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Valuation of American BasketOptions using Quasi-Monte Carlo
Methods
d-fine GmbH
Christ Church College
University of Oxford
A thesis submitted in partial fulfillment for the MSc in
Mathematical Finance
September 26, 2009
Abstract
Title: Valuation of American Basket Optionsusing Quasi-Monte Carlo Methods
Author: d-fine GmbH
Submitted for: MSc in Mathematical FinanceTrinity Term 2009
The valuation of American basket options is normally done by using the
Monte Carlo approach. This approach can easily deal with multiple ran-
dom factors which are necessary due to the high number of state variables
to describe the paths of the underlyings of basket options (e.g. the Ger-
man Dax consists of 30 single stocks).
In low-dimensional problems the convergence of the Monte Carlo valuation
can be speed up by using low-discrepancy sequences instead of pseudo-
random numbers. In high-dimensional problems, which is definitely the
case for American basket options, this benefit is expected to diminish.
This expectation was rebutted for different financial pricing problems in
recent studies. In this thesis we investigate the effect of using different
quasi random sequences (Sobol, Niederreiter, Halton) for path generation
and compare the results to the path generation based on pseudo-random
numbers, which is used as benchmark.
American basket options incorporate two sources of high dimensional-
ity, the underlying stocks and time to maturity. Consequently, different
techniques can be used to reduce the effective dimension of the valuation
problem. For the underlying stock dimension the principal component
analysis (PCA) can be applied to reduce the effective dimension whereas
for the time dimension the Brownian Bridge method can be used. We an-
alyze the effect of using these techniques for effective dimension reduction
on convergence behavior.
To handle the early exercise feature of American (basket) options within
the Monte Carlo framework we consider two common approaches: The
Threshold approach proposed by Andersen (1999) and the Least-Squares
Monte Carlo (LSM) approach suggested by Longstaff and Schwartz (2001).
We investigate both pricing methods for the valuation of American (bas-
ket) options in the equity market.
Contents
1 Introduction 1
2 Basket Options 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Different Types of Basket Options . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Plain Vanilla Basket Options . . . . . . . . . . . . . . . . . . 4
2.2.2 Exchange Options . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.3 Asian Basket Options . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 American-style Basket Options . . . . . . . . . . . . . . . . . . . . . 7
3 Monte Carlo Simulation 8
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Mathematical Foundations . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.1 Strong Law of Large Numbers . . . . . . . . . . . . . . . . . . 9
3.2.2 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2.3 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 10
3.3 Pseudo-Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Quasi-Random Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4.1 Low Discrepancy and Integration Error . . . . . . . . . . . . . 14
3.4.2 Halton Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.4.3 Sobol Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4.4 Niederreiter Sequences . . . . . . . . . . . . . . . . . . . . . . 19
3.5 The curse of dimensionality . . . . . . . . . . . . . . . . . . . . . . . 22
4 Effective Dimension Reduction 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Brownian Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
i
5 Valuation of American Basket Options 28
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Model Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Monte Carlo Path Construction . . . . . . . . . . . . . . . . . . . . . 30
5.4 Valuation of Early Exercise Feature . . . . . . . . . . . . . . . . . . . 33
5.4.1 Least-Squares Approach . . . . . . . . . . . . . . . . . . . . . 33
5.4.2 Threshold Approach . . . . . . . . . . . . . . . . . . . . . . . 35
6 Comparison of Numerical Methods 37
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.3 Software Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.4.1 Classical Put Options . . . . . . . . . . . . . . . . . . . . . . . 40
6.4.2 Exchange Options . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.4.3 Basket Options . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7 Conclusions 49
A Data 51
Bibliography 51
ii
List of Tables
6.1 Statistics for different paths generation methods for American Dax
call valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
A.1 Description Dax constituents . . . . . . . . . . . . . . . . . . . . . . 51
A.2 Correlation matrix for Dax constituents for April 30, 2009 . . . . . . 52
iii
List of Figures
3.1 Two-dimensional projections of pseudo-random sequences . . . . . . . 13
3.2 Two-dimensional projections of Halton sequences . . . . . . . . . . . 17
3.3 Two-dimensional projections of Sobol sequences . . . . . . . . . . . . 20
3.4 Two-dimensional projections of Niederreiter sequences . . . . . . . . . 21
4.1 Brownian bridge construction . . . . . . . . . . . . . . . . . . . . . . 26
6.1 Convergence properties plain-vanilla American put . . . . . . . . . . 42
6.2 Convergence properties American exchange options using Underlying-
Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Convergence properties American Dax call for different random num-
ber generators using Least-Squares approach . . . . . . . . . . . . . . 48
A.1 Convergence properties of American put for different random number
generators using Least-Squares approach . . . . . . . . . . . . . . . . 53
A.2 Convergence properties of American put for different random number
generators using Threshold approach . . . . . . . . . . . . . . . . . . 54
A.3 Convergence properties of American exchange options for different ran-
dom number generators . . . . . . . . . . . . . . . . . . . . . . . . . . 55
A.4 Convergence properties American Dax call for pseudo-random num-
bers and Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . . . 56
A.5 Convergence properties American Dax call for Niederreiter and Halton
sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
iv
Chapter 1
Introduction
The pricing and optimal exercise of options with early exercise features is one of the
most challenging problems in mathematical finance. The problem becomes even more
complex, if the model for the economy has more than one state variable, i.e. more
than one factor affecting the value of the option as well as the optimal exercise mo-
ment. American basket options are a prominent example for derivatives with early
exercise features and multiple state variables. Due to their construction principle bas-
ket options generally represent an advantageous alternative to hedge risky positions
of several assets, since trading only one basket option instead of several single-asset
options decreases transaction costs.
Usually lattice methods or finite differences will be used to price American or
Bermudan style options. However, these techniques are inefficient for high dimen-
sional problems (i.e. problems with more than three state variables) and they are
very difficult to apply on path-dependent options. For higher dimensions it would be
relatively cheap to use simulation techniques since its computational cost does not
increase exponentially as compared to other methods and they are relatively simple
to apply. However, for a long time, simulation techniques seemed unapplicable to
options with early exercise features. This is due to their forward-construction prin-
ciple and their path-by-path generation. Since contingent claims with early exercise
features are traded in all important derivative markets, many suggestions have been
made during the last years to price such options by simulation, and in particular to de-
termine the optimal early-exercise strategy. Examples can be found in Tilley (1993),
Barraquand and Martineau (1995), Carriere (1996), Broadie and Glasserman (1997),
Andersen (1999), Longstaff and Schwartz (2001), or Ibanez and Zapatero (2004).
In this case the starting point will be the application of standard Monte Carlo
simulation with pseudo-random numbers, eventually combined with some variance
1
reduction techniques such as antithetic variables. However, using pseudo-random
numbers can be quite slow, due to the relatively poor convergence rate of O(M−1/2)
for M sample paths. Here, the application of Quasi-Monte Carlo simulation (QMC),
e.g. using Sobol, Niederreiter or Halton sequences, can improve the performance of
Monte Carlo simulations dramatically, since in optimal cases the convergence rate
of QMC is O(log(M)dM−1), where d represents the dimension of the integration
problem. Therefore, for small values of d the convergence is due to the construc-
tion principle of low-discrepancy sequences significantly higher as for standard Monte
Carlo. However, for higher dimensional problems this advantage will diminish, since
the convergence strongly depends on the dimension of the integration problem for
which reason the application of QMC to price American basket option seems not ad-
vantageous. Especially for American options it cannot be expected that QMC works,
since due to regression or optimization over the sample paths the sample points have
interactions. However, Chaudhary (2005) discovered a significant speedup in the nu-
merical results for the Least-squares approach proposed by [25]. Therefore we also
expect to find this improvement for the convergence of the valuation of American
basket options when applying QMC. Moreover, earlier empirical studies for finance
problems, see among others [30, 29, 12], have shown in fact significant improvements
using QMC even in high dimensional financial pricing problems up to dimension 360.
The main reason is that many practical problems in mathematical finance have a low
effective dimension – a notion introduced by [12] – i.e. either the integrand depends
only of several variables, or the integrand can be approximated by a sum of functions
with each depending only on a small number of variables at the same time. Sev-
eral techniques to reduce the effective dimension have been successfully applied for
derivative pricing problems. The most prominent are the Brownian bridge method for
path construction (BB) first applied by Moskowitz and Caflisch (1996) and Caflisch
et al. (1997) and the principal component analysis (PCA) first used by Acworth
et al. (1997) to identify the underlyings which contribute the largest part of the total
variance. Besides this empirical studies Wang and Sloan (2007) provides a theoretical
analysis of the possible convergence order of QMC used in conjunction with BB or
PCA.
However, all these studies only analyze the convergence properties of QMC with
BB or with PCA, but, to the best of our knowledge, not the combined application of
QMC with both – BB and PCA.
In our analysis we consider the valuation of American basket options as a promi-
nent example for a high-dimensional derivative pricing problem, whereby we reduce
2
the effective dimension by applying the Brownian bridge method (BB) and the Princi-
pal Component Analysis (PCA) to reduce the effective time and underlying stock di-
mension. We focus our analysis on the convergence properties and numerical stability
by applying the different techniques as well as to develop a path construction method
which takes both effective dimension reduction techniques into account. Moreover,
we analyze two common approaches which can easily be adapted for high-dimensional
pricing problems to incorporate the early exercise features of American-style options
into the Monte Carlo method: The Threshold approach proposed by Andersen (1999)
and the Least-Squares (LS) approach suggested by Longstaff and Schwartz (2001).1
The thesis is structured as follows: The next chapter contains a general description
of basket options. The theoretical framework for Monte Carlo simulation is given in
chapter 3. Chapter 4 describes the different techniques to reduce the effective dimen-
sion of high dimensional pricing problems. The model framework for the valuation
of American basket options is outlined in chapter 5: this chapter contains the con-
struction of the Monte Carlo paths in this framework and takes into account the
effective dimension reduction techniques as well as the valuation approaches to deter-
mine the early exercise value of an American basket option are contained . Chapter
6 presents the numerical results from the application of these methods to test basket
options. A final chapter concludes with a brief summary of the results and some
recommendations for further applications of Quasi-Monte Carlo methods based on
our results.
1The hedging of American basket options and the calibration of the pricing model will not beconsidered in this thesis. We assume in our study, that all necessary market parameters are correctlygiven.
3
Chapter 2
Basket Options
2.1 Introduction
In a broader sense we summarize under the class ”basket options” all derivatives
whose payoff depends on more than one underlying. The underlying basket normally
contains different equities, commodities, or currencies. In the last years a wide range
of different basket options have emerged with special features such as early exercise
and path dependency. We describe the most prominent types of basket options in
following sections. Thereby we assume a basket of equities as underlying, since for
the analysis in this thesis we only consider this type of underlying. For the pricing
of basket options an appropriate standard model (such as Black-Scholes for equities)
for each individual asset and for the correlation between the underlying stochastic
drivers is used. The pricing model is provided in Section 5.2.
2.2 Different Types of Basket Options
2.2.1 Plain Vanilla Basket Options
Plain vanilla basket options are basket options in the true sense and give the holder
the right to buy (call) or sell (put) a specified basket of equities at an agreed price
(strike price) at the maturity date of the option (European style). The payoff, PO,
of a basket put option as function of the underlying equities, Si, is given as
POPutBasket(S1, ..., SU) = max(K −
U∑i=1
ωiSi, 0) (2.1)
where K is the strike price, U the number of underlying equities and ωi the fixed
weight of each asset in the basket, and Si the corresponding prices at the maturity
of the option. In case of physical settlement, the weights must be integers because
4
fractions of stocks cannot be delivered. For cash settlement options, which is usually
the case, this constraint can be neglected.
At first glance index options could also be considered as basket options, or vice
versa, since an index is nothing else as a basket of equities. But this is definitely
not the case in practical applications. The index weights normally change every day
depending on the free float, current stock price, etc. whereas the weights of a basket
options are fixed in the option contract. Therefore, for example, the price of a DAX
index option for a special day is not the same as the price of a basket option, where
the basket consists of the index constituents with weights valid for that special date,
and all other option parameters are the same.
Nevertheless, basket options are often priced by modeling the basket value as
single underlying, where the volatility of the fictitious underlying representing the
basket can be derived from the covariance matrix of the constituents. In that case,
standard option pricing formulas can be applied to value the option and to calculate
the greeks with respect to the basket value.
2.2.2 Exchange Options
Exchange options1 give the holder the right to exchange one asset for another, i.e.
the payoff depends on two assets. Obviously the price of such an option also depends
on the correlation between them.
POExch = max(S1 − S2, 0) (2.2)
Compared to plain vanilla options an exchange option can be interpreted as put option
on S2 with strike S1 or as call option on S1 with strike S2. Margrabe (1978) derived a
closed form solution for the European exchange option value V in the Black-Scholes
framework for the two-asset case, which in general is not possible for basket options
VExch(S1, S2, t) = S1e−D1(T−t)N(d1)− S2e
−D2(T−t)N(d2) (2.3)
where
d1 =log(S1
S2
)+ (D2 −D1 + 0.5σ2)(T − t)
σ√
(T − t)and d2 = d1 − σ
√(T − t), (2.4)
and σ =√σ2
1 + σ22 − 2ρ12σ1σ2 (2.5)
1Exchange options are also known as Margrabe options or outperformance options.
5
Here, Di represents the respective dividend yield of asset i, t the current time, T the
maturity of the exchange option, N(di) the cumulative normal distribution function
of di, σi the volatility of asset i and ρi,j the correlation between the assets i and j,
i, jε1, 2.
2.2.3 Asian Basket Options
The payoff of Asian Basket Options depends on the average value of the underlying
basket over a specific period during the lifetime of the option. Thereby, the average
used in the calculation is usually defined as arithmetic or geometric average. For
practical and legal reasons, the dates to calculate the average are sampled discretely,
e.g. only closing prices every Wednesday are used to calculate the average. In principle
continuous sampling of data would be thinkable as well. Besides the typical features
of options Asian options can be further classified into ”average rate” or ”average
strike” option.
The payoff of an average rate option is based on the difference between the basket
average value and a fixed strike
POCallAsian Rate(S1, ..., SU) = max(A−K, 0) (2.6)
with
A =T∑t=0
U∑i=1
ωiSi(t) (2.7)
is the arithmetic average over the sampling period and B =∑U
i=1 ωiSi(T ) the value
of the basket at time T . The payoff of an average strike option is based on the
difference between the current basket average value and the basket average value over
the sampling period
POCallAsian Strike(S1, ..., SU) = max(B − A, 0). (2.8)
Continuously-sampled, geometric average rate options can be valued using a mod-
ified version of the Black-Scholes formula, if the underlying value follows a lognormal
process. In this case the geometric average is also lognormal distributed. Kemna and
Vorst (1990) derived this solution by adjusting the volatility and the cost of carry
term of the Black-Scholes formula. However, this formula can only be used for Asian
basket options, if the basket value itself is modeled as underlying (compare 2.2.1).
For arithmetic average rate options a closed form solution cannot be given but there
are different approximation formulas (e.g. Turnbull and Wakeman (1991) or Levy
(1992)) to calculate the value of this option type.
6
2.3 American-style Basket Options
American-style basket options are special in the sense that they can be exercised at
any time before their maturity date. This feature is quite common in derivatives
markets and not special for basket options, e.g. most traded single equity options are
American style. As the combination of basket options with early exercise features is
one of the hardest problems to solve in mathematical finance we give some general
remarks about American-style options.
