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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 tR 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
sy 

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
CT2  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
CS , K , D   CS , K , D  1   CS , K , D  CS , 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