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Levy process models can capture the large price changes on sudden exogenous events and can better demonstrate the high peak and heavy tail characteristics of financial data. The Fourier transformation method is famous for pricing derivatives under the Levy processes beause of its efficiency, separation models from payoff function, and handling models with characteristic functions, but it is criticized for its restriction on path dependency. In this article, we propose a unified cosine willow tree method, which inherits the merits of the transformation method but overcomes the shortcomings. Moreover, the hedging Greeks can be obtained as a by-product from the tree structure with minor extra cost. Some popular variance derivatives are also discussed to demonstrate the flexibility of the proposed method in handling path dependency. Finally, the theoretical convergence is analyzed for various Levy process models. Key Findings ▪ Propsoe a recombining willow tree framework for stochastic processes with explicit characteristic functions on option pricing and hedging. ▪ Analyze the convergence of the willow tree structure on option pricing under L’evy processes. ▪ Propose efficient pricing algorithms for volatility option, variance swap with corridor and timer option.

Recently published research has identified an empirical approximation to the free exercise boundary of an American equity put option. This article examines the accuracy of this simple, readily calculated approximation through linear regression analyses of hundreds of numerically generated free boundaries. The author finds that this Little Ansatz provides an excellent estimate of the optimal boundary for all meaningful parameter values and times to expiration. A second result of this article is the mapping of the two empirical parameters as functions of risk-free rate and volatility. This mapping is necessary to enable fast option pricing and sensitivity calculations for large books of these derivative transactions. Proper calculation of the American put option is a long-standing challenge that has spawned immensely creative and admirable research. The discovery of this empirical approximation has great practical importance. TOPICS:Derivatives, options, security analysis and valuation, quantitative methods, statistical methods Key Findings ▪ The author finds that a recently published empirical expression provides an excellent estimate of the American free exercise boundary for all meaningful parameter values and times to expiration. ▪ The author determines efficient and accurate mapping of the two necessary adjustable parameters of the approximation as functions of risk-free rate and volatility. ▪ The author suggests future research and improvements for this fast, empirical solution to the American free exercise boundary.

The underlying models for bond options are often based on some linear drift functions such that the option Greeks depend crucially on the mean reversion parameters. Substantial estimation bias may arise when these parameters are estimated using standard methods such as maximum likelihood estimation, leading to a bias in estimating the Greeks. To address this issue, following Phillips and Yu (2005), a jackknife method is adopted in this article. In particular, we apply the method directly to the estimation of option Greeks, rather than the estimation of parameters. This approach is general and computationally inexpensive; hence, it is convenient in practice. The finite-sample performance is investigated in several Monte Carlo studies. At last, in dynamic Delta hedging, we show that the bias reduction in the estimation of option Greeks using the proposed method can achieve some economic value. TOPICS:Derivatives, options, fixed income and structured finance, quantitative methods, simulations Key Findings ▪ Like option pricing, the bias problem is still serious in estimating the Greeks of the options. ▪ The jackknife method is a good method to reduce the estimation bias in estimating the Greeks. ▪ The bias reduction in the estimation of option Greeks using the proposed method can achieve some economic values in option markets around the world.

A reduced-form model is proposed to disentangle the dynamics of positive-jump, diffusion, and long-term risks in VIX assets. A dissection of the three component risks in volatility is undertaken to determine their distinctive dynamics, risk prices, and market information. The dynamics of three component volatility risks play important and distinctive roles in the VIX derivatives market. The risk dissection substantially improves VIX option valuation. The component risks considerably explain VIX asset returns with a distinct exposure pattern and require separate risk premia. The risks are priced distinctively in the volatility market and are sensitive to different macroeconomic conditions. A comparison of the VIX derivatives market with the S&P 500 market reveals incremental information from the component risks beyond that provided by the traditional S&P 500 and its option market. TOPICS:Derivatives, options, factors, risk premia Key Findings ▪ A reduced-form model is used to disentangle the jump risk dynamic from the diffusion and long-term risks in the volatility market. The dynamic volatility jump risk is shown to be critical in the VIX derivatives market. ▪ The authors find unique properties in the different volatility risks. The three component risks are separately priced in the VIX market, the risk premia compose the total variance premium by different patterns, and they have unique responding for the macroeconomic conditions. ▪ They compare the pricing information in the S&P 500 and volatility markets and show that component risk dynamics in the VIX derivatives market contain incremental and critical pricing information beyond what is embedded in the S&P 500 and its option market.

Traditional plain vanilla options may be regarded as contingent claims whose value depends upon the simple returns of an underlying asset. These options have convex payoffs, and as a consequence of Jensen’s inequality, their prices increase as a function of maturity in the absence of interest rates. This results in long-dated call option premia being excessively expensive in relation to the fraction of a corresponding insured portfolio. We show that replacing the simple return payoff with the log return call option payoff leads to substantial premium savings while providing the similar insurance protection. Call options on log returns have favorable prices for very long maturities on the scale of decades. This property enables them to be attractive securities for long-term investors, such as pension funds. TOPICS:Options, pension funds Key Findings ▪ This article develops valuation and risk techniques for a log return payoff option under a Geometric Brownian Motion. ▪ A comparison is made between premium advantages of the log return contract to those of traditional European options. ▪ A pricing and optimal excise boundary formula for perpetual and finite maturity American log return options id derived. ▪ This article examines long-term insurance applications of the new contract that are prohibitively expensive for traditional options.

