More mathematical finance

251 Pages·2011·33.94 MB·English

by  Mark Suresh Joshi

PWP
The long-awaited sequel to the "Concepts and Practice of
ZRm
Mathematical Finance" has now anived. Taking up where the
first volume left off,a range of topics is covered in depth
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Extensive sections include portfolio credit derivatives. quasi
Monte Carlo, the calibration and implement.ltion of由e
L1BOR market model, the acceleration of binomial trees，由e
Mathematical
Fourier transform in option pricing and much more
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Throughout Mark Joshi brings his new unique blend of
由eo町， lucidity, practicality and experience to bear on Issues ω
同 Finance
relevant to 由e working quantit.ltive analyst. E
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Praise for the Concepts and Practice of
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Mathematical Finance: 2
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咱vershadows many 。由er books available on the same subject" 响
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"Mark Joshi succeeds admirably -an excellent s臼rting point for a
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numerate person in the field of mathematical finance."·· Risk
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Magazine
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"Very few books provide a balance between financialtheory and
practice. This book is one of the few books that strikes 由at ]OME
balance." .. SIAM Review
ISBN 978-0-9871228-0-3
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Contents
This publication is in copyright. Exceptwhere permitted by law,
noreproductionofany partmay takeplacewithoutthe written Preface Xlll
permission ofthe copyrightholder.
Chapter 1. Optionality,convexityand volatility t
, I
s
1.1. Introduction i
t
Firstpublished 2011 -
1.2. Volatility andconvexity
1.3. Convexity and optionality 句
3
1.4. Is convexity necessary? 吁
/
0
1.5. Key points 0
0
0
1.6. Furtherreading 0
0
1.7. Exercises
Chapter 2. Where does the money go? n
y
NationalLibrary ofAustralia Cataloguing-in-Publicationentry: 2.1. Introduction ny
n
υ
2.2. The money bleed tiA"
t
Author: Joshi，岛1ark， S. 2.3. Analyzingtheexamples i-..
-
2.4. Volatility convexityand theexistenceofsmiles 句
I
1
Title: More mathematical finance /MarkS. Joshi. 22..56.. FKuerythPeorinretsading 句，白付/11111
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ISBN: 9780987122803 (hbk.) 2.7. Exercises 呵，
"
2222222
Chapter3. TheBacheliermodel 4
Notes: Includes bibliographical references and index. 3
3.1. Introduction 3呵
Subjects: Finance-Mathematical models. 3.2. The pricingformula 3/呵
3.3. Approximations andcomparisons O
Business mathematics.
3.4. Key points 吁
/
3.5. Furtherreading 叮
DeweyNumber: 332.0151 I
3.6. Exercises 竹
/
AYAYAY
Chapter4. DerivingtheDelta 今
，
4.1. Introduction 由?
"
4.2. The stockmeasure ?
"
iv CONTENTS CONTENTS V
4.3. Homogeneity 30 8.3. Computing theloss distribution in asingle-factormodel 73
4.4. Othercases 31 8.4. Turningloss distributions into prices 74
4.5. Key points 32 8.5. Stochasticrecovery rates 76
4.6. Furtherreading 32 8.6. TheFouriertransform approach 77
4.7. Exercises 32 8.7. Bucketing 78
8.8. Key points 79
Chapter5. Volatility derivatives and model-free dynamic replication 33 8.9. Furtherreading 80
5.1. Introduction 33 8.10. Exercises 80
5.2. Varianceswaps 34
5.3. Pricinggeneral volatility derivatives 36 Chapter9. Impliedcorrelation for portfoliocreditderivatives 81
5.4. Hedgingavolatility derivative 38 9.1. Introduction 81
