This document discusses Monte Carlo simulation techniques for pricing derivatives. It begins with an overview of Monte Carlo simulation and its use in derivative pricing models. It then covers different types of Monte Carlo path evolution (element-wise, path-wise, etc.). The document discusses implementing Monte Carlo simulation, including generating Wiener paths using methods like Euler discretization and Brownian bridges. It provides an example applying these techniques to price an option on the geometric average rate and shows convergence with different levels of path stratification. The key steps in Monte Carlo simulation for derivative pricing are outlined as simulating asset paths, computing payoffs, and estimating the option value and associated standard error.

1. M ONTE C ARLO S IMULATION IN D ERIVATIVE
P RICING M ODELS
Kai Zhang
Numerical Algorithms Group
Warwick Business School
The Thalesians Quantitative Finance Seminar
Canary Wharf, London
December 15, 2009
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2. Outline
Monte Carlo Overview
Monte Carlo Evolution Type
Monte Carlo for Generic SDE
Wiener Path Generator
Discretization Scheme
Exact Simulation
References on Monte Carlo
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3. Derivative Pricing with Monte Carlo
Comparison with Lattice or PDE
1. Monte Carlo can deal with high-dimensional models.
2. Suitable for path-dependent options.
Issues
1. Different answer each simulation (supplied with standard error).
2. Slower convergence than PDE in low-dimension (1-2) case.
3. Special treatment needed for options with embedded decision.
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4. Principals of Monte Carlo
Estimation of Expectation
1. Simulate a sample distribution.
2. Estimate the expectation by sample mean.
3. Estimate error bound from standard deviation.
In Derivative Pricing Models
1. Simulate a path of asset values, X = (X0 , X1 , · · · , XN ).
2. Compute payoff from the path, V(X).
3. Compute numeraire from the path, N(X).
V
4. Compute option value as N0 EN .
N
Crucial step is path simulation.
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5. Monte Carlo Evolution Type I
Notations for Sample Paths
X = {Xij }j=1,2,··· ,M
i=0,1,··· ,N
1. i is the index for time steps.
2. j is the index for sample paths.
Four Evolution Types
1. Element-wise
2. Path-wise
3. Slice-wise
4. Holistic
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6. Monte Carlo Evolution Type II
Element-Wise
1 1 1 1 2 2 2 2 3 3
X0 → X1 → X2 · · · XN → X0 → X1 → X2 · · · XN → X0 → X1 · · ·
1. Low level evolution type.
2. A single value at a time.
3. Not necessarily to evolve the entire path.
4. For knock out barrier option, can stop when barrier is hit.
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7. Monte Carlo Evolution Type III
Path-Wise
ˆ1
(X0 , ··· ˆ1
, XN )
↓
ˆ j ˆj
(X0 , · · · , XN )
↓
ˆM
(X0 , · · · ˆM
, XN )
1. One path at a time.
2. Suitable for path-dependent option.
3. Suitable for valuing and hedging a book of options.
4. An application is adjoint method (Giles and Glasserman (2006)).
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8. Monte Carlo Evolution Type IV
Slice-Wise
ˆ1 ˆ ˆ ˆ
   1  1  1
X0 X2 Xi XN
 .   .   .   . 
 .  →  .  → ··· . ··· →  . 
. . . .
ˆM
X ˆM
X ˆ iM
X ˆN
XM
0 2
1. One slice at a time.
2. Do not always evolve forward in time.
3. Suitable for using variance reduction.
4. An example is Brownian bridge with stratified sampling.
5. Another application is Longstaff and Schwartz Monte Carlo.
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9. Monte Carlo Evolution Type V
Holistic
ˆ1 ˆ1
 
