This paper addresses a finite difference approximation for an infinite dimensional Black-Scholes equation obtained in Chang and Youree [5]. The equation arises from a consideration of an European option pricing problem in a market in which stock prices and the riskless asset prices have hereditary structures. Under a general condition on the payoff function of the option, it is shown that the p...

We propose a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. Our approach combines different high-order spatial discretisations with Hundsdorfer and Verwer’s ADI time...

In this work, we derive an analytical solution for the value of Parisian up-and-in calls by using the “moving window” technique developed by Zhu and Chen [15] for pricing European-style Parisian up-and-out calls. Our pricing formula can be applied to both European-style and American-style Parisian up-and-in calls, due to the fact that with an “in” barrier, the option holder cannot do or decide ...

In this paper the problem of European option valuation in a Levy process setting is analysed. In our model the underlying asset follows a geometric Levy process. The jump part of the log-price process, which is a linear combination of Poisson processes, describes upward and downward jumps in price. The proposed pricing method is based on stochastic analysis and the theory of fuzzy sets. We assu...

Different from the conventional methods of probability theory, this paper discusses the currency option pricing problem within the framework of uncertainty theory. Considering the uncertain factors in the financial market, we propose a new uncertain currency model with floating interest rates based on the assumption that the domestic interest rate, the foreign interest rate and the exchange rat...

Derivative securities, when used correctly, can help investors increase their expected returns and minimize their exposure to risk. Options offer leverage and insurance for risk-averse investors. For the more risky investors, they can be ways of speculation. When an option is issued, we face the problem of determining the price of a product which depends on the performance of another security a...

Closed form option pricing formulae explaining skew and smile are obtained within a parsimonious non-Gaussian framework. We extend the non-Gaussian option pricing model of L. Borland (Quantitative Finance, 2, 415-431, 2002) to include volatility-stock correlations consistent with the leverage effect. A generalized Black-Scholes partial differential equation for this model is obtained, together ...

This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that the stock price is log-normally distributed and that the universe is riskneutral. Then, using Ito’s Lemma, we will justify the use of the risk-neutral rate in these initial calculations. Finally, we will prove put-call parity in order to price European...

Consider an investment system with a nonnegative expected return in a one period economy. We show that, for an option with a given strike price, there exists a pricing interval [pC , pW ] such that replacing the original investment with the option will benefit judging by the Kelly criterion only when the price of the option lies outside of the interval. More specifically, buying call options wi...