Bounds and Comparisons of Quasi-Monte Carlo Methods in ... · Bounds and Comparisons of Quasi-Monte Carlo Methods in Option Pricing John R. Birge⁄ The University of Chicago Graduate

Bounds and Comparisons of Quasi-MonteCarlo Methods in Option Pricing

John R. Birge∗

The University of ChicagoGraduate School of Business

Illinois Institute of Technology10 October 2005

∗ - Joint with Xuefeng (Jennifer) Jiang, Northwestern University.

Outline

• Motivation - Story

• Integration

• Error analysis

• Option model

• Sample Results

• Why it works?

Motivation

Challenge: Find x to

minx∈X

∫

Ξφ(x, ξ)P (dξ),

where φ is found by optimization for each ξ.

Fundamental Problem: Evaluate the integral, in general, find∫

Ξφ(x, ξ)P (dξ),

where the dimension of Ξ is large (100’s or 1000’s).

Alternatives

Simplified Problem: Find

∫

[0,1]nf(x)dx.

Options:

• Quadrature

– Requires smooth f

– Low dimension

• Bounding Inequalities

– Require some properties (e.g., f convex)

– Generally quite loose

• Monte Carlo Sampling

– Confidence intervals

– Independent of dimension

– Limited to 1√N

error bounds for N samples

Monte Carlo Method

Method: Choose random samples x1, . . . , xN and estimate

I =

∫

x∈[0,1]nf(x)dx

by

I(N) =1

N

N∑i=1

f(xi).

Result: Assuming independent samples, finite variance Vf , use Central Limit Theoremto find, with probability α:

I ∈ [I(N)−√

Vf√N

Φ−1(1− α

2), I(N) +

√Vf√N

Φ−1(1− α

2)],

where Φ is the standard normal cumulative.

Note: in optimization, this becomes projection on feasible set, sometimes betterasymptotic results.

Problem 1: Without reduced region, cannot avoid error declining as 1/√

N .

Problem 2: Use pseudo-random points - problems in high dimensions (often collinear

unless care is taken in generating these points).

Alternative Strategies

Question: Can you do better if you choose where to put your sample points, x1, . . . , xN?

Alternative analysis: Numerical instead of statistical (quasi- random )

Example: Consider integration on the unit interval (n = 1) where f has bounded totalvariation (TVf). What is the error in using I(N) ?

Approach: Integrate over f and count discrepancy, DN of xi measure relative tonatural (uniform), where

DN(x1, . . . , xN) = sup0≤x≤1

N∑i=1

1[0,x)(xi)/N − x.

Error Bound Result

Result: Assume continuity and use integration by parts.

|I − I(N)| = |∫

[0,1]

f (x)dx− 1

N

N∑i=1

f (xi)|

= |f (1)−∫

[0,1]

xdf (x)− f (1) +1

N

N∑i=1

(f (1)− f (xi))|

= | −∫

[0,1]

xdf (x) +1

N

N∑i=1

∫

[0,1]

(1[0,x)(xi))df (x)|

= |∫

[0,1]

(1

N

N∑i=1

1[0,x)(xi)− x)df (x)|

≤ | supJ

(1

N

N∑i=1

(1[0,xj)(xi)− x)(f (xj+1)− f (xj)))| for partitions J

≤ DNTVf .

Impact: Error depends on discrepancy and characteristics of f . Also,

generalizes to higher dimension (independent of n).

Finding Low Discrepancy Points

Question: Where is the best place to put your sample points, x1, . . . , xN?

Example: On the unit interval, how well can you do?

0 1

Result: Best place is x1 = 1/2N, x2 = 3/2N, . . . , xN = 2N−12N so (1/6, 1/2, 5/6) for N = 3.

Higher dimensions: Want to avoid collinearity and also allow N to change (withoutshifting all the points).

Alternatives:

• Alpha: Based on irrationals, such as the roots of the first n primes.

• Halton: Based on van der Korput sequence.

• Faure

• Sobol

α-Sequence

• x = x− bxc, x ∈ <

• n1, . . . , ns: the first s prime numbers

• α sequence: x0, x1, . . ., xn,. . . with

xn = (n√n1, . . . , n√ns) ∈ Is

for all n ≥ 1.

