Numerical Methods Book Crepey

8/4/2019 Numerical Methods Book Crepey

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Numerical Methods in Finance

Stphane CRPEY, vry [email protected]

CIMPAJordanie, September 2005

Figure 1: Local volatility on the DAX index


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CIMPAJordanie September 2005

Contents

1 Random numbers 2

2 Pseudo random generators 3

2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Properties required for a good pseudo-random numbersgenerator . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Constructing pseudo-random number generators . . . . . . 6

3 Low-discrepancy sequences 8

3.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 General remarks on low discrepancy sequences . . . . . . 10

3.3 Sobol sequences . . . . . . . . . . . . . . . . . . . . . . . 11

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4 Simulation ofnon-uniform random variables or vectors 14

4.1 Inverse method . . . . . . . . . . . . . . . . . . . . . . . 14

4.2 Simulation of Gaussian standard variables . . . . . . . . . 14

4.3 Simulation of Gaussian vectors . . . . . . . . . . . . . . . 17

5 Principle of the Monte Carlo Simulation 19

5.1 Limit theorems . . . . . . . . . . . . . . . . . . . . . . . 195.2 Estimation principle . . . . . . . . . . . . . . . . . . . . . 20

5.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Variance Reduction Techniques 246.1 Antithetic Variables . . . . . . . . . . . . . . . . . . . . . 24

6.2 Control Variables . . . . . . . . . . . . . . . . . . . . . . 26

6.3 Importance Sampling . . . . . . . . . . . . . . . . . . . . 27

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6.4 Efficiency of the Monte Carlo methods . . . . . . . . . . . 29

7 Quasi Monte Carlo Simulation 317.1 General principle . . . . . . . . . . . . . . . . . . . . . . 31

7.2 Koksma-Hlawka inequality . . . . . . . . . . . . . . . . . 31

7.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8 Computing the Greeks by (Quasi) Monte Carlo 35

8.1 Finite Differences . . . . . . . . . . . . . . . . . . . . . . 35

8.2 Derivation of the payoff . . . . . . . . . . . . . . . . . . . 36

8.3 Payoff regularization . . . . . . . . . . . . . . . . . . . . 37

9 (Quasi) Monte Carlo algorithms for vanilla options 40

9.1 (Q)MC BS1D . . . . . . . . . . . . . . . . . . . . . . . . 40

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9.2 (Q)MC BS2D . . . . . . . . . . . . . . . . . . . . . . . . 57

10 Simulation of processes 8010.1 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . 80

10.2 BlackScholes model . . . . . . . . . . . . . . . . . . . . 83

10.3 General diffusions: Euler and Milshtein scheme . . . . . . 85

10.4 Heston model . . . . . . . . . . . . . . . . . . . . . . . . 87

10.5 Monte Carlo Simulation for Processes . . . . . . . . . . . 89

11 (Quasi) Monte Carlo methods for Exotic Options 91

11.1 Lookback options . . . . . . . . . . . . . . . . . . . . . . 9111.2 Andersen and Brotherton-Ratcliffe Algorithm for Look-

back Options . . . . . . . . . . . . . . . . . . . . . . . . 94

11.3 Barrier options . . . . . . . . . . . . . . . . . . . . . . . 104

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11.4 Asian options . . . . . . . . . . . . . . . . . . . . . . . . 107

12 Trees for vanilla options 11012.1 Cox-Ross-Rubinstein as an approximation to Black-Scholes 110

12.2 Algorithm (CRR) . . . . . . . . . . . . . . . . . . . . . . 116

12.3 Variants of the CRR tree . . . . . . . . . . . . . . . . . . 123

12.4 Trinomial trees . . . . . . . . . . . . . . . . . . . . . . . 126

12.5 Algorithm (Kamrad-Ritchken) . . . . . . . . . . . . . . . 131

12.6 Miscellaneous remarks . . . . . . . . . . . . . . . . . . . 138

13 Trees for exotic options 14413.1 Inaccuracy of the direct method for barrier options . . . . 144

13.2 The Ritchken algorithm for barrier options . . . . . . . . . 145

13.3 Customization of trees . . . . . . . . . . . . . . . . . . . 145

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14 Finite Differences for European Vanilla Options 149

14.1 Localization and Discretization . . . . . . . . . . . . . . . 149

14.2 The -scheme . . . . . . . . . . . . . . . . . . . . . . . 158

14.3 Explicit Method . . . . . . . . . . . . . . . . . . . . . . . 160

14.4 Implicit Methods . . . . . . . . . . . . . . . . . . . . . . 161

15 Finite Differences for American Vanilla Options 167

15.1 Variational inequality in finite dimension . . . . . . . . . . 167

15.2 Linear complementarity problem . . . . . . . . . . . . . . 168

15.3 Splitt ing methods . . . . . . . . . . . . . . . . . . . . . . 173

16 Finite Difference -scheme Algorithm for Vanilla Options 176

17 Finite Differences for 2D Vanilla Options 189

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17.1 Numerical integration by an A.D.I. Method . . . . . . . . 192

17.2 American Options . . . . . . . . . . . . . . . . . . . . . . 193

17.3 Algorithm (A.D.I. BS2D) . . . . . . . . . . . . . . . . . . 194

18 Finite Differences for Exotic Options 208

18.1 Lookback Options . . . . . . . . . . . . . . . . . . . . . . 208

18.2 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . 20918.3 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . 211

19 Dynamic tests 215

19.1 Delta-hedging . . . . . . . . . . . . . . . . . . . . . . . . 21519.2 Dynamic tests using Brownian Bridge in the BlackScholes

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

20 Conclusion 224

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This lecture is largely based on the public releases of the option pricing

software and documentation system PREMIA developed since 1999 by

the MATHFI project at INRIA www.inria.fr and CERMICScermics.enpc.fr

In particular all code samples, written in the C programming language,

are extracted from PREMIA (except for minor modifications).

The aim of the Premia project is threefold: first, to assist the R&D

professional teams in their day-to-day duty, second, to help the academics

who wish to perform tests of a new algorithm or pricing method without

starting from scratch, and finally, to provide the graduate students in the

field of numerical methods for finance with open-source examples of

implementation of many of the algorithms found in the literature

www-rocq.inria.fr/mathfi/Premia/index.html

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1 Random numbers

Sample random variables or vectors in Rd, d 1First get sequences of uniform random variables or vectors un over

[0, 1]d. Then transform the un into xn with specific distributions

Sample iid random variables over [0, 1] and group them in buckets of size

d

Or use quasi random generators (low-discrepancy sequences) over [0, 1]d

A bad generator can lead to false results for a considered simulation

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2 Pseudo random generators

Simulation of pseudo-uniform variables over [0, 1] [31, 30, 29, 36, 34, 23]

2.1 Definition

A pseudo-random number generatoris a structure G = (S, s0, T , U , G)

where S is a finite set of states, s0 S is the initial state, the mappingT : S S is the transition function, U is a finite set ofoutputs symbols,and G : S U is the output function.This definition was introduced by LEcuyer in [31] or [30] for instance.

Since S is finite, the sequence of states is ultimately periodic. The period

is the smallest positive integer such that for some integer 0 and forall n , s+n = sn. The smallest with this property is called thetranscient. When = 0, the sequence is said to be purely periodic.

The resolution of a generator is the largest number x such that all output

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values are multiples ofx. It determines the maximal number of different

values we can obtain with the generator.

2.2 Properties required for a good pseudo-randomnumbers generator

Large period length, at least 260

Good equidistribution properties and statistical independence ofsuccessive pseudorandom numbers

The generator should pass statistical tests for uniformity and

independence [23, 30]: general tests like chi-square or

Kolmogorov-Smirnov tests; specific tests like equidistribution test,

serial test, gap test, partition test, etc.. Note that since generated

sequences are deterministic, we can always find a test the sequence

will fail.