In this article, the authors discuss the use of quantum probability, that is, the probability theory of quantum mechanics, for option pricing and for finance in general. The authors discuss the motivations for applying quantum probability to finance. The critical issues are replacing random variables with operators, self-reflexivity of markets, and the existence of incompatible observations. The authors outline quantum probability theory, quantum stochastic processes, and the pricing of options in a quantum context. TOPICS:Options, portfolio theory, portfolio construction Key Findings • Quantum probability theory is a probabilistic theory of observations. Observations can change the system and be incompatible. • Quantum probability offers a more empirically faithful handling of large events and of uncertainty. • A better theory of valuation is offered by quantum probability theory than classical probability theory.

This article presents a discrete-time option pricing model that is rooted in reinforcement learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model can go model-free and learn to price and hedge an option directly from data, without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for the optimal pricing and hedging of options. Once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based reinforcement learning. Further, due to the simplicity and tractability of our model, which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relationship to the original BSM model, we suggest that our model could be used in the benchmarking of different RL algorithms for financial trading applications. TOPICS:Derivatives, options Key Findings • Reinforcement learning (RL) is the most natural way for pricing and hedging of options that relies directly on data and not on a specific model of asset pricing. • The discrete-time RL approach to option pricing generalizes classical continuous-time methods, enables tracking mis-hedging risk, which disappears in the formal continuous-time limit, and provides a consistent framework for using options for both hedging and speculation. • A simple quadratic reward function, which presents a minimal extension of the classical Black-Scholes framework when combined with the Q-learning method of RL, gives rise to a particularly simple computational scheme where option pricing and hedging are semi-analytical, as they amount to multiple uses of a conventional least-squares regression.

The QUAD method is a fast, flexible numerical pricing technique, widely applicable to many option types in its QUAD I and QUAD II versions where the underlying process has a closed-form density function or characteristic function. In its most advanced version, QUAD III, sacrificing only a little speed, it retains all the flexibility and applicability of earlier versions while covering an even greater range of underlying processes through use of approximations of the density functions. Here, we show how cases without suitable approximations can be handled by using finite difference methods for (only) that part of the calculation. We illustrate with the no arbitrage SABR model for the underlying. TOPICS:Derivatives, options Key Findings • Option pricing techniques under the umbrella term QUAD are the fastest generally applicable and flexible numerical methods we have for derivatives pricing. • Cases where there is no transition density function or characteristic function can be solved by using an approximation of the particular density function. However, in this article, an alternative approach is demonstrated, substituting finite difference calculations for the approximation. • The no-arbitrage SABR model is used as an example, since it is of special interest to practitioners.

Option pricing models traditionally have utilized continuous-time frameworks to derive solutions or Monte Carlo schemes to price the contingent claim. Typically these models were calibrated to discrete-time data using a variety of approaches. Recent work on GARCH based option pricing models have introduced a set of models that easily can be estimated via MLE or GMM directly from discrete time spot data. This paper provides a series of extensions to the standard discrete-time options pricing setup and then implements a set of various pricing approaches for a very large cross-section of equity and index options against the forward-looking traded market price of these options, out-of-sample. Our analysis provides two significant findings. First, we provide clear evidence that including autoregressive jumps in the options model is critical in determining the correct price of heavily out-of-the money and in-the-money options relatively close to maturity. Second, for longer maturity options, we show that the anticipated performance of the popular component GARCH models, which exhibit long persistence in volatility, does not materialize. We ascribe this result in part to the inherent instability of the numerical solution to the option price in the presence of component volatility. Taken together, our results suggest that when pricing options, the first best approach is to include jumps directly in the model, preferably using jumps calibrated from intraday data.

Mainstream option valuation theory relies implicitly on the assumption that latent states (such as stochastic volatility) and parameters are perfectly known, an assumption that is dubious in many ways. Computing the value of options under the assumption of perfect knowledge will typically introduce bias. Correcting for the bias is straightforward but can be computationally expensive. Fourier-based methods for computing option values are nowadays the preferred computational technique in the financial industry as a result of speed and accuracy. The author shows that the bias correction for parameter and state uncertainty for a large class of processes can be incorporated into the Fourier framework, resulting in substantial computational savings compared with Monte Carlo methods or deterministic quadrature rules previously used. In addition, the author proposes extensions, such as time varying parameters and hyperparameters, to the class of uncertainty models. The author finds that the proposed Fourier method is retaining all the good properties that are associated with Fourier methods; it is fast, accurate, and applicable to a wide range of models. Furthermore, the empirical performance of the corrected models is almost uniformly better than that of their noncorrected counterparts when evaluated on S&P 500 option data. TOPICS:Derivatives, options, factor-based models, analysis of individual factors/risk premia Key Findings • Parameter and state uncertainty in option models is often ignored but this leads to bias. • The bias correction introduced in this paper can be computed through the standard Fourier methodology, being fast and accurate. • The methodology results in better model in-sample and out-of-sample for a wide range of models, and the best results are found for parameters where the uncertainty is substantial.

