chapter 4: monte-carlo methodsrohop/spring_13/chapter4.pdfchapter 4: monte-carlo methods a...

University of Houston/Department of MathematicsDr. Ronald H.W. Hoppe

Numerical Methods for Option Pricing in Finance

Chapter 4: Monte-Carlo MethodsA Monte-Carlo method is a technique for the numerical realization of a stochastic process by

means of normally distributed random variables.

In financial mathematics, it is used for the computation of the expectation of option prices in

situations where explicit solutions like the Black-Scholes formula are not accessible (e.g., exo-tic options).

For simplicity, we consider a European plain-vanilla put option on a stock St whose value be-

haves according to the geometric Brownian motion

(⋆) d St = r St dt + σ St dWt

with a risk-free interest rate r ≥ 0, constant volatility σ > 0, and a Wiener process Wt.

We recall that the option price V(St, t) at t = 0 is given by the discounted expectation

() V(S0,0) = exp(−rT) E(V(ST,T)) .



The Four Basic Steps of the Monte-Carlo SimulationThe idea behind the Monte-Carlo method is based on the approximation of the expectation

E(V(ST,T)) in () by the simulation of M pathes St | 0 < t < T of the value of the stock:

Step 1: Simulation of the pathes

For M independent pathes, compute the solution S(k)t ,1 ≤ k ≤ M, of the geometric Brownian

motion (⋆). This requires the simulation of M independent realizations of the Wiener process

Wt and the numerical solution of the stochastic differential equation for the respective pathes.

Step 2: Computation of the payoff function

For 1 ≤ k ≤ M, compute the payoff function associated with the path S(k)t : V

(k)T = (K − S

(k)T )+.

Step 3: Computation of an estimate of the expectation

According to the law of large numbers, the arithmetic mean of equally distributed, indepen-

dent random variables converges almost surely to the expectation. Hence, an appropriate es-

timate of the expectation in () is given by V(k)T = (K − S

(k)T )+.

Step 4: Computation of an estimate of the option price

Compute an approximation of the option price by V = exp(−rT) E(VT).



4.1 Generation of Equally Distributed Random Numbers

For the simulation of a Wiener process, we use standardized normally distributed random

numbers Z ∼ N(0,1) in order to compute the increments ∆W = Z√

∆t.

Definition 4.1 (Equally Distributed Random Variables)

(i) A random variable X is called equally distributed on [a,b] ⊂ lR, (notation: X ∼ U[a,b]), if

its density function is given by f(x) = 1/(b − a) , x ∈ [a,b].

(ii) A sequence Xnn∈lN of random variables is called F-distributed, if the Xn are indepen-

dent realizations of random numbers with the distribution function F.

In the sequel, we will consider the following random number generators:

• Linear congruential generators

• The generator RANDU

• Fibonacci generators



4.1.1 Linear Congruential Generators

Linear congruential generators are obtained by the following algorithm:

Algorithm 4.1: Linear Congruential Generators

Given M ∈ lN, a,b ∈ 0,1, ...,M − 1, a 6= 0, and a seed X0 ∈ 0,1, ...,M − 1, compute

For i = 1,2, ...

Xi := (a Xi−1 + b) mod M ,

Ui := Xi/M .

where a mod M := M · (a/M − ⌊a/M⌋) with ⌊a/M⌋: largest integer less or equal a/M.

Lemma 4.1 (Properties of Linear Congruential Generators)

The random numbers generated by Algorithm 4.1 have the properties

(i) Xi ∈ 0,1, ...,M − 1.

(ii) The Xi are periodic with period less or equal M.

(iii) If b = 0 (multiplicative congruential generator), we must exclude the seed X0 = 0.

(iv) If a = 1, we obtain Xi = (X0 + ib) mod M (too easy to predict!).



Analysis of Linear Congruential Generators

The random numbers Xi and Ui, i ≥ 1, obtained by the linear congruential generator, can be

written as m-tuples (Xi,Xi+1, ...,Xi+m−1) or points (Ui, ...,Ui+m−1) ∈ [0,1)m. They are situated on

(m − 1)-dimensional hyperplanes which can be constructed as the following analysis in case

m = 2 reveals:

Xi = (a Xi−1 + b) mod M ⇐⇒ Xi = a Xi−1 + b − k M , k M ≤ a Xi−1 + b < (k + 1) M .

