constant parameters easily create difficulties for them. One of the most important extensions of the Black-Scholes paradigm is to allow stochastic volatility, but even nonstochastic timevarying volatility destroys the important property that the tree recombines, which limits the growth in the number of nodes as time advances. Stochastic volatility introduces a second random variable, which then requires adding another dimension to the tree, under the constraint that the return and volatility changes must maintain the same degree of correlation as in the data. The Heston model features correlation in return and volatility shocks, but building it into a lattice is tricky. In this article, Ruckdeschel, Sayer, and Szimayer develop a lattice method that begins with a binomial tree for the volatility and a trinomial tree for stock price, and then connects them in such a way that the empirical degree of correlation between return and volatility is maintained. Efficiency relative to existing methods is increased, and in some cases it is possible to improve performance further by matching higher moments as well.

One of the classic problems in option pricing theory is valuing the American put. It is easy to show the existence of an early exercise boundary for the price of the underlying asset such that each point in time, if the price is below the boundary exercise is optimal, while above the boundary it is not. But deriving a closed form valuation equation for this so-called free-boundary problem is not possible. Other types of options with early-exercise features, like Bermudan contracts, calls on dividend-paying stocks, and options on futures and forwards all present similar difficulties, and introducing a wider range of returns processes, such as jump-diffusions, does not make things easier. Many ideas for approximating solutions for early-exercise options have been proposed over the years. In this article, Powell explores a different approach using genetic optimization. This powerful search technique has not been used much in finance, but the results presented here suggest that it is capable of producing approximate valuations that are as good as other methods using rather simple computer technology. The purpose of the article is to illustrate the potential of genetic optimization for this class of problems, leaving possibilities for fine-tuning it to increase efficiency, for example by imposing known properties of the early-exercise boundary a priori, are left for future exploration.

The Black–Scholes model for pricing a European option written on a lognormal returns process was easy. Unfortunately, real world markets don’t try to make life easy for financial econometricians. Returns processes are more complicated than lognormal diffusions, and an array of derivative products with much more complex payoffs than European calls are now traded. Many of those are path dependent, making Monte Carlo simulation the only viable valuation technique. But, Monte Carlo can require many replications and substantial execution time to achieve reasonable accuracy for a single set of parameters. In this article, Popovic and Goldsman offer a hybrid approach that can achieve accurate pricing very efficiently. A base set of Monte Carlo runs is constructed and then a Gram–Charlier density approximation is fitted to the histogram of simulation results (which may first need to be transformed to be better adapted to the approximation). A series of examples based on pricing Asian options illustrates how the procedure performs and demonstrates its flexibility in handling non-lognormal underlying returns processes.

By introducing Brownian motion to finance, Black, Scholes, and Merton caused a quantum jump in sophistication and realism in the returns processes assumed in asset pricing models. Diffusion processes have taken us a long way, but for many markets there is considerable evidence that diffusions aren9t enough to represent real world returns accurately. We need the possibility of occasional discrete jumps, as well. Short-term interest rates are especially prone to jumps because they are strongly influenced by administered rates, such as the Fed funds rate, that are changed in discrete increments. In this article, the authors develop a model of the short rate in the spirit of the Vasicek model, but with exponential up-jump and down-jump processes appended. The pricing equations for European options on zero coupon bonds and interest rate swaps can be obtain by cumulant or Fourier transform techniques, but for American exercise the authors also present a lattice model suitable for computing theoretical values and hedging parameters. The key to capturing the jump effects within a lattice framework is to allow multiple branching from each node, such that large jump moves can occur. Convergence properties of both techniques seem quite good.

The hardest computational problems in derivatives are those that involve American exercise and path-dependent payoffs. Standard Monte Carlo simulation generates paths in a forward direction, so path-dependent payoffs can be computed easily. But optimizing the early exercise decision requires generating separate families of paths going forward from each possible exercise point. Achieving an accurate solution for a complex problem can easily require more paths than can be held in computer memory. The popular Longstaff-Schwartz approximation technique for these problems works well but suffers from the same difficulty when the number of early exercise decisions is large, or in general, when problems become too complex. As shown here, Longstaff-Schwartz with too few secondary paths induces bias in addition to inaccuracy in the valuation. In this article, Dutt and Welke introduce an innovative approach that amounts to generating Monte Carlo paths in the reverse direction. Starting from the desired probability distribution of terminal asset prices, it simulates paths backward in time in such a way as to arrive with probability 1 at the initial asset value, with the resulting paths being fully consistent with the underlying returns process. The great advantage is that the simulated prices for later dates can be dropped from memory once they have been used. The authors illustrate the power of their “Just-in-time” approach on several examples, including valuing a 30 year mortgage under CIR interest rates taking into account both the prepayment and the default options.

