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Pricing double barrier options using Laplace transforms. (English) Zbl 0940.91026

Barrier options have become very popular instruments in derivative markets as they are cheaper than standard options but offer a similar kind of protection. It is relatively straightforward to price and hedge “single barrier” options. A natural extension to “single barrier” options is double barrier options. These options have a barrier above and below the price of the underlying, and the option gets knocked in or out as soon as one of the two barriers is hit. Several papers have already analysed double knock-out and put options and derived expressions for the Laplace transform of the double barrier option price. They invert the Laplace transform numerically to obtain option prices. These papers deal, however, with one type of double barrier option: double barrier knock-out calls and puts. In this paper the author addresses the pricing of double barrier options. To derive the density function of the first-hit times of the barriers the Laplace transform is analytically inverted by contour integration. With these barrier densities pricing formulae for new types of barrier options are derived: knock-out barrier options which pay a rebate when either one of the barriers is hit. Furthermore, more complicated types of barrier options like double knock-in options are discussed.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
44A10 Laplace transform
35K05 Heat equation
91G20 Derivative securities (option pricing, hedging, etc.)
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