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Little intrinsic structureSuccessive values produced by some of the described generators have

a lattice structure in any given dimension. Efficiency, fast generation algorithm, requiring not too much memory

space

Especially if we use many generators together or in parallel

Repeatability (fixing a given seed)Very useful for practical applications. Otherwise we can use thecurrent time (computer clock) to initialize the generators

PortabilityIt means that the generator will produce exactly the same sequence

on different computers or with different compilers

UnpredictabilityIt means that we can not predict the next generated value by thealgorithm from the previous ones (though this is less important for

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finance than for other applications, cryptography in particular)

2.3 Constructing pseudo-random number generators

The simplest methods to construct random number generators are linear

methods. Linear methods use a linear recurrence relation to compute the

next value from the previous onesLinear Congruential Generators (LCG):

The nth random number is given by

un =Un

m [0, 1]

where

Un = (aUn1 + c) mod m

where m > 0, a > 0 and c are fixed integers

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Such generators produce a lot of regularity in sequences and an

unfavorable lattice structure

Random numbers generator of LEcuyer with Bayes & Durhamshuffling procedure : Combination of two short periods LCG to obtain alonger period generator

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3 Low-discrepancy sequences

Such sequences neither are random nor pseudo-random but deterministicand successive values are not independent [37, 36, 35, 32]

However they satisfy good properties of equidistribution on [0, 1]d and we

have that 1NNi=1 f(i) [0,1]

d f(u)du

3.1 Definition

Notations:

We note [|0, x|] = {y = (y1, . . . , yd) [0, 1]d, y x} ; we consider thaty

x if and only if for all j = 1, . . . , d : yj

xj .

(x) denotes the volume of[|0, x|]. (x) = x1 xd.We note Id = [0, 1]d the closed d-dimensional unit cube.

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For = (n)n1 a sequence in Id and x Id, we note :

Dn(, x) =1

n

ni=1

1[|0,x|](i) (x)

Definitions:

A sequence (n)n 1 is said to be equidistributedon [0, 1]d if

x [0, 1]d, limn+

1

n

ni=1

1[|0,x|](i) = (x)

The value Dn() defined as:

Dn() = supxId |Dn(, x)|is called the star discrepancy for the first n terms of the sequence.

The discrepancy is a very important notion for Quasi-Monte Carlo

simulation. It measures how a given set of points is distributed in

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Id = [0, 1]d. It can be viewed as a quantitative measure for thedeviation from the uniform distribution.

A sequence () is said to be a low-discrepancy sequence if itsdiscrepancy satisfies DN = O(

(logN)d

N ) or if it is asymptoticallybetter than the one of a random sequence obtained from the law ofiterated logarithm as O(( log logNN )

12 ).

3.2 General remarks on low discrepancy sequences

Quasi-random numbers combine the advantage of a random sequence thatpoints can be added incrementally, with the advantage of a lattice.

For large dimension d, the theoretical bound (log N)d/N may only bemeaningful for extremely large values ofN.

Low discrepancy sequences are very useful for low dimension. In highdimension d, a lattice can only be refined by increasing the number of

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points by a factor 2d.

Orthogonal projections: if a d-dimensional sequence is uniformlydistributed in Id, then two-dimensional sequences formed by pairing

coordinates should also be uniformly distributed. The appearance of

non-uniformity in these projections is an indication of potential problems

in using a quasi-random sequence for integration [32].

3.3 Sobol sequences

The Sobol sequence [42] is one of the most used sequences for

Quasi-Monte Carlo simulation and one of the most successful for

financial applications. Its construction is based on primitive polynomials

in the field Z2 and bitwise XOR (Exclusive Or) operations

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sobol dim 1 and 2

Figure 2: Orthogonal projection on the first two coordinates of the first

10000 points of the Sobol sequence in dimension 160

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sobol dim 159 and 160

Figure 3: Orthogonal projection on the last two coordinates of the first

10000 points of the Sobol sequence in dimension 160

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4 Simulation of non-uniform random

variables or vectors4.1 Inverse method

Simulation of a variable with known (or computable) inverse cumulative

distribution function F- First, simulate a variable u uniformly distributed on [0, 1], that is a

random number from one of the pseudo-random numbers generators or

one of the quasi-random number generators

- Then take x = F1(u)

Example: To simulate an exponential random variable with parameter ,

take 1 ln(1 U).4.2 Simulation of Gaussian standard variables

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One direct method to generate standard Gaussian variables: Box-Mller

transformation

Box-Mller transformation:If(u, v) is uniformly distributed on [0, 1]2 then x and y defined by

x =2log u sin(2v)

y =

2log u cos(2v)

are distributed as independent standard gaussians.

Proof: Let be uniform over [0, 2] and 2 be exponential with

parameter 1/2, independent from . Then the pair

(X, Y) = ( cos , sin ) is standard Gaussian. Indeed, for every

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measurable and bounded function f

Ef(X, Y) = +

0 2

0 f( cos , sin )

1

2 e

2

2

d(2

)

d

2

=

+0

20

f( cos , sin )de2

2d

2

= R R f(x, y)e x2+y22 dxdy

2

2

Remarks: This method requires two independent random values toobtain two gaussian variables

- It must not be used when random numbers u and v are generated fromtwo successive values of a one-dimensional low-discrepancy sequence,

because they are not independent in this case. To apply one of these

algorithms with Quasi Monte Carlo simulation, you should generate u

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and v independently, that is necessarily from two different

one-dimensional sequences or from one two-dimensional sequence.

Simulating gaussian variable with the inverse method: The inversecumulative distribution 1 has not an explicit form. Thus to use theinverse method to simulate gaussian variable we need an approximation of

1. A very good and quick approximation is given by Moros algorithm.

4.3 Simulation of Gaussian vectors

Simulate of a N-dimensional Gaussian vector V with zero mean and a

covariance matrix is done by the following way:

- First we compute the lower triangulary matrix obtained with the

Cholesky decomposition of, that is such that = t. We have :

ii =

ii i1

k=1(ik)2

ji =ij

Pi1k=1 ikjkii

for j = i + 1, . . . , N

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- We generate N independent gaussian standard variables gi. We noteG = (g1, . . . , gN).

- We compute V = G.Then V is distributed asN(0, ).Two-dimensional case:

Assume that the covariance matrix is expressed by:

21 1212 22 Then is given by the following matrix:

11 1221 22 = 1 02 1 22 This will be used in case of simulation of correlated two-dimensionalbrownian motions, for instance to price options in a BS2D model

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5 Principle of the Monte Carlo Simulation

A general method for evaluating an integral as an expected value, basedon the Strong Law of Large Numbers (LLN) and the Central LimitTheorem

It provides an unbiased estimator and the error on the estimate iscontrolled within a confidence interval

5.1 Limit theorems

Strong Law of Large Numbers:

1

N

N

i=1

(xi)

a.s.

E[(X)]ifxi are i.i.d to X and E[|(X)|] < +.

Central Limit Theorem: The estimator converges in law to a gaussian

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standard distribution.

N

1NN

i=1

(xi) E[(X)] L N(0, 1)where 2 = Var[(X)] and 2 < +.5.2 Estimation principle

We want to estimate the following parameter I:

I = E[(X)]

where is some function on D Rn over R and X = (X1, . . . , X n) isa n-dimensional vector of random variables with law .

I can be expressed as an integral

I =

D

(x)d(x)

An unbiased estimator ofI for N trials with the Standard Monte Carlo

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method is defined by:

N =1

N

N

i=1 (xi)with xi i.i.d to .

Variance of the estimator is given as:

2N =

2

N

with unbiased estimator:

2N =

1

N

1

1

N

N

i=12(xi) 2N

Variance decreases to 0 when N +. It means that the greater N is,the more accurate the estimator is. The speed of convergence ofN to I isN

. It is independant of the dimension n.

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A confidence interval IC = [A, B] for the threshold (confidence level)1 2 is such that P(A < I < B) = 1 2 and it is built as follows:

IC = [N zN; N + zN]where z = 1(1 ) and 1 is the inverse cumulative distributionfunction of the standard gaussian law.For instance, if the threshold is chosen to 95% then = 2, 5% and

z 1.96.5.3 Properties

We briefly summarize some advantages and disadvantages of theStandard Monte Carlo method.

Advantages:- This method does not require regularity or differentiabilityproperties for the function . Thus we can implement this methodvery easily if we are able to generate the variable X according to .

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- The estimator is unbiased.

- The error on the estimate can be controlled by the Central Limit

Theorem, and we can build a confidence interval. Disadvantages:

- We have to realize a lot of simulations to obtain an accurate

estimator. Therefore computing time can be very high.

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6 Variance Reduction Techniques

We saw that a disadvantage of the standard Monte Carlo Simulation is itsrequired computing time. Thus we are now interested in AcceleratedMonte Carlo Simulation.To reduce computing time, we can use variance reduction techniques[38]. The principle is to rewrite the parameter I in order to express it in

function of a new random variable with smaller variance 2. Then weneed a smallest number of iterations to obtain the same accuracy on theestimate.