5.5. Key points 40 9.2. Impliedcorrelations 82
5.6. Furtherreading 40 9.3. Basecorrelation 85
5.7. Exercises 40 9.4. Mappingmethodologies 87
9.5. Hedging and thecomputation ofGreeks 89
Chapter6. Creditderivatives 41 9.6. Key points 91
6.1. Introduction 41 9.7. Furtherreading 92
6.2. The basic instruments 42 9.8. Exercises 92
6.3. The philosophy ofpricingcreditderivatives 46
6.4. Hazard rates 48 Chapter 10. Alternate models for portfoliocreditderivatives 93
6.5. Pricingsimplecreditinstruments 50 10.1. Introduction 93
6.6. Key points 50 10.2. Random factor loadings 95
6.7. Furtherreading 51 10.3. Ellipticcopulas 100
6.8. Exercises 51 10.4. Multiple default processes 102
10.5. Intensity Gamma 104
Chapter7. TheMonte Carlo pricingofportfoliocreditderivatives 53 10.6. Key points 111
7.1. Introduction 53 10.7. Furtherreading 111
7.2. TheLi model 54 10.8. Exercises 111
7.3. Importancesamplingfor basketdefaultswaps 56
7.4. Tranched CDOs byMonte Carlo 59 Chapter 11. Thenon-commutativity ofdiscretization 113
7.5. The defaultdensity in theLi Model 62 11.1. Introduction 113
7.6. Thelikelihood ratio method for basketcreditderivatives 63 11.2. Discretization and risk-neutrality 113
7.7. Thepathwise method for nth-to-defaultswaps 65 11.3. Discretization and Greeks 117
7.8. Key points 67 11.4. Factorreduction 120
7.9. Furtherreading 67 11.5. Importancesampling 122
7.10. Exercises 68 11.6. Coordinatechanges 123
11.7. Calibration 124
Chapter8. Quasi-analytic methods for pricingportfoliocreditderivatives 71 11.8. Key points 127
8.1. Introduction 71 11.9. Furtherreading 127
8.2. Theloss distribution for independentdefaults 72 11.10. Exercises 127
VI CONTENTS CONTENTS Vll
l151517222025255
0
Chapter 12. Whatis afactor? 129 Chapter 15. QuasiMonteCarlo Simulation 0
0
12.1. Introduction 129 15.1. Introduction 0
0
12.2. Factors for an implementationoftheL1\在M 130 15.2. Choices and morechoices O
A
12.3. Factorreduction 132 15.3. Theproperuse ofSobol numbers Y
A
12.4. The numberofcommonfactors 136 15.4. Assessingconvergence U
n
12.5. The dimension ofthe spaceattainable 138 15.5. Key points υ
A
12.6. Markovian dimension with drifts 142 15.6. Furtherreading υ
A
12.7. Markov functional models 144 15.7. Exercises υ
12.8. 1\在atrix separability 146
Chapter 16. Pricing continuous barrier options using ajump-diffusion
12.9. Key points 148
207
model
12.10. Furtherreading 148
207
16.1. Introduction
12.11. Exercises 149
209
16.2. TheMertonjump-diffusion model
16.3. Importancesamplingandstratification 210
Chapter 13. Early exerciseandMonteCarlo Simulation 151 16.4. Thepriceconditional on nojumps occurring 211
13.1. Introduction 151 212
16.5. The algorithm
13.2. A sketchoftheleast-squares method 152 213
16.6. Numerical results
13.3. The details ofthe least-squares algorithm 153 217
16.7. Key points
13.4. Carrying outthe regression 155 218
16.8. Furtherreading
13.5. Breaking acontract 157 218
16.9. Exercises
13.6. Assessing andextending least-squares 159
13.7. Upperbounds and the seller's price 160 Chapter 17. TheFourier-Laplacetransform and option pricing 219
13.8. Recharacterising theoptimal hedge 163 219
17.1. Introduction
13.9. Upper 如ounds for breakablecontracts 165 17.2. Definitions and basic results 219
13.10. Neverexercisesub-optimally 166 17.3. Working with the logforward 228
13.11. Multiplicative upperbounds 167 17.4. TheFouriertransform in log-strikespace 233
13.12. Key points 172 17.5. Thetime-value approach 239
13.13. Furtherreading 172 17.6. The probability decomposition approach 241
13.14. Exercises 172 17.7. Workingwith characteristic functions 242
244
17.8. Known characteristicfunctions