For an arbitrary tuple (z0, z1) ∈ Z2 we obtain:

(†) z0 Xi−1 + z1 Xi = z0 Xi−1 + z1 (a Xi−1 + b − k M) =

= Xi−1 (z0 + a z1) + z1 b − z1 k M =

= M · (xi−1z0 + a z1

M− z1 k)

︸︷︷︸

=: c

+ z1 b .

Division by M yields the equation of a straight line in the (Ui−1,Ui)-plane:

(‡) z0 Ui−1 + z1 Ui = c + z1 b M−1.



Analysis of Linear Congruential Generators

Question: Does there exist a tuple (z0, z1) ∈ Z2 such that

the intersection of the straight lines with the square [0,1)2

is non-empty for only a few lines which would violate the

requirement of uniform distributions of the points?

If we choose (z0, z1) ∈ Z2 and z0 + a z1 = 0 mod M, then c ∈ Z.

Solving for c in (†) results in the maximum interval

cmin ≤ c ≤ cmax

such that the associated straight line (‡) has non-zero in-

tersection with [0,1)2.

Figure 4.1: M = 2048 , a = 1229 , b = 1 , X0 = 1. The figure displays the first 499 points

(Ui−1,Ui) obtained by the linear congruential generator. These points are lying on 5 parallel

straight lines.



4.1.2 The Random Number Generator RANDUThe generator RANDU is based on a multiplicative congruential method with the data

M = 231, a = 216 + 3 , b = 0 .

Figure 4.2 suggests that the points (Ui−1,Ui) are randomly distributed in the plane. However,

Figure 4.3 displays the tuples (Ui−2,Ui−1,Ui), and we observe that they are lying on only 15 dif-

ferent planes in the unit cube [0,1)3 which is a serious disadvantage in many applications.

Figure 4.2 Figure 4.3



4.1.3 Fibonacci GeneratorsFibonacci generators use the recursively given Fibonacci sequence according to

Xi := (Xi−1 + Xi) mod M , Ui := Xi/M , i ≥ 2 ,

where X1,X2 are given seeds.

Figure 4.4

Figure 4.4 displays the first 2000 points (Ui−1,Ui)

obtained by the Fibonacci generator for M = 2179

and X1 = X2 = 1. Significantly less than 2000 points

are seen in the unit square due to the fact that the

sequence Uii∈lN repeats any 197 entries.

Therefore, this Fibonacci generator is less suited for

random number generation.



Better and widely used random number generators are the so-called lagged Fibonacci generators:

Algorithm 4.2: Lagged Fibonacci Generators

Given M ∈ lN, µ, ν ∈ lN and initial values X1, ...,Xmax(µ,ν), for i ≥ maxµ, ν compute

Xi := (Xi−µ + Xi−ν) mod M , Ui := Xi/M .

Figure 4.5

For M = 2048 , µ = 5 , ν = 17, Figure 4.5 shows

the first 2000 points (Ui−1,Ui) obtained by the

lagged Fibonacci generator.



4.2 Transformation of Random VariablesNormally distributed random variables can be obtained by a transformation of equally distri-

buted random variables using either the inversion of the distribution function or a transforma-

tion between random variables.

4.2.1 Inversion of the Distribution Function

Theorem 4.1 (Inversion of the Distribution Function)

Assume that U ∼ U[0,1] is an equally distributed random variable and assume further that F

is a continuous, strongly monotone distribution function. Then, the random variable F−1(U)

is F-distributed.

Proof. Since P(U ≤ ξ) = ξ , ξ ∈ [0,1], we have

P(F−1(U) ≤ x) = P(U ≤ F(x)) = F(x) .

Remark: For the normal distribution Φ there are no explicit expressions neither for Φ nor for

Φ−1 which means that the nonlinear equation Φ(x) = u must be inverted numerically by, e.g., the

application of a Newton-like method. However, for u ≈ 1 the problem is ill-conditioned.



4.3.2 Marsaglia’s Polar Method

Marsaglia’s variant of the Box-Muller algorithm avoids the evaluation of trigonometric functions:

For U1,U2 ∼ U[0,1], we obviously have Wi := 2Ui − 1 ∼ [−1, +1],1 ≤ i ≤ 2. The pair (W1,W2) defines

a point in the plane. Only points inside the unit disk

K := (W1,W2) | w21 + W2

2 < 1

are accepted. A Transformation K → S := [0,1]2 is provided by means of polar coordinates

x1

x2

=

W21 + W2

21

2π arctan( W2

W1)

.