6.1 Antithetic Variables

Principle of antithetic variables is to introduce some correlation betweenthe terms of the estimate.When simulation is done by the inverse cumulative distribution function,we use uniform numbers ui on [0, 1]. For this method, we use each ui

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twice, as ui and 1 ui. These both variables have same law but are notindependant. We note xi and xi the variables generated from ui and

1 ui respectively.An unbiased estimator ofI with N trials is defined by:

N =1

2N

Ni=1

((xi) + (xi))

with xi i.i.d to .Variance of the estimator is given as:

2N =1

2N(V ar[(X)] + Cov((X), (X)))

The following theorem give sufficient conditions to obtain a variance

reduction with this method.Theorem: If is a monotone, continuous, derivable function then

(antN )2 1

2(stdN )

2

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Factor 1/2 is due to the sample size for the antithetic method: in fact, theestimator contains 2N terms.

6.2 Control Variables

Principle of this method is to introduce an other model for which we havean explicit solution and to estimate the difference between our firstparameter I and the new one.

I = E[(X)]

= E[(X) (X)] + E[(X)]where is a function such that E[(X)] = m is known.

An unbiased estimator ofI with N trials is defined by:

N =1

N

Ni=1

((xi) (xi)) + m

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with xi i.i.d to .

Variance of the estimator is given as:

2N =1NV ar[(X) (X)]

= 1N (V ar[(X)] + V ar[(X)] 2Cov((X), (X)))Variance reduction with regard to standard Monte Carlo simulation is not

guaranteed by this method. To decrease the variance, functions and

must have a large positive correlation. It implies a specific choice for the

control variate .

6.3 Importance Sampling

The basic idea of importance sampling consists in concentrating thedistribution of the sample points in the most contributive parts of the

space. For that, we introduce an new density which changes the initial

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density ofX.

I =

D(x)(x)(x)d(x)

= E (X)(X)(X)We obtain the following estimator:

N =1

N

N

i=1(xi)

(xi)

(xi)

with xi i.i.d to .Variance of the estimator is expressed as:

2N =1

NV ar

(X)

(X)

(X)

is named the importance function. It must verify that (x) > 0, x Esuch that (x)(x) > 0.

Variance reduction with regard to standard Monte Carlo simulation is not

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guaranteed by this method. It depends on the choice of, which is not an

easy step. However, the minimum of the variance is reached for the

following importance density called the optimal density.

=|(x)|(x)|(y)|(y)dy

Usually this density is unknown and it contains the term I as soon as

> 0.6.4 Efficiency of the Monte Carlo methods

We introduce a criterion to compare the efficiency of the various

simulation methods, standard simulation or with variance reduction

techniques. This criterion takes into account the computing time requiredby the simulation for each method. It is independent on the sample size.

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Efficiency of the method j with regard to the method i is defined by:

(i, j) = limNi,Nj+Ni(i)

Nj (j)tNi(i)tNj (j)The method j is considered to be more efficient than the method i if

(i, j) 1If we assume that the computing time is proportional to the sample size,

that is tNi(i) = kiNi then:

(i, j) =ij

kikj

where k is a factor, which exprimes the complexity of the algorithm for

the considered method. For instance, if(i, j) = 3, it means that method i

requires 9 times more time to obtain the same accuracy than method j. In

other words, with the same computing time, standard error for method j is

3 times smaller than the one of method i.

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7 Quasi Monte Carlo Simulation

7.1 General principle

Estimate

E[f(U)] [0,1]d

f(u)du

byNi=1 f(i) where is a d-dimensional low-discrepancy sequenceSpecial case: f F1X . Then (X)

(law)= f(U), so

E[f(U)] = E[(X)].

7.2 Koksma-Hlawka inequalityDefinitions:

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The variation of a function f on Id = [0, 1]d in the sens ofVitali isdefined by :

Vd

(f) = suppP(Id)Ap |(f, A)|where P(Id) is the set of all partitions ofId into subintervals,

p P(Id) denotes a partition and A p a subinterval.(f, A) is the alternative sum of the values off at the vertices ofA.

Vd(f) =[0,1]d

dfu1 . . . ud du1 . . . d udif the partial derivatives exist and are continuous on Id.

The variation off on Id in the sense ofHardy and Krause is defined

by :

V(f) =dr=1

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restriction off to the r dimensional face

{(x1, . . . , xd) Id/xk = 1 if k {i1, . . . , ir}}

Theorem: Iff has bounded variation V(f) on Id in the sense ofHardy-Krause, then for any 1, . . . , n [0, 1]d, we have:

n 1,

1

n

n

i=1f(i)

[0,1]d

f(x)dx

V(f)Dn()

7.3 Remarks

Through the Koksma-Hlawka inequality, we understand the need to have

sequences with discrepancy DN as small as possibleThe Koksma-Hlawka inequality gives an a priori deterministic bound for

the error in the approximation of[0,1]d

f(x)dx by the sum 1nn

i=1 f(i).

This error is expressed in term of the discrepancy of the sequence and the

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variation of the function f. Nevertheless it is often difficult to calculate or

even to estimate the variation off. Moreover, since for large dimension d,

the asymptotic bound (log N)d

/N of low-discrepancy sequences mayonly be meaningful for extremely large values ofN, and because

(log N)d/N increases exponentially with d, then the bound in

Koksma-Hlawka inequality gives no relevant information until a very

large number of points is used

In opposition to Monte Carlo simulation, Quasi-Monte Carlo doesnt

provide an confidence interval for the estimator. We cannot compute

empirical variance of the sample because successive terms are not

independent. This is due to the construction of the low-discrepancy

sequenceAnother difference in comparison with Monte Carlo is that the

convergence rate for QMC simulation depends on the dimension d of the

considered model through discrepancy Dd

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8 Computing the Greeks by (Quasi) Monte

Carlo

In risk-neutral diffusion models, the option price is given by:

P(x) = ERN[(SxT)]

where Sxt

is the unique solution of

dSt = b(St)dt + (St)dWt , Sx0 = x.

Option theory states that the option seller must hold at every date t

=

x

E[(SxT)]

units of stock in his portfolio in order to hedge the option. The goal of thissection is to propose various (Q)MC methods to compute [13, 6, 14].

8.1 Finite Differences

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C Jo da e Septe be 005

For fixed h > 0, we approach by decentered finite differences

1

h E[(Sx+hT )] E[(SxT)]or, preferably, by centered finite differences

1

2h

E[(Sx+hT )] E[(SxhT )]

The two terms of these differences can be estimated by (Q)MC methods.

In so doing, it is better to use common random numbers to estimate thetwo terms. For instance, in the special case of the BlackScholes modelSxt = x exp((r

2

2 )t + Wt), can be estimated by

1

2hM

M

i=1 ((x + h)e(r2

2 )T+Tgi)

((x

h)e(r

2

2 )T+Tgi)

where the gi are standard Gaussian (Q)MC draws.

8.2 Derivation of the payoff

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p

In cases where is regular and where we know how to derive SxT with

respect to x, can be formally computed as

= x

E(SxT) = E

x(SxT) = E

(SxT)SxTx

For instance, in the BlackScholes model, SxT

x =SxTx , so

=1

xE[(Sx

T)Sx

T]

provided the derivative of exists, for instance in the sense of

distributions.

8.3 Payoff regularization

In the BlackScholes model, we have

E[(SxT)] =

R

(y)pT(x, y) dy

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p

with

pT(x, y) =1

y

22Te 1

22T

log( y

x)(r22 )T

2

So, formally

=

R

(y)

xpT(x, y) dy

= R (y)xpT(x, y)

pT(x, y)pT(x, y) dy

= E

(SxT)

xlog(pT(x, S

xT))

.

Straightforward computation gives

x log(pT(x, SxT)) =

WT

xT

Thus

= E

(SxT)

WTxT

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p

which can be easily computed by (Q)MC.

The following result extends this methodology to more general diffusions.

Theorem: Assuming b and in class C1b with positive, let S be thesolution of

dSt = b(St)dt + (St)dWt , Sx0 = x.

Defining

Yxt =

Sxtx

then Y is the unique solution of

dYtYt

= bx(St)dt + x(St)dWt , Yx0 = 1

Moreover, for every function in class C0b

= E

(SxT)

1

T

T0

Yt(St)

dWt

.

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p

9 (Quasi) Monte Carlo algorithms for vanilla

options

9.1 (Q)MC BS1D

Description: Computation, for a Call - Put - CallSpread or DigitEuropean Option, of its Price and its Delta with the Standard Monte Carlosimulation. In the case of Monte Carlo simulation, the program providesestimations for price and delta with a confidence interval. In the case ofQuasi-Monte Carlo simulation, the program just provides estimations forprice and delta. For a Call, the implementation is based on the Call-PutParity relationship.

