Computational Finance Using C and C# (Quantitative Finance)

Regula-Series Titles Computational Finance Using C and C# The Analytics of Risk Model Validation Forecasting Expected Returns in the Financial Markets Corporate Governance and Regulatory

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Aims and Objectives

• Books based on the work of financial market practitioners and academics

• Presenting cutting-edge research to the professional/practitioner market

• Combining intellectual rigour and practical application

• Covering the interaction between mathematical theory and financial practice

• To improve portfolio performance, risk management and trading book performance

• Covering quantitative techniques

Market

Brokers/Traders; Actuaries; Consultants; Asset Managers; Fund Managers; tors; Central Bankers; Treasury Officials; Technical Analysis; and Academics for Mas-ters in Finance and MBA market

Regula-Series Titles

Computational Finance Using C and C#

The Analytics of Risk Model Validation

Forecasting Expected Returns in the Financial Markets

Corporate Governance and Regulatory Impact on Mergers and Acquisitions

International Mergers and Acquisitions Activity Since 1990

Forecasting Volatility in the Financial Markets, Third Edition

Venture Capital in Europe

Funds of Hedge Funds

Initial Public Offerings

Linear Factor Models in Finance

Computational Finance

Advances in Portfolio Construction and Implementation

Advanced Trading Rules, Second Edition

Real R&D Options

Performance Measurement in Finance

Economics for Financial Markets

Managing Downside Risk in Financial Markets

Derivative Instruments: Theory, Valuation, Analysis

Return Distributions in Finance

Series Editor: Dr Stephen Satchell

Dr Satchell is Reader in Financial Econometrics at Trinity College, Cambridge;Visiting Professor at Birkbeck College, City University Business School and Univer-sity of Technology, Sydney He also works in a consultative capacity to many firms,

and edits the journal Derivatives: use, trading and regulations and the Journal of Asset

Management.

Computational Finance Using C and C#

George Levy

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Library of Congress Cataloging-in-Publication Data

Levy, George

Computational Finance Using C and C# / George Levy

p cm – (Quantitative finance)

Includes bibliographical references and index

ISBN-13: 978-0-7506-6919-1 (alk paper) 1 Finance-Mathematical models I Title.HG106.L484 2008

332.0285’5133-dc22

2008000470

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

For information on all Academic Press publications

visit our Web site at www.books.elsevier.com

Printed in the United States of America

08 09 10 11 9 8 7 6 5 4 3 2 1

2.2 A Brownian model of asset price movements 9

2.5 Ito’s lemma for multiasset geometric Brownian motion 132.6 Ito product and quotient rules in two dimensions 15

2.9 Time-transformed Brownian motion 21

2.11 The Ornstein–Uhlenbeck bridge 27

4.2 Pricing derivatives using a martingale measure 59

4.4 Vanilla options and the Black–Scholes model 62

5.4 Grid methods for vanilla options 1355.5 Pricing American options using a stochastic lattice 172

6.2 The multiasset Black–Scholes equation 1816.3 Multidimensional Monte Carlo methods 1836.4 Introduction to multidimensional lattice methods 185

7.3 Foreign exchange derivatives 228

C.3 The variance and covariance of random variables 305C.4 Conditional mean and covariance of normal distributions 310

Appendix D: Statistical distribution functions 313D.1 The normal (Gaussian) distribution 313

D.3 The Student’s t distribution 317D.4 The general error distribution 319

E.3 The cumulative normal distribution function 322E.4 Arithmetic and geometric progressions 323

Appendix F : Black–Scholes finite-difference schemes 325

F.2 The log transformation and a uniform grid 325

Appendix G: The Brownian bridge: alternative derivation 329

H.1 Some results concerning Brownian motion 333

This book builds on the author’s previous book Computational Finance:

Nu-merical Methods for Pricing Financial Instruments, which contained

informa-tion on pricing equity opinforma-tions using C code The current book covers the lowing instrument types:

fol-• Equity derivatives

• Interest rate derivatives

• Foreign exchange derivatives

• Credit derivatives

There is also an extensive final chapter which demonstrates how a C-basedanalytics pricing library can be used by C# portfolio valuation software In ad-dition this application:

• illustrates the use of C# dictionaries, abstract classes and NET vices

InteropSer-• permits the reader to value bespoke portfolios

• allows market data to be specified via a configuration file

• contains a generic basket pricer for which the reader can specify the payofffunction

• can be freely downloaded for use by the reader

The current book also contains increased coverage of stochastic processes, Itocalculus and Monte Carlo simulation These topics are supported by practicalapplications and solved example problems

In addition the Numerical Algorithms Group (NAG) have allowed readers

to enjoy an extended trial licence for the NAG C library and associated cial routines from the following url: www.nag.co.uk/market/elsevier_glevy TheNAG C library may be called into C# and provides a large suite of mathematicalroutines addressing many areas covered in this book (random numbers, statisti-cal distributions, option pricing, correlation and covariance matrices etc.)

finan-Computational Finance Using C and C# also includes supporting software

that may be downloaded for free The software consists of executable files, figuration files and results files With these files the user can run the exampleportfolio application in Chapter 8 and change the portfolio composition andthe attributes of the deals

con-Additional upgrade software is available for purchase with Computational

Finance Using C and C# The software includes:

• Code to run all the C, C# and Excel examples in the book

• Complete C source code for the Analytics_Mathlib math library that is used

George LevyBenson, Oxfordshire, UK

2008

1 Overview of financial derivatives

A financial derivative is a contract between two counterparties (here referred

to as A and B) which derives its value from the state of underlying financial quantities We can further divide derivatives into those that carry a future oblig-

ation and those that don’t In the financial world a derivative which gives the

owner the right but not the obligation to participate in a given financial contract

is called an option We will now illustrate this using both a Foreign Exchange

Forward contract and a Foreign Exchange option

Foreign Exchange Forward—a contract with an obligation

In a Foreign Exchange Forward contract a certain amount of foreign currencywill be bought (or sold) at a future date using a prearranged foreign exchangerate

For instance, counterparty A may own a Foreign Exchange Forward which,

in one year’s time, contractually obliges A to purchase from B the sum of $200

for £100 At the end of one year several things may have happened

(i) The value of the pound may have decreased with respect to the dollar(ii) The value of the pound may have increased with respect to the dollar

(iii) Counterparty B may refuse to honor the contract—B may have gone bust,

etc

(iv) Counterparty A may refuse to honor the contract—A may have gone bust,

etc

We will now consider events (i)–(iv) from A’s perspective.

Firstly, if (i) occurs then A will be able to obtain $200 for less than the current

market rate, say £120 In this case the $200 can be bought for £100 and thenimmediately sold for £120, giving a profit of £20 However, this profit can only

be realized if B honors the contract—that is, event (iii) does not happen Secondly, when (ii) occurs then A is obliged to purchase $200 for more than

the current market rate, say £90 In this case the $200 are bought for £100 butcould have been bought for only £90, giving a loss of £10

The probability of events (iii) and (iv) occurring are related to the Credit Risk associated with counterparty B The value of the contract to A is not affected

by (iv), although A may be sued if both (ii) and (iv) occur Counterparty A

should only be concerned with the possibility of events (i) and (iii) occurring—that is, the probability that the contract is worth a positive amount in one year

and the probability that B will honor the contract (which is one minus the

probability that event (iii) will happen)

From B’s point of view the important Credit Risk is when both (ii) and (iv) occur—that is, when the contract has positive value but counterparty A defaults.

Foreign Exchange option—a contract without an obligation

A Foreign Exchange option is similar to the Foreign Exchange Forward, the

difference being that if event (ii) occurs then A is not obliged to buy dollars

at an unfavorable exchange rate To have this flexibility A needs to buy a eign Exchange option from B, which here can be regarded as insurance against

For-unexpected exchange rate fluctuations

For instance, counterparty A may own a Foreign Exchange option which, in one year, contractually allows A to purchase from B the sum of $200 for £100.

As before, at the end of one year the following may have happened:

(i) The value of the pound may have decreased with respect to the dollar(ii) The value of the pound may have increased with respect to the dollar

(iii) Counterparty B may refuse to honor the contract—B may have gone bust,

etc

(iv) Counterparty A may have gone bust, etc.