This article proposes an improved estimation and calibration method to a family of GARCH models. The suggested method fixes one parameter such that the unconditional kurtosis of the model matches the sample kurtosis. An empirical analysis using Engle and Ng’s (1993) NGARCH(1,1) model shows that the method dominates previous estimation methods in multiple ways. The optimization problem is simplified and made less sensitive to initial values. The optimization time, both when estimating based on historical returns and calibrating to option prices, is reduced by roughly 50%. The in-sample fit is barely affected, while the option pricing, both in sample and out of sample, is improved. TOPICS:Statistical methods, quantitative methods, options, derivatives

The article proposes a semi-martingale approximation to a fractional Levy processes that is capable of capturing long and short memory in the stochastic process together with fat tails. We use the semi-martingale process in option pricing and empirically compare its performance to other option pricing models, including a stochastic volatility Levy process. We contribute to the empirical literature by being the first to report the implied Hurst index computed from observed option prices using the Levy process model. Calibrating the implied Hurst index of S&P 500 option prices in a period that covers the 2008 financial crisis, we find that the risk-neutral measure is characterized by a short memory in turbulent markets and a long memory in calm markets. TOPICS:Options, statistical methods, performance measurement

This article shows the fragility of the widely-used Stochastic Volatility Inspired (SVI) methodology in option pricing. The results highlight the sensitivity of SVI to the fitting penalty function. The authors compare different weight functions and propose to use the implied vega weights. They then unveil the relationship between vega weights and the minimization task of observed and fitted price differences, and show that implied vega weights can stabilize the SVI fit to illiquid options.

The most common approach in fitting option pricing models to market data is first to make an assumption about the underlying asset’s returns process and then develop an option pricing model for that process that is tested against market option prices. The returns process is estimated from historical data, option values are computed, and then compared against a cross-section of prices from the options market. Unfortunately, this often does not work well, and plainly it is inefficient in its use of the data. However, efforts to combine returns data from the asset market and prices from the options market into a single estimation have also not had much success. In this article, Chang, Cheng, and Fuh propose a new procedure to combine data from both markets in the estimation, in which options are assumed to be subject to random pricing noise relative to model values. The additional slack gives the estimator better ability to match prices in both markets. The article contrasts the performance of the full model approach with an approach that only uses stock prices or options prices to fit an option pricing model based on an underlying GARCH process. The value of the combined approach is demonstrated both theoretically as an asymptotic result in the model and also in a Monte Carlo simulation.

An option’s market price reflects the risk-neutral probability that it will end up in the money. Research has been increasing in recent years that shows how, given a set of market prices for options covering a range of strikes, an estimate of the entire risk-neutral probability distribution can be obtained. The technique is based on the fact that the second partial derivative of the option pricing function with respect to the strike price is the risk-neutral density (discounted from option expiration). This idea is generally attributed to Breeden and Litzenberger’s 1978 paper. In this article, Zimmermann shows that the connection between the second partial derivative of the option price with respect to the exercise price and risk-neutral probabilities has a much longer history, including a little-known 1974 note by Fischer Black, and going all the way back to Bachelier in 1900.

Derivatives pricing models based on arbitrage produce a single no-arbitrage option value. In theory, trading by arbitrageurs will force the market price to equal this value. However, the real world is full of frictions and uncertainty, and there is always a range in which the market price may wander without becoming sufficiently mispriced to touch off arbitrage trades. Under weaker assumptions, as explored in this article, such a range should encompass a convex set of prices such that the net payout has positive expected value under a broad set of supporting probability distributions and is closed under scaling, which makes it a mathematical cone. Madan and Schoutens use this principle to develop a theory of conic option pricing , in which an upper and lower bound on an option’s price are obtained by applying two deformations to the probability density for returns. Where the market price is expected to fall within these bounds is based on the principle that order flow for buying and selling should balance out on average. The resulting valuation does not require a specific density, such as lognormal (this article uses the variance-gamma process). Rather, it specifies how the density chosen by the user should be modified to produce the bounds. An empirical demonstration of the approach using seven years of options data on 208 underlying assets shows that market prices obey the pricing bounds quite well. An interesting and valuable extension is the valuation of delta and delta-gamma hedged positions, which naturally produce much tighter bounds.

