Jens Carsten Jackwerth

Option Implied Risk-Neutral Distributions and Implied Binomial Trees : a Literature Review

Solving backward through an option pricing model to find the “implied volatility” (IV) that makes the model value equal the market price is a technique nearly as old as the Black-Scholes model itself. In fact, calculating the implied volatility yields the entire implied risk-neutral returns distribution: It is lognormal with mean equal to the riskless interest rate and constant volatility equal to IV. But research across many different options markets has shown clearly that neither implied nor empirical volatility is constant, and returns distributions appear to be far from lognormal. This has led to new theoretical pricing models that can incorporate non-constant volatility and more general returns distributions. It has also led to techniques for obtaining the entire risk-neutral returns distribution implied in a full set of market prices for options with different strikes and maturities. One of the most general and flexible approaches, first suggested by Rubinstein, is to construct an implied binomial tree.This article leads off our Symposium on the topic of implied distributions. In it, Jackwerth presents a comprehensive review of the literature on option-implied risk neutral distributions and implied valuation trees.

The Pricing Kernel Puzzle: Reconciling Index Option Data and Economic Theory

The pricing kernel puzzle of Jackwerth ( 2000) concerns the fact that the empirical pricing kernel implied in S&P 500 index options and index returns is not monotonically decreasing in wealth as standard economic theory would suggest. Thus, those options are currently priced in a way such that any risk-averse investor would increase his/ her utility by trading in them. 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, albeit only for parameter constellations that are not typically observed in the real world. The Capital Asset Pricing Model (CAPM) of William Sharpe and the option-pricing models of Fisher Black, Robert Merton, and Myron Scholes were seminal in developing our understanding of the pricing of financial assets; these works sparked a firestorm of research by economic theorists and empiricists. Despite the fact that the CAPM was developed to price equity shares while the option-pricing models were developed for options, these two sets of models share a common feature, namely a state-price

Recovering Stochastic Processes from Option Prices

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.

Option Implied Risk-Neutral Distributions and Implied Binomial Trees: A Literature Review

In this selective literature review, we start by observing that in efficient markets, there is information incorporated in option prices that 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 move 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, and other non-parametric methods.

Option pricing : real and risk-neutral distributions

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.

Asymmetric Volatility Risk: Evidence from Option Markets

We show how to extract the expected risk-neutral correlation between risk-neutral distributions of the market index (S&P 500) return and its expected volatility (VIX). Comparing the implied correlation with its realized counterpart reveals a significant index-to-volatility correlation risk premium. It compensates for the fear of enduring negative market returns and measures a new dimension of conditional risk not covered by other variables such as the variance risk premium or skewness. Incorporating information from both equity and volatility markets, it predicts future investment opportunities and (conditional as well as unconditional) risk.

Financial Market Misconduct and Public Enforcement: The Case of Libor Manipulation

Using comprehensive data on London Interbank Offer Rate (Libor) submissions from 2001 through 2012, we document systematic evidence consistent with banks manipulating Libor to profit from Libor related positions and, to a degree, to signal their creditworthiness during the distressed times for banks. The evidence is initially stronger for banks that were eventually sanctioned by the regulators and disappears for all banks post-2010 in the aftermath of Libor investigations. Our findings suggest that public enforcement, with the threat of large penalties and the loss of reputation, can be effective in deterring financial market misconduct.Using data on Libor submissions from 1999 to 2012, wend weak support for the hypothesis that banks manipulate submissions to appear less risky and strong support for the hypothesis that banks manipulate Libor to generate higher cash ows. Our results are stronger for the manipulation period as identied by regulators (January 2005 to May 2009), for currencies and maturities with substantial notional amounts of interest-rate derivatives outstanding, for European banks, and for banks that have already paidnes related to manipulation. We calculate the cumulative gains in bank market capitalization due to manipulation to be

Does the Ross Recovery Theorem Work Empirically

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Funding Illiquidity Implied by S&P 500 Derivatives

19 billion.

Birds of a Feather – Do Hedge Fund Managers Flock Together?

Starting with the fundamental relationship that state prices are the product of physical probabilities and the stochastic discount factor, Ross (2015) shows that, given strong assumptions, knowing state prices suffices for backing out physical probabilities and the stochastic discount factor at the same time. We find that such recovered physical distributions based on the S&P 500 index are incompatible with future realized returns and fail to predict realized returns and variances. These negative results remain even when we add economically reasonable constraints. Simple benchmark methods based on a power utility agent or on the historical return distribution cannot be rejected.