Algorithm 4.4: Marsaglia’s Polar Method

Step 1: Generate U1,U2 ∼ U[0,1] and Wi := 2Ui − 1 , 1 ≤ i ≤ 2, as long as W21 + W2

2 < 1.

Step 2: Z1 := W1

√

−2ln(W21 + W2

2)//(W21 + W2

2) and Z2 := W2

√

−2ln(W21 + W2

2)//(W21 + W2

2) are

normally distributed random variables.

Remark: Marsaglia’s polar method is more efficient than the Box-Muller algorithm.



Algorithm 4.3: Box-Muller Algorithm

Step 1: Generate U1 ∼ U[0,1] and U2 ∼ U[0,1].

Step 2: Set Θ := 2π U2 and ρ :=√

−ln(U1).

Step 3: Z1 := ρ cos(Θ) and Z2 := ρ sin(Θ) are normally distributed random variables..

Figure 4.6

The histogram of Z1 in Figure 4.6 based on 200000

random numbers shows that the algorithm indeed

approximately produces normally distributed ran-

dom variables.



4.2 Normally Distributed Random Variables

4.3.1 The Box-Muller Algorithm

We apply Theorem 4.2 in case n = 2 , A = S := [0,1]2 , f ≡ 1 on S, and h : S → lR2 given by

y1 = h1(x1,x2) =√

−2 ln(x1) cos(2πx2) , y2 = h2(x1,x2) =√

−2 ln(x1) sin(2πx2) .

Since y21 + y2

2 = −2 ln(x1) and y2/y1 = tan(2πx2), the inverse function can be easily computed

x1 = exp(−1

2(y2

1 + y22)) , x2 =

1

2π

arctan(y2

y1)

with the determinant det Jh−1(y) of the Jacobian Jh−1(y) given by

det Jh−1(y) =1

2π

exp(−1

2(y2

1 + y22))

which is the density of the standard normal distribution in lR2.



4.2.2 Transformation of Random Variables

Theorem 4.2 (Transformation of Random Variables)

Assume that X is a random variable with density function f on A := x ∈ lRn| f(x) > 0. Assume

further that the mapping h : A → B := h(A) is continuously invertible. Then, the random variable

Y = h(X) has the density function

g(y) := f(h−1(y)) |det Jh−1(y)| , y ∈ B ,

where Jh−1(y) is the Jacobian of h−1 in y.

Proof. Applying the transformation rules for multiple integrals, we obtain

P(Y = h(X) ∈ B) = P(X ∈ h−1(B)) =∫

h−1(B)

f(x) dx =∫

B

f(h−1(y)) |det Jh−1(y)| dy .



4.4 Correlated Random Variables

We consider a random variable Z = (Z1, ...,Zn) with density function f. We further denote by

µ = E(Z) = (E(Z1), ...,E(Zn)) the expectation and by Σ = (Σij)ni,j=1 with

Σij = (Cov Z)ij := E((Zi − µi)(Zj − µj))

the associated covariance matrix which is symmetric, positive definite.

We want to construct an N(µ,Σ)-distributed random variable Y.

Theorem 4.3 (Correlated Random Variables)

Under the above notations let Σ = LLT be the Cholesky decomposition of Σ with a lower trian-

gular matrix L ∈ lRn×n. Then, Y = µ + LZ is an N(µ,Σ)-distributed random variable with den-

sity function

f(x) =1

(2π)n/2

1

(det Σ)1/2exp(−

1

2(x − µ)T Σ−1 (x − µ)) .



Radical-Inverse Function and the Halton Sequence

(ii) Radical-Inverse Function

The radical-inverse function ΦB allows to define van der Corput sequences xiNi=1 with respect

to an arbitrary basis B. Using the same notation as in (i), they are defined by

ΦB(i) :=j

∑

k=0dk B−(k+1)

, xi := ΦB(i) .

(iii) Halton Sequence

Assume that p1, ...,pm are pairwise prime integers. The Halton sequence is defined as the se-

quence of vectors

xi = Φp1(i), ...,Φpm

(i) , i ∈ lN .



Van der Corput Sequence

Remark: The sequence X(N)i N

i=1 with x(N)i = i/N,1 ≤ i ≤ N, obviously satisfies DN = 1/N and

thus is of low discrepancy. However, it is not suited for Monte Carlo integration, since for

each N a new sequence has to be set up. For increasing N, it is more efficient to use already

computed numbers.