Most numerical methods for option pricing (lattice methods, Monte Carlo simulation) focus on the market’s short-run dynamics, in which instantaneous returns cumulate to a probability distribution at maturity, as the Black–Scholes model does. If the risk-neutral density is available, it is possible to jump immediately to maturity, but this only works for options without path-dependence and for a small number of returns processes. The willow tree method offers a solution between these two cases. The option’s life is split into a small number of discrete chunks, and the price dimension at each of these time steps is represented by a finite and relatively small set of possible values. Each node allows a transition to every price node at the next step; thus, the “tree” resembles a willow, with many long branches connecting the sets of prices separated by discrete time intervals. For the problems it can handle, the willow tree approach has been shown to be accurate and much faster than procedures that require modeling an asymptotically infinite number of instantaneous transitions. In this article, Lu and Xu show how to extend a willow tree to two stochastic factors, an underlying stock price and the interest rate, to apply the model to pricing convertible bonds. The procedure is amenable to many different returns processes. In a numerical illustration, the authors show that the two-factor willow tree is as accurate as a binomial or a Monte Carlo approach but requires much less computing time.

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.

For many kinds of derivative valuation problems, especially those that try for greater realism using return processes that are more consistent with empirical evidence, Monte Carlo simulation is the only feasible solution technique. Among the well-known strategies to improve its efficiency, the use of a well-chosen control variate is often very effective. But a good selection can make a lot of difference. This article explains how the mathematical concept of comonotonicity can be applied as a new way to create a control variate. A remarkable improvement in performance can be achieved using the comonotonic upper bound as the control variate. The article illustrates the power of Comonotonic Monte Carlo simulation in estimating tail value at risk and pricing arithmetic Asian options.

Securities with payoffs determined by multiple correlated stochastic factors present some of the hardest valuation problems in derivatives. Except for rare cases in which a closed-form solution is available, the “curse of dimensionality” causes serious problems for Monte Carlo simulation and other numerical methods as the number of assets, N , grows. This article presents an approach that can simplify such problems enormously with relatively little lost in terms of accuracy for common cases. The idea is quite simple: Model each variable’s risk as being composed of exposure to a single common factor (e.g., the market portfolio) plus an independent idiosyncratic shock. With this assumption, the distribution of the payoff then becomes a realization of the single common factor, which can be easily simulated numerically, plus a draw from the composite density for the sum of the independent idiosyncratic shocks. The article illustrates the approach for pricing rainbow options and the n th-best-of- N contract. The result is highly accurate and almost instantaneous.

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.

Option theory designed to value puts and calls on traded financial instruments has been profitably extended in many directions, one of which is to “real options.” Real options refer to choices that can become economically valuable in the future and whose existence can be valued in the present using (one hopes) more or less standard option pricing techniques. Most examples involve investment projects and similar choices at the firm level. In this article, Afik looks at some real options that are offered to individuals, that is, car buyers. A customer buys a new car from a dealer, who commits to buy it back as a used car in the future at a pre-specified price. The exercise price may vary according to how old the car is when the customer exercises the option. The option’s moneyness is a function of prices in the used-car market. A second related optional contract allows the customer to swap the old car for a new one of the same type at a fixed price difference. The swap contract (actually an option to exchange one asset for another) is influenced by prices for both used and new cars on the exercise date. The author presents valuation models and a simulation to illustrate valuation of these consumer real options.

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.

A bothersome empirical finding in the options market microstructure literature is that in a small but significant percentage of the cases, market prices for a call and for its underlying stock move in opposite directions. For example, the stock price falls, but the call goes up, which is a violation of options’ monotonicity property. Using detailed intraday data on the Taiwan TAIEX index and its options, the authors find violations of monotonicity in about one third of the trades. They consider two possible factors that might push the call (put) price in a different (the same) direction as the stock: stochastic volatility, which introduces a second variable with its own dynamics into option pricing, and demand pressure from trades like covered calls, which can have opposite impacts on the stock and the option. The two factors together can explain all but 18% of the monotonicity violations in the data.

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 Black–Scholes (BS) model was a major conceptual breakthrough, and it established the theoretical framework within which valuation models for all kinds of derivatives are developed. But it was quickly found to be problematic as a practical tool for traders. The ubiquitous and anomalous “volatility smile” means that a set of options written on a single underlying asset are routinely priced in the market as if it had multiple different volatilities. In practice, traders replace the theoretical model with “practitioner Black–Scholes,” which is the BS equation but with a different volatility input for each option. That is, the market uses the BS equation , not the BS model . This inconsistency led Figlewski to ask: If the BS equation is used as practitioner Black–Scholes, in a way that is inconsistent with its theoretical foundation, how much better is it, if at all, than some other equation that satisfies the basic conditions necessary to avoid profitable static arbitrage (e.g., convex shape, call value increases with asset price, etc.) and has one free parameter (like implied volatility) that can be calibrated to the market, but has no pretense of economic content beyond that? For the S&P 500, it turned out that such a model-free model performed just about as well as the BS equation. Figlewski’s equation applied only to a single maturity. In this article, Orosi extends the equation to cover the full term structure of options, with an adjustment to prevent arbitrage, and compares it against several alternatives, including a variant of practitioner Black–Scholes. In sample and out of sample, the model-free model performs comparably to the best of the BS-based alternatives.

