simulation in finance

8/9/2019 Simulation in Finance

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Ben Van Vliet

October 12, 2010

Financial ApplicationsSimulation

with


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2010 Ben Van Vliet 2

I. VBA for EXCEL..................................................... .................................................... 3A. CODE DEFINITIONS....................... ............................................................. ............ 4B. RECORDING A MACRO....................................... ................................................... 8II. INTRODUCTION TO SIMULATION .......................................................... .......... 13

III. CONTINUOUS DISTIRBUTIONS ............................................... ...................... 18A. UNIFORM DISTRIBUTION........................................................ ........................... 15B. INVERSE TRANSFORM METHOD ................................................. ..................... 18C. EXPONENTIAL DISTRIBUTION........................ .................................................. 19D. WEIBULL DISTRIBUTION............................................................................. ....... 21E. TRIANGULAR DISTRIBUTION................................................................... ......... 22F. NORMAL DISTRIBUTION ..................................................... ............................... 24G. LOGNORMAL DISTRIBUTION................................................................ ............ 28H. GENERALIZED INVERSE TRANSFORM METHOD ......................................... 29IV. DISCRETE DISTRIBUTIONS .................................................. .......................... 30A. BERNOULLI TRIALS...................................................... ....................................... 30

B. BINOMIAL DISTRIBUTION....................................................................... ........... 31C. TRINOMIAL DISTRIBUTION..................................................... .......................... 32D. POISSON DISTRIBUTION................................................................... .................. 33E. EMPIRICAL DISTRIBUTIONS..................... ......................................................... 34F. LINEAR INTERPOLATION .......................................................... ......................... 35V. GENERATING CORRELATED RANDOM NUMBERS ...................................... 35A. BIVARIATE NORMAL............................ ............................................................. .. 36B. MULTIVARIATE NORMAL.................................................. ................................ 37VI. MODELING REAL TIME FINANCIAL DATA ................................................ 39A. FINANCIAL MARKET DATA................................................ ............................... 42B. MODELING TICK DATA.......... ....................................................... ...................... 44C. MODELING TIME SERIES DATA.................................................. ...................... 511. FAT TAILS.................................. ............................................................. ................ 532. STOCHASTIC VOLATILITY MODELS ....................................................... ........ 54a. ARCH(1)..................... ....................................................... ....................................... 54

b. GARCH(1,1)..................................................... ........................................................ 54VII. MODELING OPTIONS .................................................... ................................... 56A. THE SIMULATION WAY ........................................................ .............................. 56B. THE COX-ROSS-RUBENSTEIN or BINOMIAL WAY........................................ 57C. THE BLACK-SCHOLES WAY...................................................................... ......... 58PROJECT I .................................................. ........................................................ ............. 59PROJECT II ................................................... ....................................................... ............ 60PROJECT III.......................................... ........................................................ ................... 61PROJECT IV ................................................. ....................................................... ............ 62


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I. VBA for EXCEL

We use the Visual Basic Editor create and modify Visual Basic for Applications (VBA) procedures and modules in our Excel applications. Every Excel file can contain a VBA project to which you can add modules and forms. Here is a brief overview of the VBA

Editor environment.

Project Explorer: Displays the list of projects and project items in thedocument.

Properties Window: Displays the list of the properties for the selected items

Code Window: In the code window you can write, display and edit thecode in new procedures or in existing event procedures.You can open any number of code windows on differentmodules, class modules, user forms, so that the code iseasily visible (and cut, copy and paste can be done).

The Side Bar: This is used to place the breakpoints when you debug the program.


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Object Browser: This provides quick access to list the commands, functionsthat are in the development environment.

A. SYNTAX DEFINITIONS

Now here are some definitions of a few terms:

Procedure: A procedure is a block of code enclosed in Sub and End Sub statements orin Function and End Function statements. The difference between a Suband a Function is that a Sub procedure (or Sub-routine) has no returnvalue. Both Sub routines and Functions may have input parameters withtheir types defined in the parentheses. A Function must also declare areturn type after the parentheses.

Sub MyPr ocedur e( a As Doubl e )

End Sub

Funct i on MyPr ocedur e( a As Doubl e ) As Doubl e

The r et ur n val ue i s set as f ol l ows:MyPr ocedur e = a + 5

End Funct i on

Module: A module is a code window where we write procedures.

Macro: An Excel macro is a Sub routine that contains instructions. This Subroutine can be triggered to run via button in your spreadsheet. We usemacros to eliminate the need to repeat steps of common performed tasks.

Variables: Variables are physical memory locations inside the computer. VBA hasseveral variable types to hold different kinds of data. The most commonlyused are: Boolean (true/false), Char (character), Date (Date and Time),Double (floating point number), Integer , Object (any type), String (series of characters), Variant (any type).

What about converting from one type to another?

To convert the data type from one to another type conversions are used. VBAs built-infunctions:

Function DescriptionCDbl Convert to Double


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Cint Convert to IntegerCStr Convert to StringCLng Convert to LongCDate Convert to DateChr Convert value to ascii character

Asc Convert from ascii to value

Here is some simple code to show type conversions:

Opt i on Expl i ci t

Sub t est ( )

Di m a as I nt egerDi m b as St r i ngDi m c as Doubl e

a = 5b = Hel l oc = 3. 14159

Di m d as St r i ngd = CSt r ( a )

Di m e as Doubl eE = CDbl ( a )

End Sub

Constant:

A constant is a variable whose value cannot change. A Constant is declared using thekeyword Const instead of Dim.

Const Exchange As St r i ng = CMEConst St ar t Dat e As Dat e = 05/ 05/ 2010

When declaring a constant, you must assign its value.

What is the scope and lifetime of variables and constants?

As soon as a variable or constant goes out of scope it can no longer be accessed and itloses its value. Variables and constants can be given different scopes based upon how wedeclare them.

