Transcript An application of Malliavin calculus: Reducing the

Optimal Malliavin weighting functions
for the simulations of the Greeks
MC 2000 (July 3-5 2000)
Eric Ben-Hamou
Financial Markets Group
London School of Economics, UK
[email protected]
Outline
•
•
•
•
•
Introduction & motivations
Review of the literature
Results on weighting functions
Numerical results
Conclusion
3-5 July 2000
MC 2000 Conference
Slide N°2
Introduction
• When calculating numerically a quantity
– Do we converge? to the right solution?
– How fast is the convergence?
• Typically the case of MC/QMC simulations
especially for the Greeks important measure
of risks, emphasized by traditional option
pricing theory.
3-5 July 2000
MC 2000 Conference
Slide N°3
Traditional method for the Greeks
– Finite difference approximations: “bump and recompute”
– Errors on differentiation as well as convergence!
– Theoretical Results: Glynn (89) Glasserman and Yao
(92) L’Ecuyer and Perron (94):
– smooth function to estimate:
- independent random numbers: non centered scheme:
convergence rate of n-1/4 centered scheme n-1/3
- common random numbers: centered scheme n-1/2
– rates fall for discontinuous payoffs
3-5 July 2000
MC 2000 Conference
Slide N°4
How to solve the poor convergence?
• Extensive litterature:
– Broadie and Glasserman (93, 96) found, in simple cases, a
convergence rate of n-1/2 by taking the derivative of the
density function. Likelihood ratio method.
– Curran (94): Take the derivative of the payoff function.
– Fournié, Lasry, Lions, Lebuchoux, Touzi (97, 2000)
Malliavin calculus reduces the variance leading to the same
rate of convergence n-1/2 but in a more general framework.
– Lions, Régnier (2000) American options and Greeks
– Avellaneda Gamba (2000) Perturbation of the vector of
probabilities.
– Arturo Kohatsu-Higa (2000) study of variance reduction
– Igor Pikovsky (2000):
condition on the diffusion.
3-5 July 2000
MC 2000 Conference
Slide N°5
Common link:
• All these techniques try to avoid differentiating the
payoff function:
TheGreek  EPayoff *Weight
• Broadie and Glasserman (93)

ln p ST , 

– Weight = likelihood ratio
– should know the exact form of the density function


E F ST    F S 
ln pS ,  pS , dS





 E  F ST 
ln pS , 



3-5 July 2000
MC 2000 Conference
Slide N°6
• Fournié, Lasry, Lions, Lebuchoux, Touzi (97,
2000) : “Malliavin” method
• does not require to know the density but the
diffusion.
• Weighting function independent of the payoff.
• Very general framework.
• infinity of weighting functions.
• Avellaneda Gamba (2000)
• other way of deriving the weighting function.
• inspired by Kullback Leibler relative entropy
maximization.
3-5 July 2000
MC 2000 Conference
Slide N°7
Natural questions
• There is an infinity of weighting functions:
– can we characterize all the weighting functions?
– how do we describe all the weighting functions?
• How do we get the solution with minimal variance?
– is there a closed form?
– how easy is it to compute?
• Pratical point of view:
– which option(s)/ Greek should be preferred? (importnace
of maturity, volatility)
3-5 July 2000
MC 2000 Conference
Slide N°8
Weighting function description
• Notations (complete probability space,
uniform ellipticity, Lipschitz conditions…)
dXt  bt , X t dt   t , X t dWt
• Contribution is to examine the weighting
function as a Skorohod integral and to
examine the “weighting function generator”
3-5 July 2000
MC 2000 Conference
Slide N°9
Integration by parts
• Conditions…Notations
EDt  X T    E X T  
• Chain rule
EDt  X T    E'  X T Dt X T 
• Leading to