Examples of sequences of low discrepancy:

(i) Van der Corput sequence

Let i = (dj...d0)2 :=∑j

k=0 dk2k be the representation of the integer i ∈ lN in fixed point arith-

metics with respect to the basis B = 2. Then, the sequence xilN obtained by bit inversion

according to

xi = (.d0...dj)2 :=j

∑

k=0dk 2−(k+1)

is called van der Corput sequence.



Sequences of Low Discrepancy

Definition 4.3 (Sequences of Low Discrepancy)

A sequence xiNi=1 of points or numbers xi ∈ lRm

,1 ≤ i ≤ N, is called a sequence of low discre-

pancy, if there exists a constant Cm > 0 such that for all N ∈ lN:

DN ≤ Cm(ln(N))m

N.

Remarks:

(i) For not too large m ∈ lN we have that approximately DN ≈ O(N−1). Compare this with

DN = O(N−1/2) for sequences of random numbers in general.

(ii) Let Ui ∈ 0,1/M, ..., (M − 1)/M be random numbers obtained by a linear congruential gene-

rator. These are not even uniformly distributed, since with Q′ := [1/(2M + 2),1/(M + 1)]

DN ≥ |card(Ui | Ui ∈ Q′)

N− vol(Q′)| = vol(Q′) =

1

2M + 2.



Properties of Discrepancy and Star Discrepancy

Theorem 4.4 (Properties of Discrepancy and Star Discrepancy)

Let DN and D∗N be the discrepancy and star discrepancy of a point set xilN,xi ∈ lRm

,1 ≤ i ≤ N.

Then, there holds

D∗N ≤ DN ≤ 2m D∗

N .

Proof. The proof is left as an exercise.

Upper Bound for Monte Carlo Integration

Theorem 4.5 (Theorem of Hlawka and Koksma)

Assume that Ω := [0,1]m,m ∈ lN, and f : [0,1]m → lR is a function of bounded variation BV(f).

Then, for the error εN in the Monte Carlo integration there holds

|εN| ≤ BV(f) D∗N .



4.5.2 Discrepancy

The idea behind discrepancy is that for a uniformly distributed point set x1, ...,xN ⊂ [0,1]m

the fraction of the points lying within some quadrilateral Q ⊂ [0,1]m should correspond to the

volume of the quadrilateral, i.e.,

card(xi,1 ≤ i ≤ N | xi ∈ Q)

N≈

vol(Q)

vol([0,1]m).

Definition 4.2 (Discrepancy and Star Discrepancy)

The discrepancy of a point set x1, ...,xN ⊂ [0,1]m is defined by

DN := supQ⊂[0,1]m

|card(xi,1 ≤ i ≤ N | xi ∈ Q)

N− vol(Q)| ,

where the supremum is taken with respect to all quadrilaterals Q ⊂ [0,1]m.

The star discrepancy D∗N is defined in the same way, but with the supremum taken with respect

to those quadrilaterals Q∗ ⊂ [0,1]m for which one corner is in the origin, i.e., Q∗ =∏m

i=1[0,yi]

where (y1, ...,ym) denotes the diagonally opposite corner.



If the random variables x1, ...,xN ∈ Ω are independent and uniformly distributed, the law of

large numbers implies QN → vol(Ω) E(f) =∫

Ω f(x) dx as N → ∞. Hence, for the variance of the

error εN := I − QN we obtain

Var(εN) = E(ε2) − E(εN)2 =σ

2(f)

Nvol(Ω)2 , σ

2(f) =∫

Ωf(x)2 dx − (

∫

Ωf(x) dx)2 .

The central limit theorem implies the following convergence result for the standard deviation√

Var(εN) = O(N−1/2) as N → ∞ ,

which means slow convergence (improvement of the accuracy by a factor of 10 requires to in-

crease the computational cost by a factor of 100). Another disadvantage is the lack of a strict

error bound. Before we discuss methods of variance reduction in section 4.6, we will address

the issue to choose the random points xi,1 ≤ i ≤ N, in such a way that with increasing N

the accuracy gets better but clustering is avoided.



4.5 Sequences of Random Numbers with Low DiscrepancyAs a measure for the quality of uniform distribution of random numbers we introduce the no-

tion of discrepancy and give examples of sequences of random numbers with low discrepancy.A related issue is that to improve convergence properties in relevant applications such as the

numerical integration of high-dimensional integrals.