Van der Corput sequence

• b: b ≥ 2, base of the Van der Corput sequence

• Zb = 0, 1, . . . , b− 1 : the least residue system mod b

• n =∑L(n)

i=0 ai(n)bi, where L(n) = blogb nc, n ≥ 0.

• Radical-inverse function (Glasserman [2004])

φb(n) =

L(n)∑i=0

ai(n)b−i−1;

• Van der Corput sequence in base b: x0, x1, . . .,xn,. . . with

xn = φb(n)

for all n ≥ 0.

Halton sequence

• b1, . . . , bs: the first s prime numbers

• Halton sequence: x0, x1, . . . with

xn = (φb1(n), . . . , φbs

(n)) ∈ Is

for all n ≥ 0.

• Note: The Halton sequence is acceptably uniform for lower dimensions, up toabout s = 10.

Faure sequence

• p: the smallest prime number, p ≥ s and p ≥ 2

• x1n = φp(n) =

∑L(n)i=0 ai(n)p−i−1 where

n =

L(n)∑i=0

ai(n)pi

• xkn = φk

p(n) =∑L(n)

i=0 aki (n)p−i−1 k = 1, . . . , s where

aki (n) =

L(n)∑

j=i

C ij ak−1

i (n) mod p, C ij =

j!

i!(j − i)!.

• Faure sequence: x0, x1, . . . , xn, . . . where

xn = (x1n, x

2n, . . . , x

sn)

Sobol’ sequence

• vi: “direction numbers,” vi = mi2i

• mi: odd positive integers, mi ≤ 2i

• Primitive polynomial (Knuth [1981]):

P = xd + a1xd−1 + . . . + ad−1x + 1

• recurrence formula for mi and vi:

vi = a1vi−1 ⊕ a2vi−2 ⊕ . . .⊕ ad−1vi−d+1 ⊕ vi−d

⊕[vi−d/2d], i > d,

mi = 2a1mi−1 ⊕ 22a2mi−2 ⊕ . . .⊕ 2d−1ad−1mi−d+1

⊕2dmi−d ⊕ [mi−d/2d], i > d,

Sobol’ sequence(Cont’d)

• 1-dimensional Sobol’ sequence: x0, x1, . . . , xn, . . . where

xn = b1v1 ⊕ b2v2 ⊕ . . . ,

n =

blog2 nc∑i=1

bi2i

• s-dimensional Sobol’ sequence:

– choose s different primitive polynomials to calculate s different sets of directionnumbers

Error Bounds in Option Pricing

• Traditional bounded variation bounds do not apply(Koksma-Hlawka Inequality)

| 1N

N∑n=1

f(xn)−∫

Isf dµ| ≤ V (f)D∗

N(P)

Owen(2004): C(X),P(X) are not BVHK.

• Seeking error bounds that apply to option pricing problems

Does Quasi-Random Work Well for Options and (if so) Why?

Works Well

• Standard European options results

• American option results

Towards Understanding

• Dimension is effectively lower than n

• Functions in applications are well-behaved

• Discrepancy only matters on certain sets

European Option Results - Short Horizon

European Option Results - 2

European Option Results - 3

European Option Results -4

Error Bounds(s=1)

• Consider a European call

c = e−rT EST [(ST −K)+]

= e−rT

∫ +∞

0

(ST −K)+ dν(ST )

= e−rT

∫ +∞

0

(ST −K)+g(ST ) dST ,

• Black-Scholes model:

ST = S0e(r−σ2

2 )T+σε√

T ,

c = e−rT

∫ +∞

−∞(S0e

(r−σ22 )T+σε

√T −K)+g(ε) dε,

21

• QMC approximation:

c ≈ 1N

N∑

i=1

e−rT (S0e(r−σ2

2 )T+σεi

√T −K)+

where εi = G−1(zi), G(ε) =∫ ε

−∞ g(x) dx

• Error of approximation(in 1 dimension):

|e−rT

∫ +∞

−∞(S0e

(r−σ22 )T+σε

√T −K)+g(ε) dε

− 1N

N∑

i=1

e−rT (S0e(r−σ2

2 )T+σεi

√T −K)+|.