Input parameters:

StepNumber N Generator_Type

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Increment inc

Confidence Value

Output parameters:

Price P Error Price P

Delta Error delta Price Confidence Interval: ICP =[Inf Price, Sup Price]

Delta Confidence Interval: IC =[Inf Delta, Sup Delta]

The underlying asset price evolves according to the Black and Scholes

model, that is:

dSu = Su((r d)du + dBu), STt = s

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then

ST = s exp((r d 2

2)t) exp(Bt)

where ST denotes the spot at maturity T, s is the initial spot, t is the time

to maturity.

The Price of an option at T t is:

P = E[exp(

rt)f(K, ST, R)]

where f denotes the payoff of the option, K the strike and R the rebate

(for Digit option only).

The Delta is given by:

= s

E[exp(rt)f(K, ST, R)]

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The estimators are expressed as:

P = 1N exp(rt)N

i=1

P(i)

where P(i) = f(K, ST(i), K)

=

1

Nexp(rt)

N

i=1

sP(i) =

1

Nexp(rt)

N

i=1(i)

The values for P(i) and (i) are detailed for each option.

Put: The payoff is (K ST)+. We have:P(i) = (K ST(i))+

(i) =

ST(i)s = ST(i)s if P(i) 0

0 otherwise

Call: The payoff is (ST K)+.

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The Call-Put Parity relations for price and delta are expressed by:

C = P + s exp(

dt)

Kexp(

rt)

C = P + exp(dt)where C and P respectively denotes the Call and the Put prices. They

will be used for the Call simulation (in order to limit variance).

CallSpread: The payoff is (ST

K1)

+

(ST

K2)

+.

We have:

P(i) =

(ST(i) K1)+ (ST(i) K2)+

(i) =8>>>:

ST(i)s

= ST(i)s

if ST(i) K1 and ST(i) K2

S

T

(i)s =

ST

(i)s if ST(i) K2 and ST(i) K1

0 otherwise

Digit: The payoff is R1{STK0}.

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We have:P(i) = R1{ST(i)K0}

To have an estimation of the Delta in the case of a Digit option, weneed to use the increment value inc at each iteration i as:

i =1

2s.inc[P (ST(i)(s(1 + inc)), K , R) P (ST(i)(s(1 inc)), K , R)]

Code Sample:static int MCStandard(...)

{

...

/*Value to construct the confidence interval*/

alpha= (1.- confidence)/2.;z_alpha= Inverse_erf(1.- alpha);

/*Initialisation*/

flag= 0;

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s_plus= s*(1.+inc);

s_minus= s*(1.-inc);

mean_price= 0.0;

mean_delta= 0.0;

var_price= 0.0;

var_delta= 0.0;

/* CallSpread */

K1= p->Par[0].Val.V_PDOUBLE;

K2= p->Par[1].Val.V_PDOUBLE;

/*Median forward stock and delta values*/

sigma_sqrt=sigma*sqrt(t);

forward= exp(((r-divid)-SQR(sigma)/2.0)*t);forward_stock= s*forward;

forward_delta= exp(-SQR(sigma)/2.0*t);

/* Change a Call into a Put to apply the Call-

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Put parity */

if((p->Compute) == &Call)

{

(p->Compute) = &Put;

flag= 1;

}

/*MC sampling*/

init_mc= InitGenerator(generator,simulation_dim

,N);

/* Test after initialization for the generator

*/

if(init_mc == OK){

mc_or_qmc= Rand_Or_Quasi(generator);

/* Begin N iterations */

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for(i=1 ; iCompute)(p->Par,S_T);

/*Delta*/

/*Digit*/

if ((p->Compute) == &Digit)

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{

price_sample_plus= (p->Compute)(p->

Par, U_T*s_plus);

price_sample_minus= (p->Compute)(p->

Par, U_T*s_minus);

delta_sample= (price_sample_plus-price

_sample_minus)/(2.*s*inc);

}

/* CallSpread */

else

if ((p->Compute) == &CallSpread )

{

delta_sample= 0;

if(S_T > K1)delta_sample += U_T;

if(S_T > K2)

delta_sample -= U_T;

}

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/*Call-Put*/

else

if ((p->Compute) == &Put)

{

if (price_sample>0.)

delta_sample= -U_T;

else

delta_sample= 0.0;

}

/*Sum*/

mean_price+= price_sample;

mean_delta+= delta_sample;

/*Sum of squares*/

var_price+= SQR(price_sample);

var_delta+= SQR(delta_sample);

}

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/* End N iterations */

/* Price */

*ptprice= exp(-r*t)*(mean_price/(double) N);

*pterror_price= sqrt(exp(-2.0*r*t)*var_pri

ce/(double)N - SQR(*ptprice))/sqrt(N-1);

/*Delta*/

*ptdelta= exp(-r*t)*mean_delta/(double) N;

*pterror_delta= sqrt(exp(-2.0*r*t)*(var_de

lta/(double)N-SQR(*ptdelta)))/sqrt((double)N-1);

/* Call Price and Delta with the Call Put

Parity */if(flag == 1)

{

*ptprice+= s*exp(-divid*t)- p->Par[0].Val

.V_DOUBLE*exp(-r*t);

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*ptdelta+= exp(-divid*t);

(p->Compute)= &Call;

flag = 0;

}

/* Price Confidence Interval */

*inf_price= *ptprice - z_alpha*(*pterror_p

rice);

*sup_price= *ptprice + z_alpha*(*pterror_p

rice);

/* Delta Confidence Interval */

*inf_delta= *ptdelta - z_alpha*(*pterror_d

elta);*sup_delta= *ptdelta + z_alpha*(*pterror_d

elta);

}

return init_mc;

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}

Further Comments:/* Value to construct the confidence interval */

For example if the confidence value is equal to 95% then the value zused to construct the confidence interval is 1.96.

/*Initialization*/

/*Median forward stock and delta values*/Computation of intermediate values we use several times in the program.

/* Change a Call into a Put to apply the Call-Put parity */We modify the parameters of the option; they will be reinitialized at the

end of the simulation program.

/*MC sampling*//* Begin N iterations */

/*Price*/

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At the iteration i, we obtain

ST(i) = s exp r d 2

2 t exp(Bt(i))P(i) = Payoff(ST(i), K)

from a simulation ofBt(i) with the selected generator as

tgi where gi is

a standard gaussian variable.

/*Delta*/Calculation of Delta i with formula previously detailed for each option.

/*Digit*//*CallSpread*/

/*Call-Put*/

/*Sum*/Computation of the sums

Pi and

i for the mean price and the mean

delta.

/*Sum of squares*/

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Computation of the sums

P2i and

2i necessary for the variance

price and the variance delta estimations.


/*Price*/

The price estimator is:

P = 1N exp(rt)Ni=1

P(i)

The error estimator is P with :

2

P=

1

N 1 1N exp(2rt)N

i=1 P(i)2 P2

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/*Delta*/

=1

Nexp(

rt)

N

i=1 (i)The error estimator is with:

2 =1

N

1

1

Nexp(2rt)

N

i=1(i)2 2

/* Call Price and Delta with the Call Put Parity */

We now compute the price and the delta for a call.Parameters of the option are reinitialized.

/* Price Confidence Interval */

The confidence interval is given as:ICP = [P zP; P + zP]

with z computed from the confidence value./* Delta Confidence Interval */

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The confidence interval is given as:

IC = [

z; + z]

with z computed from the confidence value.

9.2 (Q)MC BS2D

Description: Computation, for a Call on Maximum, Put on Minimum,Exchange or Bestof European Option, of its Price and its Delta with theStandard Monte Carlo or Quasi-Monte Carlo simulation. In the case ofMonte Carlo simulation, this method also provides an estimation for theintegration error and a confidence interval.

Input parameters:

StepNumber N Generator_Type

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Increment inc Confidence Value

Output parameters:

Price P Error Price P

Deltas 1, 2 Errors delta 1 , 2 Price Confidence Interval: ICP =[Inf Price, Sup Price]

Delta Confidence Intervals: ICj =[Inf Delta, Sup Delta]

The underlying asset prices evolve according to the two-dimensional

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BlackScholes risk-neutral dynamics, that is:

dS1u = S

1u((r

d1)du + 1dB

1u), S

1Tt = s

1

dS2u = S2u((r d2)du + 2dB2u), S2Tt = s2.

where SjT denotes the spot at maturity T, sj is the initial spot and

(B1u, u

0) and (B2u, u

0) denote two real-valued Brownian motions

with instantaneous correlation .