1. Procedure-level scope . Variables are declared using Dim or Const inside a procedure.


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Publ i c Sub MyPr ocedur e( )

Di m a As I nt egera = 7

End Sub

When the procedure is done running, the variable or constant goes out of scopedand is destroyed.

2. Module-level scope . Variables that are declared above the Procedure definitionsstay within scope after the procedure is done running.

Di m a As I nt eger


a = 7

End Sub

All variables with this level of scope are available to all procedures that are withinthe module.

3. Project-Level, Workbook-Level, or Public Module-Level. These variables are

declared at the top of any standard public module and are available to all procedures in all modules.

If...Then...Else:

Often we need to test for equality or inequality. We do this with an If...Then statement.The general syntax is this: if the condition to validate is true, then execute some code.


Di m a as I nt eger = 2

Di m b as I nt eger = 3I f a < b Then

MsgBox( Tr ue )End I f

End Sub


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If need to add lines of code in the case where the expression evaluates to false, we useIf...Then...Else. This syntax is:

I f a > b ThenMsgBox( Tr ue )

El se MsgBox( Fal se )End i f

Select...Case:

If multiple evaluation conditions exist, we can use a Select...Case statement. You canthink of a SelectCase statement as a series of if statements. The syntax is:

Sel ect Case a

Case I s < 0 Case 1 To 5

Case I s > 5

Case El se

End Sel ect

DoLoop, While and Until:

For repeated execution of a block of code, we can use one of several repletion or iterationstructures. The general syntax is of a Do While loop is:

Do Whi l e a < 10a = a + 1

Loop

This line of code inside the Do While loop will executed repetitively until the conditionevaluates to false. The syntax of the DoLoop While is:

Doa = a + 1

Loop Whi l e a < 10


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ForNext:

The most commonly used repetition structure is the For loop. The syntax for a For loopis:

For a = 1 t o 10 St ep 1

Next a

In this case, the Step 1 is optional because the For loop will by default increment a by 1each time through the loop. However, any other incrementation would require the Stepclause.

For a = 50 t o 0 St ep - 2

Next a

B. RECORDING A MACRO

Here is an example showing how to record a VBA macro to calculate the average of fivenumbers.

There are several ways of doing the above problem.

1. General Way . Computing the sum of the five numbers and dividing by the totalnumber of numbers.


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2. Use the built-in Excel function for Average . Get the list of built-in Excelfunctions and select AVERAGE function.

3. Write or Record a simple macro.


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Click OK, and then do the calculations your way (either by Step a/b fromabove). And Click on the STOP button as shown.

This way you have a recorded macro, the code generated in this process isgenerated in the Visual Basic Editor.


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You can open the Visual Basic Editor by either pressing Alt + F11 key, or by goingthrough the following steps shown here:

The following is the Visual Basic Editor which has the generated macro.


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II. INTRODUCTION TO SIMULATION

A model is a representation of reality. Traditionally, models are mathematical equations,which are attempts at analytical or closed form, solutions to problems of representation.

All models are wrong. Some models are useful. -Edwards Deming

These equations enable estimation or prediction of the future behavior of the systemfrom a set of input parameters, or initial conditions. However, many problems are toocomplex for closed form equations.

Simulation methods are used when it is unfeasible or impossible to develop aclosed form model. Simulation as a field of study is a set of algorithms that depend uponthe iterative generation of random numbers to build a distribution of probable outcomes.Because of their reliance on iteration, sometimes millions of them, simulation isaccomplished only through the use of computer programs.

Simulation is especially useful when applied to problems with a large number of

input distributions, or when considerable uncertainty exists about the value of inputs.Simulation is a widely-used method in financial risk analysis, and is especially successfulwhen compared with closed-form models which produce single-point estimates or humanintuition. Where simulation has been applied in finance, it is usually referred to as MonteCarlo simulation.

Monte Carlo methods in finance are often used to calculate the value ofcompanies, to evaluate investments in projects, to evaluate financial derivatives, or tounderstand portfolio sensitivities to uncertain, external processes such as market risk,interest rate risk, and credit risk. Monte Carlo methods are used to value and analyzecomplex portfolios by simulating the various sources of market uncertainty that mayaffect the values of instruments in the portfolio.

Monte Carlo methods used in these cases allow the construction of probabilisticmodels, by enhancing the treatment of risk or uncertainty inherent in the inputs. Whenvarious combinations of each uncertain input are chosen, the results are a distribution ofthousands, maybe millions, of what-if scenarios. In this way Monte Carlo simulationconsiders random sampling of probability distribution functions as model inputs to

produce probable outcomes.Central to the concept of simulation is the generation of random numbers. A

random number generator is a computer algorithm designed to generate a sequence ofnumbers that lack any apparent pattern. A series of numbers is said to be (sufficiently)random if it is statistically indistinguishable from random, even if the series was created

by a deterministic algorithm, such as a computer program. The first tests for randomnesswere published by Kendall and Smith in the Journal of the Royal Statistical Society in1938.

These frequency tests are built on the Pearson's chi-squared test, in order to testthe hypothesis that experimental data corresponded with its theoretical probabilities.Kendall and Smith's null hypotheses were that each outcome had an equal probability andthen from that other patterns in random data would also be likely to occur according toderived probabilities.


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For example, a serial test compares the outcome that one random number isfollowed by another with the hypothetical probabilities which predict independence. Aruns test compares how often sequences of numbers occur, say five 1s in a row. A gaptest compares the distances between occurrences of an outcome. If a data sequence isable to pass all of these tests, then it is said to be random.

As generation of random numbers became of more interest, more sophisticatedtests have been developed. Some tests plot random numbers on a graph, where hidden patterns can be visible. Some of these new tests are: the monobit test which is afrequency test; the WaldWolfowitz test; the information entropy test; the autocorrelationtest; the K-S test; and, Maurer's universal statistical test.


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A. UNIFORM DISTRIBUTION

Parameters a and b, the lower and upper bounds.

