Christophe Chorro - Variance reduction techniques: Around B&Schristophe.chorro.fr/docs/MC4.pdf ·...
Transcript of Christophe Chorro - Variance reduction techniques: Around B&Schristophe.chorro.fr/docs/MC4.pdf ·...
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Variance reduction techniques: Around B&S
Christophe Chorro ([email protected])
University Paris 1
July 10 2008
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 1 / 45
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Study Plan
Variance reduction techniquesIntroductionReminder on B&S ModelControl VariateAntithetic VariablesImportance samplingConditioning
Biblio: • Stochastic Calculus and Black Scholes:
http://christophe.chorro.fr/docs/CSangl.pdf
• A.G.Z Kemna and A.C.F. Vorst, A pricing method for options based onaverage asset values, J. Banking Finan., 1990, March, 113–129.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 2 / 45
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Plan
1 Variance reduction techniquesVariance reduction techniques:IntroductionReminder on B&S modelControl VariateAntithetic VariablesImportance samplingConditioning
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 3 / 45
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Variance reduction techniques: Introduction
Let (Xn)n∈N be i.i.d random variables with values in R such that E[|X1|2] < ∞.
For large n, with a confidence of 95%,
E[X1] ∈[
Snn− 1.96σ√
n,
Snn
+1.96σ√
n
]with σ2 = Var(X1).
The magnitude of the error is given by 1.96σ√n : the size of σ is fundamental forthe speed of convergence.
Idea: Reduce σ
Find Y such that E[X1] = E[Y ] and Var(X1) > Var(Y ).
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 4 / 45
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Plan
1 Variance reduction techniquesVariance reduction techniques:IntroductionReminder on B&S modelControl VariateAntithetic VariablesImportance samplingConditioning
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 5 / 45
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Brownian motion
We consider a probability space (Ω,A, P).
DefinitionStandard Brownian motion (B.M) is a stochastic process (Bt)t∈[0,T ] fulfilling :
a) B0 = 0 P-a.s.
b) B is continuous i.e t → Bt(w) is continuous for P almost all w.
c) B has independent increments: For Si t > s, Bt − Bs is independent ofFBs = σ(Bu, u ≤ s).
d) the increments of B are stationary and gaussian: For t ≥ s, Bt − Bs followsa N (0, t − s).
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 6 / 45
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Brownian motion
We consider a subdivision 0 = t0 < ... < tn = T of [0, T ]. We want to simulate
(Bt0 , ...Btn).
Idea:Btk = Btk−1 + Btk − Btk−1︸ ︷︷ ︸
N (0,tk−tk−1)⊥Btk−1 ,...,B0
.
PropositionIf (G1, ...Gn) are i.i.d N (0, 1), we define
X0 = 0, Xi =i∑
j=1
√tj − tj−1Gj i > 0.
Then(X0, ..., Xn) =
D(Bt0 , ...Btn).
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 7 / 45
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Brownian motion
0.0 0.4 0.8−
0.8
−0.
20.
40.0 0.4 0.8
−1.
5−
0.5
0.0 0.4 0.8
−0.
60.
00.
4
0.0 0.4 0.8−
0.5
0.5
Figure: 4 paths of the Brownian motion on [0, 1] generated using the precedingmethod with the regular subdivision of step 0.001.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 8 / 45
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Black Scholes model
We consider the time interval [0, T ] and r the risk free rate (supposed to beconstant) during this period.
Non-risky asset: Its dynamic is given by
S00 = 0, S0t = e
rt .
Risky asset: Under the historical probability P its dynamic is given by thefollowing SDE:
dSt = µStdt + σStdBt (1)
with initial condition S0 = x0 > 0 and where B is a standard BM under P.
Itô formula ⇒ St = x0e(µ−12 σ
2)t+σBt .
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 9 / 45
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Black Scholes model
0.0 0.4 0.81
35
sigma=1, mu=1
0.0 0.4 0.8
0.4
0.7
1.0
sigma=1, mu=−1
0.0 0.4 0.8
1.0
2.5
4.0
sigma=0.5, mu=1
0.0 0.4 0.80.
40.
71.
0
sigma=0.5, mu=−1
Figure: Simulation of a path of the risky asset in the B&S model for differentparameters
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 10 / 45
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Black Scholes model
What is, in this model, the price of a contingent claim with payoff ΦT at T?