Chapter 14. TheBrownian bridge 175 17.9. TheHeston characteristicfunction 247
14.1. Introduction 175 17.10. Numerical implementation 249
14.2. Reducing to thedriftless case 175 17.11. Key points 251
14.3. The law ofthe minimum for aBrownian bridge 177 17.12. Furtherreading 251
14.4. The distribution atintervening times 178 17.13. Exercises 251
14.5. UsingtheBrownian bridgefor path generation 180
253
14.6. The geometric bridge 181 Chapter 18. Thecos method
253
14.7. Key points 183 18.1. Introduction
253
14.8. Furtherreading 183 18.2. Cosineseries
255
14.9. Exercises 183 18.3. Cosineseries andcharacteristicfunctions
CONTENTS 1X
V11l CONT丑NTS
306
18.4. European option pricing 256 21.11. Exercises
18.5. Homogeneous models and thecos method 259
Chapter22. Adjoint and automaticGreeks 307
18.6. Bermudan options 260
307
22.1. Introduction
18.7. American options 263
22.2. Model Deltas usingthe Giles一Glassermanmethod 308
18.8. Key points 264
22.3. PathwiseVegas intheLMM using theGiles-Glasserman method 311
18.9. Furtherreading 264
313
22.4. Theadjointacceleration
18.10. Exercises 264
22.5. TheLMMas asequenceofvectoroperations 320
Chapter 19. Whatare marketmodels? 265 22.6. Thelimitations oftheadjoint method 322
19.1. Introduction 265 22.7. Forwards versus backwards 323
324
19.2. Thegeneral set-up 266 22.8. Key points
324
19.3. Drifts and martingales 267 22.9. Furtherreading
324
19.4. Calibration 268 22.10. Exercises
19.5. Products 272
Chapter23. Estimatingcorrelationfor theLffiORmarketmodel 327
19.6. Key points 279
327
23.1. Introduction
19.7. Furtherreading 280
327
23.2. The set-up
19.8. Exercises 280
328
23.3. Timeparameterization
Chapter20. Discountingin market models 281 23.4. Interactions with boot-strapping 329
331
20.1. Introduction 281 23.5. Factorreduction
332
20.2. Possible numeraires 282 23.6. Other marketmodels
332
20.3. The mostcommon choices and theirconsequences 284 23.7. Time-series step size
333
20.4. Usingthe numeraire todiscount 286 23.8. Correlation smoothing
337
20.5. Numeraire matching,variancereduction and discretization bias 288 23.9. Does itreally matter?
338
20.6. Forwarddiscountingin the spotmeasure 289 23.10. Key points
338
20.7. Key points 290 23.11. Furtherreading
338
20.8. Furtherreading 291 23.12. Exercises
20.9. Exercises 291
Chapter24. Swap-ratemarket models 341
341
Chapter21. Drifts again 293 24.1. Introduction
21.1. Introduction 293 24.2. Deducing thebond-ratiosfor theco-terminal model 342
343
21.2. Rapidcomputation ofdrifts 293 24.3. Cross-variationderivative
346
21.3. Evolving the bond 295 24.4. Swap-ratedriftcomputations
21.4. Positivityissues with bond evolution 297 24.5. Constant maturity marketmodels 348
350
21.5. Predictorcorrector 299 24.6. Co-initial swap-rates
352
21.6. Stopping predictorcorrector 299 24.7. Incremental marketmodels
21.7. Pietersz-Pelsser-Regenmortel 301 24.8. Calibratingtheco-terminal swap-rate marketmodel 356
357
21.8. Numerical comparisonsofdrift methods 303 24.9. Evolvingswap-rates
21.9. Keypoints 305 24.10. LIBORversus swapωrate marketmodels 358
359
21.10. Furtherreading 306 24.11. Key points
X CONTENTS
CONTENTS Xl
24.12. Furtherreading
360 Chapter 28. Theconvergenceofbinomial trees 407
24.13. Exercises
360 28.1. Introduction 407
28.2. Richardsonextrapolation 408
Chapter25. Calibrating marketmodels
363 28.3. Convergenceofsimple trees forEuropean options 412
25.1. Introduction
363 28.4. Convergencetheorems 414
25.2. Understandingpseudo-squareroots
365 28.5. Redesigningtrees 415
25.3. Decomposingpseudo-roots
367 28.6. The Leisen-Reimertree 417
25.4. Time dependenceandfactor maintenance
368 28.7. Higherorderconvergence 419
