代写 statistic STAD70 Statistics & Finance II

STAD70 Statistics & Finance II
6. Variance Reduction Techniques
1

Variance Reduction Techniques
 Look at 4 techniques for reducing MC estimation variance
1. Antithetic Variables
2. Stratification
3. Control Variates
4. Importance Sampling
 Different techniques give different variance reduction, depending on problem at hand
2

Antithetic Variables
 Idea: for each Normal random deviate Zi consider its antithetic variable Zi
 Both come from same distr., but are dependent  Calculate discounted payoff (call it Y) under
bothvariablesY  f(Z), Y  f(Z) iiii
 Estimate price as
Y 1n Yn Y1n YY
ii AV2ni1i i1ini12
 Balance payoffs of paths with opposite returns
3

Example
 Find asymptotic distr. of antithetic variable
Y Y ii
2
estimator in terms of moments of
4

Antithetic Variables
 Simple & broadly applicable technique, but does not always offer improvements
 Technique is helpful when payoff of original & antithetic variable is negatively related
 Proof: helpful if Var Y   Var  AV 
 
1 2n 
2n
i1
Y i

5

Example
 Antithetic Variable pricing of European call
n
Y
YAV
s.e.(Y )
s.e.(YAV )
50
5.8626
4.7646
0.7617
0.4623
250
4.7019
4.7439
0.3018
0.2362
500
4.3211
4.7834
0.2013
0.1722
2500
4.7537
4.6539
0.1017
0.0734
5000
4.6634
4.6923
0.0704
0.0531
25000
4.7503
4.7024
0.0317
0.0238
50000
4.7046
4.6941
0.0224
0.0166
true Black-Scholes price = 4.7067
6

Stratification
 Idea: split RV domain into equiprobable strata & draw equal # of deviates from each one
 E.g. (2 strata) draw equal number of independent positive & negative Zi
 Stratification ensures equal representation for every area in domain of RV
 Always reduces variance, but could be marginally  Method works best when function (payoff)
changes over RV’s domain
 Computationally difficult for multidimensional RV’s7

Stratification
 Consider #m equiprobable Normal strata {Aj} P(ZAj)1/mforZ~N(0,1),j1, ,m
 Stratified estimator of Y  f (Z)
Y 1m Y(j),whereY(j) 1n f(Z(j))
Str
LetY 1 nm f(Zi)beestimatorofEf(Z) nm i1
m j1 n i1 Z(j)~iidN(0,1|Z(j)A),j1, ,m
i
iij
 Y ( j ) is estimator within each stratum j
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Stratification
 Show that Y is unbiased estimator of E  f (Z ) Str
9

Stratification
 Show that Var[Y ]  Var[Y ] Str
10

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Example
 Stratified pricing of European call
m
n
Y Str
s.e.(Y ) Str
1
10000
4.6908
0.0712
10
1000
4.7174
0.0207
20
500
4.7303
0.0136
50
200
4.7065
0.0088
100
100
4.7062
0.0054
200
50
4.7124
0.0046
500
20
4.7047
0.0024
1000
10
4.7068
0.0021
true Black-Scholes price = 4.7067
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Control Variates
 To estimate E[Y ]  E[ f (Z )] using MC:
generate. iid Zi & use Y  n Y / n 

n f (Z ) / n
i1
 Assume there is another option with payoff g
 For our purposes,
whose price E[X] E[g(Z)] we already know
 Idea: Use MC (i.e. Y , X ) to estimate both E[Y ] and E[X ], but adjust estimate Y to take into account the error of estimate X
 E.g. ifX underestimates E[X], then increaseY
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i
i
f i1
is option’s discounted payoff

