Valuing options in Heston's stochastic volatility model: Another analytical approach

We are concerned with the valuation of European options in Heston's stochastic volatility model with correlation. Based on Mellin transforms we present new closed-form solutions for the price of European options and hedging parameters. In contrast to Fourier-based approaches where the transformation variable is usually the log-stock price at maturity, our framework focuses on transforming the current stock price. Our solution has the nice feature that similar to the approach of Carr and Madan (1999) it requires only a single integration. We make numerical tests to compare our results to Heston's solution based on Fourier inversion and investigate the accuracy of the derived pricing formulae. --Stochastic volatility,European option,Mellin transform

Pricing American options with Mellin transforms

Mellin transforms in option pricing theory were introduced by Panini and Srivastav (2004). In this contribution, we generalize their results to European power options. We derive Black-Scholes-Merton-like valuation formulas for European power put options using Mellin transforms. Thereafter, we restrict our attention to plain vanilla options on dividend-paying stocks and derive the integral equations to determine the free boundary and the price of American put options using Mellin transforms. We recover a result found by Kim (1990) regarding the optimal exercise price of American put options at expiry and prove the equivalence of integral representations herein, the representation derived by Kim (1990), Jacka (1991), and by Carr et al. (1992). Finally, we extend the results obtained in Panini and Srivastav (2005) and show how the Mellin transform approach can be used to derive the valuation formula for perpetual American put options on dividend-paying stocks. --Mellin transform,Power option,American put option,Free boundary,Integral representation

On modified Mellin transforms, Gauss-Laguerre quadrature, and the valuation of American call options

We extend a framework based on Mellin transforms and show how to modify the approach to value American call options on dividend paying stocks. We present a new integral equation to determine the price of an American call option and its free boundary using modi ed Mellin transforms. We also show how to derive the pricing formula for perpetual American call options using the new framework. A recovery of a result due to Kim (1990) regarding the optimal exercise price at expiry is also presented. Finally, we apply Gauss-Laguerre quadrature for the purpose of an efficient and accurate numerical valuation. --Modified Mellin transform,American call option,Integral representation

A series of Ramanujan, two-term dilogarithm identities and some Lucas series

We study an elementary series that can be considered a relative of a series studied by Ramanujan in Part 1 of his Lost Notebooks. We derive a closed form for this series in terms of the inverse hyperbolic arctangent and the polylogarithm. Special cases will follow in terms of the Riemann zeta and the alternating Riemann zeta function. In addition, some trigonometric series will be expressed in terms of the Clausen functions. Finally, a range of new two-term dilogarithm identities will be proved and some difficult series involving Lucas numbers will be evaluated in closed form.Comment: 26 pages, no figures or table

Balancing polynomials, Fibonacci numbers and some new series for π\pi

We evaluate some types of infinite series with balancing and Lucas-balancing polynomials in closed form. These evaluations will lead to some new curious series for $\pi$ involving Fibonacci and Lucas numbers. Our findings complement those of Castellanos from 1986 and 1989.Comment: 16 pages, 5 table

Valuing Options in Heston’s Stochastic Volatility Model: Another Analytical Approach

We are concerned with the valuation of European options in Heston's stochastic volatility model with correlation. Based on Mellin transforms we present new closed-form solutions for the price of European options and hedging parameters. In contrast to Fourier-based approaches where the transformation variable is usually the log-stock price at maturity, our framework focuses on transforming the current stock price. Our solution has the nice feature that similar to the approach of Carr and Madan (1999) it requires only a single integration. We make numerical tests to compare our results to Heston's solution based on Fourier inversion and investigate the accuracy of the derived pricing formulae