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Small Dimension PDE for Discrete Asian Options Eric BenHamou (LSE, UK) & Alexandre Duguet (LSE, UK) CEF2000 Conference (Barcelona) Plan • • • • • • Introduction How to reduce the dimension Homogeneous case Extension to non Homogeneous case Numerical results Conclusion 6-8 July 2000 CEF 2000 Conference Slide N°2 Introduction to the Asian Option • Origin and motivation – spot manipulation – periodic cash flows • Definition: – type of averaging – fixed or floating strike • Pricing problem? 6-8 July 2000 CEF 2000 Conference Slide N°3 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) 6-8 July 2000 CEF 2000 Conference Slide N°4 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 6-8 July 2000 CEF 2000 Conference Slide N°5 Notations and Assumptions • Continuous time trading economy with an infinite horizon. Complete market with absence of arbitrage • St underlying modelled by a diffusion equation: dSt rt St dt St t , St dWt Wt tR one dimensional Brownian motion • Discrete non proportional dividends D 6-8 July 2000 CEF 2000 Conference Slide N°6 Different PDEs • Traditional PDEs (Ingersoll 87 Forsyth Vetzal Zvan 98) 1 2 C t σ t, S S2 CSS rS t CS St C I rC 0 2 • Change of Variable: (adaptation of Roger an Shi 95) g t Ag 0 1 2 2 2 n y ry t 2 i 1 i ti y y 2 2 2 1 2 2 t , s s r y t 2 2 s s sy 6-8 July 2000 CEF 2000 Conference Slide N°7 Change of variable T e t • Payoff: rs ds T S s ds K | t 0 • “Homogenised” Payoff T S f t , k , St s ds k | t t St T g t , y, s e t rs ds 6-8 July 2000 f t , y, s CEF 2000 Conference Slide N°8 Rewriting of the Payoff T e t rs ds T S s ds K | t 0 T S t S rs ds e t St s ds K s ds | t 0 S t St t T T e t rs ds t S s St f t , K ds , St , St 0 S t • Ideal case: Homogeneity!!! 6-8 July 2000 CEF 2000 Conference Slide N°9 Black Scholes case (1/2) • Crucial property of Homogeneity!!! CBS S , K C S , K K K0 CBS S , K C S , K0 K0 K • So method for an Asian option on S2 S1 – Calculate a call on with strike K 2 2 – Payoff rT2 S1 S 2 e E K 2 6-8 July 2000 CEF 2000 Conference S1 S 2 2 Slide N°10 Black Scholes case (2/2) for any value at date 2 2K call exercised S1e r T2 T1 S1 r T2 T1 value at time 1 Ke 2 S1 S1 • if S1 2K call is equal to C T2 T1 , , K 2 2 • if S1 obtained because we calculate for any S CT2 T1 , S , K 6-8 July 2000 CEF 2000 Conference Slide N°11 Numerical Scheme • Θ-Schema for PDE Crank Nicholson Scheme on the Log of the underlying • Linear interpolation 50% r 5% T2 2 y T1 1y • Good Results 6-8 July 2000 Strike 100% 110% PDE 24.47 20.81 MC 24.46 20.81 CEF 2000 Conference Slide N°12 Extension to non-homogeneous Case (1) • Smile (Dupire model 93) t, St – Non homogeneous model – form of implied vol 2 K FT K ST BS BS t, St 0 1 Sm F Cu K T • vega correction C S , K , S , K C S , K , Vega BS S , K 6-8 July 2000 CEF 2000 Conference Slide N°13 Extension to non-homogeneous Case (2) • Numerical Results 3.5 20 With smile 15 Without smile 10 180% call price 25 110% call price K 180% 4 K 110% 30 3 2.5 With smile 2 Without smile 1.5 Vega correction 1 5 0.5 0 0 50 100 50 150 150 Spot Spot 6-8 July 2000 100 CEF 2000 Conference Slide N°14 Dividend case • Same sort of correction CS , K , D CS , K , D 1 CS , K , D CS , K ,0 40 35 Call price 30 25 With dividend 20 Without dividends 15 Dividend correction 10 5 0 50 100 150 Spot 6-8 July 2000 CEF 2000 Conference Slide N°15 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 6-8 July 2000 CEF 2000 Conference Slide N°16