Probability density:

ab x f = 1)(

Cumulative distribution function F(x):

aba x

x F =)(

Expected value of x:

2)(

ba x E

+=

Variance of x:

12

)()(

2ab xV

=

The Linear Congruential Generator (LCG) will generate uniformly distributed integersover the interval 0 to m - 1:

))(mod( 1 k d cuu ii += The generator is defined by the recurrence relation, where ui is the sequence of

pseudorandom values, and 0 < m, the modulus, 0 < c < k, the multiplier, and 0 < d < m,the increment. u0 is called the seed value.

Publ i c Funct i on LCG( c As Doubl e, d As Doubl e, k As Doubl e, _

u0 As Doubl e ) As Doubl eLCG = ( c * u0 + d) Mod kEnd Funct i on

A B C D1 c 6578 =LCG(B1,B2,B3,B4) =C1/($B$3-1)2 d 1159 =LCG($B$1,$B$2,$B$3,C1) =C2/($B$3-1)3 k 7825 4 u0 5684

TABLE 1

EXCEL: =MOD( c * u0 + d, k )

The real problem is to generate uniformly distributed random numbers over the interval 0to 1, what we call the standard uniform distribution, where the parameters a = 0 and b =1. A standard uniform random number, u s, can be accomplished by dividing the LCGrandom integer by k 1 as in Table 1. However, Excel and VBA already have functionsthat return standard uniform random numbers:


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EXCEL: =RAND()

VBA: =Rnd()

Publ i c Funct i on Uni f or mRand( ) As Doubl eUni f or mRand = Rnd( )

End Funct i on

Generating Uniformly Distributed Random Numbers:

Sub Gener at e( )Di m i as I nt egerFor i = 0 To Range( " A1" ) . Val ue

Range( "A2") . Of f set ( i ) . Val ue = Rnd( )Next i

End Sub

In any case, the state of the art in uniform random number generation is the MersenneTwister algorithm. Most statistical packages, including MatLab, use this algorithm forsimulation.

Turning a standard uniform random number, u s, into a uniformly distributed randomnumber, u, over the interval a to b.

EXCEL: = a + RAND() * ( b - a )

VBA:

Publ i c Funct i on Uni f or m( a As Doubl e, b As Doubl e ) As Doubl eUni f or m = a + Rnd( ) * ( b a )

End Funct i on

Generating Uniformly Distributed Random Integers:

Turning a standard uniform random number, u s, into a uniformly distributed randominteger of the interval a to b:

EXCEL: = FLOOR( a + RAND() * ( b a + 1 ), 1 )

VBA:

Publ i c Funct i on Uni f or m( a As Doubl e, b As Doubl e ) As Doubl eUni f orm = I nt ( a + Rnd( ) * ( b - a + 1) )

End Funct i on


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III. CONTINUOUS DISTIRBUTIONS

A. INVERSE TRANSFORM METHOD

The inverse transform method generates random numbers from any probability

distribution given its cumulative distribution function (cdf). Assuming the distribution iscontinuous, and that its probability density is actually integratable, the inverse transformmethod is generally computationally efficient.

The inverse transform methods states that if f(x) is a continuous function withcumulative distribution function F(x) , then F(x) has a uniform distribution over theinterval a to b. The inverse transform is just the inverse of the cdf evaluated at u:

)(1 u F x =

The inverse transform method works as follows:1. Generate a random number from the standard uniform distribution, u s.

2. Compute the value x such that F ( x) = u. That is, solve for x so that F -1

(u) = x. 3. x is random number drawn from the distribution f .


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C. EXPONENTIAL DISTRIBUTION

Parameter , the scale parameter. The exponential distribution arises when describing theinter-arrival times in a (discrete) Poisson process.


xe x f = 1)(

Derivation of the cumulative distribution function F(x):

= x

dx x f x F 0

)()(

dxe x F x x

=

0

1)(

dxe x F x x

=

0

1)(

0)(

xe x F x =

0)( += ee x F x xe x F = 1)(

Expected value of x: =)( x E

Variance of x:2)( = xV

To generate a random number from an exponential distribution:)( x F u s =

So that:)(1 su F x

= Solve for x:

x s eu

=1 x

s eu =1

xu s = )1ln(

)1ln( su x =

Notice that if u s is a uniformly distributed random number between 0 and 1, then 1 u s isalso a uniformly distributed random number between 0 and 1. Thus,

)ln( su x = is equivalent to the prior solution.


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EXCEL: = -$A$4 * LN( 1 - RAND() )

VBA:

Funct i on Random_Exp( bet a As Doubl e ) As Doubl eRandom_Exp = - bet a * Log( 1 - Rnd( ) )

End Funct i on


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D. WEIBULL DISTRIBUTION

Parameters and , the location and scale, where x 0.


)(1)( xe x x f =


)(1)( xe x F = Expected value of x:

=

1)( x E

Variance of x:

)1()21()( 121 ++= aa xV

To generate a random number from a Weibull distribution:

)( x F u s = So that:

)(1 su F x =

Solve for x: 1))1ln(( su x =

EXCEL: = $B$2 * ( -LN( 1 - RAND() ) ) ^ ( 1 / $B$1 )

VBA:

Funct i on Random_Wei bul l ( al pha As Doubl e, bet a As Doubl e ) As Doubl eRandom_Wei bul l = bet a * ( - Log( 1 - Rnd( ) ) ) ^ ( 1 / al pha )

End Funct i on


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E. TRIANGULAR DISTRIBUTION

Parameters a, b, and m, the lower and upper bounds and the mode or most likely value, sothat a m b.