Due to the early exercise feature American-style options are more valuable than
their European-style counterparts which can only be exercised at maturity. This can
be directly concluded from the early exercise criterion: this says that the option should
be exercised as soon as the option payoff is greater than the value of continuation;
mathematically:
PO(t) ≥ Et[PO(t∗)] (2.9)
where E represents the expectation operator, and t∗ the optimal future exercise point
of the option. Therefore, an American-style option is at least as valuable as its
European-style counterpart.
The optimal exercise boundary of an option is then given by all points where
equation (2.9) holds. Therefore, the optimal exercise boundary for basket options is
a function of time t and all underlying values Si, with i = 1, ..., U . If the early exercise
boundary is known, valuation of American-style options is straightforward. But this
is not the case in reality. Thus, the challenging part of valuing American-style options
is the determination of the early exercise boundary, which has to be done numerically,
since no closed form solution can be derived for the free boundary problem. Examples
for approximation formulas to value American-style options can be found in Geske
and Johnson (1984), Barone-Adesi and Whaley (1987) or in a quite recent article from
Zhu (2006). But, for basket options with several underlyings these approximations
can generally not be applied, especially when the option payoff depends at the end on
one single equity from the basket. American exchange options are one of the special
cases where a closed form approximation was derived by Bjerksund and Stensland
(1993) based on the approximation formulas for American single-equity options.2
2This is contrary to Margrabe (1978) who has argued that American exchange options are notmore valuable than European ones.
7
Chapter 3
Monte Carlo Simulation
3.1 Introduction
Monte Carlo simulation is a numerical method that is advantageous for many mathe-
matical problems where no closed-form solution is available, especially for high dimen-
sional problems. Due to its nature Monte Carlo simulation is mainly used to simulate
random variables or stochastic processes, that can be described by some given prob-
ability density function. One of the most prominent applications for Monte Carlo
is that of numerical integration, i.e. the solution of definite integrals with compli-
cated boundary conditions.1 This concept of numerical integration by Monte Carlo
simulation was originally introduced for option pricing theory by Boyle (1977).
To illustrate the idea for numerical integration consider the one-dimensional inte-
gral ∫Ω
f(x)p(x)dx = Ep[f(x)] = I(f) (3.1)
where f(x) is an arbitrary function and p(x) is a probability density function with∫Ω
p(x)dx = 1. (3.2)
Thereby, Ω denotes the range of integration and Ep[f ] is the expected value of f
with respect to p. The problem of numerical integration can be interpreted as the
approximation of an expected value. To estimate the value of I(f) a large number M
of independent and identically distributed (i.i.d) sample values xi of the probability
1Beside numerical integration another powerful and very popular application for Monte Carlois stochastic optimization. Optimization problems are by nature not stochastic. However, to findthe global optimum of a high-dimensional optimization problem where the objective function isnondifferentiable such that usual gradient methods fail, stochastic optimization can be applied.Thereby, the sample points of the multi-dimensional space will be drawn randomly and the objectivefunction is then evaluated with the drawn sample points. This procedure will be repeated as longas new optima are found outside a given error bound compared to the last found optimum.
8
density function p(x) are drawn by using a random number generator. The estimate
I(f) of I(f) is calculated as the average over all evaluations of f(x) at the random
sample points and is given by
IM(f) =1
M
M∑i=1
f(xi). (3.3)
Benefits of using Monte Carlo simulation are the very basic mathematics, the
simple modeling of correlations, and the flexibility of the method. In addition, the
increased availability of more powerful computers and software packages during the
recent years makes it easier to implement Monte Carlo methods and therefore en-
hances its popularity. The main disadvantage is that the method is slow to solve
partial differential equations for low-dimensional problems up to three or four dimen-
sions compared to other numerical methods like finite-differences. Moreover, applying
Monte Carlo in Mathematical Finance to free boundary problems, like American-style
options, is very challenging and seemed impracticable up to the mid 90s.
The rest of this chapter gives some basics of Monte Carlo simulation without
incorporating financial engineering applications to show the generality of this method.
3.2 Mathematical Foundations
3.2.1 Strong Law of Large Numbers
The strong law of large numbers (SLLN) ensures that the estimate as given in (3.3)
converges almost surely to the true value of the integral in the sense that
limM→∞
IM(f) = limM→∞
1
M
M∑i=1
f(xi)a.s.−−→ I(f) (3.4)
i.e.
P
(limM→∞
1
M
M∑i=1
f(xi) = I(f)
)= 1 (3.5)
Almost surely means that other values are theoretically possible for a given sample,
whereas for increasing sample sizes, the probability of other values converges toward
zero. On the opposite side this also means that IM(f) is itself a random variable and
not a deterministic number.
9
3.2.2 Error Estimation
Concluding from the SLLN, IM(f) is an unbiased estimator for I(f), i.e. E[IM(f)] =
I(f) which converges to the true value almost surely. We can define the integration
error, εM(f), as
εM(f) = IM(f)− I(f) (3.6)
so that it is a zero-mean random variable. Given this definition, the estimation bias
is defined as
Bias(M) = E[εM(f)] (3.7)
and the root mean square error (RMSE) as
RMSE(M) =√
E[εM(f)2]. (3.8)
Empirically, the RMSE is often estimated as the standard deviation of IM(f), since
the true value of the integral (3.1) is not known.
3.2.3 Central Limit Theorem
The central limit theorem (CLT) allows us to characterize the size and statistical
properties of the Monte Carlo integration error. Starting with the unbiased estimator
IM(f) from (3.3) with xi i.i.d. and assuming Var(xi) = σ2x < ∞ and Var(f(xi)) =
σ2f <∞, then the CLT asserts that as the sample size M increases, the standardized
estimatorIM(f)− I(f)
σf/√M
(3.9)
converges in distribution to the standard normal distribution, which is often written
as2
limM→∞
IM(f)− I(f)
σf/√M
= limM→∞
εM(f)√M
σf⇒ N (0, 1), (3.10)
which is equivalent to
εM(f) ∼ σf√M
z , as M →∞ (3.11)
where z ∼ N (0, 1) is a zero-mean unit-variance normally distributed random variable,
⇒ denotes convergence in distribution and N (x, y2) the normal distribution with
mean x and variance y2.
2The precise statement for convergence in distribution is given by
limM→∞
P
(IM (f)− I(f)σf/√M
≤ Z
)=
1√2π
∫ Z
−∞e−s22 ds.
10
Therefore, the CLT ensures that the standard error of the estimation tends to
zero at rate 1/√M , i.e. to lower the integration error by a factor of ten the sample
size has to be increased by hundred. The convergence of Monte Carlo simulation can
sometimes be improved by application of variance reduction techniques such as anti-
thetic variables, stratified sampling, control variates or moment matching methods.
See Boyle et al. (1997) or Wilmott (1999) for a detailed description of these methods
in mathematical finance.
As seen above, the CLT is usually stated as formula for the integration error for
given sample size M and known variance σ2f . However, in many practical applications,
the exact value of the integral is unknown, so errors and the variance cannot be
determined. In this case, the CLT formula will be used conversely, i.e. the empirical
sample variance, σ2f , is determined from evaluation f with M points, where M < M ,
and then the sample size for a required level of accuracy, ε, with confidence c is
determined by
M = ε2σ2fN−1(c)2, (3.12)
where N−1 is the inverse of the cumulative normal distribution function.
3.3 Pseudo-Random Numbers
The core of nearly each application of Monte Carlo simulation is a sequence of random
numbers used to run the simulation. This random numbers have to satisfy a given
probability density function, if the number of draws goes to infinity. This is typi-
cally done by transformation of independent, uniformly distributed random numbers
drawn from the interval (0,1)3. To transform the random numbers into the desired
distribution function, usually the inverse cumulative distribution function is used.4
Most programming languages and spreadsheets provide a uniform random-number
generator. Note, however, that these numbers are not random per se; they are pseudo-
random, since computers follow deterministic algorithms to produce these numbers.
Thereby, the pseudo-random sequences are made to have as many properties of ’true’
3Here, 0 and 1 are explicitly excluded from the interval, since for most probability distributionthese points maps to either −∞ or ∞, which leads to numerical problems.
4If there is no closed-form solution for the inverse cumulative distribution function, mostly anaccurate approximation is used. For standard normal variables some other transformation methodsare also available, e.g. Box-Muller or Marsaglia method.
11
random numbers as possible, i.e. they are sufficiently good at mimicking genuine
random numbers with respect to fundamental statistical tests5.
The linear congruential method is the most commonly used method to generate
uniformly distributed pseudo-random number sequences. The basic idea is to produce
integer values ci on a given interval [0, C−1], where C is some constant, and to return
a uniform (0, 1) variable xi by rescaling ci. The recurrence algorithm can be written
in the following form:
ci+1 = (aci + b) mod C (3.13)
xi+1 = ci+1/C (3.14)
where the multiplier a and b are integer constants. The initial value (seed) c0 is
an integer in the given interval [0, C − 1] and is specified by the user, to be able to
rerun a simulation with the same settings (’Reproducibility’). Once cN = c0 holds, the
sequence starts over and will repeat itself, so the period length of a linear congruential
method is an critical issue6. Beside the linear congruential method other methods
for uniform random number generators exist as well, e.g. mid-square method or the
Mersenne twister7.
There exists a series of reliable methods to produce uniform pseudo-random se-
quences from Press et al. (1992), called ran0 to ran4, that have been well tested and
documented. In our study we use ran2 to generate pseudo-random numbers.
In Figure 3.1, we show pairwise projections of the first 2048 random points gener-
ated with ran2 on several two-dimensional unit intervals. The particular dimensions
shown were selected randomly. It can be seen that clusters of points are possible,
since the points are independently drawn and therefore have a certain chance to be
very close to other points. On the other side, there are also empty spaces, where
no points are observed. Moreover, it can be seen that the patterns are independent
of the considered pair of dimension. The clumps and empty spaces are one reason
for the relatively poor convergence rate of O(M−1/2) for Monte Carlo simulations
with pseudo-random numbers. However, since the patters are independent of the di-
mension, the error bound and convergence rate is also independent of the dimension,
which makes Monte Carlo simulation valuable for high-dimensional problems.
5Niederreiter (1992) provides an overview of pseudo-random number generation and statisticaltests for randomness of uniform random numbers as well as the most common methods to transformuniform random numbers into nonuniform random numbers.
6If N = (C−1), every distinct value in the interval is produced before repeating and the methodis said to have full period. However, a large value of C does not guarantee this property, since theperiod length depends also on the choices for a and b.
7See Glasserman (2003), Jackel (2002) or Niederreiter (1992) for different methods of uniformrandom number generation.
12
3.1.1 Dimension 1 vs Dimension 2 3.1.2 Dimension 27 vs Dimension 32
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Figure 3.1: Two-dimensional projections of pseudo-random sequences for different dimensions for the first 2048points.
3.4 Quasi-Random Numbers
Quasi-random numbers8, also known as low-discrepancy sequences9, differ from or-
dinary Monte Carlo with pseudo-random numbers in the sense that they make no
attempt to mimic randomness. Instead, they are designed to increase accuracy
specifically by generating points which avoid gaps and clusters, i.e. they are too
evenly distributed to be random. Therefore, Monte Carlo Simulation with quasi-
8The term quasi-random is in some sense misleading, since low discrepancy sequences are totallydeterministic.
9A good discussion of low-discrepancy sequences can be found in Glasserman (2003) and Nieder-reiter (1992).
13
random numbers, henceforth we will call this Quasi-Monte Carlo (QMC), can im-
prove the performance of Monte Carlo simulations, offering shorter computational
times and/or higher accuracy, respectively. In optimal cases the convergence rate of
QMC is O(log(M)dM−1), where d represents the dimension of the integration prob-
lem.
For the remainder of this section, we give a short introduction to the number-
theoretical concept of QMC and introduce some common low-discrepancy sequences,
which we use in our analysis.
3.4.1 Low Discrepancy and Integration Error
The main objective of quasi-random sequences is to improve the uniformity of the
sequence, which is measured in terms of its discrepancy. This is defined by considering
the number of points in rectangular subsets R =∏d
j=1(uj, vj) of the d-dimensional
unit cube Id = [0, 1]d. Let XM = (xi)Mi=1 be a sequence of M points in Id, then for
an arbitrary subset RεR, the counting function ](XM , R), that indicates the number
of points for which xiεR, is defined as
](XM , R) =M∑i=1
χR(xi), (3.15)
where χR is the characteristic function of set R. The (extreme) discrepancy of XM is
then given by
DM = supRεR
∣∣∣∣](XM , R)
M−m(λd(R)
∣∣∣∣ , (3.16)
where λd(R) denotes the d-dimensional volume (measure) of R. If R? is the set of all
rectangles in Id of the form∏d
j=1(0, vj), the star discrepancy is defined as
D?M = sup
RεR?
∣∣∣∣](XM , R)
M− λd(R)
∣∣∣∣ . (3.17)
If a sequence XM is perfectly uniformly distributed, then DM = D?M = 0. For
the one-dimensional case, Niederreiter shows that D?M ≥ 1
2M, whereas for the multi-
dimensional case, i.e. d ≥ 2, Niederreiter states that ”it is widely believed” that any
point set XM satisfy
D?M ≥ cd
log(M)d−1
M(3.18)
and for any infinite sequence X∞ the first M elements satisfies
D?M ≥ c′d
log(M)d
M, (3.19)
14
where the constants cd > 0 and c′d > 0 depend only on d. Niederreiter [28] gives also
some other definitions of discrepancy as well as further useful properties.
The star discrepancy plays a central role for the definition of the integration error
as defined in (3.6), when quasi-random numbers are used. The key result for QMC
error bounds is known as the Koksma-Hlawka inequality10, which gives an upper
bound of the integration error by the product of two quantities – one depending only
on the integration f , and one depending only on the quasi-random sequence used.
The Koskma-Hlawka inequality says that if f has bounded variation V (f) in the sense
of Hardy and Krause on Id, then for any sequence XMεId, the following inequality
holds11 ∣∣∣∣∣ 1
M
M∑i=1
f(xi)−∫Id
f(u)du
∣∣∣∣∣ ≤ V (f)D?M . (3.20)
The variation for the one-dimensional case is V (f) =∫ 1
0|df |. For the multidimen-
sional case the variation in the sense of Hardy and Krause is given as
V (f) =d∑
k=1
∑1<i1<i2<···<ik<d
V (k)(f ; i1, ..., ik) (3.21)
where
V (d)(f) =
∫ 1
0
· · ·∫ 1
0
∣∣∣∣ ∂df
∂u1 · · ·ud
∣∣∣∣ du1 · · · dud (3.22)
which holds, whenever the indicated partial derivative is continuous on Id. Using
inequality (3.19) the integration error as given by the Koksma-Hlawka inequality
(3.20) can rewritten as∣∣∣∣∣ 1
M
M∑i=1
f(xi)−∫Id
f(u)du
∣∣∣∣∣ ≤ V (f)c′dlog(M)d
M, (3.23)
i.e. the convergence rate for numerical integration using pseudo-random sequences is
O( log(M)d
M).
Comparing the QMC integration error bound to the standard Monte Carlo inte-
gration error (3.11) leads to the following observations:
• In both cases the error bound is a product of one term depending on the inte-
grand function f and another term depending on the properties of the sequence.