25.5. Mappingbetween models and swaption approximations
368 28.8. Codefor higherordertrees 420
25.6. Cascadecalibration
371 28.9. More and more trees 422
25.7. Fittingcaplets andco-terminal swaptions
374 28.10. Choicesfortrees 425
25.8. Rescaling andLMMcalibration
381 28.11. American options 426
25.9. Periodmismatch
383 28.12. Assessing accuracy 428
25.10. Global optimization
385 28.13. Truncation choices 429
25.11. Calibration withdisplacements
386 28.14. Key points 430
25.12. Key points
387 28.15. Furtherreading 430
25.13. Furtherreading
388 28.16. Exercises 430
25.14. Exercises
388
Chapter29. Asymmetry in option pricing 433
Chapter26. Cross-currency market models 389 29.1. Introduction 433
26.1. Introduction 389 29.2. American optionality 434
26.2. Notation 390 29.3. Incomplete markets 437
26.3. Dynamics 390 29.4. Transactioncosts 439
26.4. Understandingcali也ration 393 29.5. Key points 441
26.5. Pricinggiven acalibration 395 29.6. Furtherreading 441
26.6.Approximationfoymulas fofthe voiatiiityoftheforwardFXfate 396 29.7. Exercises 441
26.7. Equity-linkednotes
397
26.8. Key points Chapter30. A perfectmodel? 443
398
26.9. Furtherreading 30.1. Introduction 443
399
26.10. Exercises 30.2. The vanillaoptions trader 444
399
30.3. Dynamic hedging with aperfectmodel 445
Chapter27. 1\在ixture models 30.4. Theportfolio 446
401
27.1. Introduction 30.5. Theexotics trader 447
401
27.2. Uncertain parametermodels 30.6. Key points 447
402
27.3. As asmoothingmethodology 30.7. Furtherreading 448
403
27.4. Theadvantages and disadvantages 30.8. Exercises 448
403
27.5. Key points
404 Chapter31. Thefundamental theorem ofasset pricing. 449
27.6. Furtherreading
405 31.1. Introduction 449
27.7. Exercises
405 31.2. Theeasy direction 450
Xll CONTENTS
31.3. Thehard direction in the discretecase
451
31.4. Attainingthe minimal price
454
31.5. Keypoints
456
31.6. Furtherreading
456
31.7. Exercises
456
AppendixA. ThediscreteFouriertransform
457
Preface
A.l. Introduction
457
A.2. Roots ofunity
457
A.3. The discreteFouriertransform
460
A.4. The fastFouriertransform Itisnow ten yearssincethefirstdraftof"theConceptsandPracticeof1\在athe­
462
A.5. ThediscreteFouriertransform andconvolutions maticalFinance" was自nished.Thevolumeofresearchpublishedduringthattime
463
A.6. ThefastFouriertransform and matrix multiplication has beenimmense. Newareas havearisenandmanyquestionshavebeenresolved.
464
A.7. Key points Somemarketssuchasportfoliocreditderivativeshavearisen,boomedandcrashed.
466
A.8. Furtherreading "MoreMathematical Finance" is therefore asequel,and itis intended t。如e asec
466
ond orthird bookon financial mathematics. In particular,rather than recall basic
Bibliography
467 theory,Iwi出1丑IrefertωO
and maximizethe amountofnew material.
Index
477
Thissequelisnotintendedtobecomprehensive.Thefieldisnowfartoolarge
for such an undertaking to bepractical. In anycase,Iam afirm believerin "write
whatyou know."Mostofthe topics in thebookarerelated to my ownresearch in
onewayoranother,andIhopetopass onsomeoftheinsightsIhavegainedfrom
using and implementing these models.τbme that is the essence ofthe book,my
objective is to give the reader my own personal perspectives on how one should
view various issues. Thus whilst most ofthe mathematics and models here pre
sented can be found somewhere in the literature,the perspectives I present often
cannot.
Muchofthebookfocusesonnumericalmethods.Apricingmodelisnotmuch
useunless itcanbeimplementedandcalibrated. TheabilitytocomputeGreeksis
anotheressential. My objective is therefore to show how the mathematics can be
translatedintoanimplementable,usablemodel.However,thisisnotarecipebook.