X0 , · · · , XN
ˆ
 X2, · · · ˆ2
, XN 
 0 
 . . 
 . . . 
.
ˆ M, · · ·
X0 ˆN
, XM
1. The whole set of sample paths at one time.
2. Suitable when the whole set of paths is needed.
3. Variance reduction: moment matching methods.
4. Has to store everything.
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10. Implementing a Monte Carlo
To Simulate a Path of Asset Value One Needs
1. A random number generator.
2. A Wiener path generator.
3. A scheme for discretizing SDE.
Simulation Procedure
1. A discrete Wiener path, W = (W0 , W1 , · · · , WN ).
2. Back out normal variables, Zi = Wi − Wi−1 , i = 1, · · · , N.
3. Choose a proper discretization scheme for SDE.
4. Use Zi to construct a path for SDE, S = (S0 , S1 , · · · , SN ).
I will talk about these in turn.
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11. Wiener Path Generator I
Definition of Wiener Path
1. Independent increment: ∆Wi = Wi − Wi−1 , i = 1, · · · , N.
2. Gaussian increment: Wi − Wi−1 ∼ N(0, ∆t).
3. Continuous in t a.s.
4. W0 = 0 a.s.
Ways of Simulating a Discrete Wiener Path
1. Euler discretization.
2. Brownian bridge.
3. Spectral decomposition (do not discuss).
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12. Wiener Path Generator II
Euler Discretization
√
1. Wi = Wi−1 + ∆tǫi , ǫi ∼ i.i.d. N(0, 1).
2. Forward evolution, W0 → W1 → W2 → · · · → WN .
3. N multiplications and N − 1 additions.
Brownian Bridge
1. Given Wi and Wk , simulate Wj , i < j < k, as
tk − tj tj − ti (tj − ti )(tk − tj )
Wj = Wi + Wk + ǫ
tk − ti tk − ti tk − ti
where ǫ ∼ N(0, 1).
2. Binary chop evolution
W0 → WN → WN/2 → WN/4 → W3N/4 → WN/8 → W3N/8 · · · .
3. 3N multiplications and 2N additions.
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13. Wiener Path Generator III
Sobol Brownian Bridge
1. First few draws determine the main skeleton of the Wiener path.
2. Fill the first draw (final point) with stratified samples.
3. Fill up the subsequent draws using Sobol numbers.
4. Can also use a mixture of Sobol and random numbers.
NAG Library Functions
1. Function G05YMF in Fortran Mark 22.
2. Generates Sobol numbers up to 50,000 dimensions.
3. Sufficient for any reasonable applications.
4. Digital scrambling Sobol generator can have standard error.
5. .DLL callable from Excel/VBA.
6. Brownian bridge is in the next release.
7. Will have GPU version of Sobol Brownian bridge.
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14. Wiener Path Generator: Numerical Example
Geometric Average Rate Option
1. Option on geometric average of values at reset dates.
2. Closed-form formula (Kemma and Vorst (1990)).
Model Parameters (GBM)
1. Initial asset value, S0 = 100.
2. Constant interest rate, r = 0.05.
3. Asset volatility, σ = 0.2.
Option Parameters
1. Maturity T = 1yr.
2. Strike X = 100.
i
3. Reset dates Ti = , i = 0, 1, · · · , 16.
16
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15. Wiener Path Generator: Convergence Results
Sobol Brownian Bridge with Different Levels of Stratifications
0.4
0 Stratification (E=3.6E−2)
1 Stratification (E=5.4E−4)
4 Stratifications (E=5.2E−5)
16 Stratifications (E=4.2E−8)
0.3
0.2
0.1
Picing Bias
0
−0.1
−0.2
−0.3
0 10 20 30 40 50 60 70 80 90 100
Replication
Figure: Convergence with Different Levels of Stratifications, ∆t = 1/16,
Efficiency E = SE2 × CPU, Explicit Value 5.8417
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16. Wiener Path Generator: Be Aware of the Deterministic Bias
x 10
−3 Fully Stratified Sobol Brownian Bridge
1
0.5
0
−0.5
−1
Pricing Bias
−1.5
−2
−2.5
−3
−3.5
−4
0 10 20 30 40 50 60 70 80 90 100
Replication
Figure: Deterministic Bias of Fully Stratified Sobol Brownian Bridge
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17. Discretization Scheme I
General Asset SDE
dX = a(t, X)dt + b(t, X) · dW
1. X = (Xt1 , · · · , Xtd )′ is a vector of factors.
2. W = (Wt1 , · · · , Wtd )′ is d-dimensional Brownian motion.
3. a = ad×1 , b = bd×d and bb′ = D, the factor covariance matrix.
Discretization
1. Chose a mesh size δ = maxi ∆ti .
2. Approximate Xt by its time discrete version Xtδ .
3. Need Lipschitz and linear growth bound conditions on a and b.
4. Measure the closeness of Xtδ to Xt by weak and strong criterions.
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18. Discretization Scheme II
Weak and Strong Convergence
1. Strong convergence
δ
E|XT − XT | Cδβ
where δ = maxi ∆ti and C independent of δ.
2. Weak convergence
δ
|E[f (XT )] − E[f (XT )]| Cδβ
where f ∈ C 2(β+1) with polynomial growth.
3. β is known as the order of convergence.
We look at Euler, Milstein and higher order Itô-Taylor schemes.
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19. Discretization Scheme III
Euler Scheme
ˆ ˆ ˆ ˆ
Xi+1 = Xi + a(t, Xi )∆t + b(t, Xi ) · ∆Wi
ˆ ˆ
1. a ≡ a(t, Xi ) and b ≡ b(t, Xi ), for t ∈ [ti , ti+1 ].
√
2. O(∆t) weak convergence, O( ∆t) strong convergence.
3. Simplest but crude.
Milstein Scheme
ˆk ˆ ∂bk,l
Xi+1 = Xik + ak ∆t + bk,l ∆Wil + bn,m Im,l
∂X n
l l,m,n
ti+1 t
where Im,l = ti ti dWs dWtl .
m
1. Weak convergence O(∆t), strong convergence O(∆t).
2. Need evaluation of Im,l .
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20. Discretization Scheme IV
Iterated Itô Integral Im,l
1. Can show
1
Im,m = (∆Wim )2 − ∆t ,
2
Im,l + Il,m = ∆Wim ∆Wil .
∂bk,l ∂bk,m 1
2. If b =
n n,m
b , ∀l, m, n, can take Im,l = ∆Wim ∆Wil.
n n,l
∂X ∂X 2
3. For weak approximations, can approximate by moment matching
Im,l ≈ ∆Wim ∆Wil + Vm,l
where P(Vm,l = ±∆t) = 1 .
2
4. For strong approximations, see Kloeden and Platen (1999).
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21. Discretization Scheme V
Strong O(∆t3/2 ) Itô-Taylor
∂bi,j2
∆X i =ai ∆t + bi,j ∆W j + bk,j1 Ij ,j
∂X k 1 2
j j1 ,j2 ,k
 
