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In this partial and selective literature review of option implied risk-neutral distributions and of implied binomial trees, we start by observing that in efficient markets, there is information contained in option prices, which might help us to design option pricing models. To this end, we review the numerous methods of recovering risk-neutral probability distributions from option prices at one particular time-to-expiration and their applications. Next, we extend our attention beyond one time-to-expiration to the construction of implied binomial trees, which model the stochastic process of the underlying asset. Finally, we describe extensions of implied binomial trees, which incorporate stochastic volatility, as well as other non-parametric methods.
Arnold, Crack and Schwartz (2010) generalize the Rubinstein (1994) risk-neutral implied binomial tree (R-IBT) model by introducing a risk premium. Their new risk-averse implied binomial tree model (RA-IBT) has both probabilistic and pricing applications. They use the RA-IBT model to estimate the pricing kernel (i.e., marginal rate of substitution) and implied relative risk aversion for a representative agent. This paper presents additional theoretical details on the use of assumed utility functions to generate discount rates in the RA-IBT and theoretical details on the propagation of risk-averse probabilities through an RA-IBT (and how this process differs from the propagation of probabilities through a Rubinstein R-IBT). We also present both no-arbitrage and CAPM-driven derivations of the certainty equivalent risk-adjusted discounting formula that is used in Arnold, Crack and Schwartz (2010) and a direct estimation routine for the RA-IBT that is similar to Rubinstein's ldquo;one-two-threerdquo; technique. This paper also presents additional empirical applications of the model, including a comparison of risk-neutral and risk-averse implied distributions, and applications of the RA-IBT to financial options trading, time series return forecasting, and a previously infeasible corporate finance real option valuation problem. We also use the RA-IBT to explore the differences between risk-neutral and risk-averse moments of returns. We also discuss practical applications of the RA-IBT model to Value at Risk and stochastic volatility option pricing models.
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.
An earlier article, quot;Implied Binomial Trees,quot; introduced a theoretical model for implying the stochastic process of an underlying asset price from the prices of associated options. This sequel provides details concerning application of the model to the full record of Samp;P 500 index options transactions from April 2, 1986 through December 31, 1993. Most prominently, it introduces a revised optimization technique for estimating expiration-date risk-neutral probability distributions which is probably theoretically superior and definitely orders of magnitude faster than the approaches outlined in the antecedent paper. This method maximizes the smoothness of the distribution while at the same time insuring that multimodalities are not unrealistically strong. With the exception of the lower left-hand tail of the distribution, alternative optimization specifications typically produce approximately the same implied distributions. Considerable care is taken to specify such parameters as interest rates, dividends, and synchronous index levels, as well as to filter for general arbitrage violations resulting implied probability distributions exhibit changes in skewness as time-to-expiration approaches which are consistent and to use time aggregation to correct for unrealistic persistent jaggedness of implied volatility smiles. The with theoretical predictions. While time patterns of skewness and kurtosis exhibit a discontinuity across the divide of the 1987 market crash, they remain remarkably stable on either side of the divide. Moreover, since the crash, the risk-neutral probability of a four standard deviation decline in the Samp;P index (-46% over a year) is 100 times more likely than would appear to be the case under the assumption of lognormality.
The focus of this volume is on the development of new approaches for the market-conform valuation of newly issued derivatives. The first chapter presents a flexible approach to construct the binomial process of the underlying asset price by using a simultaneously backward and forward induction algorithm. This framework can be used to price and hedge a wide range of plain-vanilla and exotic options. In the second chapter this new approach is compared to existing models using a sample of plain-vanilla options, American call options and European Barrier options from two competing markets. In the third chapter new methods to value American-style options via Monte Carlo simulations in accordance with given market prices are discussed. After a short introduction to Monte Carlo methods, two new approaches are proposed. These new frameworks are illustrated via pricing examples for standard American put options.
A real option on a commodity is valued using an implied binomial tree (IBT) calibrated using commodity futures options prices. Estimating an IBT in the absence of spot options (the norm for commodities) allows real option models to be calibrated for the first time to market-implied probability distributions for commodity prices. Also, the existence of long-dated futures options means that good volatility estimates may now be incorporated into capital budgeting evaluations of real options projects with long planning horizons. An example is given using gold futures options and a real option to extract gold from a mine. We include a unique out-of-sample test that shows how option pricing errors evolve on sub-trees emanating from future levels of the underlying.