Financial Derivative and Energy Market Valuation: Theory and Implementation in Matlab®

651 Pages·12.111 MB·English

by  Michael Mastro(auth.)

FINANCIAL DERIVATIVE AND
ENERGY MARKET VALUATION
FINANCIAL DERIVATIVE AND
ENERGY MARKET VALUATION
Theory and Implementation
(cid:2)
in Matlab R
Michael Mastro
U.S.NavalResearchLab
Washington,DC
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright©2013byJohnWiley&Sons,Inc.Allrightsreserved.
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LibraryofCongressCataloging-in-PublicationData:
Mastro,MichaelA.,1975–
®
Financialderivativeandenergymarketvaluation:theoryandimplementationinMatlab /
MichaelMastro.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-118-48771-6(cloth)
1. Derivativesecurities. 2. Energyderivatives. 3. MATLAB. I.Title.
HG6024.A3M37742012
332.64’57–dc23
2012031825
PrintedintheUnitedStatesofAmerica
10987654321
CONTENTS
Preface vii
1 Financial Models 1
2 Jump Models 35
3 Options 65
4 Binomial Trees 105
5 Trinomial Trees 131
6 Finite Difference Methods 167
7 Kalman Filter 231
8 Futures and Forwards 245
9 Nonlinear and Non-Gaussian Kalman Filter 295
10 Short-Term Deviation/Long-Term Equilibrium Model 349
11 Futures and Forwards Options 359
12 Fourier Transform 397
13 Fundamentals of Characteristic Functions 459
14 Application of Characteristic Functions 467
15 Levy Processes 505
16 Fourier-Based Option Analysis 547
17 Fundamentals of Stochastic Finance 585
18 Affine Jump-Diffusion Processes 605
Index 645
v
PREFACE
Energy markets and their associated financial derivatives are characterized by
sudden jumps, mean reversion, and stochastic volatility. These aspects necessitate
sophisticated models to properly describe even a subset of these traits. Moreover,
the implementation of these models itself requires advanced numerical methods.
This book establishes the fundamental mathematics and builds up all necessary
statistical, quantitative, and financial theories.
A number of theoretical topics are expanded, including the Fourier transform,
momentgeneratingfunctions,characteristicfunctions,andfiniteandinfiniteactivity
Levyprocessessuchasthealphastable,temperedstable,gamma,variancegamma,
inverse Gaussian, and normal inverse Gaussian processes. Applied mathematics
such as the fast Fourier transform and the fractional fast Fourier transform are
developed and used to generate statistical distributions and for option pricing.
Onthebasisofthisknowledge,state-of-the-artquantitativefinancialmodelsare
developed without the need to refer external sources. Seminal works are derived
andimplemented,includingtheBlack–Scholes,Black,Ornstein–Uhlenbeck,Mer-
ton Gaussian jump diffusion, Kou double exponential jump diffusion, and Heston
stochastic volatility models.
Nevertheless, these models cannot capture the true behavior of the energy
markets. The influential two-factor stochastic convenience yield model and the
short-term long-term model are derived and implemented.
It is shown that adding jumps to these models can be done in a rather ad hoc
manner; however, a thorough discussion of the affine transform formalism is pre-
sented. This provides an elegant framework to augment jumps to the two-factor
models or develop similar jump-diffusion models.
Tofitthesemodelsanddisplaytheirpredictivepower,aparticularfocusismade
indevelopingandutilizingtheKalmanfilter.ForlinearmodelswithGaussiannoise,
the Kalman filter finds an optimal recursive solution with very little computational
burden. For the nonlinear and non-Gaussian models developed in this book, we
build up and exploit the extended, Gauss–Hermite, unscented, Monte Carlo, and
particle Kalman filters.
Assuggestedbythetitle,amajorveinofthisbookistheimplementationofthese
models.Availabilityofworkingcodeofmodernfinancialmodelsislimited,andthe
implementationistypicallyonlydiscussedinanabstractsense.Thisbookdetailsthe
®
necessary steps for implementation and displays the working code. The Matlab
vii
viii PREFACE
environment was selected because it is broadly available and is simple to port to
other popular environments including C++ and C#. In addition, Matlab provides
refined graphing routines that allow our code to focus on the relevant quantitative
and financial concepts.