For no clear reason, many retail investors seem to have an unnatural taste for exotic structured products with payoffs that depend on complex and hard-to-value contingencies. Examples include barrier reverse convertibles, turbo certificates, and the object of this article, the bonus certificate. The payoff on a bonus certificate is tied to the level at expiration of an underlying variable, such as a stock index, and the path it has followed over its lifetime. At maturity, the return is the terminal value of the underlying, plus a bonus equal to the greater of a fixed exercise price less the terminal price, which is paid only if the underlying has not penetrated a prespecified barrier level during the contract’s lifetime. Valuing such a pathdependent instrument depends heavily on the nature of the return process. Baule and Tallau investigate the pricing and performance of bonus certificates in the German market, under several alternative models, with different assumptions about volatility behavior. The general conclusion, in rough terms, is that the issuers make an excess return while the buyers pay too much.

Black and Scholes assumed the returns for an option’s underlying stock followed a lognormal diffusion, which produces a lognormal probability density for the stock price at option expiration. This was mathematically very convenient but, unfortunately, not very satisfactory empirically. Observed return distributions for actual stocks, stock indexes, and most other kinds of assets exhibit distinctly non-lognormal shapes, with negative skewness and fat tails. Numerous non- Black–Scholes pricing models have been developed to try to capture these important features of real-world returns. Here, Markose and Alentorn propose the use of the generalized extreme value (GEV) distribution in place of the lognormal. Tail shape in the GEV distribution is governed by a single parameter, and it can be proven that the extreme tails of any plausible return density resemble those from a GEV. The authors develop a pricing model under GEV returns, whose form is very like Black–Scholes, along with equations for its Greek letter risks. They then demonstrate its superior ability to fit FTSE option prices from 1997 to 2009 and explore its behavior around critical market events, such as the 1997 Asian crisis, the 9/11 attack in 2001, and the collapse of Lehman Brothers in 2008.

One of the key ways in which asset returns in the real world depart from the lognormal diffusions assumed by Black and Scholes is that volatility has been found to be clearly time varying. Formal stochastic volatility models introduce a second Brownian motion variable to govern volatility changes, which leads to considerable additional complexity in solving the models and, importantly, gives up the dynamic market completeness that “no-arbitrage pricing” depends on. GARCH models occupy an intermediate position between models in which volatility is a fixed constant and those in which volatility is a second, unspanned, stochastic process, because the same Brownian motion determines both the returns and the evolution of volatility under GARCH. But current procedures for econometrically fitting a GARCH option pricing model are complex and computationally intensive. Heston and Nandi improved matters by developing a GARCH option pricing model whose characteristic function has a convenient form, but solving the model still requires considerable computation time. In this article, Mazzoni takes their approach further, starting from the cumulant-generating function and introducing a set of approximations that leads to analytic equations for an option’s value and its Greeks. The results are highly accurate, and the computation time is orders of magnitude faster than that of existing methods for pricing options under GARCH.

A model with a closed-form solution is the Holy Grail of derivatives valuation, because as computers have become increasingly powerful, exact answers to even very complicated formulas can typically be obtained almost instantaneously. Unfortunately, as derivative instruments have become increasingly complicated, closed-form valuation has become increasingly rare. Researchers are now trying to develop more efficient and accurate approximation techniques. Lattice models, such as the classic binomial model of Cox, Ross, and Rubinstein, are among the workhorses of this effort. Lattice models converge to accurate values as the number of node calculations increases, but convergence is often erratic and slow. The biggest problem is that the option payoff between two price nodes can be highly nonlinear, for example when the critical price barrier for a knock-out option falls between two layers of nodes, and a large jump in the computed option value occurs when a small change in the number of time steps causes the critical price to hop from one side of a node to the other. A variety of tricks have been proposed to deal with this problem by adapting the geometry of the lattice to make the nodes land directly on top of the critical prices. Generally this has required the additional flexibility afforded by a trinomial structure rather than the binomial, but that is costly in terms of efficiency. In this article, Dai and Lyuu introduce a new approach that achieves remarkable improvement in efficiency by combining binomial and trinomial structures. Here the trick is to construct a binomial lattice with nodes that land on top of the key prices, but to use a very small amount of trinomial lattice to connect the initial price—the model option value—to the binomial structure. This allows both the best placement of the tree relative to the critical areas and also great efficiency gains because binomial lattice probabilities for the terminal nodes can be computed directly using combinatorial results, skipping over calculations for all of the intermediate time steps.

Lattice models are workhorses of practical option pricing, especially for American options, but an important design criterion is that they need to be set up so that the interior branches recombine, otherwise, they become computationally intractable; the underlying state variable becomes path-dependent, the tree “splinters,” and the number of distinct nodes goes up exponentially rather than arithmetically as the number of time steps grows. Unfortunately, we have accumulated a great deal of evidence that volatility is time varying, which is a factor that splinters the tree. In this article, Beliaeva and Nawalkha show how to get around the problem by transforming the returns process to create two uncorrelated path-independent trees for returns and variance that can be put together into a single recombining two-dimensional lattice. The procedure is general; here they illustrate its use with the popular Heston model.

