《Path integral approach to the pricing of timer option》.pdf

Path integral approach to the pricing of timer options with the Duru-Kleinert time transformation L. Z. J. Liang,1 D. Lemmens,1 and J. Tempere1, 2 1 TQC, Universiteit Antwerpen, Universiteitsplein 1, 2610 Antwerpen, Belgium 2Lyman Laboratory of Physics, Harvard University, Cambridge, MA 02138. 1 1 (Dated: January 20, 2011) 0 2 Abstract n a In this paper, a time substitution as used by Duru and Kleinert in their treatment of the hydrogen J 9 atom with path integrals is performed to price timer options under stochastic volatility models. 1 We present general pricing formulas for both the perpetual timer call options and the finite time- ] R horizon timer call options. These general results allow us to find closed-form pricing formulas for P . n both the perpetual and the finite time-horizon timer options under the 3/2 stochastic volatility i f - model as well as under the Heston stochastic volatility model. For the treatment of timer option q [ under the 3/2 model we will rely on the path integral for the Morse potential, with the Heston 1 v model we will rely on the Kratzer potential. 3 1 7 PACS numbers: 89.65.Gh, 05.10.Gg, 02.30.Sa 3 . 1 Keywords: Duru-Kleinert transformation, timer options, path integrals 0 1 1 : v i X r a 1 I. INTRODUCTION Timer options, first introduced for sale by Societe Generale Corporate and Investment Banking (SG CIB) in 2007 [1, 2], are relatively new products in the equity volatility market. The basic principle of this option is similar to the European vanilla option, with the key distinction being the