AlthoughIpresentalgorithms,myobjectiveistogivethereaderanunderstanding
that makes the algorithms clear, rather than to present a piece of pseudo啕code
to copy out. I will, however, occasionally point to where the relevant code can
be found in the QuantLib open source library. I largely avoid presenting purely
numericaltechniques which are well known outsidefinance. Forexample,I leave
the details ofhow to carry out Gaussian integration to othertexts. However,I do
presentextensivediscussion ofhow to use Sobol numbers for quasi-Monte Carlo
simulation,sincethis seems to be amuchmisunderstoodtopic.
X111
XIV PREFACE PREFACE XV
I have restricted this text to mathematical finance in the sense ofderivatives possible.However,thereareinevitablysomedependenciesandInowdiscussthese.
pricing. A more specific butrather unwieldy title might have been "how to think
First,thecreditchaptersshouldbereadinorder.Second,theintroductiontomarket
about some numerical techniques for pricingderivativecontracts."
modelsshouldbereadbeforetheothermarketmodelschapters;theseare,however,
At some point, one must call a halt to writing, and many topics t妇at were largely independent ofeach other. Thecross-currency market model chapter(26)
considered have not made it in to this book.Tf1ese indude Levy processes, OIS does,ofcourse,assumefamiliaritywithmarketmodels.Third,thecoschapter(18)
discounting,SABR,asymptoticexpansionapproximations,solvingSDEs,numer relies on theFourier transform chapter(17). The quasi-Monte Carlochapter (15)
ical methods for solvingPDEs,shortrate models,the HJM model， commodities吨 and the importance sampling chapter (16) both depend on the Brownian bridge
power derivatives, CGMYSV, GPUs, proxy methods for Greek computation, in; chapter (14). The remaining chapters are largely stand alone and can be read in
terPolation methodologies fof interest rates, local volatility, aYITI-value modeh. anyorder.
VAR, CES, mean-variance theory, CAPM, utility theory,APT ... The list is end The websitefor this bookis
less.Even户ally，whenIagainfeel出atIhaveeno
www.markjoshi.com/more
i让t will betimetωo write "EvenMoreMathematical日Finance."
visit there for updates, questions, new editions, typos and news. Please use the
So what topics are covered? I spend four chapters on portfolio credit deriva
forum there to askquestions about the textand to inform meoftypos.
tIves since it is an area that has gone from obscurity to fame to notoriety in the
last few years-I iook at binomial trees in depth since it is a topic which is much Ihaveincludedendofchapterexercises.Thesetakeavarietyofformsranging
misunderstood:we wili see that there are at least twenty difemIlt ways to place from simple computations to complicated proofs. Many of them are more com
the nodes ofa tree,and thateach ofthese can be implemented in at leastsixteen puterprojects than exercisessinceultimately this bookis about modelling. I have
digerentways.MonteCarlotechniquesareexaminedindepthwithcf1aptersonthe not included solutions, but you are encouraged to discuss the problems on the
Brownianbridge,quasi-Montecario,theearlyexerciseproblemandstratiacation- book's website.
Market modeis for pricingexotic interestrate derivatives have been my prin翻 Various versionsofthemanuscripthavebeenreadbyaratherlargenumberof
cipai researcf1interestfor many years.This is reaected bychapters on theirapplim people and I thank them all for their comments. The readers include BarbaraLa
cability, drift computation and aPPI-oxiInation, correlation estimation, swap-rate Scala,Will Wright,Chris Beveridge,Jiun Hong Chan, Stephen Chin,Nick Den
mafket models,calibration, discounting and cross-currency market models.The son, Andrew Downes,Robert Tang, Chao Yang,Ferdinando Ametrano, Paulius
cha户户s on disc削zation， factor reduction, ql邸 Jakubenas,HarryLo,LewBurton,AgustinLebron,OhKang Kwon,AlanLewis,
S臼en四sSI山ti叽viti创es with adjoint m丑led妇lo出ds whilst written in a more gener剖a1 context are Nagulan Saravanamuttu,Graeme West,Dherminder Kainth and Lorenzo Lies饨，
a剖Isωodirectly relevantto marketmodels. as well as many others.