The lognormal diffusion process was the most convenient assumption Black and Scholes could make to capture the general features of stock price movements; it allows stochastic evolution of returns in continuous-time, and stock prices are bounded below by zero. But once we gained more empirical knowledge about returns distributions and observed the persistent volatility skew in real world option prices, alternative processes such as jump-diffusions were introduced. In this article, Câmara, Krehbiel, and Li note that, unlike an individual stock, a stock index will have a minimum value that is strictly positive, because a component stock whose price is going towards zero will be replaced in the index by a different stock. Thus a stock index should follow a displaced jump-diffusion. With this assumption, the authors derive an option pricing formula and test it on 10 years of S&P 500 Index and index option data. The model’s behavior with regard to implied jump intensity and frequency, the shape of the volatility skew, and the existence of a positive lower bound on the index are quite plausible and the goodness of fit is better than either Rubinstein’s displaced diffusion model (without jumps) or Merton’s jump-diffusion model (with lower bound of zero).

Stock options issued to employees has become a significant component of total compensation in many firms. Although option pricing technology has evolved remarkably over the years, valuing employee stock options (ESOs) still presents numerous challenges, and there is no model that is generally accepted in practice. Simply treating ESOs as if they were plain vanilla (American) stock options fails to take proper account of a number of their particular features. Along with vesting rules and the uncertainty they entail, these features include the fact that the equity of existing stockholders will be diluted when the ESOs are exercised. In that sense they resemble warrants rather than traded option contracts. Also, the fact that ESOs are issued on a regular basis means there are typically a number of outstanding ESOs with differing maturities and strike prices, which introduces interaction among them in valuation and in optimal exercise strategy. In this article, Dennis and Rendleman present a lattice technique to value ESOs that takes account of the effects of dilution and optimal exercise for a firm that has a diverse set of ESOs outstanding. Their approach increases computational efficiency substantially over the standard solution technique for a lattice model, but realistic problems still can become intractable as the number of time steps and options grows. Fortunately, the results indicate that in most cases, the effect of dilution and interaction among exercise decisions is present but relatively small.

The continuous-time framework for option pricing leads to the very desirable property that a continuous hedging strategy allows us to price options as if investors were risk neutral. But the real world doesn9t permit continuous rebalancing with no transactions costs, and discrete-time models cannot get away from risk preferences. However, it has been shown that risk neutral valuation still holds for certain specific combinations of risk preference and returns process. In this article, Vitiello and Poon substantially broaden the class of such models to returns distributions that can be expressed as a mixture of g distributions. The g distribution is obtained through a transformation of a Gaussian variable. The authors derive the pricing kernel that pairs with a given g distribution to produce risk neutral valuation, and then derive option valuation models for calls and puts. As special cases, their formulas produce the known results for lognormal and normal returns. Extending the model further, they obtain pricing formulae for mixtures of two g distributions. Finally, they illustrate the use of the new models to value options on the S&P 500 index and on Heating Degree Days (HDD). One of the key virtues of this approach is its ability to price options on underlyings like HDD, that are not susceptible to delta hedging because they are illiquid or not traded at all.

Volatility of the underlying asset is a critical input to modern option pricing models. How to get the best volatility estimate is, therefore, a matter of great importance to options traders. One technique that has been known for a long time, although it is less commonly used, involves not just daily closing prices, but the daily high and low. This actually introduces quite a bit more information into the calculation, since the high and low are functions of the entire intraday price path for the underlying, not just the price at a single point in time. In contrast, models that allow volatility to vary over time, notably GARCH, have become popular as a way to capture time-variation in the volatility process. This article introduces a GARCH-type Multiplicative Error Model (MEM) for the daily price range. Like other GARCH specifications, the MEM can be modified to accommodate different densities for the stochastic shocks, different orders of dependence on past shocks and volatilities, and even fractional integration to produce a long-memory process. The authors fit the model to the daily price ranges for the CHF/USD and USD/GBP exchange rates and show it to be quite effective in producing accurate (and profitable) volatility predictions.

More than 40 years of playing around with the Black-Scholes equation has produced a large number of insights about its properties. Some are just cute little mathematical tidbits; others may have considerable value for practitioners, even if they amount to no more than useful rules-of-thumb or computational tricks; and some actually reflect deep mathematical properties of the Black-Scholes framework. In this article, Poulsen discusses several such neat little results about the option pricing model.

Binomial, trinomial, and other lattice-based option pricing procedures are extremely useful in dealing with early exercise because solution by backward induction is possible. As one rolls back through the tree, the option value at each node depends on the current price for the underlying and the paths that might be followed going forward. But for many kinds of derivatives, the value at a given point in time depends on both the current asset price and the price path that it has followed up to the current point. Derivatives on interest rates in the Heath-Jarrow-Morton (HJM) model are one example. Another is option pricing when the underlying volatility varies over time according to a GARCH model. The most successful technique for adapting a lattice model to a path-dependent process has been to carry along a set of auxiliary variables that summarize (approximately) the necessary path information at each node. These techniques work, but they are typically very time consuming. In this article, Lin and Ritchken present a way to greatly increase the efficiency of this procedure. Some auxiliary path information must still be carried through the tree, but much less is needed. They demonstrate the sharp increase in performance this makes possible in an HJM application and for an underlying that follows a GARCH process.