E X T ,   E '  X T 
X T , 




3-5 July 2000
MC 2000 Conference
Slide N°10
Necessary and sufficient conditions



E '  X T 
X T ,   E'  X T Dt X T 



• Condition


E
X T , | X T   EDt X T | X T 
 

• Expressing the Malliavin derivative


Yt
 

E
X T , | X T   E  t , X t  1 X T  | X T 
YT
 



3-5 July 2000
MC 2000 Conference
Slide N°11
Minimal weighting function?
• Minimum variance of E X T Weight
• Solution: The conditional expectation
with respect to X T EWeight | X T 
• Result: The optimal weight does
depend on the underlying(s) involved
in the payoff
3-5 July 2000
MC 2000 Conference
Slide N°12
For European options, BS
• Type of Malliavin weighting functions:
WT 
  rT
  E e f ST 
xT 

  rT
1
  E e f ST 
2

Tx

 WT 2
1 



W

T
 T
 

  rT
 WT 2
1 
v  E e f ST 
 WT  
 

 T
  rT
 WT

  E e f ST 
 T 
 


3-5 July 2000
MC 2000 Conference
Slide N°13
Typology of options and remarks
• Remarks:


  E e  rT f ST 
WT 
xT 
– Works better on second order differentiation…
Gamma, but as well vega.
– Explode for short maturity.
– Better with higher volatility, high initial level
– Needs small values of the Brownian motion (so
put call parity should be useful)
3-5 July 2000
MC 2000 Conference
Slide N°14
Finite difference versus Malliavin
method
• Malliavin weighted scheme: not payoff
sensitive
• Not the case for “bump and re-price”

– Call option E ST

x 
K
 
  ES

x 
T

K


1/ 2



x 
x 
E  ST
 K  ST
K



1
/
2
x 
x  2 

 E ST
 ST
 2 2 E e rT WT


 2

3-5 July 2000
MC 2000 Conference
Slide N°15

• For a call

x 
E  ST
K

 O 

  S

x 
T
K

 2
1/ 2


• For a Binary option

E 1S

x 
T
 E 1S
O
3-5 July 2000
K
x 
 
T
  1S
x 
T
 K  ST x


K


2 1/ 2
1/ 2
MC 2000 Conference
Slide N°16
Simulations (corridor option)
0
Gamma Value
-0.0005
-0.001
-0.0015
Malliavin Simulation
Finite Difference
-0.002
Exact value -0.000917
Simulations Number
-0.0025
1
3-5 July 2000
101
201
301
401
MC 2000 Conference
501
601
701
Slide N°17
801
Simulations (corridor option)
-0.003
Malliavin Simulation
-0.00325
Finite Difference
-0.0035
Delta Value
Exact value -0.00411
-0.00375
-0.004
-0.00425
-0.0045
-0.00475
Simulations Number
-0.005
1
3-5 July 2000
101
201
301
401
MC 2000 Conference
501
601
701
Slide N°18
801
Simulations (Binary option)
0
1
-0.00025
2501
5001
7501
10001
Simulations Number
-0.0005
Gamma Value
-0.00075
-0.001
-0.00125
-0.0015
Malliavin Simulation
-0.00175
Finite Difference
-0.002
Exact value -0.001057
-0.00225
-0.0025
3-5 July 2000
MC 2000 Conference
Slide N°19
Simulations (Binary option)
0.0215
0.0214
0.0213
Delta Value
0.0212
0.0211
0.021
Malliavin Simulation
Finite Difference
0.0209
Exact value 0.02113
0.0208
Simulations Number
0.0207
1
3-5 July 2000
5001
MC 2000 Conference
10001
Slide N°20
Simulations (Call option)
0.0225
Gamma Value
0.02
Malliavin Simulation
0.0175
Finite Difference
Exact value 0.02007
Simulations Number
0.015
1
3-5 July 2000
1301
2601
3901
5201
MC 2000 Conference
6501
7801
9101
Slide N°21
10401
Simulations (Call option)
0.78
0.7775
0.775
0.7725
Delta Value
0.77
0.7675
0.765
Malliavin Simulation
0.7625
Finite Difference
0.76
Exact value 0.7735
0.7575
0.755
0.7525
Simulations Number
0.75
1
3-5 July 2000
1301
2601
3901
5201
MC 2000 Conference
6501
7801
9101
Slide N°22
10401
Conclusion
• Gave elements for the question of the
weighting function.
1/ n
T
n

Max0u T Su   lim n   Su du 
 0

• Extensions:
–
–
–
–
3-5 July 2000
Stronger results on Asian options
Lookback and barrier options
Local Malliavin
Vega-gamma parity
MC 2000 Conference
Slide N°23