Implied Binomial Trees: A Literature Review
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paper_title: Option Pricing: A Simplified Approach

paper_content:

This paper presents a simple discrete-time model for valuing options. The fundamental economic principles of option pricing by arbitrage methods are particularly clear in this setting. Its development requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-Scholes model, which has previously been derived only by much more difficult methods. The basic model readily lends itself to generalization in many ways. Moreover, by its very construction, it gives rise to a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.
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paper_title: Implied exchange rate distributions: evidence from OTC option markets

paper_content:

This paper uses a rich new data set of option prices on the dollar-mark, dollar-yen, and key EMS cross-rates to extract the entire risk-neutral probability density function (pdf) over horizons of one and three months. We compare three alternative smoothing methods---cubic splines, an implied binomial tree (trimmed and untrimmed), and a mixture of lognormals---for transforming option data into the pdf. Despite their methodological ifferences, the three approaches lead to a similar pdf distinct from the lognormal benchmark, and usually characterized by skewness and leptokurtosis. We then document a striking positive correlation between skewness in these pdfs and the spot rate. The stronger a currency the more expectations are skewed towards a further appreciation of that currency. We interpret this finding as a rejection that these exchange rates evolve as a martingale, or that they follow a credible target zone, explicit or implicit. Instead, this this positive correlation is consistent with target zones with endogenous realignment risk. We discuss two interpretations of our results on skewness: when a currency is stronger, the actual probability of further large appreciation is higher, or because of risk, such states are valued more highly.
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paper_title: Brownian Motion in the Stock Market

paper_content:

It is shown that common-stock prices, and the value of money can be regarded as an ensemble of decisions in statistical equilibrium, with properties quite analogous to an ensemble of particles in statistical mechanics. If Y = loge[P(t + r)/P0(t)], where P(t + r) and P0(t) are the price of the same random choice stock at random times t + r and t, then the steady state distribution function of Y is \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\varphi(Y)=\exp(-Y^{2}/2\sigma^{2}\tau)/\sqrt{2\pi \sigma^{2}\tau}$ \end{document}, which is precisely the probability distribution for a particle in Browman motion, if σ is the dispersion developed at the end of unit time. A similar distribution holds for the value of money, measured approximately by stoc...
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paper_title: Implied Binomial Trees.

paper_content:

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions. Copyright 1994 by American Finance Association.
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paper_title: Options on Leveraged Equity: Theory and Empirical Tests

paper_content:

We develop an option pricing model for calls and puts written on leveraged equity in an economy with corporate taxes and bankruptcy costs. The model explains implied Black-Scholes volatility biases by relating them to the firm's structural characteristics such as leverage and debt covenants. We test the model by comparing predicted pricing biases with biases observed in a large cross-section of firms with liquid exchange traded option contracts. Our empirical study detects leverage related pricing biases. The magnitudes of these biases correspond to those predicted by our model. We also find significant pricing biases for firms financed primarily by short-term debt. This supports our model because short-term debt introduces net-worth hurdles similar to net-worth covenants.
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paper_title: Implied Risk-neutral Probability Density Functions From Option Prices: Theory and Application

paper_content:

Due to their forward-looking nature, derivative markets provide monetary authorities with a rich source of information for gauging market sentiment. For example, a futures price gives a widely used measure of the market's views about the future value of an asset, namely its mean or expected value at the maturity date of the futures contract. Moreover, the information available from futures prices can be extended by using option prices to estimate the market's entire probability distribution of the future value of an asset. This paper develops various techniques for estimating the market's probability distribution of the future value of an underlying asset from the prices of options on that asset. It discusses the relative merits and drawbacks of each approach, and shows how our preferred approach can be applied to estimate ex ante probability distributions using LIFFE equity and interest rate options, and Philadelphia Stock Exchange currency options. The paper then illustrates the potential value of this type of information to the policy-maker in assessing monetary conditions and conducting monetary operations. Finally, the paper looks at the limitations in data availability and highlights some areas for future research.
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paper_title: Beyond implied volatility

paper_content:

After a brief review of option pricing theory, we introduce various methods proposed for extracting the statistical information implicit in options prices. We discuss the advantages and drawbacks of each method, the interpretation of their results in economic terms, their theoretical consequences and their relevance for applications.
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paper_title: Gain, Loss and Asset Pricing

paper_content:

We develop an approach to asset pricing in incomplete markets that bridges the gap between the two fundamental approaches in finance: model-based asset pricing and pricing by no arbitrage. We strengthen the absence of arbitrage assumption by precluding investment opportunities whose attractiveness to a benchmark investor exceeds a specified threshold. In our framework, the attractiveness of an investment opportunity is measured by the gain-loss ratio. We show that a restriction on the maximum gain-loss ratio is equivalent to a restriction on the ratio of the maximum to minimum values of the pricing kernel. By limiting the maximum gain-loss ratio, we can restrict the set of admissible pricing kernels, which in turn allows us to restrict the set of prices that can be assigned to assets. We illustrate our methodology by computing price bounds for call options in a Black-Scholes economy without intermediate trading. When we vary the maximum permitted gain-loss ratio, these bounds can range from the exact prices implied by a model-based pricing approach to the loose price bounds implied by the no-arbitrage approach.
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paper_title: Prices for State-contingent Claims: Some Estimates and Applications

paper_content:

In this paper, we present an approach to investment decisions under uncertainty that we hope will interest both business managers seeking to improve the efficiency of internal capital budgeting procedures and economists seeking insights to the behavior of aggregate business investment. Twenty years ago, a paper with such objectives would have been squarely within the mainstream of research in the field of finance. For a variety of reasons, however, the attention given to problems of applied capital budgeting in the serious literature has fallen substantially in recent years. In part, this dwindling of interest reflects the emergence in the middle 1960s of exciting new topic areas with great research potential, such as the capital asset pricing model (hereafter CAPM) of Sharpe-Lintner, the efficient markets doctrine, and, more recently, the theory of rational option pricing. But, in part, it reflects the judgment of the profession that the benefit/cost ratio of further research on applied capital budgeting was turning unfavorable. Such was particularly the case for research focusing on the "cost of capiThis paper presents a method for making risk adjustments in practical capital budgeting applications that has advantages over the currently available alternatives. The method relies on estimates of the "state prices" implicit in the market portfolio to compute the net present value of the expected payoffs of a proposed project. Tables of such state prices are presented, obtained by applying the BlackScholes option pricing formula to the Ibbotson-Sinquefield predictive distribution of real returns. The paper concludes with some cautions and suggestions for further research.
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paper_title: Options and Efficiency

paper_content:

This paper argues that in an uncertain world options written on existing assets can improve efficiency by permitting an expansion of the contingencies that are covered by the market. The two major results obtained are, first, that complex contracts can be "built up" as portfolios of simple options and, second, that there exists a single portfolio of the assets, the efficient fund, on which all options can be written with no loss of efficiency.
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paper_title: A Jumping Smile

paper_content:

This paper examines systematically the consequences of jump processes in the returns of the underlying asset, for the existence of the volatility smile. The key insight in this paper is the theoretical (but little-noticed) result that the presence of jumps brings major qualitative changes to option pricing. Specifically, when there are jumps it is not possible to have a single option price derived by arbitrage methods alone even at the limit of continuous trading. In a universe of investors who are risk-averse but otherwise unspecified the absence of arbitrage possibilities generates two bounds on option prices, within which the "true" option price lies. If, as is highly plausible, each observed option price at any given time is drawn from a different cohort of risk-averse investors then there is no reason for the existence of a single distribution able to price all options. The method developed in this paper is thus suited to the pricing of contingent claims when the smile is due to the presence of jumps. Two boundary risk-neutral distributions are identified, bracketing the observed option values. These disttributions are the tightest possible distributions supportable by our data, in the sense of producing the minimum spread between possible option values. In our sample the economic difference between these two distributions in the case of S&P 500 options is minor, in the sense that the percentage error in using one rather than the other is small for the key portion of the range of exercise prices. A major advantage of our approach is that it produces results almost always. In a sample of 1991 simultaneous observations on both options and the underlying index value spread over the years 1991, 1993, and 1996 our method produced results in every single instance. By contrast, a single risk-neutral distribution satisfying all observed option values did not exist for 41% of our observations. This failure of the data to support a single option-pricing distribution cannot be explained by models based on systematically-varying volatility of the underlying asset, or by transactions costs. By contrast, it is fully consistent with the existence of jumps in the price of the underlying asset. Further, while a detailed screening of the data may eliminate some of these failures as data errors or mispricings, the fact remains that such eliminations can only take place ex post, thus reducing the usefulness of the single distribution approaches for option practitioners.
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paper_title: Financial Markets as Adaptive Systems

paper_content:

We show, by studying in detail the market prices of options on liquid markets, that the market has empirically corrected the simple, but inadequate Black-Scholes formula to account for two important statistical features of asset fluctuations: `fat tails' and correlations in the scale of fluctuations. These aspects, although not included in the pricing models, are very precisely reflected in the price fixed by the market as a whole. Financial markets thus behave as rather efficient adaptive systems.
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paper_title: Valuing Exotic Options by Approximating the SPD with Higher Moments

paper_content:

The financial economic No Arbitrage assumption implies that in a complete market the price of any derivative security is the discounted value of its payoff function integrated against the appropriate state-price density (SPD). For most exotic path dependent payoffs, it is quite difficult to obtain a closed form expression for the state-price density, thus eliminating the prized possibility of an analytic expression for the option in question. In this article we value exotic options by matching moments to an approximating SPD from a very general, robust, and convenient class of density functions, known in the statistical literature as the Johnson (1949) family. The end result is an analytic expression for a wide variety of European style exotic options. Our formula is tested on Asian and Basket options and is found to be extremely robust when compared with Monte Carlo simulations and other numerical techniques.
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paper_title: Edgeworth Binomial Trees

paper_content:

This paper develops a simple technique for valuing European and American derivatives with underlying asset risk-neutral returns which depart from lognormal in terms of prespecified non-zero skewness and greater-than-three kurtosis. Instead of specifying the entire risk-neutral distribution by the riskless return and volatility (as in the Black-Scholes case), this distribution is specified by its third and fourth central moments as well. An Edgeworth expansion is used to transform a standard binomial density into a unimodal standardized discrete density -- evaluated at equally-spaced points -- with approximately the prespecified skewness and kurtosis. This density is in turn adjusted to have a mean equal to the riskless return (adjusted for the payout return, if any) and to a prespecified volatility. European derivatives are then easily valued by using this risk-neutral density to weight their possible payoffs.
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paper_title: No-Arbitrage Option Pricing: New Evidence on the Validity of the Martingale Property

paper_content:

The no-arbitrage approach to option pricing implies that risk-neutral prices follow a martingale. The validity of this property has been tested and rejected by Longstaff (1995). Since he tested the general framework, his results have far reaching and disturbing implications for contingent claims pricing. This paper proposes a new method to test the martingale property. This method is based on the Laguerre polynomial series. The tests use options and futures on the S&P 500 index. The new methodology and data show that the martingale property cannot be rejected. This result implies that the general approach is still valid and the existence of frictions only adds noise. Testing more specific pricing models is relevant again.
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paper_title: IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES

paper_content:

The Black-Scholes (1973) option pricing model is used to value a wide range of option contracts. However, the model often inconsistently prices deep in-themoney and deep out-of-the-money options. Options’ professionals refer to this phenomenon as a volatility ‘skew’ or ‘smile.’ In this paper, we apply an extension of the Black-Scholes model developed by Jarrow and Rudd (1982) to an investigation of S&P 500 index option prices. We find that non-normal skewness and kurtosis in option-implied distributions of index returns contribute significantly to the phenomenon of volatility skews.
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paper_title: CALL OPTION VALUATION FOR DISCRETE NORMAL MIXTURES

paper_content:

In this study a mixture call option pricing model is derived to examine the impact of non-normal underlying returns densities. Observed fat-tailed and skewed distributions are assumed to be the results of independent Gaussian processes with nonstationary parameters, modeled by discrete k-component independent normal mixtures. The mixture model provides an exact solution with intuitive appeal using weighted sums of Black-Scholes (B-S) solutions. Simulating returns densities representative of equity securities, significant mispricing by B-S is found in low-priced at- and out-of-the-money near-term options. the lower the variance and the higher the leptokurtosis and positive skewness of the underlying returns, the more pronounced is this mispricing. Values of in-the-money options and options with several weeks or more to expiration are closely approximated by B-S.
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paper_title: Estimating Gram-charlier Expansions Under Positivity Constraints

paper_content:

The Gram-Charlier expansion, where skewness and kurtosi directly appear as parameters, has become popular in Finance as a generalization of the normal density. We show how positivity constraints can be numerically implemented, thereby guaranteeing that the expansion defines a density. The constrained expansion can be referred to as a Gram-Charlier density. First, we apply our method to the estimation of risk neutral densities. Then, we assess the statistical properties of maximum-likelihood estimates of Gram-Charlier densities. Lastly, we apply the framework to the estimation of a GARCH model where the conditional density is a Gram-Charlier density.
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paper_title: SKEWNESS AND KURTOSIS IN S&P 500 INDEX RETURNS IMPLIED BY OPTION PRICES

paper_content:

The Black-Scholes (1973) model frequently misprices deep-in-the-money and deep-out-of-the-money options. Practitioners popularly refer to these strike price biases as volatility smiles. In this paper we examine a method to extend the Black-Scholes model to account for biases induced by nonnormal skewness and kurtosis in stock return distributions. The method adapts a Gram-Charlier series expansion of the normal density function to provide skewness and kurtosis adjustment terms for the Black-Scholes formula. Using this method, we estimate option-implied coefficients of skewness and kurtosis in S&P 500 stock index returns. We find significant nonnormal skewness and kurtosis implied by option prices.
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paper_title: Constructing Binomial Trees From Multiple Implied Probability Distributions

paper_content:

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.
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paper_title: Recovering Probability Distributions from Option Prices

paper_content:

This article derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities that minimize an objective function subject to requiring that the probabilities are consistent with observed option and underlying asset prices. Alternative optimization specifications produce approximately the same implied distributions. A new and fast optimization technique for estimating probability distributions based on maximizing the smoothness of the resulting distribution is proposed. Since the crash, the risk-neutral probability of a three (four) standard deviation decline in the index (about -36 percent (-46 percent) over a year) is about 10 (100) times more likely than under the assumption of lognormality. Copyright 1996 by American Finance Association.
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paper_title: Option Hedging Using Empirical Pricing Kernels

paper_content:

This paper develops a method for option hedging which is consistent with time-varying preferences and probabilities. The preferences are expressed in the form of an empirical pricing kernel (EPK), which measures the state price per unit probability, while probabilities are derived from an estimated stochastic volatility model of the form GARCH components with leverage. State prices are estimated using the flexible risk-neutral density method of Rosenberg (1995) and a daily cross-section of option premia. Time-varying preferences over states are linked to a dynamic model of the underlying price to obtain a one-day ahead forecast of derivative price distributions and minimum variance hedge ratios. Empirical results suggest that risk aversion over S&P500 return states is substantially higher than risk aversion implied by Black-Scholes state prices and probabilities using long run estimates of S&P500 return moments. It is also found that the daily level of risk aversion is strongly positively autocorrelated, negatively correlated with S&P500 price changes,and positively correlated with the spread between implied and objective volatilities. Hedging results reveal that typical hedging techniques for out-of-the-money S&P500 index options, such as Black-Scholes or historical minimum variance hedging, are inferior to the EPK hedging method. Thus, time-varying preferences and probabilities appear to be an important factor in the day-to-day pricing of S&P500 options.
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paper_title: A Simple Nonparametric Approach to Derivative Security Valuation

paper_content:

Canonical valuation uses historical time series to predict the probability distribution of the discounted value of primary assets' discounted prices plus accumulated dividends at any future date. Then the axiomatically rationalized maximum entropy principle is used to estimate risk-neutral (equivalent martingale) probabilities that correctly price the primary assets, as well as any predesignated subset of derivative securities whose payoffs occur at this date. Valuation of other derivative securities proceeds by calculation of its discounted, risk-neutral expected value. Both simulation and empirical evidence suggest that canonical valuation has merit. Copyright 1996 by American Finance Association.
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paper_title: Pricing Multivariate Contingent Claims Using Estimated Risk-Neutral Density Functions

paper_content:

Many asset price series exhibit time-varying volatility, jumps, and other features inconsistent with assumptions about the underlying price process made by standard multivariate contingent claims (MVCC) pricing models. This paper develops an interpolative technique for pricing MVCCs ' flexible NLS pricing ' that involves the estimation of a flexible multivariate risk-neutral density function implied by existing asset prices. As an application, the flexible NLS pricing technique is used to value several bivariate contingent claims dependent on foreign exchange rates in 1993 and 1994. The bivariate flexible risk-neutral density function more accurately prices existing options than the bivariate lognormal density implied by a multivariate geometric brownian motion. In addition, the bivariate contingent claims analyzed have substantially different prices using the two density functions suggesting flexible NLS pricing may improve accuracy over standard methods.
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paper_title: Nonparametric Density Estimation and Tests of Continuous Time Interest Rate Models

paper_content:

Nonparametric kernel density estimation has recently been used to estimate and test short-term interest rate models, but inference has been based on asymptotics. We derive finite sample properties of kernel density estimates of the ergodic distribution of the short-rate when it follows a continuous time AR(1) as in Vasicek. We find that the asymptotic distribution substantially understates finite sample bias, variance, and correlation. Also, estimator quality and bandwidth choice depend strongly on the persistence of the interest rate process and on the span of the data, but not on sampling frequency. We also examine the size and power of one of Ait-Sahalia's nonparametric tests of continuous time interest rate models. The test rejects too often. This is probably because the quality of the nonparametric density estimate depends on persistence, but the asymptotic distribution of the test does not. After critical values are adjusted for size, the test has low power in distinguishing between the Vasicek and Cox-Ingersoll-Ross models relative to a conditional moment-based specification test.
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paper_title: Implied exchange rate distributions: evidence from OTC option markets

paper_content:

This paper uses a rich new data set of option prices on the dollar-mark, dollar-yen, and key EMS cross-rates to extract the entire risk-neutral probability density function (pdf) over horizons of one and three months. We compare three alternative smoothing methods---cubic splines, an implied binomial tree (trimmed and untrimmed), and a mixture of lognormals---for transforming option data into the pdf. Despite their methodological ifferences, the three approaches lead to a similar pdf distinct from the lognormal benchmark, and usually characterized by skewness and leptokurtosis. We then document a striking positive correlation between skewness in these pdfs and the spot rate. The stronger a currency the more expectations are skewed towards a further appreciation of that currency. We interpret this finding as a rejection that these exchange rates evolve as a martingale, or that they follow a credible target zone, explicit or implicit. Instead, this this positive correlation is consistent with target zones with endogenous realignment risk. We discuss two interpretations of our results on skewness: when a currency is stronger, the actual probability of further large appreciation is higher, or because of risk, such states are valued more highly.
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paper_title: Implied Binomial Trees.

paper_content:

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions. Copyright 1994 by American Finance Association.
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paper_title: Reading Interest Rate and Bond Futures Options ' Smiles : How PIBOR and Notional Operators Appreciated the 1997 French Snap Election ¤

paper_content:

The aim of this paper is to compare various methods which extract a Risk Neutral Density (RND) out of PIBOR as well as of Notional interest rate futures options and to investigate how traders reacted to a political event. We first focus on 5 dates surrounding the 1997 snap election and several methods: Black (1976), a mixture of lognormals (as in Melick and Thomas, 1997), an Hermite expansion (as in Abken, Madan, and Ramamurtie, 1996), and a method based on Maximum Entropy (following Kelly and Buchen, 1996). By and large the various methods give similar RNDs. Yet, the Hermite expansion approach, by allowing for somewhat dirty options prices, by providing a good fit to options prices, and by being very fast is the retained method for the data at hand. We then consider a daily panel of options running from February 1997 to July 1997. After constructing standardized options, i.e. with a fixed time to maturity, we find that operators in both markets anticipated the snap election a few days before the official announcement and that a substantial amount of political uncertainty subsisted even a month after the elections. The greater liquidity of PIBOR options eases information extraction.
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paper_title: The Maximum Entropy Distribution of an Asset Inferred from Option Prices

paper_content:

This paper describes the application of the Principle of Maximum Entropy to the estimation of the distribution of an underlying asset from a set of option prices. The resulting distribution is least committal with respect to unknown or missing information and is, hence, the least prejudiced. The maximum entropy distribution is the only information about the asset that can be inferred from the price data alone. An extension to the Principle of Minimum Cross-Entropy allows the inclusion of prior knowledge of the asset distribution. We show that the maximum entropy distribution is able to accurately fit a known density, given simulated option prices at different strikes.
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paper_title: An Options-Based Analysis of Emerging Market Exchange Rate Expectations: Brazil's Real Plan, 1994-1997

paper_content:

This paper uses currency option data from the BMF, the Commodities and Futures exchange in Sao Paulo, Brazil, to investigate market expectations on the Brazilian Real-U.S. dollar exchange rate from October 1994 through July 1997. Using options data, we derive implied probability density functions (PDF) for expected future exchange rates and thus measures of the credibility of the crawling peg' and target zone ( maxiband') regimes governing the exchange rate. Since we do not impose an exchange rate model, our analysis is based on either the risk-neutral PDF or arbitrage-based tests of target zones. The paper one of the first to use options data from an emerging market, finds that target zone credibility was poor prior to February 1996, but improved afterwards. The market anticipated periodic band adjustments, but over time developed greater confidence in the Real. We also test whether devaluation intensities estimated from these option prices can be explained by standard macroeconomic factors.
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paper_title: Arbitrage-Based Tests of Target Zone Credibility: Evidence From ERM Cross-Rate Options

paper_content:

This paper introduces two arbitrage-based tests of target-zone credibility using a new data source, exchange-rate mechanism cross-rate options. Using daily option prices from September 1991 to August 1994, the authors assess the credibility of the pound-mark and mark-lira target zones that collapsed in September 1992 and the ongoing mark-French franc target zone. These tests are based on restrictions that must apply to all option prices within a credible target zone and are free from specification error and estimation error. The authors also identify a minimum 'intensity of realignment,' an expression indicating the probability-weighted average realignment size. Copyright 1996 by American Economic Association.
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paper_title: Implied Risk-neutral Probability Density Functions From Option Prices: Theory and Application

paper_content:

Due to their forward-looking nature, derivative markets provide monetary authorities with a rich source of information for gauging market sentiment. For example, a futures price gives a widely used measure of the market's views about the future value of an asset, namely its mean or expected value at the maturity date of the futures contract. Moreover, the information available from futures prices can be extended by using option prices to estimate the market's entire probability distribution of the future value of an asset. This paper develops various techniques for estimating the market's probability distribution of the future value of an underlying asset from the prices of options on that asset. It discusses the relative merits and drawbacks of each approach, and shows how our preferred approach can be applied to estimate ex ante probability distributions using LIFFE equity and interest rate options, and Philadelphia Stock Exchange currency options. The paper then illustrates the potential value of this type of information to the policy-maker in assessing monetary conditions and conducting monetary operations. Finally, the paper looks at the limitations in data availability and highlights some areas for future research.
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paper_title: The Sovereignty Option : The Quebec Referendum and Market Views on the Canadian Dollar

paper_content:

We use exchange traded options on Canadian dollar futures to estimate the market's risk-neutral distribution for the Canadian dollar in the days before and after the Quebec sovereignty referendum. We employ a relatively new technique that places little a priori structure on the estimated distribution. This lack of structure allows the estimated distribution to reflect the multi-modal nature of expectations associated with the referendum's results. The technique is especially suited to circumstances in which a particular event will reduce a large degree of uncertainty prior to the expiration date of the options. Our estimated distributions are consistent with a significant perceived probability that the Canadian dollar would move up or down by as much as 5 percent as a result of the vote.
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paper_title: Option Hedging Using Empirical Pricing Kernels

paper_content:

This paper develops a method for option hedging which is consistent with time-varying preferences and probabilities. The preferences are expressed in the form of an empirical pricing kernel (EPK), which measures the state price per unit probability, while probabilities are derived from an estimated stochastic volatility model of the form GARCH components with leverage. State prices are estimated using the flexible risk-neutral density method of Rosenberg (1995) and a daily cross-section of option premia. Time-varying preferences over states are linked to a dynamic model of the underlying price to obtain a one-day ahead forecast of derivative price distributions and minimum variance hedge ratios. Empirical results suggest that risk aversion over S&P500 return states is substantially higher than risk aversion implied by Black-Scholes state prices and probabilities using long run estimates of S&P500 return moments. It is also found that the daily level of risk aversion is strongly positively autocorrelated, negatively correlated with S&P500 price changes,and positively correlated with the spread between implied and objective volatilities. Hedging results reveal that typical hedging techniques for out-of-the-money S&P500 index options, such as Black-Scholes or historical minimum variance hedging, are inferior to the EPK hedging method. Thus, time-varying preferences and probabilities appear to be an important factor in the day-to-day pricing of S&P500 options.
---
paper_title: A Simple Nonparametric Approach to Derivative Security Valuation

paper_content:

Canonical valuation uses historical time series to predict the probability distribution of the discounted value of primary assets' discounted prices plus accumulated dividends at any future date. Then the axiomatically rationalized maximum entropy principle is used to estimate risk-neutral (equivalent martingale) probabilities that correctly price the primary assets, as well as any predesignated subset of derivative securities whose payoffs occur at this date. Valuation of other derivative securities proceeds by calculation of its discounted, risk-neutral expected value. Both simulation and empirical evidence suggest that canonical valuation has merit. Copyright 1996 by American Finance Association.
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paper_title: Reading Interest Rate and Bond Futures Options ' Smiles : How PIBOR and Notional Operators Appreciated the 1997 French Snap Election ¤

paper_content:

The aim of this paper is to compare various methods which extract a Risk Neutral Density (RND) out of PIBOR as well as of Notional interest rate futures options and to investigate how traders reacted to a political event. We first focus on 5 dates surrounding the 1997 snap election and several methods: Black (1976), a mixture of lognormals (as in Melick and Thomas, 1997), an Hermite expansion (as in Abken, Madan, and Ramamurtie, 1996), and a method based on Maximum Entropy (following Kelly and Buchen, 1996). By and large the various methods give similar RNDs. Yet, the Hermite expansion approach, by allowing for somewhat dirty options prices, by providing a good fit to options prices, and by being very fast is the retained method for the data at hand. We then consider a daily panel of options running from February 1997 to July 1997. After constructing standardized options, i.e. with a fixed time to maturity, we find that operators in both markets anticipated the snap election a few days before the official announcement and that a substantial amount of political uncertainty subsisted even a month after the elections. The greater liquidity of PIBOR options eases information extraction.
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paper_title: New Techniques to Extract Market Expectations from Financial Instruments

paper_content:

This paper is a selective survey of new or recent methods to extract information about market expectations from asset prices for monetary policy purposes. Traditionally, interest rates and forward exchange rates have been used to extract expected means of future interest rates, exchange rates and inflation. More recently, these methods have been refined to rely on implied forward interest rates, so as to extract expected future time-paths. Very recently only the means but the whole (risk neutral) probability distribution from a set of option prices.
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paper_title: Option Pricing: A Simplified Approach

paper_content:

This paper presents a simple discrete-time model for valuing options. The fundamental economic principles of option pricing by arbitrage methods are particularly clear in this setting. Its development requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-Scholes model, which has previously been derived only by much more difficult methods. The basic model readily lends itself to generalization in many ways. Moreover, by its very construction, it gives rise to a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.
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paper_title: How to Grow a Smiling Tree

paper_content:

Building a recombining binomial tree consistent with the observed volatility smile is trickier than you might have thought. Implied binomial (or multinomial) trees are constructed to enable pricing of derivative securities in consistency with the existing market. We introduce important modifications to the method proposed by Derman and Kani (1994). A new algorithm provides correct treatment of interest rate and dividends, as well as some additional practical improvements. For the purpose of testing, we reconstruct the volatility smile using the new implied tree and obtain excellent results. Our modifications become especially important if the interest rate is high.
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paper_title: Implied Binomial Trees.

paper_content:

This article develops a new method for inferring risk-neutral probabilities (or state-contingent prices) from the simultaneously observed prices of European options. These probabilities are then used to infer a unique fully specified recombining binomial tree that is consistent with these probabilities (and, hence, consistent with all the observed option prices). A simple backwards recursive procedure solves for the entire tree. From the standpoint of the standard binomial option pricing model, which implies a limiting risk-neutral lognormal distribution for the underlying asset, the approach here provides the natural (and probably the simplest) way to generalize to arbitrary ending risk-neutral probability distributions. Copyright 1994 by American Finance Association.
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paper_title: Generalized Binomial Trees

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We consider the problem of consistently pricing new options given the prices of related options on the same stock. The Black-Scholes formula and standard binomial trees can only accommodate one related European option which then effectively specifies the volatility parameter. Implied binomial trees can accommodate only related European options with the same time-to-expiration. The generalized binomial trees introduced here can accommodate any kind of related options (European, American, or exotic) with different times-to-expiration.
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paper_title: How to Grow a Smiling Tree

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Building a recombining binomial tree consistent with the observed volatility smile is trickier than you might have thought. Implied binomial (or multinomial) trees are constructed to enable pricing of derivative securities in consistency with the existing market. We introduce important modifications to the method proposed by Derman and Kani (1994). A new algorithm provides correct treatment of interest rate and dividends, as well as some additional practical improvements. For the purpose of testing, we reconstruct the volatility smile using the new implied tree and obtain excellent results. Our modifications become especially important if the interest rate is high.
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paper_title: 0 Empirical Performance of Alternative Option Pricing Models

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Substantial progress has been made in extending the Black-Scholes model to incorporate such features as stochastic volatility, stochastic interest rates and jumps.On the empirical front, however, it is not yet known whether and by how much each generalized feature will improve option pricing and hedging performance. This paper fills this gap by first developing an implementable option model in closed form that allows volatility, interest rates and jumps to bestochastic and that is parsimonious in the number of parameters. The model includes many known ones as special cases. Delta-neutral and single-instrument minimum-variance hedging strategies are derived analytically. Using S&P 500 options, we examine a set of alternative models from three perspectives: (1) internal consistency of implied parameters/volatility with relevant time-series data, (2)out-of-sample pricing and (3) hedging performance. The models of focus include the benchmark Black-Scholes formula and the ones that respectively allow for (i) stochastic volatility, (ii) both stochastic volatility and stochastic interest rates, and (iii) stochastic volatility and jumps.Overall, incorporating both stochastic volatility and random jumps produces the best pricing performance and the most internally-consistent implied-volatility process. Its implied volatility does not "smile" across moneyness. But, for hedging, adding either jumps or stochastic interest rates does not seem to improve performance any further once stochastic volatility is taken into account.
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paper_title: The Crash of ʼ87: Was It Expected? The Evidence from Options Markets

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Transactions prices of S&P 500 futures options over 1985-87 are examined for evidence of expectations prior to October 1987 of an impending stock market crash. First, it is shown that out-of-the-money puts became unusually expensive during the year preceding the crash. Second, a model is derived for pricing American options on jump-diffusion processes with systematic jump risk. The jump-diffusion parameters implicit in options prices indicate that a crash was expected and that implicit distributions were negatively skewed during October 1986 to August 1987. Both approaches indicate no strong crash fears during the 2 months immediately preceding the crash. Copyright 1991 by American Finance Association.
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paper_title: Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation

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AbstractThis paper presents a unifying theory for valuing contingent claims under a stochastic term structure of interest rates. The methodology, based on the equivalent martingale measure technique, takes as given an initial forward rate curve and a family of potential stochastic processes for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. Examples are provided to illustrate the key results.
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paper_title: Option Prices, Implied Price Processes, and Stochastic Volatility

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This paper characterizes all continuous price processes that are consistent with current option prices. This extends Derman and Kani (1994), Dupire (1994, 1997), and Rubinstein (1994), who only consider processes with deterministic volatility. Our characterization implies a volatility forecast that does not require a specific model, only current option prices. We show how arbitrary volatility processes can be adjusted to fit current option prices exactly, just as interest rate processes can be adjusted to fit bond prices exactly. The procedure works with many volatility models, is fast to calibrate, and can price exotic options efficiently using familiar lattice techniques. Copyright The American Finance Association 2000.
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paper_title: Using option prices to estimate realignment probabilities in the European Monetary System: the case of sterling-mark

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Abstract This paper describes a procedure for estimating the market's perceived probability distribution of future exchange rates from the prices of risk reversals, strangles and other currency options, and uses the procedure to estimate the risk neutral ex ante probability of a realignment of the pound sterling. The procedure for estimating the realignment probabilities relies on the jump-diffusion model of exchange rate behavior and the resulting option pricing formula. By fitting this model to market option price data, I retrieve the unobserved parameters of the jump-diffusion process. I then use these parameter estimates to estimate the ex ante probability distribution of exchange rates and thus the realignment probabilities.
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paper_title: Implied Volatility Functions: Empirical Tests

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Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) hypothesize that asset return volatility is a deterministic function of asset price and time, and develop a deterministic volatility function (DVF) option valuation model that has the potential of fitting the observed cross section of option prices exactly. Using S&P 500 options from June 1988 through December 1993, we examine the predictive and hedging performance of the DVF option valuation model and find it is no better than an ad hoc procedure that merely smooths Black-Scholes (1973) implied volatilities across exercise prices and times to expiration. Copyright The American Finance Association 1998.
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paper_title: Relative Pricing of Options with Stochastic Volatility

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Computer environment conditioning apparatus employs a first air flow path for recirculating air through an evaporative cooling element for cooling purposes. A portion of the incident air flow is diverted to a by-pass duct where water vapor is added thereto, the initially dry air readily accepting the vapor. The cooled and humidified air flow streams are then combined, and discharged into the controlled environment. The humidifying operation is reliably and efficiently accomplished without regard to the operative status of the cooling equipment.
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paper_title: Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility

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In this paper we present an arbitrage pricing framework for valuing and hedging contingent equity index claims in the presence of a stochastic term and strike structure of volatility. Our approach to stochastic volatility is similar to the Heath-Jarrow-Morton (HJM) approach to stochastic interest rates. Starting from an initial set of index options prices and their associated local volatility surface, we show how to construct a family of continuous time stochastic processes which define thearbitrage-freeevolution of this local volatility surface through time. The no-arbitrage conditions are similar to, but more involved than, the HJM conditions for arbitrage-free stochastic movements of the interest rate curve. They guarantee that even under a general stochastic volatility evolution the initial options prices, or their equivalent Black–Scholes implied volatilities, remain fair.We introduce stochastic implied trees as discrete implementations of our family of continuous time models. The nodes of a stochastic implied tree remain fixed as time passes. During each discrete time step the index moves randomly from its initial node to some node at the next time level, while the local transition probabilities between the nodes also vary. The change in transition probabilities corresponds to a general (multifactor) stochastic variation of the local volatility surface. Starting from any node, the future movements of the index and the local volatilities must be restricted so that the transition probabilities to all future nodes are simultaneously martingales. This guarantees that initial options prices remain fair. On the tree, these martingale conditions are effected through appropriate choices of the drift parameters for the transition probabilities at every future node, in such a way that the subsequent evolution of the index and of the local volatility surface do not lead to riskless arbitrage opportunities among different option and forward contracts or their underlying index.You can use stochastic implied trees to value complex index options, or other derivative securities with payoffs that depend on index volatility, even when the volatility surface is both skewed and stochastic. The resulting security prices are consistent with the current market prices of all standard index options and forwards, and with the absence of future arbitrage opportunities in the framework. The calculated options values are independent of investor preferences and the market price of index or volatility risk. Stochastic implied trees can also be used to calculate hedge ratios for any contingent index security in terms of its underlying index and all standard options defined on that index.
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paper_title: Recovering stochastic processes from option prices

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How do stock prices evolve over time? The standard assumption of geometric Brownian motion, questionable as it has been right along, is even more doubtful in light of the recent stock market crash and the subsequent prices of U.S. index options. With the development of rich and deep markets in these options, it is now possible to use options prices to make inferences about the risk-neutral stochastic process governing the underlying index. We compare the ability of models including Black–Scholes, naive volatility smile predictions of traders, constant elasticity of variance, displaced diffusion, jump diffusion, stochastic volatility, and implied binomial trees to explain otherwise identical observed option prices that differ by strike prices, times-to-expiration, or times. The latter amounts to examining predictions of future implied volatilities. ::: ::: Certain naive predictive models used by traders seem to perform best, although some academic models are not far behind. We find that the better-performing models all incorporate the negative correlation between index level and volatility. Further improvements to the models seem to require predicting the future at-the-money implied volatility. However, an “efficient markets result” makes these forecasts difficult, and improvements to the option-pricing models might then be limited.
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paper_title: A Nonparametric Approach to Pricing and Hedging Derivative Securities Via Learning Networks

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We propose a nonparametric method for estimating derivative financial asset pricing formulae using learning networks. To demonstrate feasibility, we first simulate Black-Scholes option prices and show that learning networks can recover the Black-Scholes formula from a two-year training set of daily options prices, and that the resulting network formula can be used successfully to both price and delta-hedge options out-of-sample. For comparison, we estimate models using four popular methods: ordinary least squares, radial basis functions, multilayer perceptrons, and projection pursuit. To illustrate practical relevance, we also apply our approach to S\&P 500 futures options data from 1987 to 1991.
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paper_title: The inverse problem of option pricing

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Valuation of options and other financial derivatives critically depends on the underlying stochastic process specified for a particular market. An inverse problem of option pricing is to determine the nature of this stochastic process, namely, the distribution of expected asset returns implied by current market prices of options with different strikes. We give a rigorous mathematical formulation of this inverse problem, establish uniqueness, and suggest an efficient numerical solution. We apply the method to the S&P 500 Index and conclude that the index is negatively skewed with a higher probability of the sudden decline of the US stock market.
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paper_title: Determining Volatility Surfaces and Option Values From an Implied Volatility Smile

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AbstractUsing only the implied volatility smile of a single maturity T and an assumption of path-independence, we analytically determine the risk-neutral stock price process and the local volatility surface up to an arbitrary horizon T′ ≥ T. Our path-independence assumption requires that each positive future stock price St is a function of only time t and the level Wt of the driving standard Brownian motion (SBM) for all t ∈ (0, T′). Using the T-maturity option prices, we identify this stock pricing function and thereby analytically determine the risk-neutral process for stock prices. Our path-independence assumption also implies that local volatility is a function of the stock price and time which can be explicitly represented in terms of the known stock pricing function. Finally, we derive analytic valuation formulae for standard and exotic options which are consistent with the observed T-maturity smile.
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paper_title: The equity option volatility smile: an implicit finite-difference approach

paper_content:

A color-television image-display system of the type using triplets of vertical phosphor stripes and associated index elements scanned horizontally by a cathode-ray beam to reproduce the image, in which the horizontal deflection speed of the beam is modulated at the triplet scanning rate in accordance with the received chrominance information applied to the CRT grid to cause the beam to dwell longer in positions for which more saturated colors are to be produced, and in which the index elements are positioned so that the normal unmodulated horizontal sweep of the beam will scan them at a rate which is an odd integral multiple of one-half the corresponding rate of scanning of the triplets of phosphor stripes, thereby to reduce phase errors in the index signal when the deflection speed is modulated. The amount of deflection-speed modulation is also automatically varied as a function of the particular hue being represented by the chrominance signal, thereby further to reduce tendencies for the deflection-speed modulation to introduce discrepancies in the rendition of the hue and saturation of certain colors; preferably the deflection-speed modulation at the triplet scanning rate is enhanced most when representing primaries, and an increasing amount of deflection-speed modulation at the rate of scanning of the individual stripes is introduced for hues departing further from primaries; at the same time an increasing amount of pulsing on of the CRT beam stripe-scanning frequency is preferably also provided. To accomplish this, circuits are provided for detecting hues at or near primaries and for quantizing the phase of the signal used to produce deflection-speed modulation at the triplet rate. The deflection-speed modulation may be applied to the beam by electrostatic deflection plates or by a small magnetic yoke; to prevent undesirable interaction between the conventional deflection yoke and the microdeflection yoke, the microdeflection yoke may be located at a position partially within the conventional yoke where the net field produced by the conventional yoke in the microdeflection yoke is negligible, and/or special output stages tuned to the deflection-speed modulation signal frequencies may be used to drive the microdeflection yoke. The deflection-speed modulation may also be used advantageously in a black-and-white television tube provided with vertical stripes which are non-responsive to the beam and non-reflective of room light to increase the dwell time of the beam on the beam-responsive portions of the screen, with resultant improvements in overall image brightness for a given average beam current, when compared to the brightness produced by a conventional black-and-white screen with a partially-transparent front face-plate providing the same screen reflectance to room light as does the vertically-striped tube.
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paper_title: Implied Volatility Functions: Empirical Tests

paper_content:

Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) hypothesize that asset return volatility is a deterministic function of asset price and time, and develop a deterministic volatility function (DVF) option valuation model that has the potential of fitting the observed cross section of option prices exactly. Using S&P 500 options from June 1988 through December 1993, we examine the predictive and hedging performance of the DVF option valuation model and find it is no better than an ad hoc procedure that merely smooths Black-Scholes (1973) implied volatilities across exercise prices and times to expiration. Copyright The American Finance Association 1998.
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paper_title: Calibrating Volatility Surfaces Via Relative-Entropy Minimization

paper_content:

A framework for calibrating a pricing model to a prescribed set of options prices quoted in the market is presented. Our algorithm yields an arbitrage-free diffusion process that minimizes the Kullback-Leibler relative entropy distance to a prior diffusion. It consists in solving a constrained (minimax) optimal control problem using a finite-difference scheme for a Bellman parabolic equation combined with a gradient-based optimization routine. The number of unknowns to be solved for in the optimization step is equal to the number of option prices that need to be calibrated, and is independent of the mesh-size used for the scheme. This results in an efficient, non-parametric calibration method that can match an arbitrary number of option prices to any desired degree of accuracy. The algorithm can be used to interpolate, both in strike and expiration date, between implied volatilities of traded options and to price exotics. The stability and qualitative properties of the computed volatility surface are discussed, including the effect of the Bayesian prior on the shape of the surface and on the implied volatility smile/skew. The method is illustrated by calibrating to market prices of Dollar-Deutschmark over-the-counter options and computing interpolated implied-volatility curves.
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paper_title: Reconstructing the unknown volatility function

paper_content:

Using market European option prices, a method for computing a {\em smooth} local volatility function in a 1-factor continuous diffusion model is proposed. Smoothness is introduced to facilitate accurate approximation of the true local volatility function from a finite set of observation data. It is emphasized that accurately approximating the true local volatility function is crucial in hedging even simple European options, and pricing exotic options. A spline functional approach is used: the local volatility function is represented by a spline whose values at chosen knots are determined by solving a constrained nonlinear optimization problem. The optimization formulation is amenable to various option evaluation methods; a partial differential equation implementation is discussed. Using a synthetic European call option example, we illustrate the capability of the proposed method in reconstructing the unknown local volatility function. Accuracy of pricing and hedging is also illustrated. Moreover, it is demonstrated that, using a different constant implied volatility for an option with different strike/maturity can produce erroneous hedge factors. In addition, real market European call option data on the SP stability of the approach is demonstrated.
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paper_title: Building a Consistent Pricing Model from Observed Option Prices

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This paper constructs a model for the evolution of a risky security that is consistent with a set of observed call option prices. It explicitly treats the fact that only a discrete data set can be observed in practice. The framework is general and allows for state dependent volatility and jumps. The theoretical properties are studied. An easy procedure to check for arbitrage opportunities in market data is proved and then used to ensure the feasibility of our approach. The implementation is discussed: testing on market data reveals a U-shaped form for the "local volatility" depending on the state and, surprisingly, a large probability for strong price movements.
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Title: Implied Binomial Trees: A Literature Review

Section 1: Introduction
Description 1: Provide background on the systematic method of option pricing, the issue with the Black-Scholes formula, and the rise of volatility smiles.

Section 2: Risk-Neutral Probabilities
Description 2: Discuss the concept of risk-neutral probabilities, different approaches to extract them from option prices, and relevant empirical research.

Section 3: Parametric Methods
Description 3: Detail various parametric methods to model risk-neutral distributions, including expansion methods, generalized distribution methods, and mixture methods.

Section 4: Non-parametric Methods
Description 4: Explore non-parametric methods such as kernel methods, maximum entropy methods, and curve fitting methods to fit risk-neutral distributions.

Section 5: Applications
Description 5: Review empirical applications of risk-neutral densities in pricing exotic options, assessing market sentiments around economic announcements, and implications for utility functions.

Section 6: Implied Binomial Trees
Description 6: Introduce one-factor implied binomial trees, their construction methods, limitations, and improvements through generalized weight functions and trinomial trees.

Section 7: Two-Factor Implied Trees
Description 7: Discuss the extension of implied binomial trees to incorporate multiple factors such as stochastic volatility and the methods to calibrate and apply these models.

Section 8: Other Non-Parametric Models
Description 8: Examine alternative non-parametric models like deterministic volatility models, generalized diffusion processes, and kernel methods for option pricing.

Section 9: Conclusion
Description 9: Summarize the limitations of the Black-Scholes model, the advantages of implied binomial trees, and their empirical performance compared to parametric models and trader heuristics.