This paper develops a closed-form option pricing formula for a spot asset whose variance follows a GARCH process. The model allows for correlation between returns of the spot asset and variance and also admits multiple lags in the dynamics of the GARCH process. The single-factor (one-lag) version of this model contains Heston’s (1993) stochastic volatility model as a diffusion limit and therefo...

The swelling of sponge-like bicontinuous mesophases of bilayers of surfactant (or lipid) in water as a function of dilution is analyzed. Analytic formulae for the swelling are derived assuming ii) constant aggregate thickness and (it) fixed surface area per surfactant molecule at an imaginary surface located within the bilayer. Approximate swelling laws are derived for bicontinuous films and co...

A formula is provided to compute the number of new blocks resulting from the decomposition induced by a shift of a single quadtree node of arbitrary size by an arbitrary amount. A precise calculation is also presented for the average number of BLACK nodes required to represent a square of width 2 m in a region quadtree.

In the original Black-Scholes model, the risk is quantified by a constant volatility parameter. It has been proposed by many authors that the volatilities should be modeled by a stochastic process to obtain a more realistic model. The volatility that corresponds to actual market data for option prices in Black-Scholes model is called the implied volatility. This volatility is in general depende...

Options are financial instruments designed to protect investors from the stock market randomness. In 1973, Fisher Black, Myron Scholes and Robert Merton proposed a very popular option pricing method using stochastic differential equations within the Itô interpretation. Herein, we derive the Black-Scholes equation for the option price using the Stratonovich calculus along with a comprehensive re...

The square root constant elasticity of variance (CEV) process has been paid little attention in previous research on valuation of barrier options. In this paper we derive analytical option pricing formulae of up-and-out options with this process using the eigenfunction expansion technique. We develop an efficient algorithm to compute numerical results from the formula. The numerical results are...

We study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black–Scholes formula and of the central limit theorem for martingales.

This paper presents the contemporary Fundamental Theorem of Asset Pricing as being equivalent to approaches to pricing that emerged before 1700 in the context of Virtue Ethics. This is done by considering the history of science and mathematics in the thirteenth and seventeenth century. An explanation as to why these approaches to pricing were forgotten between 1700 and 2000 is given, along with...