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Abstract:
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.

Abstract:
Rubinstein (1994) shows evidence of a significant time pattern in the shape of the volatility smile after the crash of 1987 and proposes an implied binomial tree approach to overcome the empirical limitations of the Black and Scholes model. This approach, and more generally the class of generalized deterministic volatility models, is based on the assumption that the local volatility of the underlying asset is a known function of time and of the path and level of the underlying asset price. In these economies, options are redundant assets. We use this observation as a testable restriction and ask three questions. First, is the observed dynamics of the smile consistent with deterministic volatility models? Second, if volatility is stochastic, so that two assets cannot dynamically complete the market, is volatility also priced and if so how important is to model explicitly the price of volatility in the design of risk management strategies? We address this question by testing if the returns on the underlying and on at-the-money options span the asset prices in the economy or if we need additional information from returns on other options or the riskfree rate. Third, are there any differences in the spanning properties of the option market before and after the 1987 market crash?

We cast these questions in terms of martingale restrictions on the pricing kernel and conduct tests based on daily S&P500 index options data from April 1986--December 1995. All our tests suggest that both in- and out-of-the-money options are needed for spanning purposes. This finding is even stronger in the postcrash period and suggests that returns on away-from-the-money options are driven by at least one additional economic factor compared to returns on at-the-money options. This finding is inconsistent with the implications of deterministic volatility models based on generalized deterministic volatility. The finding is consistent with explanations of the smile in which volatility is stochastic and priced in equilibrium and with models in which away-from-the-money options are used in equilibrium by a different specialized clientele, such as portfolio insurers, subject to different budget constraints and/or portfolio objectives than the typical investor in at-the-money options.

Abstract:
In a novel approach, standard and implied binomial trees are completely specified in terms of two basic inputs: the ending nodal probability distribution and a linear weight function which governs the stochastic process resulting in that distribution. Several key economic principles, such as no interior arbitrage, are intuitively related to these basic inputs. A simple and computationally efficient three-step algorithm, common to all binomial trees, is found. Noting that the currently used linear weight function is unnecessarily restrictive, a binomial tree even more versatile is introduced, the generalized binomial tree. Applications to recovering the stochastic process implied in (European, American, or exotic) options of several times-to-expiration are developed.

Abstract:
One of the central questions in financial economics is the determination of asset prices, such as the value of a stock. Over the past three decades, research on this topic has converged on a concept called the "state-price density". However, a puzzle has arisen. On the one hand, Cox, Ingersoll, and Ross (1985) and others argue that the ratio of the state-price density to the statistical probability density, which is commonly known as the pricing kernel, should decrease monotonically as the aggregate wealth of an economy rises. On the other hand, recent empirical work on options on the S&P 500 index suggests that, for a sizable range of index levels, the pricing kernel is increasing instead of decreasing. We investigate theoretical explanations to this puzzle. Our existing work has ruled out some alternative hypotheses, such as data imperfections and methodological problems. We provide a representative agent model where volatility is a function of a second momentum state variable. This model is capable of generating the empirical patterns in the pricing kernel. We estimate the model through GMM.

Derivatives, Asset Pricing, Option Pricing, Empirical Studies

Abstract:
The behavior of a hedge-fund manager naturally depends on her compensation scheme, her preferences, and constraints on her risk-taking. We propose a numerical method which can be used to analyze the impact of these influences. The model leads to several interesting and novel results concerning her risk-taking and other managerial decisions. We are able to relate our results to partial results in the literature and show how they fit in a more general context. We also allow the manager to voluntarily shut down the fund as well as enhancing the fund's Sharpe Ratio through additional effort. Both these extensions generate additional insights. Throughout the paper, we find that even slight changes in the compensation structure or the extent of managerial discretion can lead to drastic changes in her risk-taking.

Abstract:
We document widespread violations of stochastic dominance by one-month S&P 500 index call options market over 1986-2006. These violations imply that a trader can improve her expected utility by engaging in a zero-net-cost trade. We allow the market to be incomplete and also imperfect by introducing transaction costs and bid-ask spreads. Even though pre-crash option prices conform to the Black-Scholes-Merton model reasonably well, they are incorrectly priced if the distribution of the index return is estimated from time-series data even with a variety of statistical adjustments. Even though there are fewer violations by OTM calls than by ITM calls, there are still substantial violations by OTM calls, contradicting the inference drawn from the observed implied volatility smile that the problem primarily lies with the left-hand tail of the index return distribution. Most of the violations by post-crash options are not due to the smile being too steep: options are underpriced over 1988-1995 and overpriced over 1997-2006. The decrease in violations over the post-crash period 1988-1995 is followed by a substantial increase in violations over 1997-2006. These results do not support the hypothesis that the options market is becoming more rational over time.