=b xmif

mbab xb

m xaif amab

a x

x f

))(()(2

))(()(2

)(


=b xmif

mbab

xb

m xaif amab

a x

x F

))((

)(1

))(()(

)( 2

2

Expected value of x:

3)(

mba x E

++=

Variance of x:

18)(

222 bmamabmba xV

++=

To generate a random number from a triangular distribution:)( x F u =

So that:)(1 u F x =

Solve for x s is standard triangular, where a = 0 , b = 1 , and where:

abam

m s =

And, therefore:

>

= s s s s

s s s s s muif um

muif um x

)1)(1(1

So that x is triangular( a , b, m ):)( ab xa x s +=

EXCEL:

VBA:


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Funct i on STri angul ar( m As Doubl e ) As Doubl e

Di m us As Doubl eus = Rnd( )

I f us < m Then

El se

End I f

End Funct i on

Funct i on Tr i angul ar ( a As Doubl e, b As Doubl e, m As Doubl e ) As Doubl e

Di m ms As Doubl ems = ( m - a) / ( b - a)

Tr i angul ar = a + STr i angul ar ( ms ) * ( b - a)

End Funct i on


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)( z F u =

So that:)(1 u F z =

Solve for x:= z approximation?

Generating Random Numbers from the Standard Normal Distribution:

To generate a z s, a random number drawn from the standard normal distribution, = 0and = 1.

EXCEL: = NORMSINV( RAND() )

VBA:

There are three ways to generate standard normal random numbers. Here is the first way:

Funct i on Random_SNor m1( ) As Doubl eDi m u1 As Doubl eDi m u2 As Doubl eu1 = Rnd( )u2 = Rnd( )Random_SNor m1 = Sqr ( - 2 * Ln( u1) ) * Cos( 2 * 3. 1415927 * u2)

End Funct i on

Here is the second way using an approximation to the Normal Inverse CDF:

Funct i on SNor m_I nver seCDF( p As Doubl e ) As Doubl e Di ms ar e l ef t out f or br evi t y.a1 = - 39. 6968303a2 = 220. 9460984a3 = - 275. 9285104a4 = 138. 3577519a5 = - 30. 6647981a6 = 2. 5066283

b1 = - 54. 4760988b2 = 161. 5858369b3 = - 155. 6989799b4 = 66. 8013119b5 = - 13. 2806816

c1 = - 0. 0077849c2 = - 0. 3223965c3 = - 2. 4007583c4 = - 2. 5497325c5 = 4. 3746641


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c6 = 2. 938164

d1 = 0. 0077847d2 = 0. 3224671d3 = 2. 4451341d4 = 3. 7544087

p_l ow = 0. 02425p_hi gh = 1 - p_l ow

q = 0#r = 0#

Sel ect Case p

Case I s < p_l owq = Sqr ( - 2 * Log( p) )SNor m_I nver seCDF = ( ( ( ( ( c1 * q + c2) * q + c3) * q + c4) _

* q + c5) * q + c6) / ( ( ( ( d1 * q + d2) _* q + d3) * q + d4) * q + 1)

Case I s < p_hi ghq = p - 0. 5r = q * qSNor m_I nver seCDF = ( ( ( ( ( a1 * r + a2) * r + a3) * r + a4) _

* r + a5) * r + a6) * q / ( ( ( ( ( b1 * r _+ b2) * r + b3) * r + b4) * r + b5) * _r + 1)

Case I s < 1q = Sqr ( - 2 * Log( 1 - p) )SNor m_I nver seCDF = - ( ( ( ( ( c1 * q + c2) * q + c3) * q + c4) _

* q + c5) * q + c6) / ( ( ( ( d1 * q + d2) ) _* q + d3) * q + d4) * q + 1)

End Sel ectEnd Funct i on

Funct i on Random_SNor m2( ) As Doubl eRandom_SNor m2 = SNor m_I nver seCDF( Rnd( ) )

End Funct i on

Here is the third way which works because of the central limit theorem. However, the previous two ways should be preferred.

Funct i on Random_SNor m3( ) As Doubl eRandom_SNor m3 = Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + Rnd + _

Rnd + Rnd + Rnd + Rnd - 6End Funct i on


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Generating Random Numbers from a Normal Distribution:

To generate a random number drawn from a normal distribution, with parameters and:

s z z +=

Funct i on Random_Nor m( mu As Doubl e, si gma As Doubl e ) As Doubl eRandom_Nor m = mu + Random_SNor m3( ) * si gma

End Funct i on


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H. GENERALIZED INVERSE TRANSFORM METHOD

Probability density:100003.)(:.. 2 = x x x f g E


== x

xdx x x F 0

32 001.003.)(

Notice that this is a probability density because the total area under the curve from 0 to10 is 1. That is:

1003.)(10

0

2 == dx x x F For the inverse transform method, we must solve for F -1. We set u = F ( x ) and solve foru.

3

001.)( x x F u ==

31

1 )1000()( uu F x ==

To generate a random number from this probability density, if u s = .701, then:

88.8)701.1000( 31

== x

Should we, on the other hand, wish to truncate this distribution and, say, generate arandom number only between 2 and 5, then we must scale u s over the range F (2) to F (5)thusly:

))()(()( a F b F ua F u s +=

))2()5(()2( F F u F u s +=

)008.125(.008. += suu

Again, if u s = .701, then u = .090. The random number x drawn from f(x) over the range 2< x < 5 is:

48.4)090.1000( 31 == x

Thus, we can generalize the inverse transform method as:

)))()(()(()( 11 a F b F ua F F u F x s +==


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C. TRINOMIAL DISTRIBUTION

Parameters p, q and n.

The obvious way to extend the Bernoulli trials concept is to add ties. Thus, each de

Moivre trial yields success ( x = 1 ), failure ( x = -1 ), or a tie ( x = 0 ). The probability ofsuccess is p, of failure q, and of a tie is 1 p q.

Probability: y xn y x q pq p

y xn y xn

yY x X P === )1(

)!(!!!

),(

Expected value of x and y:nq ynp x E == )()(

Variance of x:)1()()1()( qnq yV pnp xV ==

To generate a random number from a de Moivre trial:

++

=

1uq pif 1q pu pif 0

puif 1

s

s

s

x

EXCEL: Series of de Moivre trials.

VBA:

Series of de Moivre Trials


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D. POISSON DISTRIBUTION

Parameter .