One of the most basic properties of rational option pricing is that a rise in the price of the underlying increases the value of a call and decreases the value of a put. But results reported in the literature have found that for options on the S&P 500 Index this principle is violated in practice on the order of 10 percent of the time. Should we interpret this as evidence that the options market often ignores a fundamental principle of option valuation? Or that new option models are needed that would allow such seemingly anomalous behavior? In this article, Perignon helps to resolve this issue using transactions-level data from five major index-options markets. He finds that wrong-way option price changes do occur in all of them a substantial fraction of the time. But looking for the explanation behind the numbers, he shows that market microstructure effects, like bid-ask bounce and rational tactics for trading in a market with a wide bid-ask spread, appear to account for much of the phenomenon.

Understanding the stochastic behavior of credit spreads and devising models for derivatives with credit spreads as underliers grows increasingly important every day. Mean reversion in spreads is clearly evident in the data, as is time-varying volatility. In this paper, Tahani models credit spreads using a mean-reverting GARCH framework. Adopting Heston and Nandi9s GARCH specification allows closed-form option valuation formulas to be obtained by inversion of the characteristic function. Moreover, Tahani9s model contains the Longstaff-Schwartz spread model as a special case. The model is then taken to the data, by fitting it to the credit spreads between Moody9s Aaa and Baa bond yields and U.S. Treasuries. The GARCH specification fits that data well, and the GARCH option model calibrated to the bond yields exhibits the same kinds of unusual behavior (e.g., option prices below intrinsic values, in some cases) as Longstaff and Schwartz found with their more restricted model.

Asian options payoffs are based on the average price of the underlying over some time period prior to and including the expiration date. This induces path–dependency, which presents problems for standard option pricing models. One basic principle, though, is that viewed from an initial date t, the standard deviation of the asset price at a future option expiration date T will be greater than the standard deviation of the average price over the period leading up to T. Thus, other things equal, an Asian option should be worth less than the equivalent European option. However, in this article, Ye shows that, in general, other things won9t be equal. In particular, the expected value of the average price will also be systematically different from that of the terminal price, as a function of the riskless rate and the dividend yield. Although the volatility will be lower for the Asian option, the pricing relationship can easily be reversed by the difference in expected values. Asian options can, indeed, be worth more than their equivalent plain vanilla counterparts.

Much has been written about how executive stock options should be valued. The options themselves are not that complicated, but they have a variety of special features that make valuing them difficult, like vesting rules and restrictions on selling them in the market. Forcing the option holder to keep a concentrated position in the firm9s equity and options, rather than diversifying into a broader portfolio, lowers the value of the options to the recipient. That, in turn, reduces the performance incentive that option grants are meant to produce, and raises the effective cost to the firm of providing that incentive. The problem is mitigated in many executive stock option valuation models, because they assume that the executive allocates his wealth portfolio between options and/or restricted firm stock, and a risk–free asset. Since there are no other equity instruments in the model, there is an extra demand for the options because they are the way the investor gains access to the equity market. In this article, Cai and Vijh broaden the range of available assets to include the ability to invest outside wealth in the market portfolio.

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.

Numerical option pricing methods discretize a continuous-time continuous-state state space and approximate the solution to the continuous problem with the results of local computations on all of the nodes of the discrete lattice. This procedure converges to the correct solution as the time and price intervals go to zero, but at the cost of larger and larger numbers of calculations. Many of the calculations on a fine lattice are irrelevant to valuing the derivative. They may occur at stock prices that have virtually no probability of being reached. Or the stock price may be reachable from the initial price, but be so far away from the strike price that the option is either (almost) certain to be out of the money at maturity, meaning it is worthless now, or (almost) certain to be exercised at maturity, meaning it can be priced as a forward now, with no need for any further calculations. Eliminating the superfluous calculations at these nodes can greatly improve the performance of a lattice valuation technique, and the possible increase in computation speed becomes much greater as the number of stochastic variables in the lattice increases. Here, Andricopoulos et al., show how to curtail the price range for a numerical valuation technique and demonstrate the substantial improvement in speed that it can produce.

Anyone using an option valuation model needs estimates of returns volatility. But volatility has been found to be hard to forecast accurately, partly because it varies randomly over time. Fortunately, the market provides a directly observable volatility estimate in the form of the implied volatility that can be backed out from an option’s market price. But for several increasingly popular types of options, the payoff distribution depends both on stock volatilities and also on the correlations among them. For such options, including basket options and options on the max or on the min, there are no dependable sources of implied correlations to use in pricing them. Correlations must be estimated from historical data, which leads to substantial estimation risk. In this article, Fengler and Schwendner describe a block bootstrap procedure that allows an investor to evaluate a multiasset option’s exposure to parameter risk from imperfectly estimated correlations. The results are translated into minimum bid-ask spreads that are required to account for this additional source of risk.

