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Option pricing when underlying stock returns are discontinuous
 Journal of Financial Economics
, 1976
"... The validity of the classic BlackScholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying ..."
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Cited by 507 (1 self)
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The validity of the classic BlackScholes option pricing formula dcpcnds on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the moregeneral cast when the underlying stock returns are gcncrated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black&holes formula in that it does not dcpcnd on investor prcfcrenccs or knowledge of the expcctsd return on the underlying stock. Morcovcr, the same analysis applied to the options can bc extcndcd to the pricingofcorporatc liabilities. 1. Intruduction In their classic paper on the theory of option pricing, Black and Scholcs (1973) prcscnt a mode of an:llysis that has rcvolutionizcd the theory of corporate liability pricing. In part, their approach was a breakthrough because it leads to pricing formulas using. for the most part, only obscrvablc variables. In particular,
Fractal geometry of financial time series
 Fractals
, 1995
"... Abstract – A simple quantitative measure of the selfsimilarity in timeseries in general and in the stock market in particular is the scaling behavior of the absolute size of the jumps across lags of size k. A stronger form of selfsimilarity entails not only that this mean absolute value, but also ..."
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Cited by 7 (0 self)
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Abstract – A simple quantitative measure of the selfsimilarity in timeseries in general and in the stock market in particular is the scaling behavior of the absolute size of the jumps across lags of size k. A stronger form of selfsimilarity entails not only that this mean absolute value, but also the full distributions of lagk jumps have a scaling behavior characterized by the above Hurst exponent. In 1963 Benoit Mandelbrot showed that cotton prices have such a strong form of (distributional) selfsimilarity, and for the first time introduced Lévy’s stable random variables in the modeling of price records. This paper discusses the analysis of the selfsimilarity of highfrequency DEMUSD exchange rate records and the 30 main German stock price records. Distributional selfsimilarity is found in both cases and some of its consequences are discussed. 1
Selfsimilarity of highfrequency USDDEM exchange rates
, 1995
"... High frequency DEMUSD exchange rate data (resolution > 2 seconds) are analyzed for their scaling behavior as a function of the time lag. Motivated by the finding that the distribution of 1quote returns is rather insensitive to the physical time duration between successive quotes, lags are measured ..."
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Cited by 3 (0 self)
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High frequency DEMUSD exchange rate data (resolution > 2 seconds) are analyzed for their scaling behavior as a function of the time lag. Motivated by the finding that the distribution of 1quote returns is rather insensitive to the physical time duration between successive quotes, lags are measured in units of quotes. The mean absolute returns over lags of different sizes, shows three different regimes. The smallest time scales show no scaling, followed by two scaling regimes characterized by Hurst exponents H = 0.45 and H = 0.56, with a crossover occuring at lags of # 500 quotes. The updown correlation coefficient, defined here, shows strong anticorrelations on scales smaller than 500. The lack of convergence to a large deviation rate function, convex tails in the logarithm of the probability distributions, strong updown correlations and H < 0.5, show that the dynamics on small scales is more complicated than random walk models with i.i.d. increments. Nevertheless, for both scaling re...
Publishing Company OPTIMAL, RULES FOR ORDERING UNCERTAIN PROSPECTS+
, 1974
"... In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all individuals with decreasing absolute risk averse utility functions. The optimal selection rule minimizes the admissible set of alternatives by discarding, from among a given set of altematives. those that ar ..."
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In this paper, we obtain the optimal selection rule for ordering uncertain prospects for all individuals with decreasing absolute risk averse utility functions. The optimal selection rule minimizes the admissible set of alternatives by discarding, from among a given set of altematives. those that are inferior (for each utility function in the restricted class) to a member of the given set. We show that the Third Order Stochastic Dominance (TSD) rule is the optimal rule when comparing uncertain prospects with equal means. We also show that in the general case of unequal means, no known selection rule uses both necessary and sufficient conditions for dominance, and the TSD rule may be used to obtain a reasonable approximation to the smallest admissible set. The TSD rule is complex and we provide an efficient algorithm lo obtain the TSD admissible set. For certain restrictive classes of the probability distributions (of returns on uncertain prospects) which cover most commonly used distributions in Enance and economics, we obtain the optimal rule and show that it reduces to a simple form. We also study the relationship of the optimal selection rule lo others previously advocated in the literature. including the more popular meanvariance rule as well as the semivariance rule. 1.