Beginning with Black and Scholes, it has been mathematically very convenient to model asset returns as logarithmic diffusions when developing pricing models for derivatives. But doing so causes a lot of trouble for instruments such as spread options, whose payoffs are a function of the arithmetic difference between two such assets, because the distribution of the price difference is not lognormal or any other familiar distribution. One could simply assume the difference is lognormal and obtain a simple model, but that would lead to an internal inconsistency with the models used for the component securities. Alternatively, one must develop numerical or approximation techniques to deal with this problem. In this article, the authors advance the technology by devising analytic formulas for spread option values, as well as their greeks, based on an approximation to the exercise boundary for one asset9s price, conditional on the other price. The result is much more efficient and accurate valuation than with previous approaches

There are limitless possibilities for new kinds of options, with payoff patterns of all shapes and contingencies related to any number of events. The simplest of these payoffs, like those for a call spread or a strangle, for example, can be constructed out of a straightforward combination of plain vanilla puts and calls. This leads to very easy formulas for pricing and hedging them, but the possible payoff patterns invariably consist of a collection of linear segments defined by the intervals between the strikes of the options in the set. This article provides a way to significantly extend the achievable payoff patterns to any polynomial function of the final asset price. The building block is no longer a plain vanilla option, but a power option, the payoff of which is a function of (Sa - X) for some exponent a. The authors first show how to price such options under several important price processes, including Heston-style stochastic volatility and a general Levy process. They then derive the formula for an option with a general polynomial payoff, as a function of power options. Finally, they offer some examples to illustrate the technique, including parabolic payoff call options.

Many MBA students are daunted by the requirement that calculus is a prerequisite for the program. But those who are already familiar with finance know that most of the “hard stuff” of calculus is rarely used in practice. The real key is to develop an intuitive understanding of what calculus is about and to know a few basic kinds of things, like taking a derivative. A similar kind of angst hit the finance profession 30 years ago, when continuous-time mathematics was introduced. But today, pretty much everyone has developed the intuition, and knows how to use It9s formula, so continuous-time models have become familiar. The methods of option pricing introduced in the 1970s led to the Black-Scholes formula and a number of extensions, but they are not able to handle things like stochastic volatility, jumps, or non-Gaussian innovations in any convenient way. In the 1990s, researchers began to model these features using powerful mathematical techniques based on Fourier transforms and similar methods, that allow closed-form valuation equations for much more general returns processes. But once again, the new technology seems pretty daunting. In this article, Cerný offers some valuable intuition on what the fast Fourier transform does (think about bicycle wheels!). He also provides some very useful details about key differences in how to do it in several common software packages.

Pricing barrier options for interest rates and interest-dependent securities is harder than for equities, because interest rate processes typically require more complicated models than stock returns. Even single-factor models need to incorporate the initial yield curve in order to prevent apparent arbitrage opportunities among observed bond prices in the market. When closed-form solutions do not exist, numerical valuation techniques are required, and both lattice-based approaches and Monte Carlo simulation are in common use. But these methods are very computationally demanding, and lattice techniques also require careful handling of price nodes near the barrier. Kuan and Webber introduce a new approach, in which they estimate the first passage time to the barrier, and then value the option conditional on having hit the barrier. This leads to considerable improvement in performance, either less computational effort to achieve a given accuracy, or greater accuracy for a given amount of work.

Rubinstein9s seminal work on implied binomial trees showed how to fit a tree to a set of market prices for options maturing on a single date. A problem with the technique is that the tree it produces cannot match option prices for multiple maturities, so it cannot simultaneously cover a full set of options traded in a given market. The tree structure needed for that task is significantly more complex than a simple binomial. Brown and Toft show how such a general tree may be implied out of a set of option market prices. They then illustrate the technique using S&P 500 index options and currency options on the deutschemark. The resulting trees even embody patterns of path dependence commonly observed in volatilities.

