European pricing with monte carlo simulation

1
European Pricing with
Monte Carlo Simulation
Giovanni Della Lunga
University of Insubria - Varese - Italy
SUMMER SCHOOL
FRONTIERS IN FINANCIAL MARKETS MATHEMATICS
“COPULA METHODS IN FINANCE”
Department of Mathematics for Economics and Social Science of the University of Bologna
Bologna - September, 13-15, 2004

FRONTIERS IN FINANCIAL MARKETS MATHEMATICS “Copula
2
The “path”
1. Scenario generation
 exact solution advancement
 generating scenarios by numerical integration of the stochastic
differential equations
2. Correlation and co-movement
 joint normals distributions by the Cholesky Decomposition Approach
 joint with copulae: an example using gaussian and t-copula
3. Quasi-random sequences
4. European Pricing with simulation
 the workflow of pricing with Monte Carlo
 Increasing simulation efficiency with variance reduction methods

4
Introduction
 Generating scenarios of the underlying processes
that determine the derivative’s price is an essential
and delicate task from an analytical perspective as
well as from a system design viewpoint;
 As we know, the main objective of scenarios in
pricing is to compute the expectation that gives us
the value of the financial instrument;

5
Scenario Nomenclature
 It’s well known that the value of a derivative security with payoffs at
a known time T is given by the expectation of its payoff, normalized
with the numeraire asset;
 The value of an European derivative whose payoff depends on a
single underlying process, S(t), is given by
where all stochastic processes in this expectation are consistent
with the measure induced by the numeraire asset B(t).






=
)(
]),([
)0(]0),0([
TB
TTSV
EBSV B

6
 We consider an underlying process S(t) described by the sde
 A scenario is a set of values that are an
approximation to the j-th realization,S j
(ti), of the solution of the sde
evaluated at times
 A scenario is also called a trajectory
 A trajectory can be visualized as a line in the state-vs-time plane
describing the path followed by a realization of the stochastic process
(actually by an approzimation to the stochastic process).
dWtSbdttSatdS ),(),()( +=
IitS i
j
,,1,)(ˆ =
IiTti ,,1,0 =≤≤

7

8
Scenario Construction
 There are several ways to construct scenario for pricing
 Constructing a path of the solution to the SDE at times ti by
exact advancement of the solution;
 This method is only possible if we have an analytical
expression for the solution of the stochastic differential
equation
 Approximate numerical solution of the stochastic differential
equation;
 This is the method of choice if we cannot use the previous
one;
 Just as in the case of ODE there are numerical techniques for
discretizing and solving SDE.

9
Exact Solution Advancement
 Example: Log-normal process with constant drift and
volatility
dWdt
S
dS
σµ +=
( ) [ ]





−+−





−= −−− )()(
2
1
exp)()( 11
2
1 iiiiii tWtWtttStS σσµ






+





−= )(
2
1
exp)0()( 2
tWtStS σσµ
Solution
Trajectories construction

10
 How to obtain a sequence of Wiener process?
 The simplest way
 Defining the outcomes of successive drawings of the random
variable Z corresponding to the j-th trajectory by Zj
i, we get the
following recursive expression for the j-th trajectory of S(t)
)1,0()()( 11 NZZtttWtW iiii ≈−+= −−
( ) ( ) 





−+−





−= −−−
j
iiiiii
j
i
j
ZtttttStS 11
2
1
2
1
exp)()( σσµ
Note. If the time spacing is uniform,
all the increments we add to
construct the Wiener path have the
same variance!
Note. If the time spacing is uniform,
all the increments we add to
construct the Wiener path have the
same variance!

11
 Some observations are in order...
 The set W(ti) must be viewed as the components of a vector of
random variables with a multidimensional distribution. This means
that for a fixed j Zj
i are realizations of a multidimensional standard
normal random variable which happen to be independent;
 Wheter we view the Zj
i as coming from a multidimensional
distribution of independent normals or as drawings from a single
one-dimensional distribution does not affect the outcome as long as
the Zj
i are generated from pseudo-random numbers;
 This distinction, however, is conceptually important and it
becomes essential if we generate the Zj
i not from pseudo-
random numbers but from quasi-random sequences.

12
Numerical Integration of SDE
 The numerical integration of the SDE by finite difference is
another way of generating scenarios for pricing;
 In the case of the numerical integration of ordinary
differential equations by finite differences the numerical
scheme introduces a discretization error that translates
into the numerical solution differing from the exact solution
by an amount proportional to a power of the time step.
 This amount is the truncation error of the numerical
scheme.