10For the one-dimensional case the inequality was derived by Koksma and generalized by Hlawkafor multi-dimensional problems.
11Compare Niederreiter (1992) for an excellent discussion of error bounds of QMC.
15
• The Koskma-Hlawka inequality (3.20) provides an strict deterministic error
bound, whereas the CLT provides only a probabilistic one.
• Both terms in (3.20) are extremely hard to compute, whereas the Monte Carlo
variance σ2f can be easily estimated.
• The Koskma-Hlawka bound provides a worst-case error bound which is often
found to overestimate the true QMC integration error, whereas the CLT pro-
vides a relatively good proxy for the integration error in a Monte Carlo estimate.
• QMC achieves with the same number of function evaluations a significantly
higher accuracy than the standard Monte Carlo method, which holds for reason-
able dimensions and for large M . However, the greater d will be, the Koskma-
Hlawka bound does not imply any advantage for QMC with moderate values
of M . However, for financial problems a much higher accuracy using QMC was
observed, even for several hundred dimensions, by Paskov and Traub (1995),
Joy et al. (1996) or Caflisch et al. (1997).
The QMC error bound as provided by the Koskma-Hlawka inequality leads to
the conclusion that quasi-random sequences with small star discrepancy should be
constructed, since they will guarantee small QMC integration error. This theoretical
insight shows the most important benefit of the star discrepancy and the Koskma-
Hlawka inequality, since for practical purposes, they have only limited applicability.
3.4.2 Halton Sequences
Halton sequences [18] have the simplest construction principle of quasi-random num-
bers in arbitrary dimensions d. The idea of Halton sequences is to represent an
arbitrary integer in different number bases for each dimension. In general, the natu-
ral choice for the integer will simply be the number m for the m-th sequence draw.
Let p1, p2, ..., pd be the first d relatively prime numbers greater than 1, then for each
required dimension i = 1, ..., d the integer m can be represented in the associated
prime number base as
m =
blogpi(m)c+1∑j=0
aijpji (3.24)
16
with all aij < pi. Now, Halton quasi-random numbers in the interval (0, 1) can be
calculated by the radical inverse function φpiof base pi, which is defined as
φpi(m) =
blogpi(m)c+1∑j=0
aijp−(j+1)i . (3.25)
The m-th d-dimensional vector of the Halton sequence is then given by
xm = (φp1(m), φp2(m), ..., φpd(m)). (3.26)
3.2.1 Dimension 1 vs Dimension 2 3.2.2 Dimension 27 vs Dimension 32
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Figure 3.2: Two-dimensional projections of Halton sequences for different dimensions for the first 2048 points.
In Figure 3.2, we show pairwise projections of the first 2048 Halton points on
several two-dimensional unit intervals. For the first two dimension, the uniformity of
17
the sequence looks quite good, i.e. we would expect a low star discrepancy. However,
for higher dimensions the uniformity decreases dramatically, which is inherent to the
construction principle. If the base is large, long monotone segments will be produced,
i.e. we observe long diagonal segments in the projections.
Therefore, we would expect a relatively bad convergence for high-dimensional
problems, since the star discrepancy will be relatively high for reasonable numbers of
simulation M . However, we will use the Halton sequence due to their easy construc-
tion principle as benchmark for our analysis, which we presume to perform not even
better than standard Monte Carlo in high-dimensional problems.
3.4.3 Sobol Sequences
Sobol sequences [32] probably are the most common used quasi-random sequence
for high-dimensional numerical integration and the first so-called (t, d)-sequences12 in
base 2, where t is a quality parameter which measures the uniformity of the sequence.
In general, the smaller the value of t, the more uniformly distributed the points of
the sequence are, i.e. the lower the discrepancy of the sequence.
Joe and Kuo [21] suggest an algorithm to construct high-dimensional Sobol se-
quences by treating Sobol sequences in d dimensions as (t, d)-sequences and then
optimizing the t-values of the two-dimensional projections. They obtain good two-
dimensional projections with relatively good convergence properties. Joe and Kuo
also provide primitive polynomials and initial direction numbers for dimensions up to
8300 and an C++ implementation based on the Gray Code implementation proposed
by Antonov and Saleev [4].13 This implementation of the Sobol sequence is used in
our analysis and will be briefly described next. A more detailed explanation of gen-
erating Sobol sequences is given in Bratley and Fox [10], Glasserman [17] or Jackel
[20].
For each dimension i = 1, 2, ..., d of the Sobol sequence a primitive polynomial14
of some degree si in the field Z2 of the form
Pi(x) = xsi + a1,ixsi−1 + a2,ix
si−2 + . . .+ asi−1,ix+ 1 (3.27)
12For a brief discussion of (t, d)-sequences as well as the closely related (t,m, d)-nets see the nextsubsection. An extensive overview of the theory is given in Niederreiter [28].
13Available via http://www.maths.unsw.edu.au/∼fkuo/sobol/14Primitive polynomials cannot be factored into polynomials of lower order using modulo 2 integer
arithmetic.
18
has to be chosen, where the coefficients a1,i, a2,i, . . . , asi−1,i are either 0 or 1. Defining
a sequence of positive odd integers mj,i by the s-term recurrence relation
mj,i = 2a1,imj−1,i⊕22a2,imj−2,i⊕· · ·⊕2si−1asi−1,imj−si+1,i⊕2simj−si,i⊕mj−si,i, (3.28)
where ⊕ is the bit-wise exclusive-or operator (XOR), the so-called direction numbers
v1,i, v2,i, . . . are defined by
vj,i =mj,i
2j. (3.29)
Thereby, the initial values m1,i,m2,i, . . . ,msj ,i can be arbitrary odd integers fulfilling
the property mj,i < 2j, such that each direction number vj,i lie strictly between 0 and
1. The k-th point in dimension i of the Sobol sequence, xk,i, is then given by
xk,i = b1v1,i ⊕ b2v2,i ⊕ · · · , (3.30)
where k =∑blog2(k)c+1
j=0 bj2j is the binary representation of k. Using the Gray code
implementation proposed by [4] the Sobol points can generated using
xk,i = gk,1v1,i ⊕ gk,2v2,i ⊕ · · · , (3.31)
instead of (3.30), where gk,j is the j-th digit from the right of the Gray code of k in
binary representation. Since in a Gray code representation, k and k+ 1 differ in only
one known position, e.g. the l-th bit (which is the rightmost zero bit of k), the Sobol
points can be constructed recursively using
xk+1,i = xk,i ⊕ vl,j, (3.32)
where l represents the in which k and k + 1 differ.
Figure 3.3 shows pairwise projections of the first 2048 Sobol points on several
two-dimensional unit intervals using the primitive polynomials and initial direction
numbers provided by Joe and Kuo. Since the choice of the direction numbers is
optimized to obtain good two-dimensional projections, we observe - as expected by
the construction principle - very uniformly distributed points. Therefore, we could
expect a low star discrepancy and thus good convergence properties, even for high-
dimensional problems.
3.4.4 Niederreiter Sequences
Niederreiter [28] develops a general construction principle for low-discrepancy se-
quences in arbitrary bases. Therefore, Niederreiter defines point sets and sequences
19
3.3.1 Dimension 1 vs Dimension 2 3.3.2 Dimension 27 vs Dimension 32
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3.3.3 Dimension 111 vs Dimension 128 3.3.4 Dimension 255 vs Dimension 256
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1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 3.3: Two-dimensional projections of Sobol sequences for different dimensions for the first 2048 points.
with a very regular distribution behavior, called (t,m, d)-nets and (t, d)-sequences,
respectively15. A (t,m, d)-net in base p is a point set X of pm points in Id such
that ](E,X) = pt for every elementary interval E in base p with λd(X) = pt−m and
0 ≤ t ≤ m are integers. A sequence of points x0,x1, . . . in Id is a (t, d)-sequence in
base p if, for all integers k ≥ 0 and m > t, the point set consisting of the xn with
kpm ≤ n ≤ (k+1)pm is a (t,m, d)-net in base p. Equipped with the theoretical frame-
work Niederreiter provides general (star) discrepancies and error bounds for (t, d)-
sequences in arbitrary bases and shows how to achieve a t parameter strictly smaller
15Originally, they were called (t,m, s)-nets and (t, s)-sequences, since Niederreiter uses s as di-mension parameter.
20
than the best t parameter for Sobol sequences for all dimensions greater than seven.
Thus, Niederreiter sequences have some theoretical superiority over Sobol sequences.
From a constructional point of view, they are based on polynomial arithmetic mod-
ulo some base p but used irreducible polynomials instead of primitive polynomials.
Details are given in [28] at great length.
3.4.1 Dimension 1 vs Dimension 2 3.4.2 Dimension 27 vs Dimension 32
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Figure 3.4: Two-dimensional projections of Niederreiter sequences for different dimensions for the first 2048points.
Figure 3.4, shows pairwise projections of the first 2048 Niederreiter points on sev-
eral two-dimensional unit intervals generated with the Niederreiter sequence generator
in base 2 implemented by John Burkardt [11] and adapted to dimension 8300 using
the primitive polynomials and initial direction numbers provided by Joe and Kuo.
For low dimensions the uniformity looks quite good whereas for higher dimensions
21
the Niederreiter sequence generator used gives patterns similar to Halton sequences.
One reason for the bad uniformity in high dimension could be the application of the
primitive polynomials and initial direction numbers optimized for Sobol sequences.
Overall we would expect that Niederreiter sequences in our implementation perform
worse than Sobol sequences and similar to Halton sequences.
3.5 The curse of dimensionality
Low-discrepancy methods are designed for numerical integration purposes and have
the potential to accelerate the convergence rate from the O( 1√M
) rate associated with
standard Monte Carlo to nearly O( 1M
) under appropriate conditions. The exact con-
vergence rate is given by O( log(M)d
M) derived in (3.23). This strong dependence on
dimensionality leads directly to a problem with low-discrepancy sequences: They are
only in their lowest dimensions significantly better than pseudo-random numbers, i.e.
the effectiveness of low-discrepancy sequences is lost for problems of high dimension-
ality. Since American basket option pricing is a high dimensional problem (each time
step represents one dimension plus the high number of state variables) QMC seems
to be inapplicable. However, many practical problems in mathematical finance have
a low effective dimension – a notion introduced by Caflisch et al.[12] – i.e. either the
integrand depends only of few variables, or the integrand can be approximated by
a sum of functions with each depending only on a small number of variables at the
same time. In our analysis we will try to reduce the effective dimension by applying
the Brownian bridge method (BB) and the Principal Component Analysis (PCA),
which will be described in the next chapter.
It should be mentioned in this discussion that on the other side, finite difference
methods or lattice approaches – which are well established for the pricing of American
options – seem inapplicable for baskets with more than three underlying assets, since
the joint dynamics of the underlying assets usually does not lead to a re-combining
lattice, so that the computational effort and the number of nodes in the lattice grows
exponentially with the dimension of the pricing problem.
22
Chapter 4
Effective Dimension Reduction
4.1 Introduction
As we have seen in the previous chapter, the convergence rate of QMC depends
on the dimension d of the integration problem. This could lead to the conclusion
that especially for high-dimensional problems the advantageous of QMC will vanish.
However, many empirical studies for high-dimensional financial pricing problems have
shown substantial improvements using QMC compared to standard Monte Carlo by
reducing the effective dimension. This can be done by applying the Brownian bridge
method (BB) and/or Principal Component Analysis (PCA). For our special pricing
problem, PCA will be used to reduce the number of state variables1 whereas BB will
be applied for discretization of the random walk, which reduces the time dimension.
The details of both methods are explained in the following sections.
4.2 Principal Component Analysis
Principle component analysis is a statistical technique which extracts those statistical
components from time series data that are most relevant for the dynamics of a multi-
dimensional process classified by their importance. It can be shown by means of this
method that most of the time more than 90% of the underlying dynamics of an basket
can be explained by the most important components. Depending on the number of
assets within the basket and the specific correlation structure between the assets the
number of the most important components is between 1 to 5 for reasonable basket
sizes.
1If the eigenvalues in the spectral decomposition of the marginal covariance matrix are sorted,most of the variance of the process is allocated to the first random factors, which improves theconvergence of the simulation.
23
PCA investigates the time series of several stochastic processes which may be
strongly correlated. In financial applications, each stochastic process represents an
underlying asset. As a typical example of such a group of U risk factors, we consider
the relative changes of the underlying asset prices of an equity basket. The variances
and pairwise correlations of these U risk factors are typically represented by the
covariance matrix Σ given as
Σ =
σ2
1 Σ12 . . . . . . Σ1U
Σ21. . . . . . Σ2U
... Σij...
.... . . . . .
...ΣU1 ΣU2 . . . . . . σ2
U
(U×U)
, (4.1)
where σi is the standard deviation of underlying i, Σij = ρijσiσj the covariance
between underlying i and j and ρij their correlation. The values of the covariance
matrix are typically calculated from historical time series data. The central idea of
PCA is now to reduce the effective dimension of the problem based on its statistical
structure given by (4.1). This reduction is achieved by finding the most important
components identified by the biggest eigenvalues of Σ. Since Σ is symmetric it can
be factorized as Σ = AAT , where again A can be written as A = BΛ1/2, i.e. Σ can
be diagonalized as
Σ = BΛBT . (4.2)
As a real symmetric U × U matrix, Σ has U real eigenvalues λ1, λ2, . . . , λU , where
each λi is nonnegative, since Σ is positive definite or semidefinite. Moreover, the set
of eigenvectors ν1, ν2, . . . , νU of Σ is orthonormal, i.e.
νTi νi = 1, νTi νj = 0, i 6= j, i, j = 1, . . . , U. (4.3)
From this it follows that B is the orthogonal matrix (BBT = I) with columns
ν1, ν2, . . . , νU and Λ is the diagonal matrix with diagonal elements λ1, λ2, . . . , λU .
Therefore,
A = BΛ1/2 =
ν1 ν2 · · · · · · νU
√λ1 √
λ2
. . . √λU−1 √
λU
(4.4)
and
Σ = AAT = BΛBT . (4.5)
24
Consider the generation of arbitrary U -dimensional normal random numbers x1, x2,
. . . , xU with covariance structure Σ, i.e. X ∼ N(0,Σ), from given multivariate stan-
dard normal random numbers Z ∼ N(0, I). Then x can be generated as Az, where
z can be interpreted as independent factors driving the processes of X, with Aij as
the factor sensitivity of zj on xi. Then the k-th principal component of X is obtained
from solving the following optimization problem:
maxωk
ωTk Σωk = maxωk
ωTk AATωk = maxωk
ωTk BΛBTωk (4.6)
subject to ωTk ωk = 1 and ω?i ωk = 0, i = 1, . . . , k− 1, where ω?i represents the optimal
vector for the i-th principal component found in previous iterations. The optimal
solution is easily obtained, if the eigenvalues of Σ are ordered as follows:
λ1 ≥ λ2 ≥ . . . ≥ λU . (4.7)
Then ω?i = νi solve the optimization problem, i.e. ω?iTΣω?i = νTi Σνi = λi. Therefore,
the first k principle components provide an optimal lower-dimensional approximation
to a random vector. Thereby, the explained variance of X is given by the ratio∑ki=1 λi∑Ui=1 λi
. (4.8)
In practical applications mostly the number of principle components is chosen in such
a way, that a given level of variance will be explained.