PropositionIn the B&S model there exists a unique probablity Q ∼ P such that the priceat t = 0 of a contingent claim with payoff ΦT at T is given by
price = e−rT EQ[ΦT ].
Moreover the dynamic of the risky asset under Q is given by
dSt = rStdt + σStdWt (µ ⇔ r) (2)
where W is a standard BM under Q.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 11 / 45
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Black Scholes model
Examples: The Black Scholes Formulas
For Call options (ΦT = (ST − K )+) one has
EQ[(ST − K)+] = S0N(d1)− Ke−rTN(d2) (3)
where
d1 =log(S0K ) + (r +
σ2
2 )T
σ√
Tand d2 =
log(S0K ) + (r −σ2
2 )T
σ√
T(4)
and where N is the distribution function of a N (0, 1).
For Put options (ΦT = (K − ST )+) one has
EQ[(K− ST)+] = −S0N(−d1) + Ke−rTN(−d2). (5)
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 12 / 45
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Black Scholes model
This formulas are fundamental because
They are easy to compute in practice
They are a Benchmark to test numerical methods (for example variancereduction techniques)
In the sequel, all the variance reduction methods will be test computing byMonte carlo simulations
E[(K − eσG)+] or E[(eσG − K )+]
where G ↪→ N (0, 1) i.e computing (up to some coefficients) the price of a call
or a put option in a B&S model where S0 = 1 and r = 0.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 13 / 45
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Black Scholes model: Numerical example
Price of a call E[(eσG − K )+] with σ = 0.5 and K = 1Exact value=0.28353Estimated value (N=100) = 0.249, confidence interval at 95% : [0.164; 0.334]Estimated value (N=1000) = 0.279, confidence interval [0.248; 0.308]Estimated value (N=10000) = 0.276, confidence interval [0.267; 0.285]
Price of a put E[(K − eσG)+] with σ = 0.5 and K = 1.Exact value=0.15038Estimated value (N=100) = 0.154, confidence interval at 95% : [0.112; 0.195]Estimated value (N=1000) = 0.155, confidence interval [0.143; 0.167]Estimated value (N=10000) = 0.149, confidence interval [0.145; 0.152]
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 14 / 45
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Black Scholes model
0.5 1.0 1.5 2.0 2.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Call (green) and put (red)
K
Sta
ndar
d de
viat
ion
Figure: Standard deviation of the preceding payoffs (σ = 0.5) for different strikes
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 15 / 45
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Black Scholes model
In practical cases, the moneyness S0K of most liquid options (in our exampleS0 = 1) belongs in general to [0.7, 1.3].
Thus, It’s better to price put than call by Monte Carlo methods.
To recover the price of the call we may use the Call-Put parity:
E[(eσG − K )+] = E[(−eσG + K )+] + eσ22 − K .
Price of a call E[(eσG − K )+] with σ = 0.5 and K = 1 computed byCall-Put parity and MC
Exact value=0.28353Estimated value (N=100) = 0.281, confidence interval at 95% : [0.240; 0.320]Estimated value (N=1000) = 0.285, confidence interval [0.272; 0.297]Estimated value (N=10000) = 0.287, confidence interval [0.283; 0.291]
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 16 / 45
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Plan
1 Variance reduction techniquesVariance reduction techniques:IntroductionReminder on B&S modelControl VariateAntithetic VariablesImportance samplingConditioning
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 17 / 45
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Controle Variate
We want to compute by Monte Carlo method E[f (X )].
Idea:E[f (X )] = E[f (X )− h(X )] + E[h(X )]
where h is chosen to ensure that
E[h(X )] may be computed explicitly
Var(f (X )− h(X )) � Var(f (X )) (intuitively we look for f (X )− h(X ) small).
Example: Call-Put parity
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 18 / 45
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Controle Variate: Basket options (Homework 1)
Aim: Price by MC method a Basket put in the B&S framework
Let G1 and G2 be two independent N (0, 1) and (σ11, σ12, σ21, σ22) ∈ R4. Wedefine
S1 = eσ11G1+σ12G2
S2 = eσ21G1+σ22G2 .