Control Variates
 Adjust Y for estimation error X  E[ X ] as Y(b)Y bX E[X]
 Coefficient b controls adjustment
 Show Y (b) is unbiased (for unbiased Y , X )
 Find variance of Y (b)
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Control Variates
 Show that optimal value of b is
b* Cov[X,Y] Var[X]
 In practice, don’t know Cov[X,Y],Var[X] , so estimate b* using sample estimates
ˆ n (XX)(YY) bi1i i
n i1
(X X)2 i
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Control Variates
Y,X  ii
Y
Y(b)
slope b
E[X]
X
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Control Variates
 Show optimal variance is
Var[Y(b*)]Var[Y] 12 
XY
 In practice, use sample estimates of Var[Y ], XY
Good control variates have high absolute correlation with option payoff
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Example
 Price European option using final asset price (ST) as control, assuming GBM w/ r,σ
XST g(Z) E[X] E[ST ]
 Guess correlation of control with following  In-the-money call:
 Out-of-the-money call:
 In-the-money put:
 Out-of-the-money put:
 
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Example
 European call with ST as control YerT(ST K)
S0K100,T.5   r  .05,   .2, n  104 

Mean
Std. Error
Simple MC
6.927122
0.0986554
Control Var.
6.914076
0.0411446
(Exact Black-Scholes price: 6.888729)
ˆXY  0.908882
X  ST
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Importance Sampling
 Idea: attempt to reduce variance by changing the distribution (probability measure) from which paths are generated
 Change measure to give more weight to important outcomes, thereby increasing sample efficiency
 E.g. for European call, put more weight to paths with positive payoff (i.e. for which we exercise)
 Performance of importance sampling relies heavily on changed measure used
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Importance Sampling
 Want to estimate   E [ f (Z)]   f (z)(z)dz z
where (z) is pdf of Z (in our case, Normal)  Simple MC: generate sample Zi ~iid ,i 1, ,n
ˆ andusen f(Z)/n
i
i1 Assumingyouhavesample Z~iid ,i1, ,n
i
from new pdf ψ, can still estimate α as follows
  f(z)(z)dz f(z)(z)(z)dzE [f(Z)(Z)]
(z)  (Z)
zz
ˆ 1n (Z)
  f(Z)
n i1 (Z) i
i
i 21

Importance Sampling
 Show importance sampling estimateˆ is unbiased (provided simple MC estimate ˆ is)
 Find variance of ˆ
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Importance Sampling
ˆ ˆ
 Show that Var[ ]  Var[ ] only if
E f2(Z)(Z)E f2(Z)
 (Z)   
23

Importance Sampling
ˆ
 Show that Var[ ]  0 if  (z)  f (z)(z),
for positive function f
Importance sampling works best when new pdf ψ “resembles” payoff×(original pdf) f×φ
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Importance Sampling
 Importance sampling can be extended to multiple random variates per path
f(Z1, ,Zm) E f(Z, ,Z )E f(Z, ,Z)(Z, ,Z)
 E.g. path-dependent option, with payoff
a function of #m variates forming discretized path
1m1m1m (Z, ,Z)
1m  If in addition, Z ~  & Z ~  , then
jjjj
    m(Z) E f(Z, ,Z)Ef(Z, ,Z) j j 

1 m  1 m jj
j1(Z)
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Example
 Importance sampling for deep out-of-the- money European call S0  50, K  65
 With simple MC, generate final prices as 2 
S Se,whereZ~N r Z2
T0 2
 What would be a good candidate for ψ?
T,T   
902 302 
Z ~ N log  T, T orN log  T, T  2 2
 50 2   50 2

26

Example
(Z)~Normal
payoff f(Z)
(Z)
Z
(Z) f (Z)
27

Example
(ST ) (log-Normal)
(ST ) f (ST )
payofff(ST)ST K
 (ST )
ST
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Example
 European call
using importance sampling with
S0 50, K65,T1, r.02, .2,n104 

902 
 (Z)  N log  T, 2T   E S  90 S2T
0
Mean
Std. Error
Simple MC
0.618857
0.02670904
Importance Sampling
0.6153817
0.007132492
(Exact Black-Scholes price: 0.6160138)
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