Rainbows and transforms: semi-analytic formulas

In this paper we show how the techniques introduced by Hurd and Zhou in 2010 can be used to derive a pricing framework for rainbow options by using the joint characteristic function of the logarithm of the underlying assets. Semi-analytic formulas will be achieved by splitting the option payoff function, applying different dampings to its constituent components and finally calculating the Fourier transforms of said components, allowing the pricing of call and put options on the maximum and minimum of multiple assets; we shall also revisit Margrabe’s classical formula. Our work extends a number of classical results that focus on geometric Brownian motion dynamics and it also provides an alternative pricing methodology to that proposed in some of the most recent literature.