Spread options, and more generally basket options, comprise a fundamental class of derivative contracts widely used in a range of financial markets. Due to the dependence of their payoff on the dynamics of more than one underlying risky asset, the valuation of spread and basket options is fundamentally different from the valuation of other types of options. Generally no closed form formulas exist, so approximation methods are used to obtain option values. One approach is to employ numerical methods such as Monte Carlo simulations, numerical integration and fast Fourier transforms. Alternatively, the price process of the spread/basket can be approximated with that of a single pseudo-asset, and then calculate the approximate price of the option using one of the closed-form formulas for the price of an option on a single underlying asset. Yet another stream of research develops analytical methods applicable to log-normal models. There is no preferred approach that is accurate, efficient and flexible enough to apply in general models. Numerical methods tend to be more accurate, but their computing times can be much longer compared to analytical approximation methods.

This research focuses on developing analytical approximations. One method for basket options is to match the first two moments of the basket with that of the pseudo-asset. In our ongoing research, we match the Laplace transform of the basket price with the Laplace transform of the price of the pseudo-asset. Another line of investigation is dedicated to deriving analytical approximations for a spread option price in the geometric Brownian motion case. A change of measure is employed to express the spread option value in terms of a related quantity based on the ratio of the underlying asset prices. Analytical approximations for the spread option price are then obtained using perturbation techniques, as well as statistical series expansion pricing methods.

Researchers: Gerald CheangJohn van der HoekMalgorzata Korolkiewicz

#### Key publications

J van der Hoek & MW Korolkiewicz, 'New analytic approximations for pricing spread options', in Stochastic Processes, Finance and Control. A Festschrift in Honor of Robert J. Elliott, S Cohen & K Siu (eds), World Scientific, pp. 259-284 (in press).

GHL Cheang, C Chiarella & Andrew Ziogas, 'On Exchange Options with Jumps',Proceedings of the Third IASTED International Conference on Financial Engineering and Applications, pp. 104 - 109, Acta Pres, 2006.

## Areas of study and research

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