Derivative pricing; volatility smile, incomplete markets, transaction costs; index options; stochastic dominance bounds

Abstract:
A relationship exists between aggregate risk-neutral and subjective probability distributions and risk aversion functions. We empirically derive risk aversion functions implied by option prices and realized returns on the S&P500 index simultaneously. These risk aversion functions dramatically change shapes around the 1987 crash: Precrash, they are positive and decreasing in wealth and largely consistent with standard assumptions made in economic theory. Postcrash, they are partially negative and partially increasing and irreconcilable with those assumptions. Mispricing in the option market is the most likely cause. Simulated trading strategies exploiting this mispricing shows excess returns even after accounting for the possibility of further crashes, transaction costs, and hedges against the downside risk.

Abstract:
A relationship exists between aggregate risk-neutral and subjective probability distributions and risk aversion functions. We empirically derive risk aversion functions implied by option prices and realized returns on the S&P 500 index simultaneously. These risk aversion functions dramatically change shapes around the 1987 crash: Precrash, they are positive and decreasing in wealth and largely consistent with standard economic theory. Postcrash, they are partially negative and partially increasing and irreconcilable with the theory. Mispricing in the option market is the most likely cause. A simulated trading strategy exploiting this mispricing shows excess returns even after accounting for the possibility of further crashes, transaction costs, and hedges against the downside risk.

Abstract:
Previous papers have argued that trading restrictions can result in a typical employee stock option having a subjective value (certainty equivalent value) that is substantially less than its Black-Scholes value. However, these analyses ignore the manager's ability to (at least partially) control the risk level within the firm. In this paper, we show how managerial control can lead to such options having much larger certainty equivalent values for employees who can exercise control. We also show that the potential for early exercise is substantially less valuable with managerial control.

The certainty equivalent value for a European option with managerial control can easily exceed the Black-Scholes value for a comparable option without control. However, it is questionable whether Black-Scholes is an appropriate benchmark for an option where the underlying process exhibits controlled volatility. We show how to obtain a risk-neutral valuation for such an option. That risk-neutral value can be substantially greater or less than the Black-Scholes value. Furthermore, the option's certainty equivalent value can also be greater or less than its risk-neutral value.

Employee Stock Options, Dynamic Optimatimization

Abstract:
American call and put options on the S&P 500 index futures that violate the stochastic dominance bounds of Constantinides and Perrakis (2007) over 1983-2006 are identified as potentially profitable investment opportunities. Call bid prices more frequently violate their upper bound than put bid prices do, while evidence of underpriced calls and puts over this period is scant. In out-of-sample tests, the inclusion of short positions in such overpriced calls, puts, and, particularly, straddles in the market portfolio is shown to increase the expected utility of any risk averse investor and also increase the Sharpe ratio, net of transaction costs and bid-ask spreads. The results are strongly supportive of mispricing.

option mispricing, futures options, derivatives pricing, stochastic dominance, transaction costs, market efficiency

Abstract:
Numerous hedge funds stop reporting to commercial databases each year. An issue for hedge-fund performance estimation is: what delisting return to attribute to such funds? This would be particularly problematic if delisting returns are typically very different from continuing funds' returns. In this paper, we use estimated portfolio holdings for funds-of-funds with reported returns to back out maximum likelihood estimates for hedge-fund delisting returns. The estimated mean delisting return for all exiting funds is small, although statistically significantly different from the average observed returns for all reporting hedge funds. These findings are robust to relaxing several underlying assumptions.

Return, Hedge Fund

Abstract:
This paper summarizes a program of research we have conducted over the past four years. So far, it has produced two published articles, one forthcoming paper, one working paper currently under review at a journal, and three working papers in progress. The research concerns the recovery of market-wide risk-neutral probabilities and risk aversion from option prices.