 ∂bi,j + ∂bi,j 1 ∂ 2 bi,j 
+ ak k + bk,j1 bl,j1 k l I0,j
∂t ∂X 2 ∂X ∂X
j k j1 ,k,l
 
∂ai 1 ∂ai ∂ai 1 2a
∂ i
+ bk,j k Ij,0 +  + ak k + bk,j bl,j k l  ∆t2
∂X 2 ∂t ∂X 2 ∂X X
j,k k j,k,l
∂bk,j2 ∂bk,j3 ∂ 2 bi,j
+ bl,j1 + bk,j2 k 3 l Ij1 ,j2 ,j3
∂X l ∂X k ∂X ∂X
j1 ,j2 ,j3 ,k,l
t+∆t s2 s1 j j j t+∆t s j
where Ij1 ,j2 ,j3 = t t t dWu1 dWs2 dWs3 , I0,j =
1 2 t t dudWs
t+∆t s j
and Ij,0 = t t dWu ds.
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22. Discretization Scheme VI
Iterated Itô Integral Ij1 ,j2 ,j3 , I0,j and Ij,0
1. Have properties
1 1 2
Ij,j,j = ∆W j − ∆t ∆W j,
2 2
I0,j = ∆W j ∆t − Ij,0 ,
1 1
Ij,0 ∼ ∆t ∆W j + √ ∆y ,
2 3
where ∆y ∼ N(0, ∆t).
2. For approximation of Ij1 ,j2 ,j3 , see Kloeden and Platen (1999).
3. For strong approximations, Ij,0 must be simulated.
4. For weak approximations, can approximate Ij,0 by
1
Ij,0 ≈ E Ij,0 |∆W j = ∆t∆W j.
2
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23. Discretization Scheme VII
Weak O(∆t2 ) Itô-Taylor
∂bi,j2
∆X i =ai ∆t + bi,j ∆W j + bk,j1 Ij ,j
∂X k 1 2
j j1 ,j2 ,k
 