13
 In the case of the numerical integration of SDE by finite
differences the interpretation of the numerical error introduced by
the discretization scheme is more complicated;
 Unlike the case of ODE where the only thing we are interested in
computing is the solution itself, when dealing with SDE there are
two aspects that interest us:
 One aspect is the accuracy with which we compute the trajectories
or paths of a realization of the solution
 The other aspect is the accuracy with which we compute functions of
the process such as expectations and moments.

14
 The order of accuracy with which a given scheme can approximate
trajectories of the solution is not the same as the accuracy with
which the same scheme can approximate expectations and
moments of functions of the trajectories;
 The convergence of the numerically computed trajectories to the exact
trajectories is called strong convergence and the order of the
corresponding numerical scheme is called order of strong convergence;
 The convergence of numerically computed functions of the stochastic
process to the exact values is called weak convergence and the related
order is called order of weak convergence.
See Kloeden and Platen For an exhaustive treatment of numerical schemes for SDE.

15
 The two most popular schemes for integrating a SDE are the Explicit
Euler scheme and the Milshtein scheme
))()()(),(ˆ()),(ˆ()(ˆ)(ˆ
11 iiiiiiii tWtWttSbtttSatStS −+∆+= ++
( )[ ]ttWtW
S
ttSb
ttSb ii
ii
ii ∆−−
∂
∂
+ +
2
1 )()(
)),(ˆ(
)),(ˆ(
2
1
dWtSbdttSatdS ),(),()( +=
EULER
MILSHSTEIN

16
 Order of Strong Convergence
 The scheme has q order of strong convergence if
exist a constant α that does not depend on ∆ such
that
( ) q
TSSE ∆≤−
→∆
α)(ˆlim
0

17
 Order of Weak Convergence
 Let f(S(t)) denote a function of the trajectory of the
solution. A finite difference scheme has q order of
weak convergence if exist a constant β that does
not depend on ∆ such that
[ ] [ ] q
TSfESfE ∆≤−
→∆
β))(()ˆ(lim
0

18
 Why the order of strong and weak convergence
differ?
 Let’s consider a simplified and intuitive analysis of the
Euler scheme applied to the standard log-normal
stochastic differential equation;
 The following analysis is not rigoruous and is meant to
show why the same scheme has different accuracy
when it is used to compute trajectories than when it is
used to compute expectations. For a complete and
formal treatment of this subject the reader is referred to
the classical work of Kloeden et al.

19
 Log-normal process with constant drift and volatility
)(
)(
)(
tdWdt
tS
tdS
σµ +=
( ))(1)(ˆ)(ˆ
1 iii tWttStS ∆+∆+=+ σµ






+





−= )(
2
1
exp)0()( 2
tWtStS σσµ
Solution
Euler scheme
( ))(1)(ˆ)(ˆ
1
0
0 ∏
−
=
∆+∆+=
k
i
ik tWttStS σµ

20
 Assuming I time steps in the interval 0 ≤ t ≤ T, the error for
strong convergence is (S0 = 1 for simplicity)
 Now, to determine how well the product in the right approximates
the exponential, we expand in Taylor series keeping terms just
past order ∆t...
( )














+





−−∆+∆+∏
−
=
)(
2
1
exp)(1 2
1
0
tWttWtE
k
i
i σσµσµ

21
( )
( ) )()(
6
1
)(
2
1
)(
2
1
)(
2
1
1
6
1
2
1
1)(
2
1
exp
233
2
22
2
322
tOtW
tWt
tW
tWt
xxxtWt
i
i
t
i
i
x
i
∆+∆
+∆∆





−
+∆
+∆+∆





−+=
++++≈










∆+∆





−
∆=
σ
σσµ
σ
σσµ
σσµ


  
( ) )()(
6
1
)(
2
1
)(
2
1
exp)(1
233
2
2
tOtW
tWt
tWttWt
i
i
ii
∆−∆−
∆∆





−−






∆+∆





−≈∆+∆+
σ
σσµ
σσµσµ

22
{
( ) }
)()()(
)(
2
1
exp
)()(
6
1
)(
2
1
)(
2
1
exp)(1
23
2
233
2
2
1
0
1
0
tIOWIOWtIO
TWT
tOtW
tWt
tWttWt
i
i
i
I
i
I
i
i
∆+∆+∆∆
+