Then we have:

S1T = s1 exp((r d1

21

2 )t)exp(11B1t )

S2T = s2 exp((r

d2

22

2

)t)exp(21B1t + 22B

2t )

where the parameters 11, 12, 21, 22 are given in the following matrix

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:

1,1 1,2

2,1 2,2 = 1 0

2 1 22 such that t = where is the covariance matrix expressed by:

21 12

1

2

22

The price of an option is

P = E

exp(rt)f(K, S1T, S2T)

where f denotes the payoff of the option. K denotes the strike. t time tomaturity.

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The Deltas are given by:

1 =s1 E[exp(

rt)f(K, S1T, S

2T)]

2 =s2 E[exp(rt)f(K, S1T, S2T)]


P =1N exp(

rt)

Ni=1 P(i)j = 1N exp(rt)Ni=1 sj P(i) = 1N exp(rt)Ni=1 j(i)

The values for P(i) and j(i) are detailed for each option.

Put on the Minimum: The payoff is (K min(S1, S2))+.

P(i) =`

Kmin(S1T(i), S

2T(i))

+

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IfP(i) 0 then:

1(i) =8


71/244

IfP(i) 0 then:

1(i) = 8


72/244

BestOf Option: The payoff is [max(S1 K1, S2 K2)]+.

P(i) = max(S1T K1, S

2T K2)

+

IfP(i) 0 then:

1(i) =

8Par[0].Val.V_PDOUBLE);

K2= (p->Par[1].Val.V_PDOUBLE);

}

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if ( (p->Compute) == &Exchange)

ratio= (p->Par[0].Val.V_PDOUBLE);

/* MC sampling */

init_mc= InitGenerator(generator, simulation_

dim,N);

/* Test after initialization for the generator*/

if(init_mc == OK)

{



for(i=1 ; i


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g_1= Gaussians[mc_or_qmc](simulation_dim, CR

EATE, 0, generator);

g_2= Gaussians[mc_or_qmc](simulation_dim, RE

TRIEVE, 1, generator);

exp_sigmaxwt1= exp(sigma11_sqrt*g_1);

S_T1= forward_stock1*exp_sigmaxwt1;

U_T1= forward1*exp_sigmaxwt1;

exp_sigmaxwt2= exp(sigma21_sqrt*g_1 + sigma22_sqrt*g_2);

S_T2= forward_stock2*exp_sigmaxwt2;

U_T2= forward2*exp_sigmaxwt2;

/*Price*/

price_sample=(p->Compute) (p->Par,S_T1, S_T2

);

/*Delta*/

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if ( price_sample > 0)

{

/*Call on on the Maximum*/

if ( p->Compute == &CallMax)

{

if (S_T1 > S_T2)

{

delta2_sample= 0.;

delta1_sample= U_T1;}

else

{

delta1_sample= 0.0;

delta2_sample= U_T2;

}

}

/*Put on on the Minimum*/

if ( (p->Compute) == &PutMin)

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{

if (S_T1 < S_T2)

{

delta2_sample= 0.;

delta1_sample= -U_T1;

}

else

{

delta1_sample= 0.0;delta2_sample= -U_T2;

}

}

/*Best of*/

if ( (p->Compute) == &BestOf)

{

if (S_T1- K1 > S_T2 - K2 )

{

delta2_sample= 0.;

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}

else

{

delta1_sample=0.0;


}

}

/*Exchange*/if ( (p->Compute) == &Exchange)

{


delta2_sample= -ratio*U_T2;

}

}

else

{

delta1_sample= 0.0;

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delta2_sample= 0.0;

}

/*Sum*/


mean_delta1+= delta1_sample;

mean_delta2+= delta2_sample;

/*Sum of squares*/var_price+= SQR(price_sample);

var_delta1+= SQR(delta1_sample);

var_delta2+= SQR(delta2_sample);

}


/* Price estimator */

*ptprice= exp(-r*t)*(mean_price/(double) N)

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;

*pterror_price= sqrt(exp(-2.0*r*t)*var_pri

ce/(double)N - SQR(*ptprice))/sqrt(N-1);

*inf_price= *ptprice - z_alpha*(*pterror_p

rice);

*sup_price= *ptprice + z_alpha*(*pterror_p

rice);

/* Delta1 estimator */

*ptdelta1= exp(-r*t)*mean_delta1/(double)

N;

*pterror_delta1= sqrt(exp(-2.0*r*t)*var_de

lta1/(double)N-SQR(*ptdelta1))/sqrt((double)N-1);

*inf_delta1= *ptdelta1 - z_alpha*(*pt

error_delta1);

*sup_delta1= *ptdelta1 + z_alpha*(*pt

error_delta1);

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*ptdelta2= exp(-r*t)*mean_delta2/(double)

N;*pterror_delta2= sqrt(exp(-2.0*r*t)*var_de

lta2/(double)N-SQR(*ptdelta2))/sqrt((double)N-1);

*inf_delta2= *ptdelta2 - z_alpha*(*pt

error_delta2);

*sup_delta2= *ptdelta2 + z_alpha*(*pt

error_delta2);

}

return init_mc;

}

Further Comments:/* Value to construct the confidence interval */

For example if the confidence value is equal to 95% then the value zused to construct the confidence interval is 1, 96.

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/*Initialization*//* Covariance Matrix */

/* Coefficients of the matrix such that t = *//*Median forward stock and delta values*/

Computation of intermediate values we use several times in the program.

/*MC sampling*/

/* Begin N iterations *//*Gaussian Random Variables*/Generation of 2 gaussian variables g1 and g2 used for the Brownian

motions

tgj .

/*Price*/

At the iteration i, we obtain

P(i) = payoff(K, S1T(i), S2T(i))

/*Delta*/

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Calculation of Delta 1(i) and 2(i) for the different cases with formula

given previously.

/*Call on the Maximum*//*Put on the Minimum*/

/*Best of*/

/*Exchange*//*Sum*/

Computation of the sumsP(i) and j(i) for the mean price and themeans delta.

/*Sum of squares*/

Computation of the sums

P(i)2 and

(j(i))2 necessary for the

variance price and the variances delta estimations.


/*Price*/

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The price estimator is:

P =

1

N exp(rt)N

i=1 P(i)The error estimator is P with :

2P =1

N

1 1

Nexp(2rt)

N

i=1 P(i)2 P2

The confidence interval is

ICP = [P zP; P + zP]


/*Delta*/


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The delta estimator is:

1 =1

N

exp(

rt)

N

i=1 1(i)The error estimator is 1 with:

21 =1

N 1

1

Nexp(2rt)

N

i=121(i) 21

The confidence interval is given as:

IC1 = [1 z1 ; 1 + z1 ]with z computed from the confidence value.

/* Delta2 estimator */The delta estimator is:

2 =1

Nexp(rt)

Ni=1

2(i)

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The error estimator is 2 with:

2

2 =

1

N 1 1N exp(2rt)N

i=1 22(i) 22The confidence interval is given as:

IC2 = [2 z2 ; 2 + z2 ]


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10 Simulation of processes

10.1 Brownian MotionDefinitions: A Brownian motion is a continuous adapted processB = Bt, Ft; 0 t < , defined on some probability space (, F , P ),with the properties that B0 = 0 a.s and for 0 s < t, the incrementBt Bs is independent ofFs and is normally distributed with mean zeroand variance t s.Simulation ofBt: Simulation ofBt is an easy step because we have thatL(Bt) = N(0, t)- first generate a gaussian standard variable g

- and then compute Bt as

tg

Simulation (discretization) of a Brownian trajectory, 0 t T: Wenow detail two approaches for simulating a Brownian path: the Forward

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one and the Backward one. Typically, for path-dependent options we have

to simulate B over = {tk; k = 0,...,M,t0 = 0, tM = T}.

Forward Simulation ofBt over is given by:

B(0) = 0

B(tk+1) = B(tk) +

tk+1 tkgkwhere (g1,...,gM) are independent gaussian standard variables. If we use

a discretization with evenly spaced intervals of size h = Tm , we have:

B(0) = 0

B(tk+1) = B(tk) +

hgk

Backward simulation with Brownian Bridge:

This other method is based on the following property for Brownian

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bridge:

L(Bu, s < u < t

|Bs = x, Bt = y) =

N( tutsx +

ust

s y,

(tu)(us)t

s )

and particularly

L(B t+s2

|Bs = x, Bt = y) = N(x+y2 , ts4 )

This scheme consists in simulating B as

B(0) = 0

B(T) =

T g1

B(T2 ) =B(0)+B(T)

2 +T4 g2

B( 3T4 ) = B(0)+B(T

2 )2 +T8 g3B(T4 ) =

B(T2 )+B(T)

2 +

T8 g4

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where (g1,...,gM) are independent gaussian standard variables.