Summarizing the explanations, PCA can enhance the effectiveness of QMC by
concentrating most of the variance in the first dimensions and by mens of that reduce
the effective dimension of the pricing problem. Each mathematical or statistical
software program contains methods to calculate eigenvalues and eigenvectors, which
makes PCA an easy to use method. In our analysis we use the routines TRED2,
TQLI and EIGSRT provided by [31] to calculate the square root matrix A of Σ, when
applying PCA.
4.3 Brownian Bridge
The Brownian motion or Wiener process, W , is the key driver for most stochastic
processes in mathematical finance. Since Brownian motion is a Markov process, the
standard discretization of the random walk to calculate further values Wti = Wt0+∆ti
from known values today Wt0 is given as
Wti = Wt0 + xi√
∆ti (4.9)
25
where xi is a standard normal random variable. Hence, the whole path is generated
by using a forward construction principle.
Brownian bridge is an alternative way to generate a sample path of W by using
the first random variable x1 to determine the terminal value WtN of the path. All
other random variables x2, . . . , xN are used to add more and more fine structure to
the path conditional on the already known realizations of the paths. Suppose, we
already know two values of the Brownian motion, Wti and Wtj , with ti < tj and want
to calculate the refinement between these two values at tk, with ti < tk < tj. The
value Wtk is then given by the general formula for the BB construction principle as
Wtk =tj − tktj − ti
Wti +tk − titj − ti
Wtj + xk
√(tk − ti)(tj − tk)
tj − ti, (4.10)
which can derived from the distributional properties of Brownian motion as shown in
[20] and is illustrated in Figure 4.1.
W
jtW
itW
it kt jt
Figure 4.1: Illustration of the Brownian bridge construction principle: Conditional on Wti and Wtj the value attk is normally distributed and can be derived using the conditional mean, which is obtained by linear interpolationbetween both values, and the conditional variance.
Comparing the BB construction principle (4.10) with the standard forward con-
struction given in (4.9) shows no direct advantages, neither in computational terms,
i.e. speed up of performance, nor in statistical meaning, since (4.10) is constructed
in such a way that it meets the same statistical characteristics as (4.9). However,
the potentially huge advantage comes in when using QMC in combination with BB:
through BB construction the first dimension of the low-discrepancy sequence will be
26
used to determine the terminal values of Brownian paths such that these are relatively
well distributed. Moreover, the general shape of the paths are already determined
by the first dimensions of the low-discrepancy sequence and the higher dimensions
will influence only the refinement of the paths, which is less important for weak path-
dependent options. Hence, much of the variance will be concentrated in the first large
steps. This property is quite useful in mathematical finance, since for many options
are extremely sensitive to the terminal value of the underlying. Therefore, the effec-
tive dimension of a high-dimensional pricing problem can be lowered by applying the
combination of BB and QMC. This is the reason why this application is very popular
in finance and widely used. In our analysis we use the C++ implementation of the
BB construction principle provided by [20].
27
Chapter 5
Valuation of American BasketOptions
5.1 Introduction
The pricing of basket options involves models of all underlying stocks. We assume
that the underlying stocks follow correlated geometric Brownian motions. Concretely,
in our analysis we apply Black-Scholes theory to solve this pricing problem. The
underlying theory is reviewed in the next section, especially the handling of multiple
underlyings. Beyond that there exist also approaches to price basket options with
single-factor models, in which the whole basket is modeled directly with a particular
stochastic process, mostly as GBM to apply ’simple’ Black-Scholes theory. These
approaches are not considered in our analysis.
The application of Monte Carlo methods to price American basket options is
probably most commonly used for it. This is mainly due to the ability of handling
numerous assets which otherwise would be difficult to evaluate with other numerical
methods like trees or finite differences. Thereby, one of the most challenging problems
in pricing and hedging basket options stems from the determination of the correlation
matrix, since the option value and the greeks are in general very sensitive to it and
the correlations themselves are not directly observable. One way of measuring the
correlations uses information from financial time series but they could be unstable and
misleading. This is due to large market fluctuations, as currently observed in financial
markets, and often depends on the chosen interval to calculate the correlations from
the time series data. Another way is to back out implied correlations from quoted
market prices. This is similar to the concept of implied volatilities for equity options
in the Black-Scholes world. In contrast to implied volatilities the market for basket
options is not such wide and liquid as for stock or index options, such that illiquid
28
prices of basket options probably lead to questionable results. Furthermore, it could
be possible for a special basket of assets that no basket option is available in the
market to back out the implied correlation.
The analysis of different calculation methods for the underlying basket correlations
is out of scope of the present study. For our purposes we take suitable correlation
matrices as given.
5.2 Model Framework
As stated above, we assume for our analysis a Block-Scholes world. The dynamic of
an underlying stock S in the Black-Scholes world follows a geometric Brownian mo-
tion with constant instantaneous volatility and is given by the stochastic differential
equationdStSt
= µ(S, t)dt+ σ(t)dWt, (5.1)
where dWt is the increment of a standard Wiener process, µ(S, t) the drift of the
stock and σ(t) its instantaneous volatility. Thereby, the changes dWt of the Wiener
process are normally distributed with zero mean and variance equal to the length of
the time interval passed during the change, i.e. with variance equal to dt.
Applying the Girsanov theorem the process can be rewritten under the risk-neutral
probability measure as
dStSt
= (rt − δ)dt+ σ(t)dWt = (rt − δ)dt+ dZt, (5.2)
where dWt is the increment of a standard Wiener process under the risk-neutral
measure, δ is the dividend yield of the stock and rt the risk-free rate at time t.
For the multi-dimensional case of U underlying stocks St = (S1(t), · · · , SU(t)) the
stochastic dynamics are then given as
dStSt
= (rt − δ)dt+ dZt, (5.3)
where the risk-neutral increments dZit and dZj
t of the underlying processes are corre-
lated via
E[dZitdZ
jt ] = σi(t)σj(t)ρij(t)dt. (5.4)
Therefore, the vector dZt represents the stochastic differentials of the underlying pro-
cesses and these are multivariate normally distributed with zero mean and covariance
29
matrix Σ(t), i.e.
dZt =
dZ1
dZ2
...dZU
t
∼ N(0,Σ(t)), (5.5)
where Σij(t) = σi(t)σj(t)ρij(t)dt. Note, that the correlation and volatility are time-
dependent in the general case. However, in our analysis we will assume constant cor-
relations and volatilities, i.e. ρij(t) = ρij and σ(t) = σ, respectively. In other words,
we assume a constant covariance matrix Σ, which is exogenously given. Equipped
with a somehow given covariance matrix we can proceed to price complex derivatives,
e.g. American basket options.
Before we go into a detailed description of different approaches to handle the
early exercise feature of American basket options, we will describe the Monte Carlo
technique to construct multi-dimensional paths taking low-discrepancy numbers, PCA
and BB into account.
5.3 Monte Carlo Path Construction
To construct Monte Carlo paths for correlated underlying stocks with multiple time
steps, we combine the different techniques described in Chapter 4. Thereby, we have
to find the right method for the particular pricing problem at hand. For Monte
Carlo path construction we have to discretise the continuous process given in (5.2)
by changing the infinitesimals dSt, dt and dZt into small changes ∆St, ∆t and ∆Zt
and, additionally, transforming the process of the stock itself to a lognormal process
of the stock price evolution by applying Ito’s lemma. This step is necessary since in
our definition above, we use dSt/St which means the current stock price would affect
the covariance matrix, i.e it will not longer be constant. After applying Ito’s lemma
the discretised stochastic process of an underlying stock can be written as
∆ log(St) = (rt − δ − 0.5σ2)∆t+ ∆Zt, (5.6)
and the future value of a stock is given by
St+∆t = St exp (α∆t+ ∆Zt) , (5.7)
with α = (rt − δ − 0.5σ2). Since the random increments ∆Zt have mean zero and
variance of σ2∆t, they can be easily simulated by random samples of σ√
∆tx, where x
is a sample from a standard normal distribution. This random sample can be derived
30
by using a pseudo random number generator (see Section 3.3) or by one of the low-
discrepancy sequences as discussed in Section 3.4 and transforming these uniform
distributed random numbers into standard normal ones by standard methods such
as inverse normal function approximations. By dividing the simulation time period,
normally the time to maturity T , into N equidistant time intervals, i.e. ∆t = T/N ,
equation (5.7) provides an easy way of simulating futures values of one underlying
stock at the end of intervals, tj = j∆t, j = 1, ..., N , using (5.7) as follows:
Stj = eztj , ztj = ztj−1+ α∆t+ σ
√∆tx, (5.8)
with zt0 = log(St). Thereby, the paths can be generated by the standard forward
construction principle or by using the Brownian bridge method described in Section
4.3. To apply the Brownian bridge method, we first generate a path for a standard
Brownian motion given in (4.9) using the Brownian bridge method and then back out
the path increments by taking the differences ∆Btj = path[j]− path[j − 1]. Each of
these increments can then be seen as a standard normal random number with zero
mean and variance ∆t, i.e. for the Brownian bridge method (5.8) can be adjusted to
Stj = eztj , ztj = ztj−1+ α∆t+ σ∆Btj . (5.9)
For the multidimensional case we have to simulate the following equation:
Stj = eztj , ztj = ztj−1+α∆t+ ∆Zt, (5.10)
where α can be easily calculated, since the dividend yield δi and volatility σi for
each individual asset i, i = 1, ..., U is given. We know from (5.5) that the vector
of increments ∆Zt is normally distributed with mean zero and covariance matrix Σ.
Note, that Σ is derived in such a way, that the time interval ∆t = T/N is already
taken into account.1 To derive ∆Zt, the covariance matrix has to be decomposed, i.e.
Σ = AAT , with an appropriate algorithm. This can be either done via the classical
Cholesky decomposition, i.e. A = C or by using PCA, i.e. A = BΛ1/2 as derived
in Section 4.2. The vector of increments can then be generated as ∆Zt = Ax, where
x is a sample from a multi-dimensional standard normal distribution with dimension
U ×N .
If we are generating multi-dimensional paths, we always start with a random
sample x ∈ RU×N of standard normal variables, which is generated with an arbitrary
1In general, the covariance matrix is calculated for a predefined time interval, e.g. one day, oneweek, one month or on year, and is than scaled to the required time interval ∆t.
31
random number generator. This vector will first be transformed to a matrix X, where
each column represents one time step and each row one underlying. This matrix can
be filled in two different ways:
1. Take the first U entries in x to fill the first column of the matrix X, then the
second U entries for the second column and so forth. Using this algorithm is
equivalent to calculate the first time step for all underlyings, then the second
time step (when applying BB, this will be the first refinement) and so on.
These algorithm seems promising when pricing basket options with weak path-
dependence. We call this algorithm Time-Transformation.
2. Take the first N entries in x to fill the first row of the matrix X, then the second
N entries for the second row and so forth. Using this algorithm is equivalent to
first calculate the whole path for the first underlying, then the whole path for
second underlying and so on. These algorithm seems promising when pricing
basket options with strong path-dependence. We call this algorithm Underlying-
Transformation.
If the BB method should be applied we take a further transformation before
generating the paths. In this case, we generate for each underlying i, i = 1, ..., U , i.e.
with the standard normal variables in each row of X, a pathi for a standard Brownian
motion given in (4.9) using the Brownian bridge method and then back out the path
increments by taking the differences XBBij = pathi[j]..pathi[j..1]. This procedure is
equal to the one-dimensional case as described above, see (5.9).
Equipped with this matrix of standard normal variables, i.e. either X or XBB, and
an appropriate decomposition of the covariance matrix Σ, i.e. A = C or A = BΛ1/2,
the multi-dimensional paths are generated as:
Stj = eztj , ztj = ztj−1+α∆t + Axj, (5.11)
where xj represents the j-th column of X or XBB, respectively.
After having constructed the multi-dimensional paths in this way, we are able to
price basket options with a given payoff profile based on this Monte Carlo paths.
We will describe the pricing of American basket options based on multi-dimensional
Monte Carlo paths in the next section in detail.
32
5.4 Valuation of Early Exercise Feature
The objective to price a American Option in the context of Monte Carlo Simulation
is to map (collapse) the true (unknown) high-dimension early exercise boundary onto
the realizations of a much smaller number (two, or, sometimes, one) representative
proxy variables. Two of the most popular approaches to the problem along these or
related lines are discussed in the following: The Least-Squares Monte Carlo (LSM)
approach suggested by Longstaff and Schwartz (2001) and the Threshold approach
proposed by Andersen (1999).
5.4.1 Least-Squares Approach
Longstaff and Schwartz (2001) suggest to estimate the continuation value F (k, tj)
as the conditional expectation of the payoff from keeping the option alive, using
the cross-sectional information in the simulation. Here, k = 1, ...,M represents a
sample path, and tj are the discrete time points at which the American option can
be exercised. Longstaff and Schwartz (2001) assume a complete probability space
(Ω,F , P ), a finite time horizon [0, T ] and the existence of an equivalent martingale
measure Q for the economy. They restrict their attention to derivatives with payoffs
in L2(Ω,F , Q). Hence the conditional expectation is also in L2(Ω,F , Q). Since L2 is
a Hilbert space any function belonging to this space can be represented as a countable
linear combination of Ftj -measurable basis functions for this vector space. Longstaff
and Schwartz (2001) choose Laguerre polynomials to represent the basis functions2
Ln(X) = exp(−X/2)eX
n!
dn
dXn(Xne−X), n = 0, ... (5.12)
where Ln defines the Laguerre polynomial of order n at point X. The choice of basis
functions, especially the number of basis functions is crucial for the performance of
the LSM. We tested different set of basis functions depending on all underlyings as
well as the application of state variables depending of the payoff function, e.g. the
basket value. Thereby, we found that the application of state variables, which results
in a much smaller number of basis functions, are stable and provide good convergence
properties.3 With this specification the continuation value on path k can be written
2Other types of basis functions include the Hermite, Legrende, Chebyshev, Gegenbauer or Jacobipolynomials.
3These convergence properties of the LSM approach for different choices of basis functions aredetailed analyzed in Clement et al. (2002) and two papers of Stentoft (2004)[34, 33]. Stentoft alsoshows the efficiency of the LSM method for high dimensional problems.
33
as
F (k, tj) =∞∑n=0
βnLn(X). (5.13)
In order to use this result in practice one needs to approximate F (k, tj) using a
finite linear combination with the first E < ∞ basis functions. We denote this
approximation as FE(k, tj). A natural approximation concept for FE(k, tj) is that of
least-squares regression where the discounted cash-flow CFk,tj+1one period ahead is
projected on a set of basis functions for the paths where the option is ’in-the-money’
at time tj.4
The LSM algorithm works as follows:
• Construct a set of multi-dimensional sample paths under the risk-neutral mea-
sure using (5.11).
• Work recursively and use at any time tj, 0 < j < N least-squares regression
to estimate the conditional expectation of the payoff FE(k, tj), given that the
basket option is kept alive. Construct a subset of M paths consisting of paths
where the basket option is in-the-money. The regression coefficients β are given
by
β = (A′A)−1A′y (5.14)
where the M -vector y represents the current holding values and the M×(E+2)-
matrix A contains the basis functions with ak,n = Ln(Xk,tj
)for n = 0, ..., E,
where Xk,tj denotes the state variable at time tj on the kth path and ak,E+1 = 1.5
Then the fitted values y = Aβ are used as a proxy for the continuation value.