+





−≈
∆−∆−
∆∆





−−






∆+∆





−≈∆+∆+ ∏∏
−
=
−
=
σσµ
σ
σσµ
σσµσµ

23
 Strong order
[ ] [ ]
( )

( )tO
WO
t
WtO
t
TE
WOWtO
t
T
E
WIOWtIOETSTSE
t
t
t
t
∆=












∆
∆
+∆∆
∆
=






∆+∆∆
∆
=
∆+∆∆=−
∆≈
∆≈
∆≈
∆≈


)(
1
)(
1
)()(
)()()()(ˆ
2/32/3
3
3
3

24
 To analyze the weak convergence consider as example the expectation
of the process itself...
[ ]
)()()(
)(
2
1
exp
)()(
2
1
exp
)(
2
1
exp)(1)()(ˆ
22
2
22
2
1
0
tOtO
t
T
tIO
TWTE
tIOTWTE
TWTEtWtETSTSE
I
i
i
∆=∆
∆
=∆=












+





−−






∆+





+





−
=












+





−−





∆+∆+=− ∏
−
=
σσµ
σσµ
σσµσµ
When we replace the expansion of the product, the terms with ∆W do
not contribute to the expectation!!!

25
 Now we can see clearly why the order of convergence is
different when we look at paths and when we look at
moments;
 When we look at properties such expectations, the odd
power of ∆W don’t contribute to the expectation because in
weak convergence we take the expectation first and then
the norm;
 When we look at the individual paths, on the other hand,
these terms are the main contributors to the difference
between the exact and the numerical solutions, this is
because we take the norm first and then the expectation.

26
Working with VBAWorking with VBA
A Simple Example of a
Stochastic Discrete Time Simulation
Stochastic Discrete Time Simulation

27
 Observations
 When the volatility is deterministic, the order of
strong convergence of the Euler scheme is 1;
 In this case there is nothing to be gained by using
the Milshstein scheme in place of the Euler scheme
to construct scenario trajectories.

28
The Brownian Bridge
 Assume you have a Wiener process defined by a set of time-
indexed random variables {W(t1), W(t2), ... , W(tn)}.
 How do you insert a random variable W(tk) where ti ≤ tk ≤ ti+1 into
the set in such a manner that the resulting set still consitutes
a Wiener process?
 The answer is: with a Brownian Bridge!
 The Brownian Bridge is a sort of interpolation tat allows you to
introduce intermediate points in the trajectory of a Wiener
process.

29
The Brownian Bridge
 Brownian Bridge Construction
 Given W(t) and W(t + ∆t1 + ∆t2) we want to find W(t + ∆t1 );
 We assume that we can get the middle point by a weighted
average of the two end points plus an independent normal
random variable:
where α, β and λ are constants to be determined and Z is a
standard normal random variable.
ZtttWtWttW λβα +∆+∆++=∆+ )()()( 211

30
The Brownian Bridge
 We have to satisfy the following conditions:





∆+=∆+
∆+=∆+∆+∆+
=∆+=∆+
11
1211
11
)](var[
)](),(cov[
),min()](),(cov[
ttttW
tttttWttW
tttttWttW





∆+=+∆+∆+++
∆+=∆+∆++
=+
1
2
21
22
121
)(2
)(
1
ttttttt
tttttt
λβαβα
βα
βα

31
The Brownian Bridge





∆+=+∆+∆+++
∆+=∆+∆++
=+
1
2
21
22
121
)(2
)(
1
ttttttt
tttttt
λβαβα
βα
βα








∆=
−=
∆+∆
∆
=
αγ
αβ
α
1
21
2
1
t
tt
t

32
The Brownian Bridge
 We can use the brownian bridge to generate a
Wiener path and then use the Wiener path to
produce a trajectory of the process we are
interested in;
 The simplest strategy for generating a Wiener path
using the brownian bridge is to divide the time span
of the trajectory into two equal parts and apply the
brownian bridge construction to the middle point. We
then repeat the procedure for the left and right sides
of the time interval.

33
The Brownian Bridge
 Notice that as we fill in the Wiener path, the additional
variance of the normal components we add has decreasing
value;
 Of course the total variance of all the Wiener increments does
not depend on how we construct the path, however the fact
that in the brownian bridge approach we use random
variables that are multiplied by a factor of decreasing
magnitude means that the importance of those variables also
decreases as we fill in the path;
 The dimension of the random variables with larger variance
need to be sampled more efficiently than the dimension with
smaller variance;

34
The Brownian Bridge
 In standard Monte Carlo this is not an issue because standard
Monte Carlo is equally efficient at sampling from any dimension;
 But standard Monte Carlo is very slow!
 As we will see, an alternative to standard Monte Carlo is the use of
deterministic or quasi-Monte Carlo or low-discrepancy sequences;
 Usually low-discrepacy sequences differs in their ability to cover
lower dimension as compared with higher dimension (this may be
not completely true!);
 The brownian bridge method for path construction reduces the
burden on the simulation from having to sample efficiently from the
higher dimension.