We will now consider events (i)–(iv) from A’s perspective.

Firstly, if (i) occurs then A will be able to obtain $200 for less than the current

market rate, say £120 In this case the $200 can be bought for £100 and thenimmediately sold for £120, giving a profit of £20 However, this profit can only

be realized if B honors the contract—that is, event (iii) does not happen Secondly, when (ii) occurs then A will decide not to purchase $200 for more

than the current market rate; in this case the option is worthless

We can thus see that A is still concerned with the Credit Risk when events

(i) and (iii) occur simultaneously

The Credit Risk from counterparty B’s point of view is different B has sold

to A a Foreign Exchange option, which matures in one year, and has already received the money—the current fair price for the option Counterparty B has

no Credit Risk associated with A This is because if event (iv) occurs, and A goes bust, it doesn’t matter to B since the money for the option has already been received On the other hand, if event (iii) occurs B may be sued by A but

B still has no Credit Risk associated with A.

This book considers the valuation of financial derivatives that carry tions and also financial options

obliga-Chapters 1–7 deal with both the theory of stochastic processes and the ing of financial instruments In Chapter 8 this information is then applied to aC# portfolio valuer The application is easy to use (the portfolios and currentmarket rates are defined in text files) and can also be extended to include newtrade types

pric-The book has been written so that (as far as possible) financial mathematicsresults are derived from first principles.

Finally, the appendices contain various information, which we hope thereader will find useful

micro-Clarkia pulchella) suspended in water, and wrote:

The fovilla or granules fill the whole orbicular disk but do not extend to theprojecting angles They are not sphaerical but oblong or nearly cylindrical,and the particles have manifest motion This motion is only visible to mylens which magnifies 370 times The motion is obscure yet certain

Robert Brown, 12th June 1827; see Ramsbottom (1932)

It appears that Brown considered this motion no more than a curiosity (he

be-lieved that the particles were alive) and continued undistracted with his

botan-ical research The full significance of his observations only became apparentabout eighty years later when it was shown (Einstein, 1905) that the motion

is caused by the collisions that occur between the pollen grains and the watermolecules In 1908 Perrin (1909) was finally able to confirm Einstein’s predic-tions experimentally His work was made possible by the development of theultramicroscope by Richard Zsigmondy and Henry Siedentopf in 1903 He wasable to work out from his experimental results and Einstein’s formula the size

of the water molecule and a precise value for Avogadro’s number His workestablished the physical theory of Brownian motion and ended the skepticismabout the existence of atoms and molecules as actual physical entities Many ofthe fundamental properties of Brownian motion were discovered by Paul Levy(Levy, 1939, 1948), and the first mathematically rigorous treatment was pro-vided by Norbert Wiener (Wiener, 1923, 1924) Karatzas and Shreve (1991) is

an excellent textbook on the theoretical properties of Brownian motion, whileShreve, Chalasani, and Jha (1997) provides much useful information concerningthe use of Brownian processes within finance

Brownian motion is also called a random walk, a Wiener process, or times (more poetically) the drunkard’s walk We will now present the three fun-

some-damental properties of Brownian motion

2.1.1 The properties of Brownian motion

In formal terms a process W = (W t : t  0) is (one-dimensional) Brownian

motion if:

(i) W t is continuous, and W0= 0,

(ii) W t ∼ N(0, t),

(iii) The increment dW t = W t +dt −W t is normally distributed as dW t ∼ N(0, dt),

so E [dW t ] = 0 and Var[dW t ] = dt The increment dW t is also independent

of the history of the process up to time t.

From (iii) we can further state that, since the increments dW t are independent

of past values W t , a Brownian process is also a Markov process In addition we shall now show that a Brownian process is also a martingale process.

In a martingale process P t , t  0, the conditional expectation E[P t +dt |F t] =

P t, whereF t is called the filtration generated by the process and contains the information learned by observing the process up to time t Since for Brownian

where we have used the fact that E[dW t ] = 0 Since E[W t +dt |F t ] = W t the

Brownian motion W is a martingale process.