For a1, a2 > 0 with a1 + a2 = 1, we are interested in the numericalcomputation (no closed form formula) of
E
K − (a1S1 + a2S2)︸ ︷︷ ︸
X
+
.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 19 / 45
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Controle Variate: Basket options (Homework 1)
First Idea: Classical MC methods
With a1 = a2 = 0.5, K = 1, σ11 = σ22 = 0.1 and σ12 = σ21 = 0.15,
Box Muller Method to generate gaussian random variables
Estimated value (N=10000) =6, 334.10−2 , confidence interval at 95% :
[6, 159.10−2; 6, 508.10−2]
Inversion Method to generate gaussian random variables:
Estimated value (N=10000) = 6, 336.10−2, confidence interval at 95% :
[6, 161.10−2; 6, 511.10−2]
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 20 / 45
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Controle Variate: Basket options (Homework 1)
Second Idea: Control variate when (σ11, σ12, σ21, σ22) are small.
We haveX ∼ eY
where
Y = a1Log(S1) + a2Log(S2) ↪→ N(0, σ2
)with
σ2 = (a1σ11 + a2σ21)2 + (a1σ12 + a2σ22)2
And we remark that
E[(K − X )+
]= E
[(K − X )+ −
(K − eY
)+
]+ C
where
C = E[(
K − eY)+
]= e
σ22 N(
log(K )σ
− σ) + KN( log(K )σ
).
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 21 / 45
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Controle Variate: Basket options (Homework 1)
We use a classical Monte Carlo Method (N = 10000) to approximate
E[(K − X )+ −
(K − eY
)+
]and deduce E
[(K − X )+
].
With a1 = a2 = 0.5, K = 1, σ11 = σ22 = 0.1 and σ12 = σ21 = 0.15,
Estimated value (N=10000) =6, 312.10−2 , confidence interval at 95% :
[6, 311.10−2; 6, 313.10−2]
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 22 / 45
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Controle Variate: Asian Options with Kemna and VorstMethodAim: Price by MC method an asian call in the B&S model
Thus we have to approximate (no closed form formula)
e−rT E
[(1T
∫ T0
Ssds − K
)+
]where ∀t ∈ [0, T ]
St = x0e(r−12 σ
2)t+σWt .
First Idea:1T
∫ T0
Ssds ≈1p
p∑k=0
S kTp
+ MC
With σ = 0.2, x0 = K = 100, r = 0.1, T = 1 and p = 50 one obtainsEstimated value (N=10000) = 6.93, confidence interval at 95% :[6.769; 7.101]
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 23 / 45
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Controle Variate: Asian Options with Kemna and VorstMethod
Second Idea: When σ and r are small Kemna and Vorst tell us that
1T
∫ T0
Ssds ≈ exp
(1T
∫ T0
log(Ss)ds
)= AT .
We are going to use the identity
e−rT E[(
1T
∫ T0 Ssds − K
)+
]= e−rT E
[(1T
∫ T0
Ssds − K
)+
− (AT − K )+
]︸ ︷︷ ︸
(1)
+ e−rT E [(AT − K )+]︸ ︷︷ ︸(2)
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 24 / 45
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Controle Variate: Asian Options with Kemna and VorstMethod
For (1) We use the first idea (Approximation by Riemann sums)
For (2) We just have to remark that
AT = exp
(1T
∫ T0
log(Ss)ds
)↪→ exp
(N ((r̃ − σ̃
2
2)T , σ̃2T )
)
where r̃ = r2 −σ2
12 and σ̃ =σ√3.
Thus we may use Black Scholes formula to compute explicitly (2).
With σ = 0.2, x0 = K = 100, r = 0.1, T = 1 and p = 50 one obtains
Estimated value (N=10000) = 6.981, confidence interval at 95% :[6.969; 6.993].
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 25 / 45
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Plan
1 Variance reduction techniquesVariance reduction techniques:IntroductionReminder on B&S modelControl VariateAntithetic VariablesImportance samplingConditioning
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 26 / 45
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Antithetic Variables
If (Xi)i∈N∗ are independent with the same distribution than X we mayapproximate E[g(X )] by
E[g(X )] ≈ 12n
2n∑k=1
g(Xk ) =1
2n
n∑k=1
g(X2k−1) + g(X2k ).
Suppose that there exists a transformation T : R → R such that
T (X ) has the same distribution than X .
In this case we may use
E[g(X )] =E[g(X ) + g(T (X ))]
2≈ 1
2n
n∑k=1
g(Xk ) + g(T (Xk )).
Question: Is Var [g(X1) + g(X2)] > Var [g(X1) + g(T (X1))]?
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 27 / 45
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Antithetic Variables
We haveVar [g(X1) + g(X2)] = 2Var [g(X1)]
and
Var [g(X1) + g(T (X1))] = 2Var [g(X1)] + 2Cov [g(X1), g(T (X1))].
