Theory of Rational Option Pricing

Merton, Robert C.

doi:10.2307/3003143

Cited by 7,453 publications

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“…The theoretical study of options was directed in finding a fair and presumably riskless price of these instruments. Without questions, the works of Black and Scholes [1] and Merton [2] are a turning-point in the study. Their method has been proven to be very useful for investors trading in option markets.…”

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confidence: 99%

Black–Scholes model under subordination

Stanislavsky¹

2003

Physica A: Statistical Mechanics and its Applications

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In this paper we consider a new mathematical extension of the Black-Scholes model in which the stochastic time and stock share price evolution is described by two independent random processes. The parent process is Brownian, and the directing process is inverse to the totally skewed, strictly α-stable process. The subordinated process represents the Brownian motion indexed by an independent, continuous and increasing process. This allows us to introduce the long-term memory effects in the classical Black-Scholes model. Key words:Continuous-time random walk, Brownian motion, Lévy process, Subordination, Fractional calculus, Econophysics PACS: 02.50.-r, 02.50.Ey, 02.50. Wp, 89.90.+n The option trading has the long history. The mission of the options as financial instruments is to protect investors from the stock market randomness. Since the early seventies the option market rapidly became very successive in development. The theoretical study of options was directed in finding a fair and presumably riskless price of these instruments. Without questions, the works of Black and Scholes [1] and Merton [2] are a turning-point in the study. Their method has been proven to be very useful for investors trading in option markets. On the other hand, the approach is fruitful for extending the option pricing theory in many ways. Therefore, nowadays the Black-Scholes (BS) model is very popular in finance.The BS equation is nothing else as a diffusion equation. In fact, their option price formula is a solution of the diffusion equation with the initial and boundary conditions given by the option contract terms. The fundamental principles governing the financial and economical systems are not completely uncovered. In recent years the physical community has started applying concepts and methods of statistical and quantum physics of complex systems to *

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Black–Scholes model under subordination

Stanislavsky¹

2003

Physica A: Statistical Mechanics and its Applications

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“…The antithetic variate technique attempts to reduce the variance by exploiting the existence of negative correlations between two estimates of the underlying option (Merton, 1973). To illustrate how to use this technique, suppose our objective is to estimate the option value defined by (7).…”

Section: Antithetic Variate Techniquementioning

confidence: 99%

“…Merton (1973) provided the first analytical formula for a down-and-out call option which was the oldest barrier option type. Reiner and Rubinstein (1991) provide closed form solutions for the prices of all four types of barrier on both call and put options with the assumption that the underlying asset price follows Geometric Brownian process.…”

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confidence: 99%

Efficient Monte Carlo Algorithm Using Antithetic Variate and Brownian Bridge Techniques for Pricing the Barrier Options with Rebate Payments

Alzubaidi¹

2016

Journal of Mathematics and Statistics

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Abstract:The down-and-out call barrier option with rebate payment on dividend-paying stock is simulated using a new version of the Monte Carlo algorithm. The standard Monte Carlo method for simulating such an option suffers from two sources of errors: Hitting time error inherent from time stepping and the Monte Carlo statistical error. We present a modified version of Monte Carlo method that can reduce these errors efficiently using the Brownian bridge technique for the hitting time error and the antithetic variate approach for the statistical error. We found that the Brownian bridge technique is responsible for improving the order of convergence in hitting time from one half to one and the antithetic variate technique can speed up the Monte Carlo simulation by reducing the variance of the computed payoff, giving almost twice as much accuracy. The standard error and the coefficient of variation are applied in order to measure the effectiveness of the volatility of the underlying option.

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“…Ici, par contre, nous pouvons facilement voir, par exemple, que l'augmentation du risque du prix aléatoire a, dans le sens formulé par Rothschild et Stiglitz (1970), va augmenter la quantité de capital K si elle n'affecte pas V 0 14 et si C(K) est supposée être une fonction convexe, comme c'est généralement le cas. Cette remarque provient du fait, démontré par Merton (1973), que la valeur d'une option augmente ceteris paribus si le risque du titre sous-jacent augmente.…”

Section: Concurrence Et Monopole Avec Demande Aléatoireunclassified

Les contributions de la théorie financière à la solution de problèmes en organisation industrielle et en microéconomie appliquée

Perrakis

1989

L'Actualité économique

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Theory of Rational Option Pricing

Cited by 7,453 publications

References 3 publications

Black–Scholes model under subordination

Black–Scholes model under subordination

Efficient Monte Carlo Algorithm Using Antithetic Variate and Brownian Bridge Techniques for Pricing the Barrier Options with Rebate Payments

Les contributions de la théorie financière à la solution de problèmes en organisation industrielle et en microéconomie appliquée

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