Using property (iii) we can also derive an expression for the covariance ofBrownian motion The independent increment requirement means that for the

ntimes 0 t0< t1< t2< · · · < t n < ∞ the random variables W t1− W t0, W t2−

W t1, , W t n − W t n−1 are independent So

Cov[W t i − W t i−1, W t j − W t j−1] = 0, i = j (2.1.1)

We will show that Cov[Ws , W t ] = s ∧ t.

Proof Using W t0 = 0, and assuming t  s we have

Cov[W s − W t0, W t − W t0] = Cov[W s , W t] = CovW s , W s + (W t − W s )From Appendix C.3.2 we have

Cov

W s , W s + (W t − W s )

= Cov[W s , W s ] + Cov[W s , W t − W s]

= Var[W s ] + Cov[W s , W t − W s]Therefore

where dW t is a random variable drawn from a normal distribution with mean

zero and variance dt, which we denote as dW t ∼ N(0, dt) Equation (2.1.3) can

also be written in the equivalent form:

where dZ is a random variable drawn from a standard normal distribution (that

is a normal distribution with zero mean and unit variance)

Equations (2.1.3) and (2.1.4) give the incremental change in the value of X over the time interval dt for standard Brownian motion.

We shall now generalize these equations slightly by introducing the extra

(volatility) parameter σ which controls the variance of the process We now

have:

where dW t ∼ N(0, dt) and dX t ∼ N(0, σ2dt) Equation (2.1.5) can also be

written in the equivalent form:

the position of a particular pollen grain at time t by X t, and set the position

at t = 0, X t0, to zero The statistical distribution of the grain’s position, X T, at

some later time t = T , can be found as follows:

Let us divide the time T into n equal intervals dt = T /n Since the position of the particle changes by the amount dX i = σ√dt dZ i over the ith time interval

dt, the final position X T is given by:

Since dZ i ∼ N(0, 1), by the Law of Large Numbers (see Appendix C.1), we have that the expected value of position X T is:

The variance of the position X T is:

Here we have used the fact (see Appendix C.3.1) that Var[a + bX] = b2Var[X],

where a = μT , and b = 1 From Eqs (2.1.9) and (2.1.10) we have:

2.2 A Brownian model of asset price movements

In the previous section we showed how Brownian motion can be used to scribe the random motion of small particles suspended in a liquid The firstattempt at using Brownian motion to describe financial asset price movementswas provided by Bachelier (1900) This, however, only had limited success be-

de-cause the significance of a given absolute change in asset price depends on the

original asset price For example, a £1 increase in the value of a share originally

worth £1.10 is much more significant than a £1 increase in the value of a share

originally worth £100 It is for this reason that asset price movements are

gen-erally described in terms of relative or percentage changes For example, if the

£1.10 share increases in value by 11 pence and the £100 share increases in value

by £10, then both of these price changes have the same significance, and spond to a 10 percent increase in value The idea of relative price changes in the

corre-value of a share can be formalized by defining a quantity called the return, R t,

of a share at time t The return R t is defined as follows:

R t = S t +dt − S t

S t = dS t

where S t +dt is the value of the share at time t + dt, S t is the value of the share at

time t, and dS t is the change in value of the share over the time interval dt The percentage return R∗over the time interval dt is simply defined as R∗= 100×R t

We are now in a position to construct a simple Brownian model of assetprice movements; further information on Brownian motion within finance can

be found in Shreve, Chalasani, and Jha (1997)

The asset return at time t is now given by:

R t = dS t

S = μ dt + σ dW t , dW t ∼ N(0, dt), (2.2.2)

or equivalently:

dS t = S t μ dt + S t σ dW t (2.2.3)

The process in Eqs (2.2.2) and (2.2.3) is termed geometric Brownian motion;

which we will abbreviate as GBM This is because the relative (rather than solute) price changes follow Brownian motion

ab-2.3 Ito’s formula (or lemma)

In this section we will derive Ito’s formula; a more rigorous treatment can befound in Karatzas and Shreve (1991)

Let us consider the stochastic process X:

dX = a dt + b dW = a dt + b√dt dZ, dZ ∼ N(0, 1), dW ∼ N(0, dt)

