The Black~scholes formula for pricing options on stocks and other securities has been generalized by Merton and Garman to the case when stock volatility is stochastic. The derivation of the price of a security derivative with stochastic volatility is reviewed starting from the first principles of finance. The equation of Merton and Garman is then recast using the path integration technique of t...

On the occasion of the twentieth-anniversary issue of the Public Opinion Quarterly in 1957, an issue devoted to twenty years of public opinion research, Herbert Hyman reflected on the existing state of theory about public opinion (Hyman, 1957; pp. 54-60). The state of theory was bleak, he commented, if one looked for a grand theory that integrated the vast empirical findings of past and contemp...

We derive results similar to Bo et al. (2010), but in the case of dynamics of the FX rate driven by a general Merton jump-diffusion process. The main results of our paper are as follows: 1) formulas for the Esscher transform parameters which ensure that the martingale condition for the discounted foreign exchange rate is a martingale for a general Merton jump-diffusion process are derived; usin...

We present a new spectral element method for solving partial integro-differential equations for pricing European options under the Black–Scholes and Merton jump diffusion models. Our main contributions are: (i) using an optimal set of orthogonal polynomial bases to yield banded linear systems and to achieve spectral accuracy; (ii) using Laguerre functions for the approximations on the semi-infi...

Four numerical methods to solve the Merton problem with transaction costs are surveyed. The methods first appear in in Tourin and Zariphopoulou (1994), Muthuraman and Kumar (2006), Budhiraja and Ross (2007), and Muthuraman and Zha (2008).

Financial derivatives have grown in importance over the last 40 years with futures and options being actively traded on a daily basis throughout world. The need to accurately price such financial instruments has, thus, also increased, which has given rise several mathematical models among is that of Black, Scholes, Merton whose wide acceptance partly justified by its ability mature well-develop...