35
Brownian Bridge Construction of Wiener Path
Brownian Bridge Construction of Wiener Path

36
2
Correlation and Co-movement

37
Joint Normals by the
Choleski Decomposition Approach
 The standard procedure for generating a set of
correlated normal random variables is through a
linear combination of uncorrelated normal random
variables;
 Assume we have a set of n independent standard
normal random variables Z and we want to build a
set of n correlated standard normals Ž with
correlation matrix Σ
AZZ =

Σ=t
AA

38
 We can find a solution for A in the form of a
triangular matrix














=
nnnn AAA
AA
A
A




21
2221
11
0
00
∑
−
=
−=
1
1
2
i
k
ikiiii aa σ






−= ∑
−
=
1
1
1 i
k
jkikij
ii
ji aa
a
a σ

39
 For a two-dimension random vector we have simply
 Thus we can sample from a bivariate distribution by setting








−
= 2
22
1
1
0
ρσρσ
σ
A
2
2
21222
1111
1 ZZX
ZX
ρσρσµ
σµ
−++=
+=

40
Simulation with Copula
 Gaussian Copula
 The copula of the n-variate normal distribution with linear correlation
matrix R is
where ΦR
n
denotes the joint distribution function of the n-variate standard
normal distribution function with linear correlation matrix R, and Φ-1
denotes the inverse of the distribution function of the univariate standard
normal distribution.
 Copulas of the above form are called Gaussian Copulas. In the bivariate
case the copula expression can be written as
( ))(,),()( 1
1
1
n
n
R
Ga
R uuC −−
ΦΦΦ= u
( )
( )
dtds
R
tstRs
R
vuC
u v
Ga
R ∫ ∫
− −
Φ
∞−
Φ
∞−






−
+−
−
−
=
)( )(
2
12
2
12
2
2/12
12
1 1
)1(2
2
exp
12
1
,
π

41
 Generation of random variates from the Gaussian
n-copula
1. Find the Cholesky decomposition A of R
2. Simulate n independent random variates z = (z1,..., zn)’
from N(0,1)
3. Set x = Az
4. Set ui = Φ(xi) with i = 1,2,...,n where Φ denotes the
univariate standard normal distribution function
5. (y1,...,yn)’ =[F1
-1
(u1),...,Fn
-1
(un)] where Fi denotes the i-th
marginal distribution.

42
 Generation of random variates from the Student T n-copula
1. Find the Cholesky decomposition A of R
2. Simulate n independent random variates z = (z1,..., zn)’ from N(0,1)
3. Simulate a random variate s from χ2
ν independent of z
4. Set y = Az
5. Set x = (ν/s)1/2
y
6. Set ui = Tν(xi) with i = 1, 2, ..., n and where Tν denotes the univariate
Student t distribution function
7. (g1, ..., gn)’ =[F1
-1
(u1), ..., Fn
-1
(un)] where Fi denotes the i-th marginal
distribution

43
Generation of random variates from the
Gaussian and Student t bivariate copula
Generation of random variates from the
Gaussian and Student t bivariate copula

45
 Microsoft Knowledge Base #828795
 The RAND function in earlier versions of Excel used a pseudo-random number generation
algorithm whose performance on standard tests of randomness was not sufficient (...) the pseudo-
random number generation algorithm that is described here was implemented for Excel 2003. It
passes the same battery of standard tests.
 The battery of tests is named Diehard (see note 1). The algorithm that is implemented in Excel
2003 was developed by B.A. Wichman and I.D. Hill (see note 2 and note 3). (...) It has been shown
by Rotz et al (see note 4) to pass the DIEHARD tests and additional tests developed by the
National Institute of Standards and Technology (NIST, formerly National Bureau of Standards).
 Notes
 The tests were developed by Professor George Marsaglia, Department of Statistics, Florida State
University and are available at the following Web site:
 http://www.csis.hku.hk/~diehard
 Wichman, B.A. and I.D. Hill, Algorithm AS 183: An Efficient and Portable Pseudo-Random Number
Generator, Applied Statistics, 31, 188-190, 1982.
 Wichman, B.A. and I.D. Hill, Building a Random-Number Generator, BYTE, pp. 127-128, March
1987.
 Rotz, W. and E. Falk, D. Wood, and J. Mulrow, A Comparison of Random Number Generators
Used in Business, presented at Joint Statistical Meetings, Atlanta, GA, 2001.
Random Number Generator in
Microsoft Excel

