The optimal geometric mean return is an important property of an asset. As a derivative of the underlying asset, the option also has this property. In this paper, we show that the optimal geometric mean returns of a stock and its option are the same from Kelly criterion. It is proved by using binomial option pricing model and continuous stochastic models with self-financing assumption. A simula...

We analyse the use of options for online advertisement publishers. By providing a discount or rewards to advertisers, publishers can utilise their uncertain service capacity, page-views, more efficiently. We use Generalised Nash Bargaining to study the feasibility of the option contract and solve for an optimal value for the option price. We compare the revenues and benefits from advertisements...

Volatility modelling has become a significant area of research within Financial Mathematics. Wiener process driven stochastic volatility models have become popular due their consistency with theoretical arguments and empirical observations. However such models lack the ability to take into account long term and fundamental economic factors e.g. credit crunch. Regime switching models with mean r...

In this paper we will develop a methodology for obtaining pricing expressions for financial instruments whose underlying asset can be described through a simple continuous-time random walk (CTRW) market model. Our approach is very natural to the issue because it is based in the use of renewal equations, and therefore it enhances the potential use of CTRW techniques in finance. We solve these eq...

The Fourier transform of the density for the logarithm of the stock price has seen numerous …nancial applications. For a theoretical perspective we cite as examples Du¢ e, Pan and Singleton (2000), and Bakshi and Madan (2000). This transform has become a standard calibration engine following the methods of Carr and Madan (1999) who invoked the Fast Fourier transform for its speed. Direct Fourie...

Monte Carlo simulation is a popular method for pricing financial options and other derivative securities because of the availability of powerful workstations and recent advances in applying the tool. The existence of easy-to-use software makes simulation accessible to many users who would otherwise avoid programming the algorithms necessary to value derivative securities. This paper presents ex...

This paper provides a short introduction into the mathematical foundations of the theory of valuing derivative securities. We discuss the mathematical setting of option price theory and derive the relationship between no-arbitrage and mar-tingales. We provide examples how this theory can be applied to equity and foreign-exchange derivatives. We also explain how the theory can be applied to inte...

We study the approximation of certain stochastic integrals with respect to a d-dimensional diffusion by corresponding stochastic integrals with piece-wise constant inte-grands i.e. an approximation of the form d k=1 T 0 N k s dX k s ≈ d k=1 n i=1 N k t i−1 (X k t i − X k t i−1). In finance this corresponds to replacing a continuously adjusted portfolio by discretely adjusted one. The approximat...

This paper develops a nonparametric option pricing theory and numerical method for European, American and path-dependent derivatives. In contrast to the nonparametric curve fitting techniques commonly seen in the literature, this nonparametric pricing theory is more in line with the canonical valuation method developed Stutzer (1996) for pricing options with only a sample of asset returns. Unli...