The original Black-Scholes (BS) model introduced financial economists to continuous-time mathematics. But evidence quickly accumulated that the BS constant volatility lognormal diffusion is not adequate to describe the behavior of real world stock prices or, especially, interest rates. Recent work by Duffie and Singleton, and others, extended the returns processes for which options could be valued in more or less closed form to the affine jump-diffusion (AJD) class, which allows an arbitrary number of stochastic factors and a broad range of possible behavior. And yet, such models still impose significant limits on the ways that volatility can be made stochastic. This article shows how to break out of the AJD class to a much broader range of price processes. Chourdakis’s approach is to construct a continuous-time Markov chain that can approximate nearly any plausible state process. Moreover, beyond allowing option pricing for a richer set of price processes, the technique also offers improved computational efficiency for AJD-class models, relative to alternative techniques.

Electricity is one of the most important commodities for both consumers and producers, and it is subject to sharp variations in supply and demand, sometimes within a single day. But it is also one of the most difficult commodities on which to trade derivatives, because electricity is virtually unstorable. This has given rise to contracts such as the “swing option,” which sets minimum and maximum values for both the amount of power to be purchased in each period and also the cumulative amount of electricity purchased over the life of the contract. A swing option thus entails complex timing options with regard to fulfillment of the required minimum cumulative quantity purchased, and price-related options that let the holder buy more when the market price is above the option9s strike price, all constrained by the limits on maximum instantaneous and cumulative consumption. In this article, Keppo describes the decision process and presents a swing option valuation model based on simple electricity forwards and options. The use of the approach is illustrated with a couple of examples based on Nord Pool electricity prices.

Grants of stock options as part of total employee compensation are now commonplace, but how such options should be valued and reported in firm financial statements is still an unsettled issue. The Black-Scholes (BS) model is the best-known and most widely accepted approach to option valuation as a benchmark for accounting and legal purposes, so it is seems like a natural place to begin. However, employee stock options (ESOs) are subject to several additional contingencies that are not in the BS equation, but significantly alter the possible payoffs. Specifically, severance with or without cause, including by death of the option holder, typically alters the payoff on an ESO. For example, termination with cause may also entail forfeiture of the employee’s options; termination without cause may simply advance the option’s maturity date and require the departing employee to exercise immediately, or not at all. In this article, Pandher presents a risk-neutral valuation approach for pricing ESOs under a realistic set of severance possibilities and examines the impact on valuation and on expected exercise dates. He shows that proper treatment of these additional features can make a substantial difference in the ESO value relative to Black-Scholes.

Applying the technology of option pricing and contingent claims modeling to credit risk is one of the major growth areas in derivatives research these days. It is an old idea to treat default as a firm9s management rationally exercising the shareholders9 option to go bankrupt rather than to make a required payment to the debtholders. But only in the last few years has this insight been extended and widely applied to practical bond valuation problems. Convertible bonds have also become much more popular in recent years, as the weak and volatile stock market has combined with low interest rates to create an environment in which the conversion option can be very attractive. One critical uncertainty in credit risk analysis that may not be fully appreciated by those who have not worked in this area, is what the payoff to bondholders will actually be in the event of a default. In this article, Ayache, Forsyth, and Vetzal explore the interactions among the multiple options embedded in convertibles that are subject to default risk, call provisions, and put provisions, in addition to the conversion option. Along with results that illustrate the interactions among these multiple options, the authors provide a straightforward valuation technique for convertibles, by applying principles of linear complementarity to the discretization of the fundamental PDE of contingent claims pricing.

Barrier options have become commonplace in the option market, and a variety of other financial contracts may also be thought of in terms of barrier options. But the existence of a price barrier can significantly complicate the option valuation problem when volatility is time-varying, or the barrier itself moves over time, or the barrier is only monitored at discrete intervals. In this article, Duan et al. present a new Markov chain technology for pricing barrier options that readily handles all of these problems. Out-and-in options can be valued within their framework even when volatility follows a GARCH process and a discretely monitored time-varying barrier is present.

Ederington and Guan’s article calls into question the informational efficiency of option pricing in the real world, by showing that easily constructed positions earn higher returns than they should. Whaley’s article adds evidence of a somewhat different kind to this conclusion. The Chicago Board Options Exchange has recently created an index to measure the returns to a basic covered call strategy that is long the S&P 500 index portfolio (SPX) and short an (approximately) at-the-money one-month SPX call option. In this article, Whaley describes the new index and then uses it to examine the returns to a basic covered call strategy from 1988 to 2001. Measuring the risk of a covered call presents some issues, because it is highly non-normal. Dealing with these, Whaley shows that under several standard measures of investment performance, the covered call strategy has been unusually successful, earning almost as much as the S&P index, with substantially lower risk.