 ∂bi,j + ∂bi,j 1 ∂ 2 bi,j 
+ ak k + bk,j1 bl,j1 k l I0,j
∂t ∂X 2 ∂X ∂X
j k j1 ,k,l
 
∂ai 1  ∂ai ∂ai 1 ∂ 2 ai  2
+ bk,j k Ij,0 + + ak k + bk,j bl,j k l ∆t
∂X 2 ∂t ∂X 2 ∂X X
j,k k j,k,l
1. Do away with Ij1 ,j2 ,j3 .
2. I0,j ≈ Ij,0 ≈ 1 ∆t∆W j.
2
3. Ij1 ,j2 ≈ ∆W j1 ∆W j2 + Vj1 ,j2 where P(Vm,l = ±∆t) = 1 .
2
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24. Discretization Scheme VIII
O(∆t) Predictor-Corrector
ˆ ˆ ˆ
Xi+1 =Xi + [p · a(t, Xi+1 ) + (1 − p) · a(t, Xi )]∆t
ˆ
+ [q · b(t, Xi+1 ) + (1 − q) · b(t, Xi )] · ∆Wi ,
where
ˆ ˆ ˆ
Xi+1 = Xi + a(t, Xi )∆t + b(t, Xi ) · ∆Wi ,
and
∂bi,k
ai = ai − q bj,k .
∂X j
j,k
∂bi,k
1. Easy to implement when = 0, ∀i, j, k.
∂X j
2. Important scheme for LIBOR market model.
We apply various schemes to GBM and Vasicek processes.
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25. Discretization Scheme - GBM Example I
GBM Process
dX = rXdt + σXdW
◮ Euler scheme
∆X = X(r∆t + σ∆W).
◮ Milstein scheme
1 1
∆X = X r − σ 2 ∆t + σ∆W + σ 2 ∆W 2 .
2 2
◮ Strong O(∆t3/2 ) Itô-Taylor
1 1 1
∆X = r − σ 2 X∆t + σX∆W + σ 2 X∆W 2 + σ 3 X∆W 3
2 2 6
1 2 1 2
+ r X∆t2 + (r − σ )σX∆W∆t.
2 2
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26. Discretization Scheme - GBM Example II
GBM Process
dX = rXdt + σXdW
◮ Weak O(∆t2 ) Itô-Taylor
1 1 1
∆X = X r + r2 ∆t − σ 2 ∆t + σ(1 + r∆t)∆W + σ 2 ∆W 2 .
2 2 2
◮ O(∆t) Predictor-Corrector
∆X = X + 1 + p(r − qσ 2 )∆t + qσ∆W X − qσ 2 X∆t,
where
X = X(1 + r∆t + σ∆W).
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27. Discretization Scheme - Vasicek Example I
Vasicek Process
dXt = α(µ − Xt )dt + σdWt
◮ Euler and Milstein
∆X = α(µ − X)∆t + σ∆W.
◮ Strong O(∆t3/2 ) Itô-Taylor
1 1
∆X = 1 − ∆t [α (µ − X) ∆t + σ∆W] − √ σα∆t∆y,
2 2 3
where ∆y ∼ N(0, ∆t).
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28. Discretization Scheme - Vasicek Example II
Vasicek Process
dXt = α(µ − Xt )dt + σdWt
◮ Weak O(∆t2 ) Itô-Taylor
1
∆X = 1 − α∆t [α (µ − X) ∆t + σ∆W] .
2
◮ O(∆t) Predictor-Corrector
∆X = (1 − pα∆t)X,
where
X = X + α(µ − X)∆t + σ∆W.
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29. Discretization Scheme - Numerical Example I
Strong Convergence − GBM
4
Euler
Milstein
2 Predictor−Corrector
1.5 Strong
2.0 Weak
0
−2
−4
log Err
−6
2
−8
−10
−12
−14
−16
0 1 2 3 4 5 6 7 8 9 10
log2N
Figure: Strong Convergence - GBM, 105 Paths with Brownian Bridge 29 / 35

