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Search results: 9.
Implied Binomial Trees
Published: 07/01/1994 | DOI: 10.1111/j.1540-6261.1994.tb00079.x
MARK RUBINSTEIN
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.
Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978
Published: 06/01/1985 | DOI: 10.1111/j.1540-6261.1985.tb04967.x
MARK RUBINSTEIN
The tests reported here differ in several ways from those of most other papers testing option pricing models: an extremely large sample of observations of both trades and bid‐ask quotes is examined, careful consideration is given to discarding misleading records, nonparametric rather than parametric statistical tests are used, reported results are not sensitive to measurement of stock volatility, special care is taken to incorporate the effects of dividends and early exercise, a simple method is developed to test several option pricing formulas simultaneously, and the statistical significance and consistency across subsamples of the most important reported results are unusually high. The three key results are: (1) short‐maturity out‐of‐the‐money calls are priced significantly higher relative to other calls than the Black‐Scholes model would predict, (2) striking price biases relative to the Black‐Scholes model are also statistically significant but have reversed themselves after long periods of time, and (3) no single option pricing model currently developed seems likely to explain this reversal.
A Simple Formula for the Expected Rate of Return of an Option over a Finite Holding Period
Published: 12/01/1984 | DOI: 10.1111/j.1540-6261.1984.tb04920.x
MARK RUBINSTEIN
Under conditions consistent with the Black‐Scholes formula, a simple formula is developed for the expected rate of return of an option over a finite holding period possibly less than the time to expiration of the option. Under these conditions, surprisingly, the expected future value of a European option, even prior to expiration, is shown equal to the current Black‐Scholes value of the option, except that the expected future value of the stock at the end of the holding period replaces the current stock price in the Black‐Scholes formula and the future value of a riskless invesment of the striking price replaces the striking price. An extension of this result is used to approximate moments of the distribution of returns from an option portfolio.
Displaced Diffusion Option Pricing
Published: 03/01/1983 | DOI: 10.1111/j.1540-6261.1983.tb03636.x
MARK RUBINSTEIN
This paper develops a new option pricing formula that pushes the underlying source of risk back to the risk of individual assets of the firm. The formula simultaneously encompasses differential riskiness of the assets of the firm, their relative weights in determining the value of the firm, the effects of firm debt, and the effects of a dividend policy with both constant and random components. Although this setting considerably generalizes the Black‐Scholes [1] analysis, it nonetheless produces a formula via riskless arbitrage arguments that, given estimated inputs, is as easy to use as the Black‐Scholes formula.
THE STRONG CASE FOR THE GENERALIZED LOGARITHMIC UTILITY MODEL AS THE PREMIER MODEL OF FINANCIAL MARKETS
Published: 05/01/1976 | DOI: 10.1111/j.1540-6261.1976.tb01906.x
Robert Litzenberger, Mark Rubinstein
This paper begins by comparing the available well‐developed micro‐economic models in finance which recognize uncertainty. It is argued that models whose distinctive simplifying assumption restricts utility functions are superior to those which instead restrict probability distributions, both with respect to the realism of their assumptions and richness of their conclusions. In particular, the most successful model, based on generalized logarithmic utility (GLUM), is a multiperiod consumption/portfolio and equilibrium model in discrete‐time which (1) requires decreasing absolute risk aversion; (2) tolerates increasing, constant, or decreasing proportional risk aversion; (3) assumes no exogenous specification of the contemporaneous or intertemporal stochastic process of security prices; (4) tolerates heterogeneity with respect to wealth, lifetime, time‐and risk‐preference and beliefs; (5) results in a complete specification of consumption/portfolio decision and sharing rules which include nontrivial multiperiod separation properties and explains demand for default‐free bonds of various maturities and options; (6) leads to a solution to the aggregation problem; (7) results in a complete specification of the contemporaneous and intertemporal process of security prices which reveals necessary and sufficient conditions for an unbiased term structure and the market portfolio to follow a random walk as a natural outcome of equilibrium; (8) provides an empirically testable aggregate consumption function relating per capita consumption to per capita wealth and the present value of a perpetual default‐free annuity which does not require inferences of ex ante beliefs from ex post data; (9) provides a nontrivial multiperiod extension of popular single‐period security valuation models which is empirically testable; (10) yields a simple multiperiod valuation formula for an uncertain income stream even when this income is serially correlated over time.
Recovering Probability Distributions from Option Prices
Published: 12/01/1996 | DOI: 10.1111/j.1540-6261.1996.tb05219.x
JENS CARSTEN JACKWERTH, MARK RUBINSTEIN
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.