The work is built on the idea that risk-neutral probabilities (or their related state-contingent prices) are an amalgam of consensus subjective probabilities and consensus risk aversion. The most significant departure of this work is that it reverses the normal direction of previous research, which typically moves from assumptions governing subjective probabilities and risk aversion, to conclusions about risk-neutral probabilities. Our work is partially motivated by the
conspicuous failure of the Black-Scholes formula to explain the prices of index options in the post-1987 crash market.

First, we independently estimate risk-neutral probabilities, taking advantage of the now highly liquid index option market. We show that, if the options market is free of general arbitrage opportunities and we can at least initially ignore trading costs, there are quite robust methods for recovering these probabilities.

Second, with these probabilities in hand, we use the method of implied binomial trees to recover a consistent stochastic process followed by the underlying asset price.

Third, we provide an empirical test of implied trees against alternative option pricing models (including "naive trader" approaches) by using them to forecast future option smiles.

Fourth, we argue that realized historical distributions will be a reliable proxy for certain aspects of the true subjective distributions. We then use these observed aspects together with the option-implied risk-neutral probabilities to estimate market-wide risk aversion.

Abstract:
The central premise of the Black and Scholes [Black, F., Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy 81, 637-659] and Merton [Merton, R. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141-184] option pricing theory is that there exists a self-financing dynamic trading policy of the stock and risk free accounts that renders the market dynamically complete. This requires that the market be complete and perfect. In this essay, we are concerned with cases in which dynamic trading breaks down either because the market is incomplete or because it is imperfect due to the presence of trading costs, or both. Market incompleteness renders the risk-neutral probability measure non unique and allows us to determine the option price only within a range. Recognition of trading costs requires a refinement in the definition and usage of the concept of a risk-neutral probability measure. Under these market conditions, a replicating dynamic trading policy does not exist. Nevertheless, we are able to impose restrictions on the pricing kernel and derive testable restrictions on the prices of options.We illustrate the theory in a series of market setups, beginning with the single period model, the two-period model and, finally, the general multiperiod model, with or without transaction costs.We also review related empirical results that document widespread violations of these restrictions.

Abstract:
We document that the leverage-adjusted returns on S&P 500 index calls and puts are decreasing in their strike-to-price ratio over 1986-2007, contrary to the prediction of the Black-Scholes-Merton model; and the leverage-unadjusted returns on S&P 500 index calls are decreasing in their strike-to-price ratio, contrary to the prediction and empirical results of Coval and Shumway (2001). Several factor models are tested and fail to explain the cross-section of option returns. Two option-specific factors, the change in monthly OTM put volume and the change in the VIX index, have some explanatory power when the factor premia are estimated from the universe of options but large alphas remain when the premia are estimated from equities. The three Fama-French factors leave large alphas even when the premia are estimated from options.

index options, option mispricing, derivatives, risk premia, market efficiency

Abstract:
We model a firm's value process controlled by a manager maximizing expected utility from restricted shares and employee stock options. Conditioning on his optimal behavior, we value derivatives on that controlled process. Control results in a longer expected time to exercise for the manager's stock options. It also reduces the percentage gap between his certainty equivalent and the firm's fair value for his compensation, but that gap remains substantial. Employees without control capabilities will generally face larger gaps; and for such employees, option compensation can be very inefficient. Managerial control also causes traded options to exhibit an implied volatility smile.

Abstract:
This paper examines the use of second-order stochastic dominance as both a way to measure portfolio performance relative to a benchmark and also as a technique for constructing portfolios. Using in-sample data, we construct portfolios such that their dominance over a typical pension fund benchmark is maximized. The empirical results based on 20 years of daily data suggest that this portfolio choice technique significantly outperforms the benchmark portfolio out-of-sample. Moreover, its performance is superior to mean-variance and equally weighted portfolio choice approaches.

Second-order stochastic dominance, portfolio choice, out-of-sample performance

Abstract:
We model a firm's value process controlled by a manager maximizing expected utility from restricted shares and employee stock options. The manager also dynamically controls allocation of his outside wealth. We explore interactions between those controls as he partially hedges his exposure to firm risk. Conditioning on his optimal behavior, control of firm risk increases the expected time to exercise for his employee stock options. It also reduces the percentage gap between his certainty equivalent and the firm's fair value for his compensation, but that gap remains substantial. Managerial control also causes traded options to exhibit an implied volatility smile.