Number of events that occur in an interval of time when events are occurring at a

constant rate, say exponentially.

Probability:

,...}1,0{!

)( =

x for x

e x P

x

Cumulative Distribution Function:

0!

)(0

= =

xif i

e x F x

i

i

Expected value of x and y: =)( x E

Variance of x: =)( xV

To generate a random number from a Poisson distribution:

Step 1 : Let a = e - , b = 1, and i = 0.Step 2 : Generate u s and let b = bu s. If b < a, then return x = 1.

Otherwise, continue to Step 3.Step 3 : Set i = i + 1. Go back to Step 2.

EXCEL: NA

VBA:


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E. EMPIRICAL DISTRIBUTIONS

A cumulative distribution function gives the probability than a random variable X is lessthat a given value x. So,

)()( x X P X F =

An empirical distribution uses actual data rather than a theoretical distribution function.In the table below, n = 1000 observations are made with i = 4 outcomes, xi = { 100, 200,1000, 5000 } . The outcomes are sorted from low to high. Then calculated is the

probability of each outcome, where the empirical probability, P(x i ), is the ratio of thecount of each outcome, n x, to the total number of trials.

i xi n x P(x i) F(x i) If u s = .71,1 100 500 .50 .502 200 100 .10 .603 1000 250 .25 .85 x = 10004 5000 150 .15 1.0

= 1000 = 1.0

VBA:

Funct i on Empi r i cal ( ) As Doubl e


Di m x As Doubl e

Sel ect Case usCase I s < 0. 5

x = 100Case 0. 5 To 0. 6

x = 200Case 0. 6 To 0. 85

x = 1000Case 0. 85 To 1

x = 5000End Sel ect

Empi r i cal = x

End Funct i on


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F. LINEAR INTERPOLATION

Linear interpolation is a method of curve fitting using linear polynomials. If the coordinates oftwo points are known, the linear interpolant is the straight line between these points. For a pointin the interval, the coordinate values, x and y, along the straight line are given by:

01

0

01

0 x x x x

y y y y

=

V. GENERATING CORRELATED RANDOM NUMBERS

What does correlated mean? Correlation is the tendency for two series of randomnumbers to diverge in tandem, above or below, from their respective means.

21

21

2

21

2

1,2,1

,,

)))(((

x x

n

i xi xi

x x

x x x x

x x

i

i

=

==


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B. MULTIVARIATE NORMAL

Random multivariate normal numbers (i.e. correlated random numbers for normaldistributions) can be generated by multiplying a vector of standard normal randomnumbers, Z s, by the Cholesky decomposition, L, of the correlation matrix, C, as follows:

s s LZ Z =*

The Cholesky decomposition is in the lower left triangle and main diagonal of a squarematrix. The elements in the upper right triangle are 0.

Funct i on Chol esky( mat As Range ) As Doubl e( )Di m A As Var i ant , L( ) As Doubl e, S As Doubl eDi m n As Doubl e, m As Doubl eA = matn = mat . Rows. Count

m = mat . Col umns. CountI f n m ThenChol esky = " ?"Exi t Funct i on

End I f

ReDi m L( 1 To n, 1 To n)For j = 1 To n

S = 0For K = 1 To j - 1

S = S + L( j , K) ^ 2Next K

L( j , j ) = A( j , j ) S

I f L(j , j )


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=

1803.183.803.1503.

183.503.1C

The Cholesky decomposition matrix of C is:

=

538.823.183.0864.503.00000.1

L

Given a vector of standard normal random numbers (i.e. z s):

=

151.516.1517.

s Z

The vector of correlated standard normal random variables, Z s*, is:

==

234.1050.1

517.

* s s LZ Z

To convert the individual z s*s in Z s* to z is for each z i ~ N( i, i ):

iiii z z *+=


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Brownian motion is another continuous-time stochastic process useful formodeling the prices of stocks and other financial instruments. In Brownian motion, theexpected value of the next price is equal to the last price. Geometric Brownian motion isa continuous-time stochastic process where the logarithm of the random variable followsBrownian motion. It is also called a Wiener process. A stochastic process S t is to be

geometric Brownian motion if it satisfies the following stochastic differential equation:t t t t dW S dt S dS +=

where W t is a Wiener process and is the drift rate and the volatility. For an initialvalue S 0 the equation we can find a random value at some time t in the future:

t W t t eS S

+= )2/(02

Where S t is a log-normally distributed with expected value:

t t eS S

0)( =

and variance:

)1(222

0

2 = t t S

eeS t

This conclusion can be verified using It 's lemma, where the continuous rate of return r =ln(S t /S 0 ) is normally distributed. A Wiener process W t has independent, normallydistributed changes such that:

),0(~0 t N W W W t t = (1)

That is, the expected value and variance of a Wiener process is:

0)( =t W E and t W V t =)( (2)

This variance is important because is shows that the standard deviation is the square rootof t, and so it is that stock volatility scales with the square root of time.

The following proof shows the connection to between dt from geometricBrownian motion and the normally distributed, continuously compounding drift term ( - 2 ) t. Given the assumption that stock price follows geometric Brownian motion,with Wiener process W:

)(0 t t t t dW dt S dS S S +== (3)

This says that the change in the price of the stock is equal to the price of the stock times amean drift rate times the change in time plus the standard deviation times some randomvariable.

The reason we use geometric Brownian motion to model stock price paths is becauseit encapsulates two widely observed phenomena in financial markets:

1. The long term trends of markets, represented by the mean term.


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2. The white noise or random price movements or volatility around the trend,represented by the standard deviation term, which scales with the squareroot of time.


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A. FINANCIAL MARKET DATA

Exchanges provide for continuous quoting of bids and offers in shares or contracts listed on theexchange. From an automated trading perspective, a full quote consists of seven things: thesymbol, the quantity on the highest bid price, the highest bid price, the lowest ask price, thequantity on the lowest ask price, the last price traded, and the quantity of the last trade (or the sumof the quantities of the consecutive trades on that price).