For this algorithm, we have to choose M as a power of 2. The first step is

directly for 0 to T. Intermediates steps are filled by taking successive sub-divisions of the time intervals into halves. It can be adapted with

subdivisions of different length by considering the conditional law of

brownian bridge between s and t.

Remark on these two schemes for MC and QMC simulations: Weneed a vector of size M of independent gaussian variables. For MC, these

M variables can be simulated from the same pseudo random numbers

generator. However, for a QMC simulation we need to use a

M-dimensional low-discrepancy sequence to keep independence

property.10.2 BlackScholes model

In the Black and Scholes model [3], the underlying asset price St follows

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the diffusion:

dSt = Stdt + StdBt

and then the price is a geometric Brownian process:

St = S0 exp(( 22 )t + Bt)In this particular case for which we have an explicit solution of the

diffusion process, simulation of price paths is based on simulation of

Brownian motion described in the last section. As for Brownian pathsimulation, we present the forward and backward approaches.

Forward simulation: We have:

S(tk+1) = S(tk) exp((

2

2 )(tk+1

tk) + (B(tk+1)

B(tk)))

and for a discretization with evenly spaced intervals of size h, we simply

have:

S(tk+1) = S(tk) exp(( 22 )h +

hgk)

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Backward simulation: To construct this scheme, we use Backwardsimulation for Brownian path described int the previous point. To express

this scheme, we note yt = log(St), that is yt = y0 + (

2

2

)t + Bt

yT = y0 + ( 22 )t +

T g1

yT2

= y0+yT2 +

T4 g2

y 3T4

=y0+yT

2

2 +T8 g3

yT4

= yT2 +yT2 +T8 g4

We endly take Stk = exp(ytk)

10.3 General diffusions: Euler and Milshtein schemeWe consider the general diffusion process:

dXt = b(Xt)dt + (Xt)dBt

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If we dont have any explicit solution for Xt (like for Black and Scholes

model), we have to use approximation schemes with a discretization of

the process. The both most known schemes are Euler and Milshtein. They

both take into account a time discretization of step h.

The Euler approximation scheme for this diffusion is expressed as

Xtk+1 = Xtk + b(Xtk)h + (Xtk)(Btk+1 Btk)

Simulation is obtained with a forward algorithm by:

Xtk+1 = Xtk + b(Xtk)h + (Xtk)

hgk

for k = 0,...,M 1.

The Milshtein approximation scheme for this diffusion is given by

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(space dimension = 1)

Xtk+1 = Xtk + (b(Xtk)

1

2

(Xtk)(Xtk))h+

+(Xtk)(Btk+1 Btk) +1

2(Xtk)(Xtk)(Btk+1 Btk)2

Simulation is obtained with a forward algorithm by:

Xtk+1 = Xtk + (b(Xtk)

1

2

(Xtk)(Xtk))h+

+(Xtk)

hgk +1

2(Xtk)(Xtk)hg

2k

for k = 0,...,M 1.10.4 Heston model

The best known Stochastic Volatility model is the Heston model [16],which postulates, for the instantaneous variance vt, a CIR Bessel process(square root process) correlated with the driver ofS, namely, in

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risk-neutral form

dSt = St(rdt +

vtdWt)

dvt = (vt v)dt + vtdZt(1)

where- dW, Z = dt- is the speed of mean-reversion of the instantaneous variance

- v is the long-term variance mean- 2v

is the volatility of the instantaneous volatility VolOfVol.

The Heston model is complete if we have a vanilla option in thereplication portfolio.

If the Bessel process vt eventually reaches 0 (which is possible under

some sets of parameter values, and will typically be the case for values ofthe model parameters calibrated on realistic equity index skews [18]) thenthe embedded drift pushes it back to the positive side. So vt is alwaysnonnegative, as it should be. Mean reversion ofvt is intended to lead to

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realistic properties for the term structure of the implied volatility. Vanilla

option prices and Greeks are available in this model by Fourier transform.

A classic discretization for process 1 on page 87 consists in discretizingln(S) by the Euler scheme and v by the Milshtein scheme, as

ln(S) = (r v

2)h +

vh(G +

1 2G)

v = ((vt v) + 2

4 )h +

vhG + h2G2

4

where (G, G) is a standard Gaussian pair. The interest of using theMilshtein scheme for the v component is to benefit from the increased

rate of trajectorial convergence of the Milshtein scheme with respect to

the Euler scheme. So the Milshtein scheme is better suited to reproduce

the trajectorial properties ofvt (like nonnegativity).

10.5 Monte Carlo Simulation for Processes

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In the case of Monte Carlo simulation for processes, we have

E[(X)] 1

N

N

Xi=1

(xi) =

(E[(X)] E[(Xh)]) + (E[(Xh)] 1

N

NXi=1

(xhi ))

So the error is the sum of two terms: the discretization error and the

Monte Carlo error. For usual discretization schemes such as the Euler orthe Milshtein scheme, the weak convergence rate is linear in h so that the

discretization error is as O(h) the strong convergence rate of the

Milshtein scheme is quadratic in h, but this is irrelevant here. As for the

Monte Carlo error, after scaling by 1/

N, it is asymptotically distributed

asN(0, Var[(Xh)]). Therefore in order to balance the two terms in theerror a natural choice is to take N of the order ofM2.

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11 (Quasi) Monte Carlo methods for Exotic

Options

11.1 Lookback options

We consider an option with payoff(ST, mT) where (St, t 0) is thesolution of the following diffusion equation

dSt = b(St)dt + (St)dWt

and mt = sup0st Ss. The idea to (Q)MC estimate this option is toevaluate the maximum along the trajectory of the continuous Eulerscheme given by

mt = max0stSs

where for kh t (k + 1)hSt = Skh + b(Skh)(t kh) + (Skh)(Wt Wkh)

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i h W di i d b h d f h i i l i


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with Wt conditioned on both ends of each time interval (Brownian

Bridges). The following result gives a way of simulating the law ofmtconditionally on (Skh, 0

k

M).

Theorem: The law ofmk+1 = maxkht(k+1)h St conditionally on Skhand S(k+1)h can be simulated by

1

2 Skh + S(k+1)h +

(Skh S(k+1)h)2 22(Skh)h log(Uk)

where (Uk, k 0) is a sequence of iid uniform random variables on [0, 1].Proof: Let mt = sup0stWs. Il is well known [21] that the law of thepair (Wt, mt) has density

p(x, y) = 1y01yx2(2y x)

2t3exp

(2y x)2

2t

.

Denoting Wt = Wt + bt, one can show by the CameronMartin formula

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that the law of the pair (Wt, sup0stWs) has density

p(x, y) = p(x, y) exp(bx

b2

2t).

So the law ofsup0stWs conditional on Wt = x does not depend on b.By change of variables z = (2y x)2 x2, one then shows that thevariable

(2 sup0st

Ws

Wt)

2

W

2

t

is, conditionally on Wt, 12t -exponentially distributed.Therefore, conditionally on Snh and S(k+1)h

supkht(k+1)h

St(law)

=1

2

Skh + S(k+1)h +q

(S(k+1)h Skh)2 + (Skh)2Y

where Y is an 12h -exponential variable. 2

So each simulation run consists in generating by the Euler scheme atrajectory Skh, 1 k M and a trajectory mkh, 1 k M, by using

S. Crpey Page 93


2M d d N h if QMC d h ld


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2M random draws. Note that if QMC was used, one should use a

2M-dimensional low-discrepancy sequence. Except for special cases such

as BlackScholes (see below), this would lead to use high-dimensional

low-discrepancy sequence, which is not recommended in general.

In the case of the BlackScholes model, the Euler discretization is exact,

provided one works in returns variable y = ln(S). So one can take M

equal to one and use a 2-dimensional low-discrepancy sequence.

11.2 Andersen and Brotherton-Ratcliffe Algorithm forLookback Options

Description : Computation, for a Lookback Option on Maximum, ofits Price and its Delta with the Standard Monte Carlo Simulation [1]. This

method also provides an estimation for the integration error and a

confidence interval.

