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An American option gives the holder the right, but not the obligation, to buy/sell an underlying asset from/to the writer at an agreed strike price at any time on or before the expiry date. Options are mainly used for speculation and hedging. The pricing of options is a fundamental problem in mathematical finance. One of the attractions of options is that they can be used to construct a wide range of trading strategies characterized by different payoff functions. As a preliminary step in the valuation of American options for a variety of trading strategies, in this article the pricing of perpe...

The implied volatility in the Black-Scholes framework is not a constant but a function of both the strike price (“smile/skew”) and the time to expiry. A popular approach to recovering the volatility surface is through the use of deterministic volatility function models via Dupire's equation. A new method for volatility surface calibration based on the Mellin transform is proposed. An explicit formula for the volatility surface is obtained in terms of the Mellin transform of the call option price with respect to the strike price, and a numerical algorithm is provided. Results of numeri...

With more looming uncertainties in our present financial climate and environment, models with jump–diffusion more than ever are necessary. They are suited to reproduce the large and sudden fluctuations in the level of the underlying variable, and mimic various statistical properties in observed time series. The jump–diffusion modelling setup, however, brings complexity to the valuation and hedging of derivative securities. This paper delves into the subject of hedging along with the illustration of hedging's intimate interplay with pricing. We harness the power of the Mellin transform...