Review of Derivatives Research

, Volume 15, Issue 2, pp 99–128 | Cite as

Unifying exotic option closed formulas



This paper aims to unify exotic option closed formulas by generalizing a large class of existing formulas and by setting a framework that allows for further generalizations. The formula presented covers options from the plain vanilla to most, if not all, mountain range exotic options and is developed in a multi-asset, multi-currency Black–Scholes model with time dependent parameters. It particular, it focuses on payoffs that depend on the distributions of the underlyings prices at multiple but set time horizons. The general formula not only covers existing cases but also enables the combination of diverse features from different types of exotic options. It also creates implicitly a language to describe payoffs that can be used in industrial applications to decouple the functions of payoff definition from pricing functions. Examples of several exotic options are presented, benchmarking the closed formulas’ performance against Monte Carlo simulations. Results show a consistent over performance of the closed formula reducing calculation time by double digit factors.


Exotic options Mountain range Discrete lookback Closedformula Payoff language Multi-asset multi-currency model 

JEL Classification



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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Carlos Veiga
    • 1
  • Uwe Wystup
    • 2
  • Manuel L. Esquível
    • 3
  1. 1.Centro de Matemática e Aplicações da Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa (CMA/FCT/UNL)CaparicaPortugal
  2. 2.MathFinance AGWaldemsGermany
  3. 3.Departamento de Matemática FCT/UNL & CMA/FCT/UNLQuinta da TorreCaparicaPortugal

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