22

Error Bounds(s=1)(Cont’d)

Let M to be a smallest number such that all εi, i = 1, . . . , N arelocated in [−M, M ]. Then, we define a truncated function. Let

Q(ε) =

(S0e(r−σ2

2 )T+σM√

T −K)+ if ε > M,

(S0e(r−σ2

2 )T+σε√

T −K)+ if ε ∈ [−M, M ],

(S0e(r−σ2

2 )T−σM√

T −K)+ otherwise;

23


Then the error is

|e−rT

Z +∞

−∞(S0e

(r−σ22 )T+σε

√T −K)+g(ε) dε− 1

N

NXi=1

e−rT Q(εi)|

≤ |e−rT

Z +∞

−∞(S0e

(r−σ22 )T+σε

√T −K)+g(ε) dε− e−rT

Z +∞

−∞Q(ε)g(ε) dε|

+ |e−rT

Z +∞

−∞Q(ε)g(ε) dε− 1

N

NXi=1

e−rT Q(εi)|.

24


Lemma 0.1. If M ≥ 0, then∫ +∞

M

e−x22 dx ≤

√π

2e−

M22 .

Lemma 0.2. If g(x) = 1√2π

e−x22 and M ≥ σ

√T , then

∫ +∞

M

(S0e(r−σ2

2 )T+σx√

T −K)+g(x) dx ≤ 12

S0 erT e−(M−σ

√T )2

2 .

Lemma 0.3. If M is assumed as above, then

|e−rT

∫ +∞

−∞(S0e

(r−σ22 )T+σε

√T −K)+g(ε) dε−e−rT

∫ +∞

−∞Q(ε)g(ε) dε|

≤ S0e− (M−σ

√T )2

2 .

25


Lemma 0.4. If M > 0, G(x) =∫ x

−∞ g(ε)dε where

g(ε) = 1√2π

e−ε22 , then there exists an α =

√2π e

M22 such that

G−1(G(M)− 1N

) ≥ M − α1N

.

Lemma 0.5. If M is assumed as above, and P is a (N , ν)-uniformpoint set such that N includes subintervals [ i−1

N , iN ), i = 1, . . . , N ,

then

|e−rT

Z +∞

−∞Q(ε)g(ε) dε− 1

N

NXi=1

e−rT Q(εi)| ≤√

2πT σS0e−σ2

2 T eσ√

TM+ M22

N,

where Q(x) is defined as above.

26


Theorem 0.6. Assuming a uniform point set as given in Lemma0.5, there exists a number N0 and C such that for all N > N0,

|e−rT

∫ +∞

−∞(S0e

(r−σ22 )T+σε


− 1N

N∑

i=1

e−rT (S0e(r−σ2

2 )T+σεi

√T −K)+| ≤ CN− 1

3 ,

where C is a constant that only depends on S0, σ and T .

27


Corollary 0.1. Under the uniform point set conditions as given inLemma 0.5, for any k ∈ N, there exists a number N0 such that forall N > N0,

|e−rT

∫ +∞

−∞(S0e

(r−σ22 )T+σε


− 1N

N∑

i=1

e−rT (S0e(r−σ2

2 )T+σεi

√T −K)+| ≤ CN− k

2k+1 ,


28


Corollary 0.2. Under the uniform point set conditions as given inLemma 0.5, given any δ > 0, there exists a number N0 such thatfor all N > N0,

|e−rT

∫ +∞

−∞(S0e

(r−σ22 )T+σε


− 1N

N∑

i=1

e−rT (S0e(r−σ2

2 )T+σεi

√T −K)+| ≤ CN− 1

2+δ,


29

Error Bounds(s=n)

Simulating the path followed by the price process S,

S(t +T

s) = S(t)e(r−σ2

2 ) Ts +σεt

√Ts , t = 0,

T

s, . . . ,

(s− 1)Ts

,

where εt ∼ φ(0, 1).

The European call price is then given by

c = e−rT EST[(ST −K)+]

= e−rT

∫...

∫

Rs

(ST −K)+ dν(ST ).

30

Error Bounds(s=n)(Cont’d)

Denote X = Rs, εi = (εi1, ε

i2, ..., ε

is). The error of the approximation

is then

|e−rT

∫...

∫

Rs

(ST −K)+ dν(ST )

− 1N

N∑

i=1

e−rT (S0e(r−σ2

2 )T+σ√

Ts (Ps

j=1 εij) −K)+|.