4.5.1 Monte Carlo Integration

Given a bounded domain Ω ⊂ lRm and an integrable function f : Ω → lR, the Monte Carlo integra-

tion provides an approximation of the m-dimensional integral of f over Ω by the weighted sum

of the values of the integrand f at randomly chosen points xi,1 ≤ i ≤ N, in the domain of integra-

tion Ω according to

I :=∫

Ω⊂lRm

f(x) dx ≈vol(Ω)

N

N∑

i=1f(xi) =: QN .



Algorithm 4.5: Correlated Random Variables

Step 1: Compute the Cholesky decomposition Σ = LLT.

Step 2: Compute Z ∼ N(0,1) component-wise, i.e., Zi ∼ N(0,1) , 1 ≤ i ≤ n, e.g., by Marsaglia’spolar algorithm.

Step 3: Y = µ + LZ is N(µ,Σ)-distributed.

MATLAB program correlated.mrandn(′state′,1)

Sigma = [543;454;345]

mu = [−5010]′;N = 10000;

L = chol(Sigma);

X = zeros(3,N);

for i = 1 : N

X(: .i) = mu + L ∗ randn(3,1);

end



Proof. Setting x = Lz and observing dx = |det L| dz, we obtain

f(z) dz =1

(2π)n/2exp(−

zTz

2) dz =

1

(2π)n/2exp(−

(L−1x)TL−1x

2) dz =

=1

(2π)n/2exp(−

xT(LLT)−1x

2) dz =

1

(2π)n/2|det L|exp(−

xTΣ−1x

2) dx =

=1

(2π)n/2(det Σ)1/2exp(−

xTΣ−1x

2) dx .

This shows that X = LZ is N(0,Σ)-distributed which readily gives the assertion.



Properties of van der Corput Sequences

Theorem 4.6 (Properties of van der Corput Sequences)

For the discrepancy of van der Corput sequences there holds

lim supN→∞

N DN

ln(N)=

B2

4(B+1) ln(B), B even

,B−1

4 ln(B), B odd

.



4.6 Numerical Integration of Stochastic Differential Equations4.6.1 The Euler-Maruyama Method

In case of an ordinary differential equation

x′(t) = a(t,x(t)) , T > 0 ,

where a ∈ C(lR+, lR) satisfies a Lipschitz condition in the second argument, the explicit Euler

method with uniform step-size h > 0 is given by

yi+1 = yi + h a(ti,yi) , ti := i h , i ≥ 0 .

It is well known that it has the order of convergence 1, i.e.

sup |yi − x(ti)| ≤ C h .

i ≥ 0

For the stochastic differential equation

dxt = a(t,xt) dt + b(t,xt) dWt

the counterpart of the explicit Euler method is the Euler-Maruyama method

yi+1 = yi + h a(ti,yi) + b(ti,yi) ∆Wi , ∆Wi := Wti+1− Wti .



4.6.2 Strong and weak convergence

Definition 4.4 (Strong and Weak Convergence)

Assume that xht is an approximation of the solution xt of the stochastic differential equation

dxt = a(t,xt) dt + b(t,xt) dWt .

(i) The approximation xhT is said to converge strongly to xT of order γ > 0, if there exists a con-

stant C > 0, such that for all sufficiently small h > 0

E(|xT − xhT|) ≤ C hγ

.

(ii) The approximation xhT is said to converge weakly to xT of order κ > 0, if there exists a con-

stant C > 0, such that for all sufficiently small h > 0

|E(xT) − E(xhT)| ≤ C hκ

.

Theorem 4.7 (Strong and Weak Convergence of the Euler-Maruyama Method)

The Euler-Maruyama method is strongly convergent of order γ = 0.5 and weakly convergent

of order κ = 1.Proof. We refer to P. Kloeden and E. Platen; Numerical Solution of Stochastic Differential

Equations. Springer, Berlin-Heidelberg-New York, 1999



4.6.3 The Method of MilsteinWe recall Ito’s Lemma which states: If Xt is an Ito process, then for sufficiently smooth f we

have that f = f(t,Xt) is an Ito process as well which satisfies

df = (∂f

∂t+ a

∂f

∂x+

1

2b2 ∂

2f

∂x2) dt + b

∂f

∂xdWt .