Dynamic Control of Firm Value Process, Managerial Incentives, Derivative Pricing Implications, External Wealth management

Abstract:
American call and put options on the S&P 500 index futures that violate the stochastic dominance bounds of Constantinides and Perrakis (2007) over 1983-2006 are identified as potentially profitable investment opportunities. Call bid prices more frequently violate their upper bound than put bid prices do, while evidence of under priced calls and puts over this period is scant. In out-of-sample tests, the inclusion of short positions in such overpriced calls, puts, and, particularly, straddles in the market portfolio is shown to increase the expected utility of any risk averse investor and also increase the Sharpe ratio, net of transaction costs and bid-ask spreads.

option mispricing, futures options, derivatives pricing, stochastic dominance, transaction costs, market efficiency

Abstract:
Widespread violations of stochastic dominance by 1-month S&P 500 index call options over 1986-2006 imply that a trader can improve expected utility by engaging in a zero-net-cost trade net of transaction costs and bid-ask spread. Although precrash option prices conform to the Black-Scholes-Merton model reasonably well, they are incorrectly priced if the distribution of the index return is estimated from time-series data. Substantial violations by postcrash OTM calls contradict the notion that the problem lies primarily with the left-hand tail of the index return distribution and that the smile is too steep. The decrease in violations over the postcrash period of 1988-1995 is followed by a substantial increase over 1997-2006, which may be due to the lower quality of the data but, in any case, does not provide evidence that the options market is becoming more rational over time.

G10, G13

Abstract:
Widespread violations of stochastic dominance by one-month S&P 500 index call options over 1986-2006 imply that a trader can improve expected utility by engaging in a zero-net-cost trade net of transaction costs and bid-ask spread. Although pre-crash option prices conform to the Black-Scholes-Merton model reasonably well, they are incorrectly priced if the distribution of the index return is estimated from time-series data. Substantial violations by post-crash OTM calls contradict the notion that the problem primarily lies with the left-hand tail of the index return distribution and that the smile is too steep. The decrease in violations over the post-crash period 1988-1995 is followed by a substantial increase over 1997-2006 which may be due to the lower quality of the data but, in any case, does not provide evidence that the options market is becoming more rational over time.

Abstract:
Murphy, Koehler, and Fogler [1997] gave in the last issue of the Journal of Portfolio Management an account of how to raise a neural net's IQ. The purpose of this reply is to point out some of the general difficulties with neural nets. Also, I would like to mention an alternative method, namely Pade approximants, which does not suffer from these difficulties.

Abstract:
Analysts are accustomed to using prices for the information they contain. A stock price, for example, can be thought of as an expected value of future cash flows. Each futures price and option price tells the analyst a bit more about the probability distribution under which those expectations should be accepted. In this Research Foundation monograph, the author describes what can and cannot be learned from option prices for applications in financial analysis and provides examples for each step so that the reader can actually apply the concepts.

Abstract:
We investigate incentive effects of a typical hedge-fund contract for a manager with power utility. With a one-year horizon, she displays risk-taking that varies dramatically with fund value. We extend the model to multiple yearly evaluation periods and find her risk-taking is rapidly moderated if the fund performs reasonably well. The most realistic approach to modeling fund closure uses an endogenous shutdown barrier where the manager optimally chooses to shut down. The manager increases risk-taking as fund value approaches that barrier, and this boundary behavior persists strongly with multiyear horizons.

Abstract:
The volatility smile changed drastically around the crash of 1987 and new option pricing models have been proposed in order to accommodate that change. Deterministic volatility models allow for more flexible volatility surfaces but refrain from introducing additional risk-factors. Thus, options are still
redundant securities. Alternatively, stochastic models introduce additional risk-factors and options are then needed for spanning of the pricing kernel. We develop a statistical test based on this difference in spanning. Using daily S&P500 index options data from 1986-1995, our tests suggest that both in- and
out-of-the-money options are needed for spanning. The findings are inconsistent with deterministic volatility models but are consistent with stochastic models which incorporate additional priced risk-factors such as stochastic volatility, interest rates, or jumps.

Abstract:
An earlier article, "Implied Binomial Trees," 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 S&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 S&P index (-46% over a year) is 100 times more likely than would appear to be the case under the assumption of lognormality.

Abstract:
This paper derives underlying asset risk-neutral probability distributions of European options on the S&P 500 index. Nonparametric methods are used to choose probabilities which 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% (-46%) over a year) is about 10 (100) times more likely than under the assumption of lognormality.