Michael Mastro
1
Financial Models
1.1. INTRODUCTION
The movement of financial assets and products generally displays some type
of expected return, even over a short period. This expected return trends at a
predictable rate that may be positive, indicating growth; negative, indicating a
decline; or zero. Additionally, there are random movements that are individually
unpredictable; however, the general distribution of these fluctuations is predictable
based on historical movements. The common approach to model randomness is to
assume a single- or multi-component Gaussian process. The generalized format to
describe a time-dependent stochastic process is
dS =α(S,t)dt +σ(S,t)dW ,
t t
where the drift α and volatility σ are functions of time t and asset price S, and
W isaWienerprocess.Ifthedriftα(S,t)=μandvolatilityσ areconstants,then
t
the process
dS =μdt +σdW
t t
isknownasarithmeticBrownianmotion.Thisprocessbyitselfstatesthatthestock
price S will increase (or decrease) without bound at a rate that is not dependent
on the current stock price. Clearly, this does not describe the typical behavior for
an asset, but modified versions of arithmetic motion are useful in finance and are
revisited later in the text.
1.2. GEOMETRIC BROWNIAN MOTION
A more appropriate description of a stock price process is that the movements in
the stock are proportional to the value of the stock. A specific description is that
the overall drift, α(S,t)=μS , is the product of an expected return μ and the
t
current asset price S . Adding a stochastic movement that is also proportional to
t
the current price level gives
dS =μSdt +σSdW ,
t t
®
FinancialDerivativeandEnergyMarketValuation:TheoryandImplementationinMatlab ,
FirstEdition.MichaelMastro.
©2013JohnWiley&Sons,Inc.Published2013byJohnWiley&Sons,Inc.
1
2 FINANCIALMODELS
which is the well known geometric Brownian motion process. A crude discrete
approximation of the stochastic differential equation for geometric Brownian
motion given by
(cid:4)S
t =μ(cid:4)t +σ(cid:4)W
S t
t
isonlyvalidovershorttimeintervals.Thisformdoeshighlightthatthepercentage
change in the stock price (cid:4)S over a sho√rt time interval is normally distributed
S
with mean μ(cid:4)t and standard deviation σ (cid:4)t, where μ is the drift and σ is the
volatility. The shorthand for a normal distribution is
(cid:2) √ (cid:3)
(cid:4)S
∼N μ(cid:4)t,σ (cid:4)t .
S
(cid:4) (cid:5)
Thevarianceofthisstochasticreturnisproportionaltothetimeinterval,var (cid:4)S =
S
σ2(cid:4)t (Hull,2006).OnebenefitofgeometricBrownianmotionisthatnegativeasset
pricesarenotpossiblebecauseanypricechangeisproportionaltothecurrentprice.
Bankruptcy could drive an asset price down to but not past the natural absorbing
barrier at zero (Chance, 1994).
The discrete approximation of the geometric Brownian(cid:4) m(cid:5)otion stochastic
equation is composed of a trend (or expectation) term E dS =μdt and an
S
uncertainty (of deviation) term. The uncertainty term is given by the Wiener
increment √
dW =ε dt,
t t
with E(dW )=0, where ε is the standard normal distribution. It turns out that
t
the variance of dW is equal to the time interval dt. The variance of the Wiener
t
increment was found by evaluating
var(dW )=E[dW −μ ]2 =E(dW 2)−E(dW )2 =E(dW2).
t t dWj t t t
√
Theexpectedvalueof(dW )2 is,bydefinition,E[(dW )2]=E[(ε dt)2](Chance,
t t t
2005). Pulling the dt factor out of the expectation gives
E(dW2)=E(ε2dt)=(E(ε2))dt.