Hence, it is easy to decide at time tj, whether it is optimal to exercise the
basket option at path k. If the value of immediate exercise is greater than the
continuation value, set CFk,t∗k equal to the immediate exercise value and t∗k = tj,
otherwise leave CFk,t∗k and t∗k unchanged. Here, t?k represents the current optimal
exercise date at path k.
• Once t0 is reached and the optimal exercise strategy is determined, the value of
the basket option is calculated under the risk-neutral measure as follows
V (t0) =1
M
M∑k=1
CFk,t∗k , (5.15)
4Using only ’in-the-money’ paths allows a better estimate of the conditional expectation functionin the relevant exercise region and improves the efficiency significantly.
5Douady (2001) also proposes to use the exercise value as further ’basis function’ in the regression.Then the matrix A consists of (E+3) columns, where the last column holds the exercise value.
34
where V (t0) represents the estimated price of the basket option.
The Longstaff and Schwartz (2001) approach is a simple technique to approximate
the value of basket options. It is easy to implement, since only the least-squares
regression and simulation techniques are required. The disadvantage of the approach
is that there is no unique procedure how to specify the structure and the form of
the basis functions. Nonetheless, it is a very powerful pricing method which values
American basket options by simulation for given parameter specification of the pricing
model.
5.4.2 Threshold Approach
Andersen (1999) proposes a simple method to price Bermudan swaptions in the Li-
bor market model. The early exercise feature of a Bermudan swaption depends on all
state variables of the LMM which leads to a high-dimensional optimization problem.
To reduce the complexity of the problem Andersen proposes a one-dimensional opti-
mization by maximizing the value of the option at each time step searching the early
exercise boundary parameterized in intrinsic value and the value of still-alive swap-
tions. This is done by introducing a Boolean function with a single, time-dependent
parameter which is determined in the optimization routine for each time step. As
a result the method is simple to implement, fast and robust and produces a lower
bound for Bermudan swaption prices.
Douady (2001) shows that the objective function in the optimization routine is
not smooth. Therefore, not every algorithm to maximize a function of one variable
guarantees convergence to the true maximum. Hence, Douady extends Andersen’s
approach by introducing ’exercise probabilities’ which depends on the fuzziness para-
meter α, and leads to a ’fuzziness’ of the threshold. Thereby, the new objective
function is smooth enough for an appropriate ’fuzziness’ parameter α to identify the
maximum of the function uniquely.6
The method of Andersen as well as the extended version of Douady can easily
be applied to other model classes, especially to equity models. In this case only
the market variables change, but both the algorithm and the general idea remain
unchanged. Therefore, the algorithm to price equity basket options contains the
following steps:
6A non fuzzy threshold corresponds to α → +∞. The choice of α should be tested numericallydepending on the functional form of the exercise probabilities. For details see [15].
35
• Construct a set of multi-dimensional sample paths under the risk-neutral mea-
sure using (5.11).
• Starting at tN−1, maximize the value of the option by searching for the opti-
mal threshold θj for each discrete time point j = 1, ..., N − 1 using backward
induction and solve the following optimization problem:
maxθj
M∑k=1
IVk,jf(θj) +Hk,j(1− f(θj)) . (5.16)
Here IVk,j denotes the intrinsic value of the basket option at path k for immedi-
ate exercise at time tj and Hk,j is the value if the basket option is not exercised
at that date. Using the Andersen approach
Hk,j = e−r(t?k−tj)IVk,t?k (5.17)
where t?k represents the current optimal exercise date at path k and
f(θj) = 1IVk,j≥θj (5.18)
is the indicator function which controls the optimal exercise at tj. In the ex-
tended version of Douady (2001) the holding value Hk,j and the exercise prob-
ability f(θj) are defined as
Hk,j = e−r(tj+1−tj)[IVk,j+1f(θj+1) +Hk,j+1(1− f(θj+1))] (5.19)
and
f(θj) =1
1 + e−α(IVk,j−θj), (5.20)
where α measures the fuzziness of the threshold. Note that the smoothing in the
extended version is due to replacement of the indicator function by a continuous
function in θj.
• Use the completely specified (optimal) exercise strategy τ ∗ = (θ1, ..., θN−1) to
price the American-style option by an independent Monte Carlo simulation with
M ′ M paths.
The price of a basket option in the Threshold approach is derived by determin-
ing the early-exercise strategy by a low-dimensional parameterization, consisting of
a small number of key variables (here only one) while on the contrary the whole
underlying universe is simulated.
36
Chapter 6
Comparison of Numerical Methods
6.1 Introduction
The convergence properties and the accuracy of different pricing methods, i.e. the
application of various random number generators and combinations of path generation
methods, is analyzed in this chapter. To assess the performance of the different
methods the following criteria are used:
• Convergence diagrams over the number of paths M for the empirical option
price, i.e. CM = 1L
∑Li=1C
Mi , to get an impression of the accuracy and conver-
gence behavior of each pricing method, where L is the number of repetitions of
the simulation algorithm.
• Log-Log plots of the empirical root mean squared error (RMSE) which is defined
as the square root of the average squared deviations of the model prices CMi
from the empirical option price CM given as
RMSE =
√√√√ 1
L
L∑i=1
(CMi − CM
)2. (6.1)
Note, that for Monte Carlo simulations the L repetitions are statistically inde-
pendent, at least when using pseudo-random number generators. For quasi-random
numbers we use different nonoverlapping sub-sequences for each value of M for the L
repetitions, whereas we should keep in mind that these sub-sequences are not samples
from any population. For our analysis, we will always use 25 repetitions, i.e. L = 25.
Furthermore, for each random number generator we additionally apply antithetic
sampling (ANT), i.e. when using M paths with antithetic sampling we generate M/2
paths with the random number vector x and M/2 paths with −x. As reported by
37
Caflisch et al. (1997) the application of antithetics with QMC significantly improves
the convergence of QMC, wherefore we also expect an improvement for the pricing of
American basket options.
Before we discuss the results we give a description of the data set we used to
estimate the covariance matrix as well as a brief description of the software imple-
mentation.
6.2 Data
We base our analysis on market data as of April 30, 2009 mainly collected from
Bloomberg. Therefore, we use the Bloomberg field EQY DVD YLD 12M1 as proxy
for the current dividend yield p.a. and the average of the historical implied volatility
of calls and puts, i.e. σi = (HIST CALL IMP VOL + HIST PUT IMP VOL)/2 as
proxy for the volatility. The historical implied volatilities are calculated as weighted
average of the implied volatilities of the three nearest at-the-money calls or puts,
respectively. We use the 6-month rate of the Bloomberg EUR Benchmark curve as
proxy for the risk-free rate which was 0.815 % p.a. The Dax constituents weights are
taken from Deutsche Borse. The values are given in Table A.1 in the Appendix.
To calculate the correlation matrix we use daily log-returns based on close prices
for the 30 Dax index constituents from Xetra for the period from January 2, 2004,
through April 30, 2009. The cleaned time series data are collected from Bloomberg,
i.e. the data are already corrected for corporate actions. The resulting correlation
matrix is given in Table A.2 in the Appendix. We chose a combination of historical
correlations and implied volatilities to derive the covariance matrix Σ since implied
volatilities are observable parameters in liquid option markets which is not possible
for correlations, especially for more exotic baskets. Therefore, correlations must be
calculated from historical observations.
6.3 Software Implementation
The pricing algorithm for our analysis was programmed in C++ and was run on a
Centrino Duo Core processor with 2.20 GHz and 2 GB RAM. The available RAM
memory restricts the number of repetitions and/or the number of simulations and/or
1The sum of gross dividend per share amounts that have gone ex-dividend over the prior 12months, divided by the current stock price.
38
the dimensionality of the problem. The problem is especially immanent when using
QMC, since sometimes the algorithm to draw the quasi random numbers is such
that the whole sample is simulated at once, i.e. inclusive all repetitions, instead of
generating the random numbers path by path or for each repetition separately. We
tried to mitigate the problem by first generating a complete set of random numbers
and saving it in the file system. Afterwards we were able to read each vector draw
for a sample path separately.
The following C++ routines are used within our implementation:
• The routines TRED2, TQLI and EIGSRT provided by [31] to calculate the ma-
trices B and Λ of Σ, when applying PCA.
• The routine CHOLDC provided by [31] to calculate the square root matrix A of
Σ, when applying cholesky decomposition.
• The Brownian bridge construction principle provided by [20] to back out the
normal path increments when applying the Brownian bridge method.
• The routine RAN2 provided by [31] to generate uniformly distributed pseudo
random numbers.
• Sobol points are generated by using the primitive polynomials and initial direc-
tion numbers provided by [21] as well as their routine sobol.cc.
• Halton numbers are generated with the algorithm provided by [20].
• The generation of Niederreiter sequences is based on the algorithm provided
by [11] adapted to dimension 8300 using the primitive polynomials and initial
direction numbers provided by [21].
• To transform the uniform pseudo or quasi random numbers into standard nor-
mal ones we use an approximation of the inverse cumulative standard normal
distribution function provided by [1]. Especially for quasi random numbers an
approximation of the inverse cumulative standard normal distribution function
should be used, since other algorithms, e.g. Box-Muller or Marsaglia method,
can destroy the properties of low-discrepancy sequences, since they require two
neighborhood random variables for the transformation.
• The routine SVDFIT provided by [31] to estimate the vector of regression coef-
ficients in the Least-Squares approach given in (5.14).
39
• The routines MNBRAK and GOLDEN provided by [31] to solve the optimization
problem given in (5.16).
6.4 Results
6.4.1 Classical Put Options
Our first analysis deals with the pricing of American put options on a single underlying
equity. This is mainly done for two reasons. First, we want to test the implementation
of our pricing routines with the parameters given in the Longstaff and Schwartz paper
[25]. Second, we can get the first results for time dimension reduction by using BB
without taken multiple underlyings into account.
We price an American put option with strike price K = 40 and time to maturity
T = 1y. The current stock price is 40, i.e. we price an at-the-money option, the
volatility is 40% and the short-term interest rate is 6%. According to Longstaff /
Schwartz the theoretical option value (THEO VALUE) is 5.312. We choose 50 time
steps per year for the discretization of the process, i.e. ∆t = 1.0y/50. Moreover, this
means that our pricing problem is already of dimension 50. The continuous exercise
features for the American put option are thus approximated with roughly one exercise
decision per week. When generating the sample paths we skip the first 1,000 vector
draws for each random number generator, which is especially recommended for quasi-
random numbers, since they could contain undesirable correlations, especially in their
initial sets, and especially in higher dimensions2.
For the Least-Squares approach we have to choose the basis functions to approx-
imate the continuation value. We follow Longstaff and Schwartz (2001) and use a
constant term and the first three Laguerre polynomials n = 0, 1, 2 given in equation
(5.12) evaluated at the underlying stock price. To avoid numerical overflow errors and
to get as precise results as possible, the payoffs CFi,tk and the stock price Xk,tj are
normalized by the strike price K as recommended by Longstaff and Schwartz (2001).
The (optimal) exercise strategy τ ∗ for the Threshold approach is determined gener-
ating max(100.0,M/5) paths and applying the extended version proposed by Douady
(2001) [15], i.e. we use equation (5.19) and (5.20) in the optimization problem (5.16).
Numerical tests indicated that α = 100.0 is a good choice for the fuzziness parameter.
Additionally, we also use normalized values for IVi,k and Hi,k, i.e. we divide by the
strike price K. To calculate the option value we use the same (optimal) exercise
strategy in all repetitions.
2Henceforth, this skipping is applied for all pricing problems in our analysis.
40
The convergence results for the different random number generators are given in
detail in the Appendix – for the Least squares approach in Figure A.1 and for the
Threshold approach in Figure A.2. The first important differences between the valu-
ation approaches for the American exercise feature is the convergence of the empiri-
cal option price to the theoretical option value (THEO VALUE). The Least-Squares
approach converges to the theoretical option value from above and the Threshold ap-
proach converges from below. The latter behavior was already noticed by Andersen in
his original work [3], where the generated prices are also biased towards lower values.
However, it should be noted that the prices are sometimes smaller than prices of the
European equivalent option which is impossible theoretically. This could be due to
the separate calculation of the optimal exercise strategy and the option price itself,
since it is only observed for some prices of the 25 repetitions for a certain sample
size. This convergence behavior appears inherent in the valuation approaches since
it is independent of the chosen random number generator and the application of the
Brownian bridge method.
Without using the Brownian bridge (BB) method the convergence of all low-
discrepancy sequences is for both valuation approaches fairly bad. This can espe-
cially be seen for Niederreiter sequences which have the worst performance. The best
convergence is observed by applying pseudo-random numbers (PRN) with antithetic
sampling (ANT). Sobol sequences already show good results.
When additionally applying BB the results change drastically which becomes ap-
parent especially for Niederreiter sequences. The convergence to the theoretical op-
tion value increases and the RMSE for the paths generated with QMC are evidently
lower as for PRN. For example, for a number of 10,000 paths the RMSE of the
PRN with antithetic sampling (... ANT BB) is approximately twice the error of the
Sobol (... SOB BB) or Halton (... HAL BB) sequences. Furthermore, it is remarkable
that the Halton sequences with BB shows reasonable convergence for both valuation
approaches, since it is the simplest multi-dimensional quasi-random generator with
long monotone segments in the two-dimensional projections as shown in Figure 3.2
(already for our 50 dimension pricing problem!).
Another surprising result is the marginal impact of using antithetics in combina-
tion with QMC. Especially for Niederreiter and Halton sequences the effect of using
antithetics in negligible compared to the effect of using BB. Moreover, when consider-
ing the convergence to the theoretical value (left hand sub-figures) the additional use
of antithetics for these sequences becomes worse. We will discuss these issues more de-
tailed when looking to the results for American basket options in section 6.4.3. Only
41
for Sobol sequences our expectation is met – in this case the application of BB and
antithetics shows the best convergence properties for QMC. In contrast to the results
for QMC it can be seen for pseudo-random number generators that the application of
BB has no influence to the convergence properties, since PRN exhibits the usual√M
behavior independent of the dimension of the problem such that effective dimension
reduction techniques as BB have no effect. On the other hand, the application of
antithetics for PRN leads to a significant variance reduction as expected.
6.1.1 Mean values using Least-Squares approach 6.1.2 RMSE using Least-Squares approach
5,45
5,55
5,65
5,75 THEO_VALUE EUR_VALUE LS_ANT LS_NIE_BB LS_HAL_BB LS_SOB_ANT_BB
5,05
5,15
5,25
5,35
5,05
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
LS_ANT LS_NIE_BB LS_HAL_BB LS_SOB_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
6.1.3 Mean values using Threshold approach 6.1.4 RMSE using Threshold approach
5,15
5,25
5,35
5,45
THEO_VALUE EUR_VALUE THRES_ANT
THRES_NIE_BB THRES_HAL_BB THRES_SOB_ANT_BB
4,75
4,85
4,95
5,05
4,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
THRES_ANT THRES_NIE_BB THRES_HAL_BB THRES_SOB_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
Figure 6.1: The left hand figures show the empirical option price of an American put as function of the number ofpaths used in the simulation using the Underlying-Transformation whereas the right hand figures compare the empiricalRMSE as function of the number of paths. The values are shown for pseudo random numbers with antithetic variables(ANT), Sobol sequences (SOB), Niederreiter sequences (NIE) and Halton sequences (HAL). The dotted lines represent√M RMSE behavior.