So,
Cov [g(X1), g(T (X1))] < 0 ⇒ Var [g(X1) + g(X2)] > Var [g(X1) + g(T (X1))] .
Under some simple conditions on g and T ,
Cov [g(X1), g(T (X1))] < 0 holds.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 28 / 45
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Antithetic Variables
LemmaIf X is a random variable and f , h : R → R two non decreasing (or nonincreasing) mappings,
E[h(X )f (X )] ≥ E[h(X )]E[f (X )].
CorollaryIf g is monotonic and T non increasing,
Cov [g(X1), g(T (X1))] ≤ 0.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 29 / 45
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Antithetic Variables
Proof of the lemma: Let Y be a random variable independent from X withthe same distribution.
FromE[(h(X )− h(Y ))(f (X )− f (Y ))] ≥ 0
we deduce
E[h(X )f (X )] + E[h(Y )f (Y )] ≥ E[h(X )f (Y )] + E[h(Y )f (X )].
Using independence and equi-distribution
E[h(X )f (X )] ≥ E[h(X )]E[f (X )]
thus
Cov [f (X ), h(X )] ≥ 0.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 30 / 45
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Antithetic Variables for call options in B&S
We want to apply the preceding method to
c = E[(eσG − K )+].
Here we have to remark that
g(x) = (eσx − K )+ is non decreasing (σ > 0),
T (x) = −x is non increasing
G and −G have the same distribution,
thus, we may use the following approximation
E[g(G)] ≈ 12n
n∑k=1
g(Gk ) + g(T (Gk )).
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 31 / 45
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Antithetic Variables for call options in B&S
Numerical results for σ = 0.5 and K = 1
Without antithetic variablesExact value=0.28353Estimated value (n=100) = 0.238, confidence interval at 95% : [0.151; 0.324]Estimated value (n=10000) = 0.285, confidence interval [0.275; 0.295]
With antithetic variablesExact value=0.28353Estimated value (n=100) = 0.283, confidence interval at 95% : [0.227; 0.338]Estimated value (n=10000) = 0.283, confidence interval [0.276; 0.289]
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 32 / 45
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Plan
1 Variance reduction techniquesVariance reduction techniques:IntroductionReminder on B&S modelControl VariateAntithetic VariablesImportance samplingConditioning
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 33 / 45
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Importance sampling
Let X having the distribution fX (x)dx , we want to approximate
E[g(X )].
We consider another random variable Y with distribution fY (y)dy . One has
E[g(X )] =∫
Rg(x)fX (x)dx =
∫R
g(x)fX (x)fY (x)fY (x)
dx = E[g(Y )
fX (Y )fY (Y )
].
If we are able to simulate de distribution of Y we may use MC method toapproximate E
[g(Y ) fX (Y )fY (Y )
].
Question: If Z = g(Y ) fX (Y )fY (Y ) , is Var(Z ) < Var(g(X )?
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 34 / 45
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Importance sampling
But
Var(Z ) =∫
Rg(x)2
f 2X (x)fY (x)
dx − E[g(X )]2.
When g ≥ 0, fY (x) = g(x)fX (x)E[g(X)] ⇒ Var(Z ) = 0
Problem: E[g(X )] is unknown...
Idea: Take fY (x) ≈ cste | fX (x)g(x) |.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 35 / 45
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Importance sampling: example 1
We want to compute by MC methods∫ 10
cos(πx2
)dx = E[cos(πU2
)] whereU ↪→ U([0, 1]).
g(x) = cos(πx2 ) is odd, g(0) = 1, g(1) = 0 thus we take
fY (x) = cste (1− x2) 1[0,1](x).
fY density ⇒ cste = 32
If Y has the density fY ,∫ 10
cos(πx2
)dx = E[Z ] where Z =2cos(πY2 )3(1− Y 2)
.
In this case
Var(Z ) ≈Var(cos(πU2 ))
100...
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 36 / 45
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Importance sampling: example 1
Remark: In the preceding example we are able to simulate Y with density
fY (x) =32
(1− x2) 1[0,1](x)
using the next result we have already seen (rejection method...)
PropositionLet fY be a density on R and
DfY = {(x , u) ∈ R× R+ | 0 ≤ u ≤ fY (x)}.
Let (Y , U) be a random variable on R× R+, then,
(Y , U) ↪→ U(DfY ) ⇔ Y has density fY and ∀x ∈ R+, U | x ↪→ U([0, fY (x)]).