Many exotic derivatives do not have closed-form valuation equations, and must be priced using approximation methods. Where they can be applied, standard lattice techniques based on binomial and trinomial trees will achieve correct valuations asymptotically. They can also generally handle American exercise. But for many problems, including pricing barrier options, convergence may be slow and erratic, producing large errors even with thousands of time steps and millions of node calculations. Options with price barriers that are only monitored at discrete points in time present additional difficulty for lattice models. Standard tree methods increase accuracy by shrinking the time and price step size throughout the lattice, but this increases the number of calculations sharply and much of the additional computation is in regions of the tree where it makes little difference to accuracy. A previous paper, Figlewski and Gao [1999], introduced the Adaptive Mesh Model (AMM), a very flexible approach that greatly increases efficiency in trinomial lattices. Coarse time and price steps are used in most of the tree, but small sections of finer mesh are constructed to improve resolution in specific critical areas. This paper presents an especially effective AMM structure for pricing options with discrete barriers. In a basic example, an AMM with 60 time steps is ten times more accurate than a 5000-step trinomial, but runs more than 1000 times faster.

The option delta plays a key role in eliminating the risk in continuously rebalanced hedges. This need not be the case in discretely rebalanced hedges. Robins and Schachter (Management Science, June 1994) show how the minimum variance discretely rebalanced (European call) option hedge ratio deviates from the option delta. In this paper, we use the binomial option pricing framework to calculate an analytical approximation to the return variance of a discretely rebalanced (American call) option hedge. We then conduct an empirical examination of the differences between delta-neutral and non-delta neutral (i.e., minimum variance) hedges. We use OEX option quotes for the first 42 trading days of 1986. We construct hedges (buying options at the ask and selling at the bid) with rebalancing intervals of from 1 to 10 days. We categorize realized hedge returns according to rebalancing interval, option time to expiration and degree to which the option is in- or out-of- the-money. Using the heteroskedasticity and autocorrelation consistent test statistic developed by Mizrach (1992) we document statistically significantly smaller hedge variances for the non-delta-based hedges across most categories.

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

A ccelerating the pricing of American options in binomial and multinomial lattices in a Black-Scholes environment has been addressed in the literature (Kim and Byun [1994] (KB), Breen [1991]), but the method presented here (hereafter the diagonal method, or DM) is substantially more efficient and is applicable to a wider variety of cases. In particular, while the KB and Breen techniques yield an acceleration factor of three or four for the American put option, the DM results in acceleration factors in excess of an order of magnitude, while arriving at identical values as those obtained from the standard Cox, Ross, and Rubinstein [1979] (CRR) binomial method. The method is specific to CRR trees in that, starting at a fixed underlying price, an up move (u) followed by a down move (d) results in an underlying price equal to the original price, i.e., d = l/u. The DM also has potential for speeding the computation of American options on interest rate instruments and options involving multiple assets.

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.

Implied volatility is obtained from an option’s observed market price by solving backward through the pricing model to find the volatility that would produce the same value from the model. Unfortunately, this typically leads to the familiar volatility smile: different implied volatilities from different options on the same underlying. But from market prices for the full set of a given stock’s options, one can extract not just volatility, but the entire risk-neutral probability distribution, from which volatility and any other moment can easily be obtained, and the result does not depend on any particular pricing model. One problem, however, is that because the risk-neutral density is over the stock price at expiration day, the options involved need to be European so that exercise can only take place on that date. But exchange-traded options are nearly all American, in the U.S. at least. In this article, Tian shows how to obtain the risk-neutral density from American option prices. The trick is to build an implied lattice model from the American options then use it to find European option prices that would be consistent with the observed American prices in the market. The density can then be extracted from these implied European prices in the standard way.

We present efficient simulation procedures for pricing barrier options when the underlying security price follows a geometric Brownian motion with jumps. Metwally and Atiya [2002] developed a simulation approach for pricing knock-out options in the same setting, but no variance reduction was introduced. We improve upon Metwally and Atiya9s method by innovative applications of well-known variance reduction techniques. We also show how to use simulation to price knock-in options. Numerical examples show that our proposed Monte Carlo procedures lead to substantial variance reduction as well as a reduction in computing time.