46
 Results in Earlier Versions of Excel
 The RAND function in earlier versions of Excel was fine in practice for users who did not require a lengthy sequence
of random numbers (such as a million). It failed several standard tests of randomness, making its performance an
issue when a lengthy sequence of random numbers was needed.
 Results in Excel 2003
 A simple and effective algorithm has been implemented. The new generator passes all standard tests of
randomness (?????).
The RAND function returns negative numbers in Excel 2003
SYMPTOMS
When you use the RAND function in Microsoft Office Excel 2003, the RAND function may return negative numbers.
CAUSE
This problem may occur when you try to use a large number of random numbers, and you update the RAND function multiple times.
For example, this problem may occur when you update your Excel worksheet by pressing F9 ten times or more.
RESOLUTION
How to obtain the hotfix
This issue is fixed in the Excel 2003 Hotfix Package that is dated January 12, 2004. For additional information, click the following
article number to view the article in the Microsoft Knowledge Base:
833618 Excel 2003 hotfix package released January 12, 2004
STATUS
Microsoft has confirmed that this is a problem in the Microsoft products that are listed in the "Applies to" section of this article.
Random Number Generator in
Microsoft Excel

47
Low-discrepacy Sequences
 The basic problem with standard Monte Carlo is that, unless special
techniques are used (which we will briefly discuss forward) it is
intrinsecally slow!
 The reason why regular MC is so slow is because randomly sampling
from a multidimensional distribution does not fill in the space with the
regularity that would be desirable;
 Random points tend to cluster and this clustering limits the efficiency
with which regular MC can capture payoff features.
 Quasi-random sequences are a deterministic way of filling in
multidimensional unit intervals in a way that garanties a higher
uniformity;
 The concept of higher uniformity in filling a generic intervall is captured
by the concept of discrepancy.

48
Discrepancy
 A measure for how inhomogeneously a set of d-dimensional vectors
{ri} is distributed in the unit hypercube is the so called discrepancy;
 A simple geometrical interpretation of this concept is as follows:
 Generate a set of N multivariate draws {ri} from a selected uniform number
generation method of dimensionality d ;
 All of this N vectors descrive the coordinates of points in the d-dimensional
unit hypercube [0,1]d
;
 Now select a sub-hypercube S(y) by choosing a point y delimiting the
upper right corner of the hyper-rectangular domain from 0 to y;
 In other words the sub-hypercube S can be written as
),[),[)( d1 y0y0yS ××= 


49
Discrepancy
 Next let nS(y) denote the number of all those draws that are in S(y)
 In the limit N → ∞ we require perfect homogeneity from the sequence
generator which means
for all y in [0,1]d
{ } { }∑∏∑ = =
≥
=
∈ ==
N
1i
d
1k
ry
N
1i
ySryS ikki
n 11 )()(

∏=
∞→
=
d
1i
i
yS
N
y
N
n )(
lim


50
Discrepancy
 The previous equation simply results from the fact that for a
perfectly omogeneous and uniform distribution on a unit
hypercube the probability of being in a subdomain is equal to
the volume of that subdomain, and the volume V of S( y ) is
given by the right-hand side of the previous equation;
 Then we can now compare
for all y
))((e
)(
ySV
N
n yS 

51
 Choosing L∞ -norm we have the followind definition for the discrepancy
 For example in the 1-dimensional case
we have simply
∏=∈
−=
d
1k
k
yS
10y
d
N y
N
n
D
d
)(
],[
)(
sup


10 21 ≤≤≤≤≤ nxxx 
n2
1k2
x
n2
1
D k
n1k
1
n
−
−+=
= ,
)(
max
Discrepancy

52
 We now arrive at the number-theoretical definition of low-
discrepancy sequences;
A sequence in [0,1]d
is called a low-discrepancy sequence if for all
N > 1 the first N points in the sequence satisfy
for some constant c(d) that is only a function of d.
( )
N
N
dcD
d
d
N
ln
)()(
≤
Discrepancy

