1. Small Dimension PDE for Discrete Asian Options Eric BenHamou (LSE, UK) & Alexandre Duguet (LSE, UK) CEF2000 Conference (Barcelona)

2. 6-8 July 2000CEF 2000 ConferenceSlide N°2 Plan Introduction How to reduce the dimension Homogeneous case Extension to non Homogeneous case Numerical results Conclusion

3. 6-8 July 2000CEF 2000 ConferenceSlide N°3 Introduction to the Asian Option Origin and motivation –spot manipulation –periodic cash flows Definition: –type of averaging –fixed or floating strike Pricing problem?

4. 6-8 July 2000CEF 2000 ConferenceSlide N°4 Different methods Closed forms solutions –Geometric approximations (Vorst 92 96) –density distributions assumptions (Turnbull Wakeman 91, Levy 92 Jacques 95, Milevsky Posner 97) –Laplace Transform (Geman Yor 93) (Madan Yu 95) Numerical methods –Monte Carlo ( Kemma Vorst 90 96) –Fast Fourier Transform PDE (Caverhill Clewlow 92 Benhamou 2000) –PDE (Roger Shi 95 He and Takashi 96 Alziary et al. 97 Forsyth et al. 98)

5. 6-8 July 2000CEF 2000 ConferenceSlide N°5 Motivations Find a numerical method consistent with: –Smile model: (Dupire 93 Derman Kani 94) –Discrete non proportional dividend Use PDE method to solve this problem: –determination of the PDE –dimension reduction problem

6. 6-8 July 2000CEF 2000 ConferenceSlide N°6 Notations and Assumptions Continuous time trading economy with an infinite horizon. Complete market with absence of arbitrage S t underlying modelled by a diffusion equation: one dimensional Brownian motion Discrete non proportional dividends

7. 6-8 July 2000CEF 2000 ConferenceSlide N°7 Different PDEs Traditional PDEs (Ingersoll 87 Forsyth Vetzal Zvan 98) Change of Variable: (adaptation of Roger an Shi 95)

8. 6-8 July 2000CEF 2000 ConferenceSlide N°8 Change of variable Payoff: “Homogenised” Payoff

9. 6-8 July 2000CEF 2000 ConferenceSlide N°9 Rewriting of the Payoff Ideal case: Homogeneity!!!

10. 6-8 July 2000CEF 2000 ConferenceSlide N°10 Black Scholes case (1/2) Crucial property of Homogeneity!!! So method for an Asian option on –Calculate a call on with strike –Payoff

11. 6-8 July 2000CEF 2000 ConferenceSlide N°11 Black Scholes case (2/2) if call exercised for any value at date 2 value at time 1 if call is equal to obtained because we calculate for any

12. 6-8 July 2000CEF 2000 ConferenceSlide N°12 Numerical Scheme Θ-Schema for PDE Crank Nicholson Scheme on the Log of the underlying Linear interpolation Good Results

13. 6-8 July 2000CEF 2000 ConferenceSlide N°13 Extension to non-homogeneous Case (1) Smile (Dupire model 93) –Non homogeneous model – form of implied vol vega correction

14. 6-8 July 2000CEF 2000 ConferenceSlide N°14 Extension to non-homogeneous Case (2) Numerical Results

15. 6-8 July 2000CEF 2000 ConferenceSlide N°15 Dividend case Same sort of correction

16. 6-8 July 2000CEF 2000 ConferenceSlide N°16 Conclusion Method efficient for these more realistic cases –smile –non proportional discrete dividends Extensions –to other options like Ratchet and path dependent options –control of the error