Analytical Finance: Volume I: The Mathematics of Equity Derivatives, Markets, Risk and Valuation

509 Pages·2017·11.59 MB·English

by  Jan R. M. Röman (auth.)

The Mathematics of Equity Derivatives,
Markets, Risk and Valuation
ANALYTICAL
FINANCE
VOLUME I
JAN R. M. RÖMAN
Analytical Finance: Volume I
JanR.M.Röman
Analytical Finance:
Volume I
The Mathematics of Equity Derivatives,
Markets, Risk and Valuation
JanR.M.Röman
Västerås,Sweden
ISBN978-3-319-34026-5 ISBN978-3-319-34027-2 (eBook)
DOI10.1007/978-3-319-34027-2
LibraryofCongressControlNumber:2016956452
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–
To my soulmate, supporter and love
Jing Fang
Preface
This book is based upon lecture notes, used and developed for the course
Analytical Finance I at Mälardalen University in Sweden. The aim is to cover
the most essential elements of valuing derivatives on equity markets. This
will also include the maths needed to understand the theory behind the
pricing of the market instruments, that is, probability theory and stochastic
processes. We will include pricing with time-discrete models and models in
continuous time.
First, in Chap. 1 and 2 we give a short introduction to trading, risk and
arbitrage-free pricing, which is the platform for the rest of the book. Then a
number of different binomial models are discussed. Binomial models are impor-
tant, not only to understand arbitrage and martingales, but also they are widely
usedtocalculatethepriceandtheGreeksformanytypesofderivative.Binomial
models are used in trading software to handle and value several kinds of
derivative, especially Bermudan and American type options. We also discuss
how to increase accuracy when using binomial models. We continue with an
introductiontonumericalmethodstosolvepartialdifferentialequations(PDEs)
and Monte Carlo simulations.
InChap.3,anintroductiontoprobabilitytheoryandstochasticintegration
fi
is given. Thereafter we are ready to study continuous nance and partial
fi
differential equations, which is used to model many nancial derivatives. We
–
focus on the Black Scholes equation in particular. In the continuous time
model,therearenosolutionstoAmericanoptions,sincetheycanbeexercised
fi
during the entire lifetime of the contracts. Therefore we have no well-de ned
boundary condition. Since most exchange-traded options with stocks as
vii
viii Preface
underlying are of American type, westill need to use descrete models, such as
the binomial model.
–
We will also discuss a number of generalizations relating to Black Scholes,
suchasstochastic volatilityandtime-dependentparameters. Wealsodiscussa
number of analytical approximations for American options.
A short introduction to Poisson processes is also given. Then we study
diffusion processes in general, martingale representation and the Girsanov
fi –
theorem.Before nishingoffwithageneralguidetopricingviaBlack Scholes
we also give an introduction to exotic options such as weather derivatives and
volatility models.
fi
As we will see, many kinds of nancial instrument can be valued via a
discountedexpectedpayoffofacontingentclaiminthefuture.Wewilldenote
this expectation E[X(T)] where X(T) is the so-called contingent claim at time
T.Thisfuturevaluemustthenbediscountedwitharisk-freeinterestrate,r,to
givethepresentvalueoftheclaim.Ifweusecontinuouscompoundingwecan
write the present value of the contingent claim as
XðtÞ ¼e(cid:2)rðT(cid:2)tÞE½XðTÞ(cid:3):
In the equation above, T is the maturity time and t the present time.
Example: If you buy a call option on an underlying (stock) with maturity
TandstrikepriceK,youwillhavetheright,butnottheobligation,tobuythe
stock at time T, to the price K. If S(t) represents the stock price at time t, the
contingent claim can be expressed as X(T) ¼ max{S(T) – K, 0}. This means
that the present value is given by
XðtÞ ¼e(cid:2)rðT(cid:2)tÞE½XðTÞ(cid:3) ¼e(cid:2)rðT(cid:2)tÞE½maxfSðTÞ (cid:2)K, 0g(cid:3):
The max function indicates a price of zero if K (cid:4) S(T). With this condition
youcanbuytheunderlyingstockatalower(same)priceonthemarket,sothe
option is worthless.
By solving this expectation value we will see that this can be given
– –
(in continuous time) as the Black Scholes Merton formula. But generally
we have a solution as
XðtÞ ¼Sð0Þ:Q1ðSðTÞ>KÞ (cid:2)e(cid:2)rðT(cid:2)tÞK:Q2ðSðTÞ>KÞ;
> >
where Q (S(T) K) and Q (S(T) K) make up the risk neutral probability
1 2
“
for the underlying price to reach the strike price K in different reference
” fi – –
systems . This can be simpli ed to the Black Scholes Merton formula as
Preface ix
XðtÞ ¼Sð0Þ:Nðd1Þ (cid:2)e(cid:2)rðT(cid:2)tÞK:Nðd2Þ:
Here d and d are given (derived) variables. N(x) is the standard normal
1 2
distributionwithmean0andvariance1,soN(d )representtheprobabilityfor
2
thestocktoreachthestrikepriceK.Thevariablesd andd willdependonthe
1 2
initial stock price, the strike price, interest rate, maturity time and volatility.
fi
Thevolatility isa measure ofhowmuch the stock pricemay vary inaspeci c
period in time. Normally we use 252 days, since this is an approximation of
the number of trading days in a year.
Also remark that by buying a call option (i.e., going long in the option
contract), as in the example above, we do not take any risk. The reason is that
wecannotlosemoremoneythanwhatweinvested.Thisisbecausewehavethe
fi
right,butnottheobligation,toful lthecontract.Theseller,ontheotherhand,
takestherisk,sincehe/shehastoselltheunderlyingstockatpriceK.Soifhe/she
’
doesn t own the underlying stock he/she might have to buy the stock at a very
high price and then sell it at a much lower price, the option strike price K.
Therefore,asellerofacalloption,whohavetheobligationtoselltheunderlying
stocktotheholder,takesariskypositionifthestockpricebecomeshigherthan
the option strike price.
Acknowledgements
Iwouldliketothankallmystudentsfortheircommentsandquestionsduring
mylectures.SpecialthanksgotoMaiXinwhoaskedmetotranslatemynotes
into English. I would also like to thank Professor Dmitrii Silvestrov, who
encouragedmetoteachAnalyticalFinance,andProfessorAnatolyMalyarenko
for his assistance and advice.
xi
Contents
1 Trading Financial Instruments 1
1.1 Clearing and Settlement 1
1.2 About Risk 7
1.3 Credit and Counterparty Risk 10
1.4 Settlement Risk 12
1.5 Market Risk 12
1.6 Model Risk 13
2 Time-Discrete Models 21
2.1 Pricing via Arbitrage 21
2.2 Martingales 23
2.3 The Central Limit Theorem 25
2.4 A Simple Random Walk 27
2.5 The Binomial Model 29
2.6 Modern Pricing Theory Based on Risk-Neutral Valuation 38
2.7 More on Binomial Models 43
2.8 Finite Difference Methods 65
2.9 Value-at-Risk (VaR) 78
3 Introduction to Probability Theory 91
3.1 Introduction 91
3.2 A Binomial Model 91
3.3 Finite Probability Spaces 93
3.4 Properties of Normal and Log-Normal Distributions 121
xiii