Journal of Computational Finance

Risk.net

Rainbows and transforms: semi-analytic formulas

Norberto Laghi

  • We introduce semi-analytic formulae to price call and put options on the maximum and minimum of a set of risky assets.
  • We use a link between best of asset options and best of call options originating in Margrabe’s work to show that the latter can be prices by using the former.
  • Building on these ideas we prove a novel pricing formula for best of asset options that leads to the same dimensionality reduction as in Margrabe’s work, but without requiring any change of measure or distributional assumption besides knowledge of the characteristic function.
  • We further show that the geometric basket option problem is essentially one-dimensional independently of distributional assumptions, and provide an appropriate formula.

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.

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