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 37 / 45
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Importance sampling: Call option in B&S
We want to use the preceding method to compute
c = E[(eσG − 1)+]
where G ↪→ N (0, 1).
Idea: Since σ is small, eσx − 1 ≈ σx thus
c =∫
R+(eσx − 1) 1√
2πe−
x22 dx =
∫R+
(eσx − 1)cste σx
cste σx√2π
e−x22︸ ︷︷ ︸
≈fY (x)
dx
y = x2,
c =∫
R+
(eσ√
y − 1)√2πy
e−y2
dx2
= E
[(eσ
√Y − 1)√2πY
]where Y ↪→ E(0.5).
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 38 / 45
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Importance sampling: Call option in B&S
Numerical results for σ = 0.5 :
Without importance samplingExact value=0.28353Estimated value (N=100) = 0.249, confidence interval at 95% : [0.164; 0.334]Estimated value (N=10000) = 0.276, confidence interval [0.267; 0.285]
With importance samplingExact value=0.28353Estimated value (N=100) = 0.287, confidence interval at 95% : [0.275; 0.300]Estimated value (N=10000) = 0.284, confidence interval [0.283; 0.285]
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 39 / 45
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Plan
1 Variance reduction techniquesVariance reduction techniques:IntroductionReminder on B&S modelControl VariateAntithetic VariablesImportance samplingConditioning
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 40 / 45
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Conditioning
We want to compute by M.C methods
E[g(X , Y )].
If h(X ) = E[g(X , Y ) | X ] we have from classical properties that
E[g(X , Y )] = E[h(X )]
From Jensen inequality for conditional expectation:
h2(X ) ≤ E[g2(X , Y ) | X ], so
Var [g(X , Y )] ≥ Var [h(X )].
Pb: MC for E[h(X )] but we have to know explicitly h.....
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 41 / 45
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ConditioningSuppose that the pair (X , Y ) has the distribution fX ,Y (x , y)dxdy . We know inthis case that:
The distribution of X is given by fX (x)dx where fX (x) =∫
R fX ,Y (x , y)dy .
The distribution of Y is given by fY (y)dy where fY (y) =∫
R fX ,Y (x , y)dx .
The conditional distribution of Y given X = x is given by fY |X (x , y)dywhere
fY |X (x , y) = 1fX (x)>0fX ,Y (x , y)
fX (x).
Thus
h(X ) = E[g(X , Y ) | X ]
where
h(x) =∫
Rg(x , y)fY |X (x , y)dy .
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 42 / 45
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Conditioning: Best of call option
We want to approximate
p = E[(Max(eσ1G1 , eσ2G2)− K )+]
where G1 and G2 are two i.i.d N (0, 1).
First idea: Classical MC...
Second idea: Conditioning by G1 + MC...
We have p = E[h(eσ1G1)] where
h(x) = E[(Max(x , eσ2G2)− K )+]
where h may be explicitly known.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 43 / 45
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Conditioning: Best of call option
In fact h(x) = E[(Max(x , eσ2G2)− K
)+] but
(Max(x , eσ2G2)− K
)+
= (eσ2G2 − K )+ 1{x≤K}
+(x − K + (eσ2G2 − x)+
)1{x>K}.
Thus
h(x) =(
eσ222 N(− log(K )σ2 + σ2)− KN(−
log(K )σ2
)
)1{x≤K}
+(
x − K + eσ222 N(− log(x)σ2 + σ2)− xN(−
log(x)σ2
)
)1{x>K}.
h is explicit.
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 44 / 45
-
Conditioning: Best of call option
With σ1 = 0.2, σ2 = 0.5, K = 1,
Without conditioning (Exact value 0.338)
Estimated value (N=100) = 0.318, confidence interval at 95% : [0.219; 0.417]
Estimated value (N=10000) = 0.327, confidence interval at 95% :[0.318; 0.336]
With conditioning (Exact value 0.338)
Estimated value (N=100) = 0.344, confidence interval at 95% : [0.323; 0.365]
Estimated value (N=10000) = 0.339, confidence interval at 95% :[0.338; 0.341]
Christophe Chorro ([email protected]) (University Paris 1)Introduction to Monte Carlo Methods (Lecture 3 and 4) July 10 2008 45 / 45
Variance reduction techniquesVariance reduction techniques:IntroductionReminder on B&S modelControl VariateAntithetic VariablesImportance samplingConditioning