53
Halton Sequence
( ) 21212020:1 012
/⇒⋅+⋅+⋅
( ) 2012
21202120:2 /⇒⋅+⋅+⋅
( ) 2012
2121212120:3 // +⇒⋅+⋅+⋅
( ) 3012
21202021:4 /⇒⋅+⋅+⋅
( ) 3012
2121212021:5 // +⇒⋅+⋅+⋅
( ) 32012
2121202121:6 // +⇒⋅+⋅+⋅
( ) 32012
212/121212121:7 // ++⇒⋅+⋅+⋅

54
Halton Sequence
 Halton sequence can be generated by reflecting the
expansion in base b about the decimal point
 At step j-th write j in base b (with b prime number)
 When we “reflect” j about the decimal point we obtain the
number of the sequence H(j)
01
10 bdbdbdj n
nn
+++= −

1
0
2
1
1 −−−
−
−
+++= n
nnj bdbdbdH 

55
Halton Sequence
 The new numbers that are added tend to fill in the gaps in the
existing sequence;
 Example with b = 2
125.02120202020214
75.0212121213
25.0212020212
5.021211
321
4
012
21
3
01
21
2
01
1
1
0
=×+×+×=⇒×+×+×=
=×+×=⇒×+×=
=×+×=⇒×+×=
=×=⇒×=
−−−
−−
−−
−
H
H
H
H

56
VBA code for Halton Sequence
Public Function Halton(n As Integer, _
x As Integer) As Double
' n è il numero da trasformare
' x è la base
' H è il numero generato dall'algoritmo
Dim H As Double
Dim z As Double
Dim m As Integer
Dim na As Integer
Dim nb As Integer
H = 0
na = n
z = 1 / x
While (na > 0)
nb = Int(na / x)
m = na - nb * x
H = H + m * z
na = nb
z = z / x
Wend
Halton = H
End Function
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.2 0.4 0.6 0.8 1

57
Generation of
Halton Sequence
Generation of
Halton Sequence

58
4
Derivatives Pricing Applications

59
European Pricing with MC
 In this case we are interested in computing (or
estimating) an Expectation;
 The expectation we are interested in is
where B(.) is the numeraire and V(T) the known
payoff at maturity.






=
)(
)(
)0()0(
TB
TV
EBV B

60
The Workflow of
Monte Carlo Pricing
 In applying MC to pricing a European derivative, we face two
challenges
 How do we construct the function V(T)/B(T) from the underlying
process?
 How do we estimate the expectation efficiently and accurately?
 The first item is important because we may not have an
analytical formula for V(T)/B(T) as a function of the
underlying process;
 The second item is important because we must get the
answer with known error bounds (if possible!) and within
time constraints.

61
The Workflow of
Monte Carlo Pricing
 In computing the distribution of discounted payoff, we must
work with processes hat are specified in an appropriate
measure;
 In its simplest form MC pricing is carried out in the pricing
measure used to derive the derivative price;
 However since we are interested only in the expectation we
do not necessarily have to carry out our simulation in the
pricing measure;
 We may be able to carry out our simulation in a different
measure than the pricing measure, a measure which is more
suitable for speed and accuracy;

62
The Workflow of
Monte Carlo Pricing
 The workflow
 In its simplest form MC pricing works by evaluating the payoff function
repeatedly and taking the average of these evaluation. Each
evaluation is called a MC Cicle;
 Each evaluation is preceeded by the computation of the underlying asset
or process;
 The underlying price needed to evaluated the payoff function is captured
by the concept of scenario.
 Each MC cycle gives us a number which is the realization of a random
variable. In some implementation of MC the random variables that get
realized at each cycle may not be independent.
 The objective of MC in pricing is to infer primarily the mean from the
properties of the sample generated by the simulation.

63
Why Monte Carlo Pricing?
 In numerical analysis there is an informal concept known as
the curse of dimensionality. This refers to the fact that the
computational load (CPU time, memory requirements, etc...)
may increase exponentially with the numer of dimensions of
the problem;
 The computational work needed to estimate the expectation
through MC does not depend explicitly on the dimensionality
of the problem, this means that there is no curse of
dimensionality in MC computation when we are only
interested in a simple expectation (this is the case with
european derivatives, thinghs are more complicated with early
exercise features).

64
Simulation Efficiency
 The efficiency of a simulation refers to the computational cost
of achieving a given level of confidence in the quantity we are
trying to estimate;
 Both te uncentainty in the estimation of the expectation as
well as the uncertainty in the error of our estimation depend
on the variance of the population from which we sample;
 However whatever we do to reduce the variance of the
population will most likely tend to increase the computational
time per MC cycle;
 As a result in order to make a fair comparison between
different estimators we must take into account not only their
variance but also the computational work for each MC cycle.

