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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
Error Bounds(s=1)(Cont’d)
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
Error Bounds(s=1)(Cont’d)
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
Error Bounds(s=1)(Cont’d)
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
Error Bounds(s=1)(Cont’d)
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
√T −K)+g(ε) dε
− 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
Error Bounds(s=1)(Cont’d)
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
√T −K)+g(ε) dε
− 1N
N∑
i=1
e−rT (S0e(r−σ2
2 )T+σεi
√T −K)+| ≤ CN− k
2k+1 ,
where C is a constant that only depends on S0, σ and T .
28
Error Bounds(s=1)(Cont’d)
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
√T −K)+g(ε) dε
− 1N
N∑
i=1
e−rT (S0e(r−σ2
2 )T+σεi
√T −K)+| ≤ CN− 1
2+δ,
where C is a constant that only depends on S0, σ and T .
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
Error Bounds(s=n)(Cont’d)
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
Error Bounds(s=n)(Cont’d)
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
Error Bounds(s=n)(Cont’d)
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
Error Bounds(s=n)(Cont’d)
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
Error Bounds(s=n)(Cont’d)
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
Error Bounds(s=n)(Cont’d)
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