Abstract

options in the Black-Scholes model as used frequently to quote the theoretical value

(TV). These formulae allow a fast computation; all involve the normal density and

cumulative distribution function.

Key words. barrier options, digital options, touch options, lookback options, forward start

options, compound options, instalment options, pricing formulae for rst generation exotic

1 Pricing Formulae for Foreign Exchange Options 2

1.1 General Model Assumptions and Abbreviations . . . . . . . . . . . . . . . . . 2

1.2 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Digital and Touch Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Digital Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 One-Touch Options . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.3 Double-No-Touch Options . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4.2 Discrete Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Forward Start Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.1 Product De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.2 The Value of Forward Start Options . . . . . . . . . . . . . . . . . . . 15

1.5.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.6 Compound and Instalment Options . . . . . . . . . . . . . . . . . . . . . . . 16

1.6.1 Valuation in the Black-Scholes Model . . . . . . . . . . . . . . . . . . 16

1.6.2 The Curnow and Dunnett Integral Reduction Technique . . . . . . . . 17

1.6.3 A Closed Form Solution for the Value of an Instalment Option . . . . . 18