65
 Suppose we want to compute a parameter P, for example the price
of a derivative security, and that we have a choice between two
types of Monte Carlo estimates which we denote by
 Suppose that both are unbiased, so that
 but
niPi ,...,1,ˆ
1 = niP i ,...,1,ˆ
2 =
[ ] PPE =1
ˆ [ ] PPE =2
ˆ
21 σσ <

66
 A sample mean of n replications of P1 gives a more precise
estimate of P than does a sample mean of n replications of P2;
 This oversimplifies the comparison because it fails to capture
possible differences in the computational effort required by
the two estimators;
 Generating n replications of P1 may be more time-consuming
than generating n replications of P2;
Smaller variance is not sufficient grounds for preferring one
estimator over another!

67
 To compare estimators with different computational
requirements, we argue as follows;
 Suppose the work required to generate one replication of Pj is
a constant, bj (j= 1,2);
 With computing time t , the number of replications of Pj that
can be generated is t / bj;
 The two estimators available with computing time t are
therefore:
∑=
1/
1
11 ˆ
bt
i
iP
t
b
∑=
2/
1
22 ˆ
bt
i
iP
t
b

68
 For large t these are approximately normally distributed with
mean P and with standard deviations
 Thus for large t the first estimator should be preferred over the
second if
 The important quantity is the product of variance and work per
run;
t
b
t
b 2
2
1
1 σσ
2
2
21
2
1 bb σσ <

69
Increasing Simulation Efficiency
 If we do nothing about efficiency, the number of MC
replications we need to achieve acceptable pricing acccuracy
may be surprisingly large;
 As a result in many cases variance reduction techiques are a
practical requirement;
 From a general point of view these methods are based on two
principal strategies for reducing variance
 Taking advantage of tractable features of a model to adjust or
correct simulation output
 Reducing the variability in simulation input

70
Increasing Simulation Efficiency
 The most commonly used strategies for variance
reduction are the following
 Antithetic variates
 Control variates
 Importance sampling
 Stratification
 Low-discrepancy sequences

71
Antithetic variates
 In this case we construc the estimator by using
two brownian trajectories that are mirror images
of each other;
 This causes cancellation of dispersion;
 This method tends to reduce the variance
modestly but it is extremely easy to implement
and as a result very commonly used;

72
Antithetic variates
 To apply the antithetic variate technique, we generate standard normal
random numbers Z and define two set of samples of the underlying price
 Similarly we define two sets of discounted payoff samples
 At last we construct our mean estimator by averagin these samples
ZTTr
eSST
σσ +−+
= )2/(
0
2
)()2/(
0
2
ZTTr
eSST
−+−−
= σσ
[ ]0,)(max KTSVT
−= ++
[ ]0,)(max KTSVT
−= +−
( )∑=
−+
+=
n
j
jj VV
n
V
1 2
11
)0(

73
Antithetic variates
 For the antithetic method to work we need V+
and V-
to be negatively correlated;
 This will happen if the payoff function is a
monotonic function of Z;
 Attemps to combine this method with other
methods tipically don’t work well!

74
Control Variates
 The method of “Control Variate” is among the
most effective methods of the first kind;
 It exploits information about the errors in
estimates of known quantities to reduce the
error in an estimate of an unknown quantity.

75
Control Variates
 To describe the method, let’s suppose that Y1,….,Yn are outputs fron n
replications of a simulation;
 Suppose that the Yi are iid and that our goal is to estimate E[Yi ] , the
usual estimator is the sample mean <Y> = (Y1+…+Yn)/n. This estimator
is umbiased and converges with probability 1 as n →∞;
 Suppose now that on each replication we calculate another output Xi
along with Yi, let te pairs (Xi,Yi) iid and the expectation of Xi be E[X]
(known!);
 Then we calculate from the i-th replication…
[ ]( )XEXbYbY iii −−=)(

76
Control Variates
 … and then compute the sample mean
 This is a control variate estimator;
 The observed error <X>-E[X] serves as a control in estimating E[Y];
 The control variate estimator has smaller variance than the standard
estimator if
( ) ( )( )∑ −−=−−=
i
ii XEXbY
N
XEXbYbY ][
1
][)(
XYYX bb ρσσ 2<

77
Control Variates
 The effectiveness of a control variate
 is mainly determined by the strength of the correlation between
the quantity of interest, Y, and the control X.
 Can vary widely with the parameters of a problem!
 With a single control variable the optimal coefficient b* is
given by
)var(
),cov(
*
X
YX
=β

78
Control Variate
 As in all the situations when the parameters determining the
result are calculated from the same simulation, this can
introduce a bias that is difficult to estimate;
 In the limit of very large numbers of iterations this bias
vanishes but the main goal of variance reduction techniques
is to require fewer simulations;
 The efficiency of this method is strongly dependent on the
correlation between the variable of which we have to compute
the estimator and the control variate itself.
 Finding efficient control variate is more an art than a science!
 In the following slides we give an example of the use of
copula function in finding control variate.