Symbol Bid Qty Bid Price Ask Price Ask Qty Last Price Last QtyES 195 125025 125050 456 125025 33

Quote 1

TICK

The term tick has different meanings in finance. The tick size is the minimum priceincrementthe minimum bid-ask spreadthat is the difference between the highest bidand the lowest offer. In the example above the inside market (i.e. the highest bid and

lowest ask) is 1250.25 to 1250.50, where the tick size is .25. The value of the minimumtick increment is not, in fact 25 cents; the contract size is much larger. In this sense, if atrader makes a tick profit on a trade, he has made .25 of a point. For the S&P 500 E-minicontract the value of a tick is $12.50. The whole-point value is often referred to as thehandle. In the above example, 1250 is the handle. If the contract increases to 1254, wesay the price is up four handles. In the case of this contract, a tick is a quarter handle.

A tickas in tick data or uptick or downtickrefers to trade that has occurred onan exchange as been broadcasted to market particpants. In Quote 1 above, a tick would

be reflected by a change in the Last Price and Last Qty fields. A days worth of tick datawould contain all the information about all the trades that occurred that day. A tick inthis sense contains at least four things:

Ticker symbol Price Volume or quantity Time

Now, some of these data elements may be represented by a value, by a change from the previous value, or by a predefined increment. For example, given data representing atrade as follows:

Ticker Symbol Price Quantity TimeES 125025 33 13:43:12.100

Trade 1

Then, to save bandwidth, the next tick may show the trade data in an abbreviated format:

Ticker Symbol Price Quantity TimeES -1 10 12

Trade 2


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The new price of the new trade is 125000, because -1 means subtract 1 increment (e.g..25) from the previous price. The time of the trade is 13:43:12.112, the time of the

previous trade plus 12 milliseconds.


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B. MODELING TICK DATA

The goal is to simulate the arrival of tick data. Clearly, there are 3 stochastic processes atwork: the price, the quantity, and the arrival time interval. To do this, we will use theexponential distribution to simulate the inter-arrival times of new ticks. We will use a

trinomial distribution to simulate price movementsup, down, or sideways. And, wewill use an empirical distribution to simulate quantities.

Funct i on deMoi vr e( p As Doubl e, q As Doubl e ) As Doubl e


Di m v As Doubl e

Sel ect Case us

Case I s < pv = 1Case p To p + q

v = - 1Case p + q To 1. 0

v = 0End Sel ect

deMoi vr e = v

End Funct i on

Funct i on Exponent i al ( bet a As Doubl e ) As Doubl e

Exponent i al = I nt ( - bet a * Log( 1 - Rnd( ) ) * 1000 + 1)

End Funct i on

Funct i on Empi r i cal ( ) As Doubl e


Di m v As Doubl e

Sel ect Case usCase I s < 0. 4

v = 100Case 0. 4 To 0. 5

v = 200Case 0. 5 To 0. 6

v = 300


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Case 0. 6 To 0. 7v = 400

Case 0. 7 To 0. 8v = 500

Case 0. 8 To 0. 9v = 1000

Case 0. 9 To 0. 925v = 1500

Case 0. 925 To 0. 95v = 2000

Case 0. 95 To 0. 975v = 5000

Case 0. 975 To 1v = 10000

End Sel ect

Empi r i cal = v

End Funct i on

The initial time is 0, the initial price 50.00, and the initial quantity is 0. The Excelformula to generate millisecond arrival intervals is:

=Exponential( 0.5 )

The Excel formula to generate price movements is:

=B3 + deMoivre( 0.333, 0.333 ) * 0.01

The Excel formula to generate random quantity:

=Empirical()

These formulae generated the following random tick data:

Times Price Qty0 50.00 0

667 50.00 3001018 49.99 100

38 49.99 100084 49.98 100

651 49.98 20001599 49.98 100

47 49.98 300100 49.97 100861 49.96 100287 49.97 100

68 49.98 50095 49.97 100

503 49.96 200


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The tick data produces the following chart:

49.70

49.75

49.80

49.85

49.90

49.95

50.00

50.05

50.10

50.15

50.20

1 61 121 181 241 301 361 421 481 541 601 661 721 781 841 901 961

Random Tick Data

Now, what we would like to do is convert the random tick data (with random inter-arrivaltimes) into a time series of bars (with constant time intervals), where a bar contains fourdata elements over the time interval:

Open Price High Price Low Price Closing Price

Using the tick data, there are a random number of ticks that occur over each one minuteinterval. (The number of ticks per interval follows a Poisson distribution.) By summingthe inter-arrival times until the minute is over, i.e. Times > 60000, we can find theExcel range of the ticks that occurred in that minute. The bar data is then:

Open Price = first price in the range High Price = MAX( range ) Low Price = MIN( range ) Closing Price = last price of the range

The tick data generated the following bars:

Open High Low Close50.00 50.09 50.05 50.0750.06 50.29 50.01 50.2450.23 50.23 50.06 50.1050.11 50.13 50.03 50.0950.08 50.23 50.08 50.2150.21 50.27 50.14 50.2050.20 50.24 50.10 50.1150.10 50.24 50.08 50.22


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50.21 50.35 50.20 50.35 The bar data produces the following chart:

49.80

49.90

50.00

50.10

50.20

50.30

50.40

1 2 3 4 5 6 7 8 9

Bar Data

This data can be used to find the continuous returns for each minute interval as per:

)ln( 1= iii P P r

Where P i equals the closing price for minute i.

Close Returns50.0750.24 0.003450.10 -0.002850.09 -0.000250.21 0.002450.20 -0.000250.11 -0.001850.22 0.002250.35 0.0026

These rates of return, r, are approximately normally distributed:

),(~ 2 N r

To demonstrate the approximation of returns to normality, we need to generate a lot morethan 10 returns. The following code will generate around 250 bars.