We select M to be such that all (εi1, . . . , ε

is), i = 1, . . . , N are

located in [−M, M ]s.

Let ε = (ε1, . . . , εs). From now on, we assume M is large enough

that S0e(r−σ2

2 )T−σsM√

Ts ≤ K.

31


We can then define the truncated function as below:

Q(ε) =

8>>><>>>:

(S0e(r−σ2

2 )T+sσMq

Ts −K)+ if

Psj=1 εj > sM,

(S0e(r−σ2

2 )T+Ps

j=1 εjσq

Ts −K)+ if −sM ≤Ps

j=1 εj ≤ sM,

0 otherwise.

The error of the approximation is then

|e−rT

Z...

Z

Rs

(ST −K)+ dν(ST )− 1

N

NXi=1

e−rT Q(εi)|

≤ |e−rT

Z...

Z

Rs

(ST −K)+ dν(ST )− e−rT

Z...

Z

Rs

Q(ε) dν(ε)|

+ |e−rT

Z...

Z

Rs

Q(ε) dν(ε)− 1

N

NXi=1

e−rT Q(εi)|.

32


Note that

|∫

...

∫

Rs

(ST −K)+ dν(ST )−∫

...

∫

Rs

Q(ε) dν(ε)|

≤∫

...

∫

Rs\[−M,M]s(ST −K)+ dµ(ST ).

Lemma 0.7. There exists a constant L that only depends on S0, r,T , σ, and s such that

e−rT

∫...

∫

Rn\[−M,M]s(ST −K)+ dµ(ST ) ≤ LeσM

√Ts −M2

2 .

33


Let b1, . . . , bs−1 be the first s− 1 prime numbers. For thes-dimensional case here, we use the following construction (see,e.g., Deak(1990)), where

xi = (φb1(i), . . . , φbs−1(i),i

N− δ),

and 0 < δ < 1N is an arbitrary parameter to maintain the sequence

in [0, 1).Lemma 0.8. For the quasi-random sequence defined above, thereexists a constant L′ that only depends on S0, r, T , σ, and s suchthat

|e−rT

Z...

Z

Rs

Q(ε) dν(ε)− 1

N

NXi=1

e−rT Q(εi)| ≤ L′esσM

qTs

+ M22

N.

34


Combining Lemma 0.7 and Lemma 0.8, we have the followingtheorem.Theorem 0.9. For the quasi-random sequence defined above, givenany δ > 0, there exists a number N0 such that for all N > N0,

|e−rT

Z...

Z

Rn

(ST −K)+ dµ(ST )− 1

N

NXi=1

e−rT (S0eεi −K)+| ≤ CN− 1

2+δ,

where C only depends on S0, r, σ and s.

35


Theorem 0.10. If a derivative has a payoff function g(ST ) suchthat ∃ a constant K0 s.t. |g(ST )| < K0ST and, over anyHj = Πi[a

ji , b

ji ) ∈ H in the uniform point set condition,

Gj(g(ST ))− gj(g(ST )) ≤ K0(ST (bj)− ST (aj)), then, for thesequence defined above, given any δ > 0, there exists a number N0

such that for all N > N0

|e−rT

∫...

∫

Rn

g(ST ) dµ(ST )− 1N

N∑

i=1

e−rT g(S0eεi)| ≤ CN− 1

2+δ,

where C is a constant that only depends on K0, S0, r, σ, T and s.

36


Corollary 0.3. If a derivative has a payoff function g(ST ) suchthat ∃ a constant K0 s.t. |g(ST )| < K0ST and, over anyHj = Πi[a

ji , b

ji ) ∈ N , Gj(g(ST ))− gj(g(ST )) ≤ K0(ST (bj)− ST (aj)),

then, for a quasi-random sequence with discrepancy D∗N < (R(N)/N)

decreasing in N , given any δ > 0, there exists a number N0 such that, for

all N > N0,

|e−rT

Z...

Z

Rn

g(ST ) dµ(ST )− 1

N

NXi=1

e−rT g(S0eεi)| ≤ Cb N

R(N)c− 1

2+δ

,

where C is a constant that only depends on K0, S0, r, σ, s, and T .