The associated integral form is

(∗) f(xt) = f(xt0) +t∫

t0

(f ′(xs) a(xs) +1

2f ′′(xs) b(xs)

2) ds +t∫

T0

f ′(xs) b(xs) dWs .

In particular, for f(x) = x we obtain

(∗∗) xt = xt0 +t∫

t0

a(xs) ds +t∫

t0

b(xs) dWs .

Choosing f = a and f = b in (∗) and inserting it into (∗∗) results in

xt = xt0 +t∫

t0

(a(xt0) +s∫

t0

(a′a +1

2a′′b2) dz +

s∫

t0

a′b dWz) ds +

+t∫

t0

(b(xt0) +s∫

t0

(b′a +1

2b′′b2) dz +

s∫

t0

b′b dWz) dWs ,

where a = a(xz) , b = b(xz) etc.



Modifying the double integral w.r.t. dWzdWs by replacing the integrand b′(xz)b(xz)

b′(xt0)b(xt0), we can write

xt = xt0 + a(xt0) (t − t0) + b(xt0)t∫

t0

dWs + b′(xt0) b(xt0)t∫

t0

s∫

t0

dWz dWs + R ,

where the remainder is of the (higher) order R = O(h3/2).

The double integral can be evaluated as follows:t∫

t0

s∫

t0

dWz dWs =t∫

t0

(Ws − Wt0) dWs =t∫

t0

Ws dWs − Wt0

t∫

t0

dWs =

=1

2(W2

t − W2t0

) −t − t0

2− Wt0 (Wt − Wt0) =

1

2((Wt − Wt0)

2− (t − t0)) .

This leads to the method of Milstein

yi+1 = yi + h a(ti,yi) + b(ti,yi) ∆W +1

2b′(ti,yi) b(ti,yi) ((∆W)2 − h) ,

where ∆W := Z√

h , Z ∼ N(0,1).

Theorem 4.8 (Convergence of the Method of Milstein)

The method of Milstein is strongly convergent of order γ = 1.



4.7 Reduction of VariancesThe slow convergence of Monte-Carlo simulations can be explained as follows:

We assume that θn =∑n

k=1 Φ(Xk)/n is some stochastic approximation of a stochastic integral

θ = E(Φ(x)), where Φ(x) := g(x)/f(x) and Xk,1 ≤ k ≤ n, are independent samples of an F-distri-

buted random variable with F′ = f .Under the assumption E(Φ(Xk)) = θ , Var(Φ(Xk)) = σ

2, 1 ≤ k ≤ n, it follows that

E(θn) =1

n

n∑

k=1E(Φ(Xk)) = θ , Var(θn) =

1

n2

n∑

k=1Var(Φ(Xk)) =

σ2

n.

Using the Chebychev inequality for square integrable random variables Y

P(|Y − E(Y)|) ≥ δ) ≤Var(Y)

δ2 , δ := σ/

√ε n ,

we obtain the error estimate

P(|θn − θ| ≥σ

√εn

) ≤ ε =⇒ P(|θn − θ| <σ

√εn

) ≥ 1 − ε .

This means that the error gets the smaller the larger the number n of samples is chosen. But

in order to reduce the error by a factor of 10, the number of samples must be increased by a

factor of 100.



4.7.1 Antithetic Variables

Theorem 4.9 (Variance reduction by an Antithetic Variable)

Assume that the approximation θn has been generated by a random variable Z ∼ N(0,1) and

generate an approximation θ−

n by means of −Z ∼ N(0,1) such that Var(θn) = Var(θ−

n ).

Introduce the antithetic variable θn according to θn := 12 (θn + θ

−

n ). Then, there holds

Var(θn) < Var(θn) .

Proof. The fundamental relation

Var(X ± Y) = Var(X) + Var(Y) ± 2 Cov(X,Y)implies

Var(θn) =1

4Var(θn + θ

−

n ) =1

4(Var(θn) + Var(θ−

n ) + 2 Cov(θn, θ−

n )) .

Taking into account thatCov(X,Y) ≤

1

2(Var(X) + Var(Y)) ,

we thus obtain

Var(θn) ≤1


n )) , if Cov(θn, θ−

n ) ≤ 0 ,

Var(θn) <1


n )) , if Cov(θn, θ−

n ) > 0 ,

which gives the assertion observing Var(θn) = Var(θ−

n ).

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