We summarize the convergence results for the empirical option price and empirical
RMSE in Figure 6.1 for the Least-Squares approach (upper sub-figures) and the
Threshold approach (lower sub-figures). Thereby, we show for each random number
generator our recommended choice when using it for the valuation of one-dimensional
American options, i.e. PRN with antithetics, Sobol with BB and antithetics, and
Halton and Niederreiter with BB and without antithetics.
42
6.4.2 Exchange Options
In order to get some intuition for the effect of the Time- or Underlying-Transformation
(TIME or UND, resp.) as described in Section 5.3 we present some calculations for the
price of American Exchange options with two underlyings (see 2.2.2) in this section.
The pricing of this relatively simple basket option allows us to analyze the effect of
the chosen transformation algorithm separately of the decomposition algorithm of
the VCV-matrix Σ, since the decomposition of the 2× 2-matrix for pricing exchange
options is unique. Afterwards we turn to the more sophisticated basket options with
multiple underlyings.
We price an American exchange option with time to maturity T = 0.5y, where
we choose 32 time steps for the discretization of the underlying price processes, i.e.
our pricing problem for the American exchange is of dimension 64. As underlying
we use the stocks of Deutsche Bank (DBK GY Equity = S1) and Munich Re
(MUV2 GY Equity = S2) with the market parameters as given in Table A.1 and
A.2. We set the current stock price of both underlyings to 100.0 instead of using
different weights for both assets3. As proxy for the risk-free rate we use the 6-month
rate of the Bloomberg EUR Benchmark curve which was 0.815 % p.a. Using
the closed form approximation derived by Bjerksund and Stensland [6], i.e. setting
K = S2, r = D2 and σ as given in (2.5), the theoretical value of the American
exchange options is 13.6347 and the value of the European counterpart is 13.2573
(EUR VALUE) according to (2.3).
For the Least-Squares approach we use a constant term and the first three Laguerre
polynomials n = 0, 1, 2 given in equation (5.12) evaluated at the difference between
both stock prices, i.e. at X = S1−S2, which was more stable and higher performance
as using Laguerre polynomials for each underlying as well as the mixture of both.
The (optimal) exercise strategy τ ∗ for the Threshold approach was derived with the
same settings as described above for the classical put option.
Since the convergence properties for the different valuation approaches are the
same as observed for classical put options, i.e. Threshold approach from below and
Least-Squares from above, we do not show all results again. We also find in our results
for American exchange options the superiority of the BB in combination QMC as well
as that BB has no influence to the performance of PRN. For Sobol sequences we found
for American exchange options that the additional application of antithetics is not
advantageously, since the convergence results are slightly worse as without antithetics.
3Normally, the basket weights are chosen such that at the beginning of the option life bothunderlyings have the same market value.
43
Therefore, we report only the results for QMC in combination with BB as well as
only the results of PRN with antithetic sampling (ANT) without BB.
6.2.1 Mean values using Least-Squares approach 6.2.2 RMSE using Least-Squares approach
14,25
14,75
15,25
15,75
THEO_VALUE EUR_VALUE LS_ANT_UND
LS_SOB_BB_UND LS_NIE_BB_UND LS_HAL_BB_UND
11,75
12,25
12,75
13,25
13,75
11,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
10,00%
100,00%
LS_ANT_UND LS_SOB_BB_UND LS_NIE_BB_UND LS_HAL_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
6.2.3 Mean values using Threshold approach 6.2.4 RMSE using Threshold approach
14,25
14,75
15,25
15,75
THEO_VALUE EUR_VALUE THRES_ANT_UND
THRES_SOB_BB_UND THRES_NIE_BB_UND THRES_HAL_BB_UND
11,75
12,25
12,75
13,25
13,75
11,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
10,00%
100,00%
THRES_ANT_UND THRES_SOB_BB_UND THRES_NIE_BB_UND THRES_HAL_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
Figure 6.2: The left hand figures show the empirical option price of an American exchange option as functionof the number of paths used in the simulation using the Underlying-Transformation whereas the right hand figurescompare the empirical RMSE as function of the number of paths. The values are shown for pseudo random numberswith antithetic variables (ANT), Sobol sequences (SOB), Niederreiter sequences (NIE) and Halton sequences (HAL).The dotted lines represent
√M RMSE behavior.
In Figure A.3 in the appendix we compare the results for using the Underlying- or
Time-Transformation algorithm (UND and TIME). Most surprisingly the application
of the Underlying-Transformation algorithm to generate the multi-dimensional paths
shows better convergence properties as the Time-Transformation algorithm, which is
contrary to our expectation for weak path-dependent options. Since the differences for
PRN with antithetic sampling (ANT) are relatively small whether using the TIME
or UND algorithm, we infer from this behavior that when already using the BB
method in combination with QMC the additional priority to generate the first time
steps for all underlyings is negligible, especially when the number of underlyings is
small. Instead, the successive path generation with BB for each underlying should be
applied. This effect becomes directly observable for Niederreiter sequences in A.3.5
and A.3.6. Moreover, for Niederreiter and Halton sequences the convergence for the
44
Threshold approach in combination with the TIME algorithm is quite high up to
10,000 paths but for higher paths numbers it declines significantly, which is not the
case when using the UND algorithm.
Summarizing these effects: we recommend the application of Underlying-Transfor-
mation algorithm (UND) in combination with QMC for American exchange options.
The empirical option price and empirical RMSE for different RNG using this trans-
formation algorithm is shown in Figure 6.2. The upper sub-figures show the conver-
gence properties for the Least-Squares approach whereas the lower sub-figures show
the same for the Threshold approach. Overall it can be seen that the convergence of
the Least-Squares approach to the theoretical value (THEO VALUE) is much more
stable and closer as compared to the Threshold approach. Furthermore, the valua-
tion with QMC in this fashion leads to significantly smaller RMSE compared to PRN
independent of the valuation approach, whereby the Sobol sequences show the best
convergence behavior among the quasi-random number generators. For instance, the
RMSE using the LS approach for Sobol sequences is on average over all sample sizes
roughly 2.4-times lower as compared to ANT, whereas this ratio is roughly 1.7 for
Halton and Niederreiter sequences.
6.4.3 Basket Options
In the last section of this chapter we present results for the valuation of an American
Dax call option, i.e. the underlying will be an equity basket with 30 underlyings itself.
Thereby we consider the effects of the different valuation approaches (Least-Squares
(LS) vs. Threshold (THRES)), transformation algorithms (Underlying (UND) vs.
Time (TIME)), VCV decomposition methods (Cholesky (CHD) vs. PCA), path gen-
eration methods (with / without BB) and random number generators (pseudo with
antithetic sampling (ANT) as well as Sobol (SOB), Niederreiter (NIE) and Halton
(HAL) with ( ANT) and without antithetics.).
We price an American Dax call with strike price K = 100 and time to maturity
T = 0.5y, where we choose 16 time steps for the discretization of the underlying
price processes, i.e. our pricing problem is of dimension 480. Again, we normalize
the current stock price of all underlyings to 100.0 and as proxy for the risk-free
rate the 6-month rate of the Bloomberg EUR Benchmark curve. If we use the
average dividend yield and the weighted volatility, i.e. ωTΣω, of the Dax, which was
5.758% and 30.126%, resp., the theoretical value of the American Dax call calculated
with finite differences is 7.4093 and the value of the European counterpart is 7.1850
45
(EUR VALUE) calculated with the Black-Scholes formula. These values will be used
as benchmark values.
Again, for the Least-Squares approach we use a constant term and the first three
Laguerre polynomials n = 0, 1, 2 given in equation (5.12) evaluated at the current
basket value, i.e.∑U
i=1 ωiSi. The (optimal) exercise strategy τ ∗ for the Threshold
approach was derived with the same settings as described above for the classical put
option.
Table 6.1: Statistics for different paths generation methods for American Dax callvaluation
Mean is used as proxy for the empirical option price calculated as average over 25 repetitions using30,000 paths. RMSE is the root mean squared error averaged over all 25 repetitions. The Intervalis calculated as Mean times (1.0∓ RMSE).
Least-Squares Approach Threshold ApproachMean RMSE Interval Mean RMSE Interval
ANT CHD TIME 7.400 0.621% [7.354;7.446] 7.349 0.732% [7.295;7.403]ANT CHD UND 7.424 0.655% [7.376;7.473] 7.376 0.683% [7.326;7.427]ANT PCA TIME 7.407 0.514% [7.369;7.445] 7.356 0.686% [7.306;7.407]ANT PCA UND 7.404 0.623% [7.357;7.450] 7.287 0.576% [7.245;7.329]
HAL CHD BB TIME 7.607 6.572% [7.107;8.107] 7.276 5.344% [6.887;7.665]HAL CHD BB UND 7.388 0.900% [7.322;7.455] 7.190 1.189% [7.104;7.276]HAL PCA BB TIME 7.444 1.299% [7.347;7.540] 7.350 0.995% [7.277;7.423]HAL PCA BB UND 7.407 0.704% [7.355;7.459] 7.327 0.581% [7.284;7.369]HAL ANT CHD BB TIME 7.675 7.474% [7.102;8.249] 7.394 6.725% [6.897;7.891]HAL ANT CHD BB UND 7.438 1.417% [7.332;7.543] 7.252 1.706% [7.128;7.376]HAL ANT PCA BB TIME 7.472 2.005% [7.322;7.622] 7.322 1.766% [7.192;7.451]HAL ANT PCA BB UND 7.407 0.926% [7.338;7.475] 7.343 0.822% [7.283;7.404]
NIE CHD BB TIME 8.345 11.572% [7.379;9.311] 7.417 9.332% [6.725;8.109]NIE CHD BB UND 7.560 2.641% [7.360;7.759] 7.345 2.770% [7.141;7.548]NIE PCA BB TIME 7.605 2.791% [7.393;7.818] 7.246 2.509% [7.065;7.428]NIE PCA BB UND 7.447 1.284% [7.352;7.543] 7.393 1.276% [7.299;7.487]NIE ANT CHD BB TIME 8.830 16.474% [7.376;10.285] 7.643 11.238% [6.784;8.502]NIE ANT CHD BB UND 7.674 4.055% [7.363;7.985] 7.397 4.366% [7.074;7.720]NIE ANT PCA BB TIME 7.764 4.310% [7.429;8.099] 7.215 2.674% [7.022;7.408]NIE ANT PCA BB UND 7.492 1.588% [7.373;7.611] 7.408 1.949% [7.264;7.552]
SOB CHD BB TIME 7.405 0.229% [7.388;7.421] 7.358 0.294% [7.337;7.380]SOB CHD BB UND 7.401 0.293% [7.380;7.423] 7.358 0.201% [7.344;7.373]SOB PCA BB TIME 7.404 0.191% [7.390;7.418] 7.374 0.199% [7.359;7.389]SOB PCA BB UND 7.398 0.246% [7.380;7.416] 7.366 0.240% [7.348;7.384]SOB ANT CHD BB TIME 7.406 0.313% [7.383;7.429] 7.348 0.285% [7.327;7.369]SOB ANT CHD BB UND 7.407 0.346% [7.382;7.433] 7.326 0.222% [7.309;7.342]SOB ANT PCA BB TIME 7.396 0.253% [7.378;7.415] 7.370 0.229% [7.353;7.387]SOB ANT PCA BB UND 7.393 0.242% [7.375;7.411] 7.354 0.202% [7.339;7.369]
Table 6.1 shows the statistics of the 25 repetitions for 30,000 paths generated with
different RNGs, whereas the associated plots of the results for different sample sizes
of paths are given in the appendix in Figures A.4 and A.5. Herwig (2006) [19] reports
for at-the-money Dax Eurex options with more than 60 days time to maturity
typical bid-ask spread of 2.6% to 2.8%, i.e. if we use the THEO VALUE as mid price
46
of the true option value, the bid-ask prices would be 7.306 and 7.513 respectively.
Therefore, the prices for Sobol sequences will generally lie within the bid-ask spread
for reasonable sample sizes, since independent of the chosen path generation methods
and the valuation approach the interval is covered by the typical bid-ask spread. For
all other methods this does not hold in general. For standard Monte Carlo methods, in
our case PRN with antithetic sampling, the bid-ask spread will only cover the interval
in Table 6.1 when using the Least squares approach. For the Threshold approach it
can occur that for a sample of 30,000 paths the simulated price is below the bid
price, especially when we take into account that we only use one sigma to calculate
the intervals given above. Therefore, we would recommend to use the Least-Squares
approach for the valuation of American or Bermudan style options.
When considering the results for Niederreiter and Halton sequences, it becomes
obvious that QMC for high dimensional problems, in our case 480 dimensions, can
fail, i.e. QMC performs worse than PRN with antithetic sampling also if effective
dimension reduction techniques are used. Thereby, it is noticeable that Niederreiter
sequences perform worse than Halton sequences which is a little bit surprising. How-
ever, due to the bad two-dimensional projections for higher dimensions shown in Sec-
tion 3.4 in Figures 3.4 and 3.2 these general bad convergence properties of Niederreiter
and Halton sequences were expected. Nevertheless, for both sequences the effects of
the application of the different path generation methods are much more obvious than
for Sobol sequences. For both sequences the RMSE decreases substantially when
using PCA instead of CHD ceteris paribus. Moreover, the Underlying-Transformation
algorithm is also for both sequences superior compared to the Time-Transformation
algorithm which confirms our findings for American exchange options. In summary,
PCA, BB and the Underlying-Transformation algorithm show the best convergence
properties for Niederreiter and Halton sequences whereby in both cases the RMSE is
higher than using PRN with antithetic sampling.
Moreover, it can be seen in the results above and in Figures A.4 and A.5 that
antithetic sampling in combination with QMC for the pricing of American basket
options should not be used. This result is quite surprising since for the pricing of
MBS – as considered by [12] – the application of antithetics significantly improves the
results.4 In our results we find for antithetics in combination with Sobol sequences
only a minor effect which does not make a significant difference or more precisely
4For the European equivalent options, which we also priced as crosscheck for our implementationand which are not reported in this thesis, we also found a positive effect of the convergence andvariance when using antithetics in combination with QMC.
47
the results get slightly worse. However, the already worse convergence results for
Niederreiter and Halton sequences get even worse when applying them in combination
with antithetics. At this point it becomes apparent that the benefit of QMC for
American options is not be given per se, since sample points will interact with each
other when regressing over all sample paths. This was already pointed out by [13]
who showed first that QMC (without antithetic variables) works fine for American
options. We believe that the reason for this is the rather limited interaction between
the paths when applying QMC. We guess when antithetics will be additionally used
in combination with QMC that this limited interaction between the paths increases
and the results become worse. Since this worse convergence is especially seen for
Halton and Niederreiter sequences, which both show bad 2-dimensional projections
for higher dimension, we take that as an indicator for our conjecture. Moreover,
for the lower dimensional pricing problems we have not found such worsening of
results when using QMC with antithetics, since for this low dimension we have seen
reasonable 2-dimensional projections.