(2.3.1)

where a and b are constants We want to find the process followed by a function

of the stochastic variable X, that is φ(X, t) This can be done by applying a Taylor expansion, up to second order, in the two variables X and t as follows:

where φ∗is used to denote the value φ(X + dX, t + dt), and φ denotes the value

φ(X, t) We will now consider the magnitude of the terms dX2, dX dt, and dt2

where we have used the fact that, since dZ ∼ N(0, 1), the variance of dZ,

E [dZ2], is by definition equal to 1 Using these values in Eq (2.3.3) and

substi-tuting for dX from Eq (2.3.1), we obtain:

important applications in the pricing of financial derivatives Here the function

φ(S, t) is taken as the price of a financial derivative, f (S, t), that depends on the value of an underlying asset S, which is assumed to follow GBM In Chapter 4

we will use Eq (2.3.6) to derive the (Black–Scholes) partial differential equationthat is satisfied by the price of a financial derivative

We can also use Eq (2.3.3) to derive the process followed by φ = log(S t ) Wehave:

These results can easily be generalized to include time varying drift andvolatility Now instead of Eq (2.2.3) we have

dS t = S t μ t dt + S t σ t dW t (2.3.12)which results in



(2.4.1)

where W P

t is Brownian motion (possibly with drift) under probability measure

P, see Baxter and Rennie (1996) Under probability measure Q we have:

W t Q = W P

0

where W t Qis also Brownian motion (possibly with drift)

We can also write

Girsanov’s theorem thus provides a mechanism for changing the drift of aBrownian motion

2.5 Ito’s lemma for multiasset geometric Brownian motion

We will now consider the n-dimensional stochastic process:

where A and B are n-element vectors respectively containing the constants,

a i , i = 1, , n, and b i , i = 1, , n The stochastic vector dX contains the

n stochastic variables X i , i = 1, , n.

We will assume that the n element random vector dZ is drawn from a

mul-tivariate normal distribution with zero mean and covariance matrix C That is,

The diagonal elements of C are C ii = Var[dW i ] = dt, i = 1, , n, and

off-diagonal elements are

C = E[dW dW ] = ρ dt, i = 1, , n, j = 1, , n, i = j

As in Section 2.3 we want to find the process followed by a function of the

stochastic vector X, that is the process followed by φ(X, t) This can be done by applying an n-dimensional Taylor expansion, up to second order, in the variables

where φ∗is used to denote the value φ(X + dX, t + dt), and φ denotes the value

φ(X, t) We will now consider the magnitude of the terms dX i dX j , dX i dt, and

dt2as dt → 0 Expanding the terms dX i dX j and dX i dt we have:

where dφ = φ∗− φ.

Now

E [dX i dX j ] = E[b i b j dt dZ i dZ j ] = b i b j dtE[dZ i dZ j ] = b i b j ρ ij dt where ρ ij is the correlation coefficient between the ith and j th assets.

Using these values in Eq (2.5.5), and substituting for dX i from Eq (2.5.1),

where μ i is the constant drift of the ith asset and σ i is the constant volatility of

the ith asset, then substituting X i = S i , a i = μ i S i , and b i = σ i S i into Eq (2.5.7)yields:

2.6 Ito product and quotient rules in two dimensions

We will now derive expressions for the product and quotient of two stochastic

processes In this case φ → φ(X1, X2), with

2.6.1 Ito product rule

Here φ = φ(X1X2), and the partial derivatives are as follows:

dφ = X2dX1+ X1dX2+2E [dX1dX2]

2

and the product rule is

d(X1X2) = X2dX1+ X1dX2+ E[dX1dX2] (2.6.2)

Brownian motion with one source of randomness

For the special case where X1is Brownian motion and X2has no random term

2.6.2 Ito quotient rule

Here φ = φ(X1/X2)and the partial derivatives are as follows:

2.7 Ito product in n dimensions

Using Eq (2.5.7) we will now derive an expression for the product of n chastic processes In this case φ → n

sto-i=1X i, and the partial derivatives are asfollows:

2.8 The Brownian bridge

Let a Brownian process have values W t0 at time t0and W t1 at time t1 We want

to find the conditional distribution of W t , where t0 < t < t1 This distribution

will be denoted by P (W t |{W t0 , W t1 }), to indicate that W t is conditional on the

end values W t0 and W t1 We now write W t0 and W t1 as

W t = W t0 +√t − t0X t , X t ∼ N(0, 1), (2.8.1)

W t1 = W t +√t1− tY t , Y t ∼ N(0, 1), (2.8.2)

where X t and Y t are independent normal variates

Combining Eqs (2.8.1) and (2.8.2) we have

Now P (W t |{W t0, W t1}) = P (X t |Z t ) , the probability distribution of X t

condi-tional on Z t From Bayes law

Since X t , Y t and Z t are Gaussians we can write

Therefore P (X t |Z t )is a Gaussian distribution with:

The variate X t = E[X t] +√Var[X t ]Z t has the same distribution as P (X t |Z t )

So we can substitute X t for X t in Eq (2.8.1) to obtain:

W t = W t0 +√t − t0



E [X t] +Var[Xt ]Z t

which gives:

An alternative derivation of the Brownian bridge is given in Appendix G

2.9 Time-transformed Brownian motion

Let us consider the Brownian motion:

and also the scaled and time-transformed Brownian motion

where the scale factor, a t , is a real function and the time transformation, f t, is a

continuous increasing function satisfying f  0; see Cox and Miller (1965)

Using Ito’s lemma,

2.9.1 Scaled Brownian motion

We will prove that W t defined by

dY t =c2dt dZ t

which gives

dY t = c√dt dZ t = c dW t

Therefore W t = dY t /cis Brownian motion

2.9.2 The Ornstein–Uhlenbeck process

We will now show that the Ornstein–Uhlenbeck process (see Section 2.10) can

be represented as follows:

Y W,t = exp(−αt)W ψ t where ψ t = σ2exp(2αt )

Proof From Eqs (2.9.2) and (2.9.7) we have:

f t =σ2exp(σ2exp(2αt ))

2α and a t = exp(−2αt) (2.9.8)Therefore

f

t dt =σ2exp(2αt ) = σ exp(αt)√dt (2.9.11)Thus

dY W,t = −αY W,t dt + exp(−αt)σ exp(αt)√dt dZ (2.9.12)which means that

dY W,t = −αY W,t dt + σ dW t (2.9.13)From Eq (2.9.13) it can be seen that conditional mean and variance are

E [dY W,t |F t ] = αY W,t dt (2.9.14)Var[dY W,t |F t ] = σ2dt (2.9.15)



where α > 0 and t → ∞

(2.9.16)So

(2.9.18)

Y W,s = exp(−αs)W ψ s where ψ s =

σ2exp(2αs) 2α

(2.9.19)The covariance is:

Cov[YW,s , Y W,t ] = E[Y W,t Y W,s ] − E[Y W,t ]E[Y W,s]

since E [Y W,s ] = E[Y W,t] = 0

Shortening the notation of Y W,t to Y t we obtain

Cov[Ys , Y t ] = Eexp( −αt)W ψ t exp( −αt)W ψ s



= exp−α(t + s)E

{W ψ t W ψ s}From Eq (2.1.2)

The Ornstein–Uhlenbeck process is often used to model interest rates because

of its mean reverting property It is defined by the equation

Using the integrating factor exp(αt) we have:

exp(αt ) dX t = −αX t exp(αt ) dt + σ exp(αt) dW t

So from Eqs (2.10.2) and (2.10.3) we obtain

So substituting the above expression into Eq (2.10.8)

(2.10.11) allow us to write the distribution of X t as

where K = 2α/σ2and γ = exp(−α(t − t ))

Ornstein–Uhlenbeck stochastic paths can thus be simulated using

2.11 The Ornstein–Uhlenbeck bridge

Let an Ornstein–Uhlenbeck process have value X t0 at time t0and X t1 at time t1

We are interested in the distribution of X t at an intermediate point, that is

V t =(1 − exp(−2α(t − t0)))(1 − exp(−2α(t1− t)))

2α(1 − exp(−2α(t1− t0))) (2.11.2)

where γ = exp(−α(t − t ))

Ornstein–Uhlenbeck stochastic paths can thus be simulated using< /p>

2.11...