t t t
To evaluate the E(ε2) term requires the computational formula for the variance
t
var(ε )=E[(ε −E(ε ))2]=E[(ε2−2ε E(ε )+E(ε ))2]
t t t t t t t
var(ε )=E(ε2)−2E(ε )E(ε )+E(ε )2 =E(ε2)−2E(ε )2+E(ε )2
t t t t t t t t
var(ε )=E(ε2)−E(ε )2
t t t
→E(ε2)=var(ε )−E(ε )2.
t t t
GEOMETRICBROWNIANMOTION 3
The variance term of a standard normal variable is one, var(ε )=1. The expected
t
value of a standard normal variable E(ε ) and the square of the expected value
t
E(ε )2 are zero. Therefore,
t
E(ε2)=var(ε )−E(ε )2 =1−0,
t t t
the expected value of a squared standard normal variable is one. The value for the
expectation of the Wiener increment squared is given by
E(dW2)=(E(ε2))dt =dt.
t t
This important result states that the square of a Wiener process equals the time
interval, dt =dW2. In other words, the Weiner process is unpredictable but the
t
square of the Weiner process is predictable.
The percentage price change of the stochastic representation of dSt is normally
St
distributed because the stochastic differential equation, written as
dS
t =μdt +σdW ,
S t
t
is a linear transformation of the normally distributed variable dW . The relative
t
return S0+dSt =1+ dSt = St overatimeperiodT istheproductoftheintervening
S0 S0 S0
price changes as displayed by
ST = S1S2 ... St−1 ST ,
S0 S0S1 St−2St−1
where each increment inrelative return Si iscapable of being furthersubdivided
Si−1
in time. A logarithm of the product converts the product series into a summation
series as given by
(cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7) (cid:6) (cid:7)
ln ST =ln S1 +ln S2 +···+ln St−1 +ln ST .
S0 S0 S1 St−2 St−1
Thecentrallimittheoremstatesthatthesummationofalargenumberofidentically
distributed and independent random variables, each with finite mean and variance,
will be approximately normally distributed (Rice, 1995).
1.2.1. Lognormal Stochastic Differential Equation
Theinsightofthepreviousparagraphprovidesamotivationtorecastthegeometric
Brownian motion
dS =μSdt +σSdW,
4 FINANCIALMODELS
as a lognormal diffusion stochastic differential equation. The alteration is accom-
plished by using the function
G=lnS,
and its derivatives
∂G 1 ∂2G 1 ∂G
= =− =0.
∂S S ∂S2 S2 ∂t
Application of Ito’s lemma gives
Infinitesimally
sm(cid:8)a(cid:9)l(cid:10)l→(cid:11)0
∂G 1∂2G ∂G 1∂2G
dG= dS+ (dS)2+ dt + (dt)2
∂S 2 ∂S2 ∂t 2 ∂t2
1 1 1
dG= dS− (dS)2+0
S 2S2
1 1 1
dG= (μSdt +σSdW)− (μSdt +σSdW)2
S 2S2
dG=(μdt +σdW)
⎛ ⎞
⎜ d(cid:8)W(cid:9)2(cid:10)=(cid:11)dt Infis(cid:8)nmita(cid:9)els(cid:10)l→im(cid:11)0ally Infisnm(cid:8)ita(cid:9)els(cid:10)l→i(cid:11)m0ally⎟
1 1 ⎜ ⎟
− ⎜σ2S2 dW2 +μσS2 dtdW +α2S2 dt2 ⎟
2S2 ⎝ ⎠
1
dG=(μdt +σdW)− σ2dt
2
⎛ ⎞
(cid:8) =(cid:9)(cid:10)η (cid:11)
⎜ ⎟
dG=⎜⎝μ− 1σ2⎟⎠dt +σdW,
2
where we have introduced a lognormal return drift factor, η, which is the con-
tinuously compounded return. A solution to the log return stochastic differential
equation is found by integrating
(cid:18) (cid:18) (cid:18)
dG = ηdt + σdW .
u u
The deterministic drift term can be integrated similar to an ordinary differential
equation. The σ coefficient is taken as a time-invariant constant, which greatly
simplifies the stochastic integral to
(cid:18)t
σdW =σ(W −W )=σ(W −0),
u t 0 t
0