Input parameters:

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N b f it ti N
http://slidesfinumscrepey.pdf/http://slidesfinumscrepey.pdf/


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Number of iterations N Generator_Type Confidence Value

Output parameters:

Price P

Error Price P Delta Error delta

Confidence Interval: [Inf Price, Sup Price]

Confidence Interval: [Inf Delta, Sup Delta]The underlying asset price evolves according to the Black and Scholes

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d l th t i


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model, that is:

dSu = Su((r d)du + dBu), STt = sthen

ST = s exp

(r d

2

2)t

exp(Bt)

ST denotes the spot at maturity T, s is the initial spot, t the time to

maturity.

We note mT = max[Tt,T](Su) the maximum reached before maturity.The Price of an option is:

P = E[exp(rt)f(K, ST, mT)]

where f denotes the payoff of the option and K the strike.The Delta is given by:

=

sE[exp(rt)f(K, ST, mT)]

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Th ti t d


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P = 1N exp(rt)

Ni=1 P(i) = 1N exp(rt)Ni=1 sP(i) = 1N exp(rt)Ni=1 (i)

The values for P(i) and (i) are detailed for each option.

Call Fixed Euro: The payoff is (max[Tt,T](Su) K)+

P(i) = (mT(i) K)+

i =

mT(i)s =

mT(i)s if P(i) 0

0 otherwise

Put Floating Euro: The payoff is (max[T

t,T](Su)

ST)

P(i) = (mT(i) ST(i))

i =mT(i)

s ST(i)

s=

mT(i) ST(i)s

=P(i)

s

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Simulation of the maximum : The conditional la ind ced b


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Simulation of the maximum mT: The conditional law induced by mT,given the observation ST, has a very simple form, so that we can easilysimulate the maximum ofS over the time interval [t, T].

Setting Wu = ln Su, then obviously mT is the exponential of themaximum ofW, and the conditional probability distribution function ofthe maximum ofW, given WTt = w1 and WT = w2, is:

Gmax(w; w1, w2) =1 exp 22t (w w1)(w w2) if w max(w1, w2)0 otherwise

In the Monte Carlo algorithm, the associated inverse function G1max is

used in order to simulate values for mT. We have the followingexpression:

G1max(y; w1, w2) =1

2

w1 + w2 +

(w1 w2)2 22t ln(1 y)

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h i if [0 1]


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where y is uniform on [0, 1].

Then, at step i, mT(i) is simulated as follows:

- ST(i) is generated as s exp(r d 22 )t exp(Bt(i)) withBt(i) =

tgi and gi is a standard gaussian variable;

- y(i) is generated as a uniform variable on [0, 1];- wTt = ln s and wT(i) = ln ST(i);

- WT(i) is computed as G1

max(y(i); wTt, wT(i));- Finally, mT(i) = exp(WT(i)).

Code Sample

static double inverse_max(double s1, double s2,

double h,double sigma,double un)

{

return ((s1+s2)+sqrt(SQR(s1-s2)-2*SQR(sigma)*h*

log(1.-un)))/2.;

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}


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}

static int LookBackSup_AndersenMontecarlo(...)

{

...

/* Monte Carlo sampling */

init_mc= InitGenerator(generator, simulation_

dim,N);/* Test after initialization for the generator

*/

if(init_mc == OK)

{

/* Initialization of the model just allows

to use Monte Carlo method */


for(i=1; i


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{

/* Simulation of a gaussian variable accor

ding to the generator type */

gs= Gaussians[mc_or_qmc](1, CREATE, 0, gen

erator);

/* Simulation of the maximum */

un= Uniform(generator);exp_sigmaxwt= exp(sigma_sqrt*gs);

S_T= forward_stock*exp_sigmaxwt;

max_log_norm=inverse_max(log_s, log(S_T), t,

sigma, un);

S_max= exp(max_log_norm);

/* Price and Delta */

/* CallFixedEuro */

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if (p->Compute == &Call OverSpot2)


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if (p >Compute == &Call_OverSpot2)

{

price_sample= (p->Compute)(p->Par, strik

e, S_max);

delta_sample= 0;

if(log_strike>max_log_norm)

delta_sample=0.;

else delta_sample=S_max/s;

}else

/* PutFloatingEuro */

if (p->Compute == &Put_StrikeSpot2)

{

price_sample= (p->Compute)(p->Par, S_T, S_

max);

delta_sample=S_T/s;

}

S. Crpey Page 102


/*Sum*/


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/ Sum /


mean_delta+= delta_sample;

/*Sum of squares*/

var_price+= SQR(price_sample);

var_delta+= SQR(delta_sample);

}

/* End N iterations */...

}

return init_mc;

}

Further Comments

/* Test after initialization for the generator */Test if the dimension of the simulation is compatible with the selected

S. Crpey Page 103


generator In this implementation initialization of the model just allows


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generator. In this implementation, initialization of the model just allowsto use Monte Carlo method. Note that if QMC was used, one should use a

2D low-discrepancy sequence.

/* Begin N iterations *//* Simulation of a gaussian variable according to the generator type,

that is Monte Carlo or Quasi Monte Carlo. */Call to the appropriate function to generate a standard gaussian variable.

/* Simulation of the maximum */Computation ofST(s) and max[Tt,T] Su(s) from the initial spot value s.

The maximum value is simulated according to the conditional lawinduced by s and ST.


11.3 Barrier options

Barrier options are a special case of lookback options in which the

S. Crpey Page 104


quantity


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quantity

E

f(ST, mT)|Skh, 0 k M

can be computed explicitely.A barrier option is defined by the following payoff

f(ST)1{mTL}.

An approximation of the option price is given by

E

f(ST)1{mTL}

where

mT = max0

k

M

mk with mk = supkh

t

(k+1)h

St

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Then


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Then

E

f(ST)1{mTL}|Slh, 0 l M

= f(ST)M1k=0

E(1{mkL}|Slh, 0 l M)

= f(ST)

M1

k=0E(1{mkL}|Skh, S(k+1)h)

= f(ST)

M1k=0

h(Skh, S(k+1)h)

where

h(x, y) = 1 exp( 2(x)2h (L x)(L y))1L>xy

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So


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So

Ef(ST)1{mTL} = Ef(ST)M1

k=0 h(Skh, S(k+1)h) .Note that, in this case, no random draws are needed other than those usedfor simulating ST, namely M random draws by simulation run whereM can be taken equal to one, in the special case of the BlackScholesmodel.

11.4 Asian options

Asian options have payoffs of the following kind

ST, 1T T

0 Su du .A standard example is given by (x, y) = (y x)+. PuttingAt =

t0

Su du, the pair (St, At) is Markovian.

S. Crpey Page 107


Straightforward application of the Euler scheme suggests to approximate


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Straightforward application of the Euler scheme suggests to approximate

AT by AhT = h(S0 + S1 + . . . + S(M1)h), which corresponds to aRiemann sum for the integral AT. Nevertheless, this discretization works

poorly in practice. There do exist better discretizations of the integral, inparticular in the BlackScholes model, by using the law of the Brownian

Bridge between 0 and h:

Wu =u

h

Wh + Zu

where Zu is a centered Gaussian variable with variance uh (h u). Thesediscretizations can be used in association with appropriate variance

reduction techniques [28]. For instance, in the BlackScholes model and

for (x, y) = (y K)+, an idea by Kemna and Vorst [22] is thefollowing one: ifr and are small, then

1

T

T0

St dt is close to exp

1

T

T0

log(St) dt

.

S. Crpey Page 108


This suggests to choose


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This suggests to choose

erT(exp(Z) K)+

where Z = 1TT0 log(St) dt, as a control variable. Z is Gaussian, so one

knows EerT(exp(Z) K)+ explicitely.

S. Crpey Page 109


12 Trees for vanilla options


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12 Trees for vanilla options

12.1 Cox-Ross-Rubinstein as an approximation toBlack-Scholes

The Cox-Ross-Rubinstein model is the Markov chain obtained by

replacing the Black-Scholes dynamics under the risk-neutral probability

by the CRR dynamics

Sn+1 (h) =

u Sn (h) with proba p

d Sn (h) with proba 1 p

with h = TN

, where T is the time to maturity of an option (ie the current

date is taken to be zero), N is the number of time steps in the tree, and

u = eh, d = e

h, p =

eh du d

S. Crpey Page 110


with r . Note that p is the risk-neutral probability in the


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w p p y

generalized CRR scheme.