37

Numerical Experiment

• Model: duality approach to American option valuation

– Rogers (2001)

– Haugh & Kogan (2001)

– Andersen & Broadie (2001)

• Numerical comparisons

38

Duality approach to American option valuation— Rogers and Haugh & Kogan(2001)

• Economy : (Ω,F ,Q)

• State : Xt ∈ Rd : t ∈ [0, T ]• Payoff of the option: ht = h(Xt)

• Primal problem:

– Price of American option:

V0 = supτ∈Γ

Eτ [hτ

Bτ]

39

Duality approach(Cont’d)

• Dual function: F (t,M)

F (t,M)B(t)

= E[ maxt≤τ≤T

(Z(τ)−M(τ))] + M(t)

• Dual problem:

U(0) = infM

F (0,M) = infM

E[ maxt≤τ≤T

(Z(τ)−M(τ))] + M(0)

• Upper bound

F (0,M) = E[ max0≤t≤T

(Z(t)−M(t))] + M(0) ≥ V (0)

M(t) = B(t)−1Veuro[S(t),K, σ, T − t, r]− Veuro[S(0),K, σ, T, r]

40

Numerical Comparisons

Simulation prices of standard American puts. Parameter valueswere K = 100, r = 0.06, T = 0.5, and σ = 0.4. N = 50, 000. (50batches with batch size 1000)

S0 True Price Pseudo alpha Halton Faure Sobol’

80 21.6059 21.7061 21.7056 21.7057 21.7081 21.7198

85 18.0374 18.1202 18.1195 18.1207 18.1205 18.1312

90 14.9187 14.9779 14.9767 14.9784 14.9755 14.987

95 12.2314 12.2749 12.2745 12.2768 12.2729 12.2826

100 9.9458 9.97603 9.97482 9.97786 9.97283 9.98254

105 8.0281 8.04813 8.046 8.04835 8.04494 8.05182

110 6.4352 6.44818 6.44715 6.44829 6.44616 6.4507

115 5.1265 5.13394 5.13443 5.13532 5.13385 5.13647

120 4.0611 4.06736 4.06659 4.06723 4.06597 4.06808

Avg. RE 0.00% 0.30% 0.29% 0.31% 0.29% 0.36%

41

Numerical Comparisons(Cont’d)

Standard errors of simulations.

S0 Pseudo alpha Halton Faure Sobol’

80 0.001564 0.00067456 0.00113523 0.00281853 0.000682138

85 0.001353 0.00058157 0.00099498 0.00232266 0.000650659

90 0.001607 0.00084140 0.00132556 0.00227021 0.000778371

95 0.001327 0.00075605 0.00107099 0.00194825 0.000729888

100 0.001717 0.00090967 0.00165845 0.00218293 0.000873421

105 0.001334 0.00083873 0.00139874 0.00178361 0.000802426

110 0.001059 0.00075968 0.00113792 0.00145889 0.000701484

115 0.000952 0.00066225 0.00096773 0.00117762 0.00062149

120 0.00083 0.00055593 0.00082419 0.00099835 0.000548867

Avg. >

Min. 85.66% 4.06% 65.41% 169.71% 1.44%

42

Numerical Comparisons(Cont’d)

Simulation times (seconds).

S0 Pseudo alpha Halton Faure Sobol’

80 6.812 8.234 21.999 35.093 47.734

85 6.453 7.906 21.609 34.702 47.702

90 6.015 7.468 21.281 34.296 47.719

95 5.453 6.953 20.703 33.75 47.718

100 4.703 6.282 19.952 33.03 47.734

105 4.063 5.765 19.328 32.39 47.75

110 3.547 5.203 18.828 31.843 47.75

115 3.156 4.75 18.421 31.469 47.749

120 2.844 4.516 18.141 31.374 47.797

Avg. > Min. 0.00% 36.28% 349.29% 648.84% 993.15%

43

Conclusion

• Error bounds

– Deterministic bounds at order of Monte Carlo methodwithout bounded variation property

• Numerical experiment:

– alpha and Sobol’ most consistent results with low variation

– alpha more efficient to simulate

44

Documents

Bounds and Comparisons of Quasi-Monte Carlo Methods in ... · Bounds and Comparisons of Quasi-Monte Carlo Methods in Option Pricing John R. Birge⁄ The University of Chicago Graduate