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Opt i on Expl i ci tOpt i on Base 1

Publ i c Type Ti ck Ti me As Doubl ePr i ce As Doubl eQuant i t y As Doubl e

End Type

Sub Run_Si mul at i on( )

Appl i cat i on. Scr eenUpdat i ng = Fal se

Randomi ze

Di m t i cks( 30000) As Ti ckDi m p As Doubl ep = 50

Di m i As I nt egerDi m j As I nt eger

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' Cr eat e r andom t i cks ' '' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

For i = 1 To 30000t i cks( i ) . Ti me = Exponent i al ( 0. 5)t i cks( i ) . Pri ce = p + deMoi vre( 0. 333, 0. 333) * 0. 01t i cks( i ) . Quant i t y = Empi r i cal ( )p = t i cks ( i ) . Pr i ce

Next i

' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '' ' Cr eat e bar s f r om t i cks ' '' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' ' '

Range( " A1") . val ue = " Open"Range( " B1" ) . val ue = " Hi gh"Range( " C1") . val ue = " Low"Range( " D1" ) . val ue = " Cl ose"

Di m count As Doubl eDi m cur sor As Doubl eDi m t i me_sum As Doubl e

cur sor = 1

For j = 1 To 30000

t i me_sum = t i me_sum + t i cks(j ) . Ti me

I f ( t i me_sum > 60000) ThenRange( "A2") . Of f set ( count ) . val ue = t i cks( cur sor ) . Pr i ceRange( "B2") . Of f set ( count ) . val ue = Max( t i cks, cur sor , j - 1)Range( "C2") . Of f set ( count ) . val ue = Mi n( t i cks, cur sor , j - 1)


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48.5

49

49.5

50

50.5

51

1 13 25 37 49 61 73 85 97 109 121 133 145 157 169 181 193 205 217 229 241

Randomly generated bar data

The histogram of the log returns of the one minute time series appear to be approximatelynormally distributed:

Histogram

0

5

10

15

20

25

30

35

- 0 . 0 0 5

0

- 0 . 0 0 4

0

- 0 . 0 0 3

0

- 0 . 0 0 2

0

- 0 . 0 0 1

0

0 . 0 0

0 0

0 . 0 0

1 0

0 . 0 0

2 0

0 . 0 0

3 0

0 . 0 0

4 0

0 . 0 0

5 0

Bin

F r e q u e n c y

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

Distribution of log returns

The mean return was very close to 0 at .000005, and the standard deviation was .0017.Given more bar data, we would see the normal distribution even more clearly.


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C. MODELING TIME SERIES DATA

Here is an actual chart using real daily closing price data of Boeing stock (symbol BA)from 10-02-2008 to 10-02-2009.

0

10

20

30

40

50

60

1 13 25 37 49 61 73 85 97 109 121 1 33 145 157 169 181 193 205 217 229 241

Price path of BA stock

The histogram for the daily log returns for BA over the year also appears to beapproximately normally distributed:

Histogram

0

5

10

15

20

25

30

35

40

45

- 0 . 1

- 0 . 0 8

- 0 . 0 6

- 0 . 0

4 - 0

. 0 2 0

0 . 0 2

0 . 0 4

0 . 0 6

0 . 0 8 0 .

1

Bin

F r e q u e n c y

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

Distribution of BA log returns

The mean log return is .00011 with a standard deviation .0335.


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1. FAT TAILS

The assumption in the previous model is that the standard deviation remains constantover the price path, which is almost certainly not representative of the real world. Stockstypically exhibit periods of low volatility and periods of high volatility, usually for

exogenous or fundamental reasonsmarket shocks, earnings reports, etc. Fat tails are a property of financial data distributions relative to normal distributions which have thintails. In an attempt incorporate extreme events with greater probability than a normaldistribution would imply, mixture models that mix distributions together have beendeveloped. As a simple example, the code below implements a jump process, where 1%of the time the width of the distribution generating the random returns increases from onestandard deviation, or volatility, to another.

Now, for simplicity, lets set 0, so that: s z

t t eS S 1= VBA:

Funct i on Mi xt ur e_of _Nor mal s( pr i ce As Doubl e, vol 1 As Doubl e, _vol 2 As Doubl e ) As Doubl e


I f us < 0. 99 ThenMi xt ur e_of _Normal s = Random_Pr i ce( pr i ce, vol 1)

El seMi xt ur e_of _Normal s = Random_Pr i ce( pr i ce, vol 2)

End I f

End Funct i on

Funct i on Random_Pr i ce( pr i ce As Doubl e, vol As Doubl e ) As Doubl e

Random_Pr i ce = pr i ce * Exp( mu + Appl i cat i on. Nor mSI nv( Rnd( ) ) * vol )

End Funct i on

A B C D1 Mean 0 502 StDev 1 0.01 49.507343 StDev 2 0.10 49.438784 49.74915 49.68668

In cell D2 and copied down is the following formula:

EXCEL: = Mixture_Of_Normals( D1, $B$2, $B$3 )

This code produced the following sample price path:


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0

10

20

30

40

50

60

1 60 119 178 237 296 355 414 473 532 591 650 709 768 827 886 945

Random price path with jumps

A histogram of the log returns will show a normal-like distribution but with fatter tails.