6.3.1 Mean values using Least-Squares approach 6.3.2 RMSE using Least-Squares approach
7,85
7,95
8,05
8,15
8,25
THEO_VALUE LS_ANT_PCA_TIME LS_ANT_PCA_UND
LS_HAL_PCA_BB_UND LS_NIE_PCA_BB_UND LS_SOB_CHD_BB_TIME
LS_SOB_CHD_BB_UND LS_SOB_PCA_BB_TIME LS_SOB_PCA_BB_UND
7,35
7,45
7,55
7,65
7,75
7,35
0 5000 10000 15000 20000 25000 30000
10,00%
100,00%
LS_ANT_PCA_TIME LS_ANT_PCA_UND LS_HAL_PCA_BB_UND
LS_NIE_PCA_BB_UND LS_SOB_CHD_BB_TIME LS_SOB_CHD_BB_UND
LS_SOB_PCA_BB_TIME LS_SOB_PCA_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
Figure 6.3: The left hand figures show the empirical option price of an American Dax option as function of thenumber of paths used in the simulation whereas the right hand figures compare the empirical RMSE as function ofthe number of paths. The dotted lines represent
√M RMSE behavior.
Summarizing our findings Figure 6.3 shows the results for the Least-Squares ap-
proach for the best performing generation method for Niederreiter and Halton se-
quences, the antithetic sampling with PCA for both transformation algorithms as well
as the results for Sobol sequences for all meaningful generation methods. Here it can
also be seen that Sobol sequences in combination with BB and PCA is superior to all
other methods. Noticeable at this point is that the choice of the transformation algo-
rithm for Sobol sequences is not such clear as for the other QRN generators, since the
Time-Transformation algorithm is slightly better than the Underlying-Transformation,
which is in line with our expectation for weak path-dependent options.
48
Chapter 7
Conclusions
In this thesis, we analyzed different approaches to derive the prices of American basket
options by Monte Carlo simulation: The Threshold approach proposed by Andersen
(1999) and the Least-Squares Monte Carlo (LSM) approach suggested by Longstaff
and Schwartz (2001). Our main observations with regard to the valuation approaches
are as follows.
• The Threshold approach converges from below to the theoretical option value
whereas the Least square approach converges from above. Thereby, it occurs
that in some cases the threshold value for the American option is below the
European equivalent option value.
• Convergence of the Least-Squares approach to the theoretical value is much
more stable and closer as compared to the Threshold approach.
Moreover, we have compared different techniques to reduce the effective dimension
of the pricing problem, namely the Brownian bridge method and Principal component
analysis, as well as different random number generators: Sobol, Niederreiter and Hal-
ton sequences out of the class of quasi-random number generators and pseudo-random
number generators in combination with variance reduction techniques (antithetic sam-
pling). Based on our developed numerics we observed the following results:
• Effective dimension reduction techniques (BB, PCA) have no influence to the
convergence properties for pseudo-random numbers, since they exhibit the usual√M behavior independent of the dimension of the problem.
• For quasi-random number generators convergence can be significantly improved
when applying the Brownian bridge method.
49
• The RMSE decreases substantially when using principal component analysis
instead of Cholesky decomposition.
• The Underlying-Transformation algorithm in combination with Brownian bridge
to generate multi-dimensional paths shows better convergence properties as the
Time-Transformation algorithm.
• The best convergence properties for each quasi-random number generator can
be observed when applying the Brownian bridge method in combination with
PCA and using the Underlying-Transformation algorithm.
• Surprisingly, Halton sequences show good convergence properties also for higher
dimensional pricing problems (up to 50 dimensions).
• For very high-dimensional problems, i.e. dimension greater than 300, Nieder-
reiter and Halton sequences perform worse than pseudo-random numbers even
if effective dimension reduction techniques are applied.
• Antithetic sampling should not be used in combination with QMC for the high-
dimensional pricing problems with early exercise features. The main reason for
that is from our point of view the increasing discrepancy in higher dimensions
for some quasi-random number generators. For the same reason antithetics
should be used in combination with pseudo-random numbers since here the
discrepancy is independent of the dimension.
• Sobol sequences in combination with BB and PCA show among all random
number generators the best convergence behavior, whereby the effect of using
BB is much stronger than from PCA.
Therefore, we would recommend to use the Least-Squares approach for the val-
uation of American or Bermudan style options within the Monte Carlo framework
independent of the chosen random number generator or the application of other tech-
niques investigated in this thesis. In doing so the number of basis function should be
as small as possible to avoid overfitting of the exercise boundary for which reason an
adequate state variable for the pricing problem should be used. Moreover, to speed
up convergence in high-dimensional pricing problems we recommend to use Sobol se-
quences instead of pseudo-random numbers in combination with the Brownian bridge
and Principal component analysis to reduce the effective dimension of the pricing
problem. Generally, when generating Monte Carlo paths with quasi-random number
generators the Brownian bridge method should always be applied.
50
Appendix A
Data
Table A.1: Description Dax constituents
Dividend yield, volatility and weight of Dax constituents at April 30, 2009. The last row containsthe values for the Dax, which are given as the weighted average of the dividend yields and theweighted average sum of the covariance matrix.
Bloomberg Ticker dividend yield δi volatility σi weight ωi
ADS GY Equity 1.747% 41.865% 1.273%ALV GY Equity 12.905% 50.880% 7.645%BAS GY Equity 7.188% 44.155% 6.032%BAY GY Equity 3.473% 32.250% 6.608%BEI GY Equity 2.247% 23.825% 0.584%BMW GY Equity 4.038% 58.785% 1.940%CBK GY Equity 19.380% 79.150% 0.787%DAI GY Equity 2.210% 49.125% 5.465%DB1 GY Equity 3.750% 52.340% 2.510%DBK GY Equity 11.070% 68.510% 5.251%DPW GY Equity 17.143% 44.915% 1.806%DTE GY Equity 8.534% 33.010% 6.258%EOAN GY Equity 5.330% 34.545% 10.748%FME GY Equity 1.815% 28.040% 1.281%FRE3 GY Equity 1.716% 35.225% 0.723%HEN3 GY Equity 2.582% 31.985% 0.864%HNR1 GY Equity 9.361% 58.925% 0.339%LHA GY Equity 7.243% 41.310% 1.097%LIN GY Equity 2.818% 31.480% 2.337%MAN GY Equity 4.256% 54.975% 1.124%MEO GY Equity 3.660% 47.715% 0.819%MRK GY Equity 2.210% 34.380% 1.033%MUV2 GY Equity 5.257% 39.630% 4.980%RWE GY Equity 8.243% 33.825% 5.547%SAP GY Equity 1.723% 31.525% 5.595%SDF GY Equity 1.098% 47.990% 1.289%SIE GY Equity 3.135% 41.000% 9.565%SZG GY Equity 5.562% 59.645% 0.473%TKA GY Equity 8.000% 63.950% 1.247%VOW GY Equity 0.807% 80.275% 4.781%
DAX Index 5.758% 30.126%
51
Table A.2: Correlation matrix for Dax constituents for April 30, 2009
Correlations based on daily returns for the period from January 2, 2004, through April 30, 2009
ADS ALV BAS BAY BEI BMW CBK DAI DB1 DBK DPW DTE EOAN FME FRE3ADS 1.000 0.541 0.534 0.449 0.340 0.493 0.484 0.576 0.498 0.575 0.488 0.396 0.490 0.230 0.265ALV 0.541 1.000 0.704 0.536 0.422 0.652 0.672 0.708 0.550 0.780 0.627 0.551 0.583 0.260 0.311BAS 0.534 0.704 1.000 0.660 0.385 0.631 0.511 0.679 0.502 0.669 0.585 0.521 0.635 0.288 0.284BAY 0.449 0.536 0.660 1.000 0.356 0.469 0.389 0.524 0.429 0.484 0.455 0.482 0.557 0.299 0.294BEI 0.340 0.422 0.385 0.356 1.000 0.378 0.318 0.386 0.358 0.364 0.330 0.360 0.381 0.268 0.302BMW 0.493 0.652 0.631 0.469 0.378 1.000 0.508 0.756 0.462 0.670 0.542 0.439 0.451 0.253 0.258CBK 0.484 0.672 0.511 0.389 0.318 0.508 1.000 0.558 0.517 0.752 0.555 0.403 0.421 0.230 0.253DAI 0.576 0.708 0.679 0.524 0.386 0.756 0.558 1.000 0.530 0.703 0.601 0.480 0.552 0.229 0.310DB1 0.498 0.550 0.502 0.429 0.358 0.462 0.517 0.530 1.000 0.565 0.468 0.400 0.432 0.243 0.260DBK 0.575 0.780 0.669 0.484 0.364 0.670 0.752 0.703 0.565 1.000 0.642 0.466 0.529 0.209 0.260DPW 0.488 0.627 0.585 0.455 0.330 0.542 0.555 0.601 0.468 0.642 1.000 0.455 0.515 0.250 0.247DTE 0.396 0.551 0.521 0.482 0.360 0.439 0.403 0.480 0.400 0.466 0.455 1.000 0.561 0.381 0.339EOAN 0.490 0.583 0.635 0.557 0.381 0.451 0.421 0.552 0.432 0.529 0.515 0.561 1.000 0.357 0.331FME 0.230 0.260 0.288 0.299 0.268 0.253 0.230 0.229 0.243 0.209 0.250 0.381 0.357 1.000 0.505FRE3 0.265 0.311 0.284 0.294 0.302 0.258 0.253 0.310 0.260 0.260 0.247 0.339 0.331 0.505 1.000HEN3 0.457 0.509 0.496 0.450 0.383 0.476 0.406 0.516 0.448 0.476 0.450 0.406 0.441 0.282 0.277HNR1 0.359 0.560 0.476 0.335 0.331 0.431 0.470 0.487 0.363 0.541 0.436 0.394 0.402 0.219 0.263LHA 0.470 0.641 0.514 0.421 0.347 0.565 0.557 0.611 0.464 0.624 0.510 0.471 0.416 0.229 0.293LIN 0.466 0.582 0.666 0.548 0.376 0.525 0.435 0.571 0.480 0.545 0.495 0.433 0.528 0.220 0.253MAN 0.524 0.633 0.636 0.485 0.341 0.623 0.567 0.661 0.515 0.629 0.548 0.366 0.488 0.225 0.299MEO 0.511 0.596 0.536 0.421 0.342 0.502 0.469 0.571 0.440 0.578 0.549 0.415 0.477 0.170 0.199MRK 0.297 0.358 0.371 0.399 0.254 0.316 0.296 0.329 0.296 0.355 0.307 0.342 0.347 0.268 0.306MUV2 0.479 0.720 0.596 0.480 0.335 0.522 0.524 0.592 0.481 0.611 0.495 0.505 0.494 0.242 0.286RWE 0.437 0.542 0.584 0.529 0.349 0.429 0.418 0.507 0.404 0.500 0.494 0.540 0.790 0.320 0.292SAP 0.502 0.577 0.534 0.476 0.332 0.457 0.441 0.565 0.450 0.551 0.454 0.445 0.481 0.233 0.257SDF 0.392 0.476 0.509 0.425 0.310 0.381 0.351 0.442 0.428 0.404 0.352 0.280 0.414 0.163 0.224SIE 0.556 0.698 0.705 0.561 0.389 0.624 0.555 0.690 0.562 0.672 0.591 0.518 0.580 0.258 0.266SZG 0.489 0.570 0.605 0.465 0.398 0.503 0.500 0.591 0.500 0.568 0.504 0.368 0.472 0.234 0.298TKA 0.516 0.626 0.667 0.496 0.366 0.583 0.550 0.650 0.509 0.622 0.553 0.433 0.531 0.234 0.281VOW -0.102 -0.181 -0.143 -0.105 0.022 0.040 -0.139 -0.056 -0.063 -0.152 -0.239 -0.122 -0.199 0.004 0.036
HEN3 HNR1 LHA LIN MAN MEO MRK MUV2 RWE SAP SDF SIE SZG TKA VOWADS 0.457 0.359 0.470 0.466 0.524 0.511 0.297 0.479 0.437 0.502 0.392 0.556 0.489 0.516 -0.102ALV 0.509 0.560 0.641 0.582 0.633 0.596 0.358 0.720 0.542 0.577 0.476 0.698 0.570 0.626 -0.181BAS 0.496 0.476 0.514 0.666 0.636 0.536 0.371 0.596 0.584 0.534 0.509 0.705 0.605 0.667 -0.143BAY 0.450 0.335 0.421 0.548 0.485 0.421 0.399 0.480 0.529 0.476 0.425 0.561 0.465 0.496 -0.105BEI 0.383 0.331 0.347 0.376 0.341 0.342 0.254 0.335 0.349 0.332 0.310 0.389 0.398 0.366 0.022BMW 0.476 0.431 0.565 0.525 0.623 0.502 0.316 0.522 0.429 0.457 0.381 0.624 0.503 0.583 0.040CBK 0.406 0.470 0.557 0.435 0.567 0.469 0.296 0.524 0.418 0.441 0.351 0.555 0.500 0.550 -0.139DAI 0.516 0.487 0.611 0.571 0.661 0.571 0.329 0.592 0.507 0.565 0.442 0.690 0.591 0.650 -0.056DB1 0.448 0.363 0.464 0.480 0.515 0.440 0.296 0.481 0.404 0.450 0.428 0.562 0.500 0.509 -0.063DBK 0.476 0.541 0.624 0.545 0.629 0.578 0.355 0.611 0.500 0.551 0.404 0.672 0.568 0.622 -0.152DPW 0.450 0.436 0.510 0.495 0.548 0.549 0.307 0.495 0.494 0.454 0.352 0.591 0.504 0.553 -0.239DTE 0.406 0.394 0.471 0.433 0.366 0.415 0.342 0.505 0.540 0.445 0.280 0.518 0.368 0.433 -0.122EOAN 0.441 0.402 0.416 0.528 0.488 0.477 0.347 0.494 0.790 0.481 0.414 0.580 0.472 0.531 -0.199FME 0.282 0.219 0.229 0.220 0.225 0.170 0.268 0.242 0.320 0.233 0.163 0.258 0.234 0.234 0.004FRE3 0.277 0.263 0.293 0.253 0.299 0.199 0.306 0.286 0.292 0.257 0.224 0.266 0.298 0.281 0.036HEN3 1.000 0.348 0.458 0.485 0.470 0.455 0.290 0.437 0.410 0.429 0.328 0.510 0.394 0.433 -0.099HNR1 0.348 1.000 0.487 0.429 0.443 0.412 0.248 0.540 0.361 0.372 0.291 0.478 0.430 0.474 -0.073LHA 0.458 0.487 1.000 0.478 0.563 0.485 0.267 0.560 0.411 0.503 0.317 0.562 0.476 0.533 -0.052LIN 0.485 0.429 0.478 1.000 0.547 0.445 0.331 0.487 0.490 0.473 0.447 0.592 0.548 0.581 -0.074MAN 0.470 0.443 0.563 0.547 1.000 0.478 0.323 0.483 0.427 0.480 0.462 0.650 0.625 0.664 -0.037MEO 0.455 0.412 0.485 0.445 0.478 1.000 0.308 0.465 0.431 0.432 0.333 0.539 0.447 0.477 -0.217MRK 0.290 0.248 0.267 0.331 0.323 0.308 1.000 0.324 0.330 0.308 0.242 0.349 0.313 0.335 -0.139MUV2 0.437 0.540 0.560 0.487 0.483 0.465 0.324 1.000 0.501 0.479 0.344 0.556 0.451 0.504 -0.072RWE 0.410 0.361 0.411 0.490 0.427 0.431 0.330 0.501 1.000 0.446 0.370 0.534 0.409 0.485 -0.185SAP 0.429 0.372 0.503 0.473 0.480 0.432 0.308 0.479 0.446 1.000 0.354 0.590 0.476 0.497 -0.101SDF 0.328 0.291 0.317 0.447 0.462 0.333 0.242 0.344 0.370 0.354 1.000 0.498 0.587 0.542 -0.003SIE 0.510 0.478 0.562 0.592 0.650 0.539 0.349 0.556 0.534 0.590 0.498 1.000 0.604 0.659 -0.155SZG 0.394 0.430 0.476 0.548 0.625 0.447 0.313 0.451 0.409 0.476 0.587 0.604 1.000 0.729 -0.005TKA 0.433 0.474 0.533 0.581 0.664 0.477 0.335 0.504 0.485 0.497 0.542 0.659 0.729 1.000 -0.034VOW -0.099 -0.073 -0.052 -0.074 -0.037 -0.217 -0.139 -0.072 -0.185 -0.101 -0.003 -0.155 -0.005 -0.034 1.000
52
A.1.1 Mean values using pseudo-random numbers A.1.2 RMSE using pseudo-random numbers
5,45
5,55
5,65
5,75 THEO_VALUE EUR_VALUE LS_MC LS_ANT LS_MC_BB LS_ANT_BB
5,05
5,15
5,25
5,35
5,05
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
LS_MC LS_ANT LS_MC_BB LS_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
A.1.3 Mean values using Sobol sequences A.1.4 RMSE using Sobol sequences
5,45
5,55
5,65
5,75
THEO_VALUE EUR_VALUE LS_SOB LS_ANT
LS_SOB_ANT LS_SOB_BB LS_SOB_ANT_BB
5,05
5,15
5,25
5,35
5,05
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
LS_SOB LS_ANT LS_SOB_ANT LS_SOB_BB LS_SOB_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
A.1.5 Mean values using Niederreiter sequences A.1.6 RMSE using Niederreiter sequences
5,45
5,55
5,65
5,75
THEO_VALUE EUR_VALUE LS_NIE LS_ANT
LS_NIE_ANT LS_NIE_BB LS_NIE_ANT_BB
5,05
5,15
5,25
5,35
5,05
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
LS_NIE LS_ANT LS_NIE_ANT LS_NIE_BB LS_NIE_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
A.1.7 Mean values using Halton sequences A.1.8 RMSE using Halton sequences
5,45
5,55
5,65
5,75
THEO_VALUE EUR_VALUE LS_HAL LS_ANT
LS_HAL_ANT LS_HAL_BB LS_HAL_ANT_BB
5,05
5,15
5,25
5,35
5,05
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
LS_HAL LS_ANT LS_HAL_ANT LS_HAL_BB LS_HAL_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
Figure A.1: The left hand figures show the empirical option price of an American put as function of the numberof paths used in the simulation using the Least-Squares approach (LS) to approximate the early exercise value whereasthe right hand figures compare the empirical RMSE as function of the number of paths. The dotted lines represent√M RMSE behavior.