79
Pricing Rainbow Options
with Monte Carlo
 Rainbow options are extensively used in structured finance
equity linked products, such as Everest, Altiplanos and the
like.
 Very often Monte Carlo simulation is the only pricing
technique available, even though it turns out to be pretty
costly and slow, particularly for high dimensional problems.
 Copula methods, and particularly Fréchet bounds, provide
approximated closed form solutions to the problem.

80
Copula functions and the pricing of
structured financial products
 Digital products structured with digital bivariate
options:
 Coupon = 10% if Nasdaq > K1 and Nikkey > K2
 Coupon = 10% C(Q(Nasdaq > K1),Q(Nikkey > K2))
 Call options on the minimum of a basket of assets
(Everest)
( ) [ ] ηηηη dTSQTSQTSQCTtP
TKSSSCall
K
NN
N
∫
∞
>>>
=
))((),...)((),)((,
),),,...,(min(
2211
21

81
Super-replication (lower bound)
 With perfect negative dependence we have
with K** defined in such a way that Q1(K**) + Q2(K**) = 1.
 So, super-replication requires two call spreads and a debt position
for an amount K**- K.
( ) ( ) [ ]
( ) ( )[ ] ( ) ( )[ ]
[ ] 





−−
−+−
=
=






−−>+>=
>
> ∫∫
KKB
KtSCKtSCKtSCKtSC
KKBdSQBdSQBCALL
KK
K
K
K
K
KKMax
**
;,**;,;,**;,
**
2211
**
**
22
**
11**
1
1 αηηη

82
Super-replication (upper bound)
 With perfect positive dependence
with K* such that Q1(K*) = Q2(K*).
 Super-replication requires a call spread on the first asset and a call
option on the second one
( ) ( )
( ) ( )[ ]
( ))*,max(;,
;,*;,
2
11*
*],max[
2
*
11* 2
KKtSC
KtSCKtSC
dSQBdSQBCMax
KK
KK
K
K
KK
+
+−=
=>+>=
>
∞
> ∫∫
1
1 ηηηη

83
The indipendence case
 In the independence case
 …where numerical integration is required.
( ) ( )∫
∞
>>=
K
Ind dSQSQBC ηηη 211 2

84
Fréchet pricing
 Using the closed form solutions for the extreme dependence cases, we
may come up with an approximated evaluation of a rainbow option with
a rank correlation equal to ρ
Cρ = (1 – ρ)Cind + ρCmax
 Conjecture:
Cρ should be a good control variate for any Monte
Carlo Simulation of a two colour Rainbow Option
with correlation ρ.

85
Efficiency of Reduction
In the Money
Rho = 0.25
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Standard
Monte Carlo
"Fréchet"
Reduction
Variate
Expiration 0.25
Asset 1 110
Asset 2 105
Volatility Asset 1 0.25
Volatility Asset 2 0.25-0.4
Strike 95
Risk Free Rate 0.05
Option Data

86
In the Money
Rho = 0.5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Expiration 0.25
Asset 1 110
Asset 2 105
Strike 95
Risk Free Rate 0.05
Option Data

87
In the Money
Rho = 0.75
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Expiration 0.25
Asset 1 110
Asset 2 105
Strike 95
Risk Free Rate 0.05
Option Data

88
Bibliography
 U. Cherubini, E. Luciano and W. Vecchiato
 Copula Methods in Finance, Wiley Finance (2004)
 P. Glasserman
 Monte Carlo Methods in Financial Engineering, Springer (2004)
 P. Jackel
 Monte Carlo Methods in Finance, Wiley Finance (2002)
 P. Kloeden, E. Platen and H. Schurz
 Numerical Solution of SDE through Computer Experiments,
Springer (1994)
 D. Tavella
 Derivatives Pricing, An Introduction to Computational Finance,
Wiley Finance (2002)
 P. Wilmott
 Derivatives, Wiley (1998)

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