2. STOCHASTIC VOLATILITY MODELS

Stochastic volatility models treat volatility as a random process, governed by statevariables, and may include a serial correlation and a tendency to revert to a long-runmean value. Stochastic volatility models are one way to resolve the shortcoming of manyfinancial models that assume constant volatility over some time horizon, which iscertainly a contradiction of widely observed phenomena. Stochastic volatility models areused to value and manage risk associated with derivative instruments.

a. ARCH(1)

An autoregressive conditional heteroscedasticity (ARCH) model considers the varianceof the current period return, (i.e. the error term) to be a function of the prior periodsquared errors. We use it finance, to account for volatility clustering, i.e. periods of highvolatility tending to be followed by periods of low volatility. The ARCH(1) equation is:

221 t t r +=+

b. GARCH(1,1)

An generalized autoregressive conditional heteroscedasticity (GARCH) model considersthe variance of the current period return, (i.e. the error term) to be a function of the prior

period squared errors and the prior period estimated of variance. We use it finance, toaccount for volatility clustering, i.e. periods of high volatility tending to be followed by

periods of low volatility. The GARCH(1,1) equation is:


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2221 t t t r ++=+

Here is an Excel implementation of the GARCH(1,1) formula:

A B C D E F

1 gamma 0.0001 Price Return GARCH Vol2 alpha 0.4 503 beta 0.6 51 0.0198 0.01 0.14 53.44 0.0467 0.0062 0.07885 53.97 0.0098 0.0044 0.06696 50.57 -0.0651 0.0028 0.0531

Where, cells C2 and C32 contain initial prices, cell D3 (and copied down) contains theformula = LN( C3 / C2 ), cell E3 is an initial value for the GARCH forecast and columnF is the square root of column E. The GARCH equation in E4 and copied down is:

EXCEL: = $B$1 + $B$2 * D3 ^ 2 + $B$3 * E3The following chart, of the data in column F, shows the kind of volatility clusteringcommonly observed in markets.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

1 5 9 13 1 7 21 25 2 9 33 37 4 1 45 4 9 5 3 57 6 1 65 6 9 73 77 8 1 85 8 9 93 9 7

Volatility Clustering


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VII. MODELING OPTIONS

Before we begin to discuss option pricing, lets quickly review an importantrelationshipput-call parity. Put-call parity states that the value of a call at a given strikeimplies a certain value for the corresponding put. If there is divergence from parity, then

an arbitrage opportunity would exist and trading could take risk-free profits until thedivergence from parity is eliminated.Put-call parity states that the price of the stock, the price of a call at a given strike

and expiration, and the price of a put with that same strike and expiration, are allinterrelated and simultaneously solved according to:

0S P XeC rt +=+

Where, C is the call price, P the put price, X the strike, S the stock price, r the interestrate and t the time till expiration.

A. THE SIMULATION WAY

Recall that the t time ahead price can be simulated in the following way:t z t

t seS S += 0

Also, the call option payoff at time t can be expressed as:

)0,max( X S C t =

Where X is the strike price of the option. Using VBA we generate many simulations of

the stock price to generate many call option payoffs.

VBA:

Funct i on Si m_Eur _Cal l ( S As Doubl e, X As Doubl e, r f As Doubl e, _t As Doubl e, si gma As Doubl e, si ms As Doubl e ) _As Doubl e

r f = ( r f - 0. 5 * si gma ^ 2) * t

sum_payof f s = 0#

For n = 1 To si msST = s * Exp( r f + Appl i cat i on. NormSI nv( Rnd) * s i gma * Sqr ( t ) )sum_payof f s = sum_payof f s + Max( ST - X, 0#)

Next n

Si m_Eur _Cal l = Exp( - r f * t ) * ( sum_payof f s / si ms)End Funct i on


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B. THE COX-ROSS-RUBENSTEIN or BINOMIAL WAY

VBA:

Funct i on Bi n_Am_Cal l ( S As Doubl e, X As Doubl e, r f As Doubl e, _t As Doubl e, si gma As Doubl e, n As Doubl e ) _As Doubl e

dt = t / n

u = Exp( si gma * Sqr ( dt ) )d = Exp( - si gma * Sqr ( dt ) )

r = Exp( r f * dt )qu = ( r - d) / ( r * ( u - d) )qd = 1 / r - qu

Di m Opt i onVal uesEnd( ) As Doubl e

ReDi m Opt i onVal uesEnd( n)For i = 0 To n

Opt i onVal uesEnd( i ) = Max( s * u ^ i * d ^ ( n - i ) - X, 0)Next i

For i = n - 1 To 0 St ep - 1

ReDi m Opt i onVal uesPr i or( i - 1) As Doubl e

For j = 0 To iOpt i onVal uesMi ddl e( j ) = Max( s * u ^ j * _

d ^ ( i - j ) - X, _qd * Opt i onVal uesEnd( j ) + _qu * Opt i onVal uesEnd( j + 1) )

Next j

ReDi m Opt i onVal uesEnd( i - 1)

For j = 0 To iOpt i onVal uesEnd( j ) = Opt i onVal uesMi ddl e( j )

Next j

Next i

Bi n_Am_Cal l = Opt i onVal uesMi ddl e( 0)End Funct i on


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PROJECT I

1. Create a VBA function that generates random numbers from a triangulardistribution. In Excel, generate 1000 random numbers using this function and show thedistribution of outcomes in a histogram.

2. Create a VBA function that generates random integers greater than 0 using anexponential distribution.

3. Here is a probability density:

500064.)( 3 = x x x f

Create a VBA function that generates random numbers from this distribution using theinverse transform method. Also, create a VBA function that generates random numbersfrom a to b, a truncation (or sub-range) of the density.


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PROJECT II

1. Generate 100 x 1s and x 2s using a bivariate normal distribution (BVN) and VBA.

2. Generate 100 xs using empirical data in bins: 100, 500, 1000, 5000, 10000. The probabilities are .2, .3, .1, .2, and .2 respectively. Use linear interpolation in VBA tocalculate values of the xs from within these bins.

3. Generate 100 x 1s to x ns using a multivariate normal distribution (MVN).


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PROJECT III

1. Generate a random stock price path for 1000 days using ARCH(1) and jumps inVBA. For the coefficients, use:

0.9 0.0001

Create a histogram of the log returns.

2. Generate a random stock price path for 1000 days using GARCH(1,1) and jumpsin VBA. For the coefficients, use:

0.5 0.4

0.0001 Create a histogram of the log returns.


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PROJECT IV

1. Create an Excel application with functions in VBA for Black Scholes puts andcalls, plus functions for the Greeks: Delta, Gamma, Rho, Theta and Vega. Be sure toshow that these functions work properly in by calling them in Excel.

simulation in finance

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