Convergence of the marginal law at maturityWithout loss of generality assume S0 = 1. For R

EP

[exp(i ln SN (h))]

= EP

exp

i lnN1n=0

Sn+1 (h)

Sn (h)

=

EP

exp

i ln

S1 (h)

S0

N= p expih+ (1 p)expihN

S. Crpey Page 111


h d 1

22


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and since p = ehdud 12 +

2

2

h + O (h)

EP

[exp(i ln SN (h))] 1 + i 22 2 22 TNN

exp

i

2

2

2

2

2

T

= Eexpi

2

2 T + BT= EP

BS [exp(i ln ST)]

where PBS is the risk-neutral Black-Scholes probability. Therefore underthe risk neutral measure

SN (h) SBST (2)in law as N .This grants the convergence of the price of standard european options

S. Crpey Page 112


with payoffs continuous and bounded, e.g. Put options. The convergence


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p y , g p g

of the Call prices follows by Call-Put parity, since the CRR scheme

satisfies the Call-Put parity relationship.

Some remarks about the limit N The limiting law depends only on pei ln(u) + (1 p) ei ln(d) throughits Taylor expansion up to o (h) . Thus u, d or/and p could be altered aslong as the involved terms of the development are not modified.

The upper and lower value of the spot at maturity are uN = eTN and

dN = eTN whereas the ratio of 2 successive points is

ud = e

2h = e2

TN . Thus at the same time, the scan of the law of

SN (h) goes to R+ and the grid gets more and more dense. It is easy to

show that the points visited by the process S(h) will eventually becomedense in [0, t] R+.

S. Crpey Page 113


Convergence of the delta


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g

Observe that

0S0 = Pu,N1 Pd,N1(u d)

= er(N1)hEP [ (uS0X)] EP [ (dS0X)]

u dwhere X is a random variable independent from S0 so that

0S0 = er(N1)hEP

(uS0X) (dS0X)

u d

Assume now that is a C1 function. Then

(uS0X) (dS0X) = ud

S0X (aS0X) da

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so


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0S0 = er(N1)h 1

(u

d)

u

d

EP [S0X (aS0X)] da

= er(N1)hEP [S0X (a (h) S0X)]

for some point a (h) in [d, u] by the mean value property. Assuming nowthat (y) y (y) is a Lipschitz function, we know that so is the CRRprice associated to the payoff, with the same Lipschitz constant which

doesnt depend on h. Therefore since [d, u] [1, 1] as h 0limh0

EP [S0X (a (h) S0X)] = lim

h0EP [ (S0X)] (3)

Now by the standard convergence result, assuming that y(y) is bounded

limh0 EP

[ (S0X)] = EPBS

[ (St)]

= erTS0BS0

Therefore the delta does converge towards the Black-Scholes delta, at

S. Crpey Page 115


least for sufficiently smooth payoffs. For a Call (or Put) option the


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argument to get ( 3) must be slightly modified, but it works.

12.2 Algorithm (CRR)Description: This is the archetype of a Tree routine. It is described in[9]. The dynamics of the underlying under the risk-neutral probability inthe BS1D model:

St

= S0

e(r)t+Wt

is replaced by the following: for i = 0, 1, . . . , N 1S(i+1) = Sie

h

where h = TN, T is the maturity of the option, = 1 with probability p,

= 1 with probability 1 p where the (conditional) risk-neutralprobability p is chosen so that

E

e(r)hS(i+1)|Si

= Si (4)

S. Crpey Page 116


where is the dividend rate.


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Input parameters:

StepNumber NOutput parameters:

Price Delta

The price of the european option P satisfies

E

erhP(i+1)|Pi

= Pi (5)

whereas the price of the american option is given in any state j at time iby

Pi = max , EerhPi+1|Pi (6)where is the payoff of the option.The algorithm is a backward computation of the option price, based on

S. Crpey Page 117


the Dynamic Programming equations ( 5) or ( 6), after a forward


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computation of the possible values of the underlying at maturity SjN for

j = 0, 1, . . . , N . Since the tree is a flat tree, it is easily seen that the value

of the underlying at time i and level j (starting from below) is the same asthat at time i + 2 and level j + 1. In particular there are only 2N + 1

possible values of the underlying between time 0 and time N. For

computational purpose it is clever in the american case to compute only

once the corresponding value of the intrinsic value of the option, at the

beginning of the algorithm.

Code Sample:

static int CoxRossRubinstein_79(...)

{

...

/*Price, intrisic value arrays*/

P=(double *)malloc((N+1)*sizeof(double));

S. Crpey Page 118


iv=(double *)malloc((2*N+1)*sizeof(double)


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/*Up and Down factors*/

h=t/(double)N;a1= exp(h*(r-divid));

u = exp(sigma*sqrt(h));

d= 1./u;

/*Risk-Neutral Probability*/

pu=(a1-d)/(u-d);

pd=1.-pu;

if ((pd>=1.) || (pd


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for (i=0;iPar,stock);

stock*=d;

}

/*Terminal Values*/

for (j=0;j


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P[j]=pu*P[j]+pd*P[j+1];

if (am)

P[j]=MAX(iv[i+2*j],P[j]);}

/*Delta*/

*ptdelta=(P[0]-P[1])/(s*u-s*d);

/*First time step*/

P[0]=pu*P[0]+pd*P[1];

if (am)

P[0]=mAX(iv[N],P[0]);

/*Price*/

*ptprice=P[0];

...

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return OK;


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}

Further Comments:

/*Up and Down factors*/Here u = e

h,d = e

h.

/*Risk-Neutral Probability*/

Computation ofp = e(r)hdud

(the value computed from ( 4 on

page 116)).

/*Intrinsic Value computation*/Storage of the 2N + 1 possible values of the intrinsic value.

/*Price initialization*/The price of the option at maturity. It involves only the values iv [2 j].

/*Backward Resolution*/

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Note that we dont re-compute the intrinsic value.


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/*Delta*/The delta here is the right hedging delta in the binomial model, namely

n =Pn+1 (uSn) Pn+1 (dSn)

uSn dSn .

There may be a more clever way to approximate the continuous-time

Black&Scholes delta./*First time step*/

/*Price*/

12.3 Variants of the CRR tree

To achieve the convergence in law ( 2 on page 112), many other choicesofu and d and q (denoting the probability inside the tree) may be done,

S. Crpey Page 123


regardless of any arbitrage or financial consideration: the tree algorithm


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becomes a numerical approximation algorithm among other ones, the only

purpose is to get a good convergence to the limiting price and delta [26]

All the following trees will remain recombining trees since this is true as

soon as u and d remain constant within the tree. Only the choice ofu, d

and the probability is at hand here.

The Random Walk scheme As long as

ST = S0 exp 22 T + BT a very natural choice is toapproximate the Brownian motion B by the standard Random Walk. This

leads to

u = e

22

h+

h

, d = e

22

hh

and q = 12 .

The algorithm to get the option price is the straightforward discretized

S. Crpey Page 124


version of the risk-neutral expectation:


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Pn = erh

1

2Pu,n+1 +

1

2Pd,n+1

The convergence may be proved in the same way as before.

Notice that the discretized process is not a martingale.

The matching-3-moments scheme

An alternative route to convergence is the Central Limit Theorem. Thisleads to the idea of matching the mean and variance of the conditional

laws of the approximating chain with those of the continuous process.

These are denoted the local consistency conditions [24, 25]. The

equations that u,d,q should satisfy are

qu + (1 q) d = eh

qu2 + (1 q) d2 e2h = e2h

e2h 1

S. Crpey Page 125


Since one degree of freedom remains, a natural idea is to match also the


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third moment, which gives the equation

qu3

+ (1 q) d3

= e3h

e32h

The solution of this system is

u =ehQ

2 1 + Q +

Q2 + 2Q 3

d = ehQ21 + Q Q2 + 2Q 3

q =eh d

u dwith Q = e

2h.

Notice that ud = e2hQ2 > 1 : this tree is not symmetric.

12.4 Trinomial trees

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Along this line there is no need any longer to remain stucked with the


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discrete-time no-arbitrage constraint one node-two sons. We may wellchoose a 3-points scheme or p-points scheme or even a number of points

depending on N (this is useful for other kinds of limiting continuous-timedynamics, like Lvy processes for instance [8]). From the previouscalculation its easy to see that the points and probabilities of the chosenscheme should be constrained by:

pj exp(i ln uj) = 1 +

i

22

2 22

h + o (h)

in the sense that these conditions ensure the convergence ( 2 onpage 112). Well see later that these conditions are

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Numerical Methods Book Crepey