53
A.2.1 Mean values using pseudo-random numbers A.2.2 RMSE using pseudo-random numbers
5,15
5,25
5,35
5,45
THEO_VALUE EUR_VALUE THRES_MC THRES_ANT THRES_MC_BB THRES_ANT_BB
4,75
4,85
4,95
5,05
4,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
THRES_MC THRES_ANT THRES_MC_BB THRES_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
A.2.3 Mean values using Sobol sequences A.2.4 RMSE using Sobol sequences
5,15
5,25
5,35
5,45
THEO_VALUE EUR_VALUE THRES_SOB THRES_ANT
THRES_SOB_ANT THRES_SOB_BB THRES_SOB_ANT_BB
4,75
4,85
4,95
5,05
4,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
THRES_SOB THRES_ANT THRES_SOB_ANT THRES_SOB_BB THRES_SOB_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
A.2.5 Mean values using Niederreiter sequences A.2.6 RMSE using Niederreiter sequences
5,15
5,25
5,35
5,45
THEO_VALUE EUR_VALUE THRES_NIE THRES_ANT
THRES_NIE_ANT THRES_NIE_BB THRES_NIE_ANT_BB
4,75
4,85
4,95
5,05
4,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
THRES_NIE THRES_ANT THRES_NIE_ANT THRES_NIE_BB THRES_NIE_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
A.2.7 Mean values using Halton sequences A.2.8 RMSE using Halton sequences
5,15
5,25
5,35
5,45
THEO_VALUE EUR_VALUE THRES_HAL THRES_ANT
THRES_HAL_ANT THRES_HAL_BB THRES_HAL_ANT_BB
4,75
4,85
4,95
5,05
4,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
1,00%
10,00%
100,00%
THRES_HAL THRES_ANT THRES_HAL_ANT THRES_HAL_BB THRES_HAL_ANT_BB
0,01%
0,10%
1,00%
0,01%
100 1000 10000
Figure A.2: The left hand figures show the empirical option price of an American put as function of the number ofpaths used in the simulation using the Threshold approach (THRES) to approximate the early exercise value whereasthe right hand figures compare the empirical RMSE as function of the number of paths. The dotted lines represent√M RMSE behavior.
54
A.3.1 Mean values using antithetic sampling A.3.2 RMSE using antithetic sampling
14,75
15,75
16,75
THEO_VALUE EUR_VALUE LS_ANT_TIME
LS_ANT_UND THRES_ANT_TIME THRES_ANT_UND
10,75
11,75
12,75
13,75
10,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
10,00%
100,00%
LS_ANT_TIME LS_ANT_UND THRES_ANT_TIME THRES_ANT_UND
0,10%
1,00%
0,10%
100 1000 10000
A.3.3 Mean values using Sobol sequences A.3.4 RMSE using Sobol sequences
14,25
14,75
15,25
15,75
THEO_VALUE EUR_VALUE LS_SOB_BB_TIME
LS_ANT_UND LS_SOB_BB_UND THRES_SOB_BB_TIME
THRES_ANT_UND THRES_SOB_BB_UND
11,75
12,25
12,75
13,25
13,75
11,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
10,00%
100,00%
LS_SOB_BB_TIME LS_ANT_UND LS_SOB_BB_UND
THRES_SOB_BB_TIME THRES_ANT_UND THRES_SOB_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
A.3.5 Mean values using Niederreiter sequences A.3.6 RMSE using Niederreiter sequences
14,25
14,75
15,25
15,75
THEO_VALUE EUR_VALUE LS_NIE_BB_TIME LS_ANT_UND
LS_NIE_BB_UND THRES_NIE_BB_TIME THRES_ANT_UND THRES_NIE_BB_UND
11,75
12,25
12,75
13,25
13,75
11,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
10,00%
100,00%
LS_NIE_BB_TIME LS_ANT_UND LS_NIE_BB_UND
THRES_NIE_BB_TIME THRES_ANT_UND THRES_NIE_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
A.3.7 Mean values using Halton sequences A.3.8 RMSE using Halton sequences
14,25
14,75
15,25
15,75
THEO_VALUE EUR_VALUE LS_HAL_BB_TIME
LS_ANT_UND LS_HAL_BB_UND THRES_HAL_BB_TIME
11,75
12,25
12,75
13,25
13,75
11,75
0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000
10,00%
100,00%
LS_HAL_BB_TIME LS_ANT_UND LS_HAL_BB_UND
THRES_HAL_BB_TIME THRES_ANT_UND THRES_HAL_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
Figure A.3: The left hand figures show the empirical option price of an American exchange option as functionof the number of paths used in the simulation using the Threshold (THRES) or Least-Squares approach (LS) toapproximate the early exercise value whereas the right hand figures compare the empirical RMSE as function of thenumber of paths. The dotted lines represent
√M RMSE behavior.
55
A.4.1 Mean values using antithetic sampling A.4.2 RMSE using antithetic sampling
8,5
9
9,5
THEO_VALUE EUR_VALUE LS_ANT_CHD_TIME
LS_ANT_CHD_UND LS_ANT_PCA_TIME LS_ANT_PCA_UND
THRES_ANT_CHD_TIME THRES_ANT_CHD_UND THRES_ANT_PCA_TIME
THRES_ANT_PCA_UND
6,5
7
7,5
8
6,5
0 5000 10000 15000 20000 25000 30000
10,00%
100,00%
LS_ANT_CHD_TIME LS_ANT_CHD_UND LS_ANT_PCA_TIME
LS_ANT_PCA_UND THRES_ANT_CHD_TIME THRES_ANT_CHD_UND
THRES_ANT_PCA_TIME THRES_ANT_PCA_UND
0,10%
1,00%
0,10%
100 1000 10000
A.4.3 Mean values using Sobol sequences A.4.4 RMSE using Sobol sequences
8,5
9
9,5
THEO_VALUE EUR_VALUE LS_SOB_CHD_BB_TIME
LS_SOB_CHD_BB_UND LS_SOB_PCA_BB_TIME LS_SOB_PCA_BB_UND
THRES_SOB_CHD_BB_TIME THRES_SOB_CHD_BB_UND THRES_SOB_PCA_BB_TIME
THRES_SOB_PCA_BB_UND
6,5
7
7,5
8
6,5
0 5000 10000 15000 20000 25000 30000
10,00%
100,00%
LS_SOB_CHD_BB_TIME LS_SOB_CHD_BB_UND LS_SOB_PCA_BB_TIME
LS_SOB_PCA_BB_UND THRES_SOB_CHD_BB_TIME THRES_SOB_CHD_BB_UND
THRES_SOB_PCA_BB_TIME THRES_SOB_PCA_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
A.4.5 Mean values using Sobol sequences with anti-thetics
A.4.6 RMSE using Sobol sequences with antithetics
8,5
9
9,5
THEO_VALUE LS_SOB_ANT_CHD_BB_TIME LS_SOB_ANT_CHD_BB_UND
LS_SOB_ANT_PCA_BB_TIME LS_SOB_ANT_PCA_BB_UND THRES_SOB_ANT_CHD_BB_TIME
THRES_SOB_ANT_CHD_BB_UND THRES_SOB_ANT_PCA_BB_TIME THRES_SOB_ANT_PCA_BB_UND
6,5
7
7,5
8
6,5
0 5000 10000 15000 20000 25000 30000
10,00%
100,00%
LS_SOB_ANT_CHD_BB_TIME LS_SOB_ANT_CHD_BB_UND LS_SOB_ANT_PCA_BB_TIME
LS_SOB_ANT_PCA_BB_UND THRES_SOB_ANT_CHD_BB_TIME THRES_SOB_ANT_CHD_BB_UND
THRES_SOB_ANT_PCA_BB_TIME THRES_SOB_ANT_PCA_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
Figure A.4: The left hand figures show the empirical option price of an American Dax option as function of thenumber of paths used in the simulation using the Threshold (THRES) or Least-Squares approach (LS) to approximatethe early exercise value whereas the right hand figures compare the empirical RMSE as function of the number ofpaths. The dotted lines represent
√M RMSE behavior.
56
A.5.1 Mean values using Niederreiter sequences A.5.2 RMSE using Niederreiter sequences
8,5
9
9,5
THEO_VALUE EUR_VALUE LS_NIE_CHD_BB_TIME
LS_NIE_CHD_BB_UND LS_NIE_PCA_BB_TIME LS_NIE_PCA_BB_UND
THRES_NIE_CHD_BB_TIME THRES_NIE_CHD_BB_UND THRES_NIE_PCA_BB_TIME
THRES_NIE_PCA_BB_UND
6,5
7
7,5
8
6,5
0 5000 10000 15000 20000 25000 30000
10,00%
100,00%
LS_NIE_CHD_BB_TIME LS_NIE_CHD_BB_UND LS_NIE_PCA_BB_TIME
LS_NIE_PCA_BB_UND THRES_NIE_CHD_BB_TIME THRES_NIE_CHD_BB_UND
THRES_NIE_PCA_BB_TIME THRES_NIE_PCA_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
A.5.3 Mean values using Niederreiter sequences withantithetics
A.5.4 RMSE using Niederreiter sequences with antithet-ics
8,5
9
9,5
THEO_VALUE LS_NIE_ANT_CHD_BB_TIME LS_NIE_ANT_CHD_BB_UND
LS_NIE_ANT_PCA_BB_TIME LS_NIE_ANT_PCA_BB_UND THRES_NIE_ANT_CHD_BB_TIME
THRES_NIE_ANT_CHD_BB_UND THRES_NIE_ANT_PCA_BB_TIME THRES_NIE_ANT_PCA_BB_UND
6,5
7
7,5
8
6,5
0 5000 10000 15000 20000 25000 30000
10,00%
100,00%
LS_NIE_ANT_CHD_BB_TIME LS_NIE_ANT_CHD_BB_UND LS_NIE_ANT_PCA_BB_TIME
LS_NIE_ANT_PCA_BB_UND THRES_NIE_ANT_CHD_BB_TIME THRES_NIE_ANT_CHD_BB_UND
THRES_NIE_ANT_PCA_BB_TIME THRES_NIE_ANT_PCA_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
A.5.5 Mean values using Halton sequences A.5.6 RMSE using Halton sequences
8,5
9
9,5
THEO_VALUE EUR_VALUE LS_HAL_CHD_BB_TIME
LS_HAL_CHD_BB_UND LS_HAL_PCA_BB_TIME LS_HAL_PCA_BB_UND
THRES_HAL_CHD_BB_TIME THRES_HAL_CHD_BB_UND THRES_HAL_PCA_BB_TIME
THRES_HAL_PCA_BB_UND
6,5
7
7,5
8
6,5
0 5000 10000 15000 20000 25000 30000
10,00%
100,00%
LS_HAL_CHD_BB_TIME LS_HAL_CHD_BB_UND LS_HAL_PCA_BB_TIME
LS_HAL_PCA_BB_UND THRES_HAL_CHD_BB_TIME THRES_HAL_CHD_BB_UND
THRES_HAL_PCA_BB_TIME THRES_HAL_PCA_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
A.5.7 Mean values using Halton sequences with anti-thetics
A.5.8 RMSE using Halton sequences with antithetics
8,5
9
9,5
THEO_VALUE LS_HAL_ANT_CHD_BB_TIME LS_HAL_ANT_CHD_BB_UND
LS_HAL_ANT_PCA_BB_TIME LS_HAL_ANT_PCA_BB_UND THRES_HAL_ANT_CHD_BB_TIME
THRES_HAL_ANT_CHD_BB_UND THRES_HAL_ANT_PCA_BB_TIME THRES_HAL_ANT_PCA_BB_UND
6,5
7
7,5
8
6,5
0 5000 10000 15000 20000 25000 30000
10,00%
100,00%
LS_HAL_ANT_CHD_BB_TIME LS_HAL_ANT_CHD_BB_UND LS_HAL_ANT_PCA_BB_TIME
LS_HAL_ANT_PCA_BB_UND THRES_HAL_ANT_CHD_BB_TIME THRES_HAL_ANT_CHD_BB_UND
THRES_HAL_ANT_PCA_BB_TIME THRES_HAL_ANT_PCA_BB_UND
0,10%
1,00%
0,10%
100 1000 10000
Figure A.5: The left hand figures show the empirical option price of an American Dax option as function of thenumber of paths used in the simulation using the Threshold (THRES) or Least-Squares approach (LS) to approximatethe early exercise value whereas the right hand figures compare the empirical RMSE as function of the number ofpaths. The dotted lines represent
√M RMSE behavior.
57
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