30. Discretization Scheme - Numerical Example II
Weak Convergence − GBM
10
Euler
Milstein
Predictor−Corrector
1.5 Strong
5 2.0 Weak
0
log Err
−5
2
−10
−15
−20
0 1 2 3 4 5 6 7 8 9 10
log N
2
Figure: Weak Convergence - GBM, 105 Paths with Brownian Bridge, Weak
Function f = [XT − E(XT )]2 30 / 35

31. Discretization Scheme - Numerical Example III
Strong Convergence − Vasicek
−10
Euler (Milstein)
1.5 Strong
2.0 Weak
Predictor−Corrector
−15
−20
log Err
−25
2
−30
−35
−40
0 1 2 3 4 5 6 7 8 9 10
log2N
Figure: Strong Convergence - Vasicek, 105 Paths with Brownian Bridge 31 / 35

32. Discretization Scheme - Numerical Example IV
Weak Convergence − Vasicek
−15
Euler (Milstein)
1.5 Strong
2.0 Weak
Predictor−Corrector
−20
−25
log Err
−30
2
−35
−40
−45
0 1 2 3 4 5 6 7 8 9 10
log2N
Figure: Weak Convergence - Vasicek, 105 Paths with Brownian Bridge,
Weak Function f = [XT − E(XT )]2
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33. Exact Simulation I
Analytical Distribution
1. For some SDE the analytical asset distribution is known.
2. In GBM asset value XT has lognormal distribution,
1 √
XT = X0 exp r − σ 2 T + σ Tǫ , ǫ ∼ N(0, 1).
2
3. For Vasicek
σ2
XT = X0 e−αT +µ(1−e−αT )+ǫ (1 − e−2αT ), ǫ ∼ N(0, 1).
2α
4. For CIR, XT has non-central χ2 distribution.
5. For Heston, see Broadie and Kaya (2006), Andersen (2007), etc.
6. Can simulate directly from XT in one step.
Requires other random number generators.
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34. Exact Simulation II
Random Numbers
1. For GBM, needs lognormal.
2. For Vasicek, needs Gaussian.
3. For CIR, needs Gaussian, Poisson and Gamma.
4. For Heston, needs Gaussian, Poisson, Gamma and Bessel.
NAG Random Number Generators
1. Chapter G05 is on random number generators.
2. Available in NAG Fortran Mark 22 and C Mark 9.
3. NAG Toolbox for MATLAB.
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35. References on Monte Carlo
Monte Carlo in Finance
1. Peter Jäckel (2002), a good introduction.
2. Paul Glasserman (2003), deeper in theory.
Discretization Schemes
Kloeden and Platen (1999), numerical solutions of SDE.
Computation
1. Mark Joshi (2008), a good introduction on C++ OOP.
2. Nick Webber (Wiley forthcoming), more on OOP C++ and VBA.
3. NAG library documentation on Monte Carlo components.
Benchmark with Closed-Form Formulae
1. NAG library Chapter S30 (Black-Scholes, Merton, Heston, etc).
2. Available in NAG Fortran Mark 22 and C Mark 9.
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