I also include afew chapters on more philosophical questions. These include
chapters on asymmetry,evaluating a perfect model, the fundamental theorem of MarkJoshi
asset pricing,convexity,mixture models and the money bleed.
Melbourne,August2011
Certainchapters have been includedsimply becauseI thinktheresults and/or
the mathematics are neatandIwantthem tosharethem withthereader. Thesein
cludeachapteronhowtodifferentiatetheBlack-Scholesformula,
CHAPTER 1
,
Optionality convexity and volatility
1.1. Introduction
Forcertaincontracts,increasingvolatilitywillalways increaseprices. Forex
ample,in theBlack-Scholes model theVegaofacalloption is always positiveso
price must be an increasing function ofvolatility. In the Merton jump-diffusion
model,priceisanincreasingfunction ofjumpintensitysoagainincreasinguncer
tainty leads to an increase in price.
However, there are certainly plenty of contracts with positive pay-offs for
which increasing volatility can reduce price. Forexample,ifa digital call option
is in-the-money, zero volatility will clearly result in a maximal price. It is also
well-knownthatfor barrieroptions,an increasein volatilitycanresultinahigher
probability ofknock-o时， and therefore a decrease in value. Similarly, in ajump
diffusionmodelincreasingjumpintensitycanresultineitheradecreaseorincrease
in pricefor adigital call option,seeChapter 15 of[105].
Ofcourse,one cannotexpect general monotonicity results to hold for digital
options,asadigitalcallplusadigitalputofthesamestrikeissimplyazero-coupon
bond,which has apriceindependentofvolatility orany otheruncertainty.
Weshallseethatconvexityistheimportantpropertyofacalloptionthatmakes
monotonicity results work, and that in a sense convexity and optionality are the
sameconcept.
1.2. Volatility and convexity
We want a more general conceptofincreasing uncertainty thatencompasses
both increasing volatility in the Black-Scholes model and increasing intensity in
ajump-diffusionmodel.
We shall say thata positiverandom variable X is more uncertain than a pos
itive random variable Y ifthey can be defined on the same probability space in
2 1.OPTIONALITY,CONVEXITYAND VOLATILITY
1.3.CONVEXITYANDOPTIONALITY 3
such a way that
X,defining thefinal stockprice in A is more uncertain than therandom variable,
X=YZ, Y,for B. We needto show that
withY and Z independent. We also require Z to bepositive.
JE(f(X)) 注 JE(f(Y)).
Wehave
logX = logY +logZ, By definition and risk-neutrality, we can write X = YZ. We write JEy for
expectation overY and similarlyfor Z. Wethen have
so
Var(logX) = Var(logY) 十 Var(logZ). (1.2.1) JE(f(X)) =JE(f(YZ)),
NotethatifZ isnotconstantthis immediatelyimpliesthatlogX has highervari (1.2.2) =JEy(JEz(f(YZ))).
ancethaniogY.ThisdeanMOIlincludesincreasingvolatilityintheBiacKJcholes
Observe that f (Ys) is a convex function of s for each value ofY,so applying
world since we have for σ1 注 σ2， that a normal ofstandard deviation is dis
0"1 Jensen's inequality to theexpectationin Z,we have that
甘ibuted as a normal ofstandard deviation ofσ2plusm independentomdsun
dard deviation飞/付 -4. (1.2.3) JE(f(X)) 注 JEy(f(YEz(Z))).
Forthejump-diffusioncase,increasingjumpintensitycanbeseenintermsof However,we know thattheexpectation ofZ is one and weconclude
muitiplying by a random vafiable genefated by thejumps arising fkom a process
with intensityequal to thedifference. (1.2.4) JE(f(X)) 注 JEy(f(Y))，
th NotethatwhenPI-icingderivatives,theincreasein uncertainty Will notafeet which is whatwe wantedto prove.
e meansincethepriceofabrwardcontractmustbeinvariant.Thisimpliesthat
Note that we have shown that theBlack-Scholes price ofany derivative with
the variable Z must have mean equalt01and we impose this restriction for the
a convex pay-offis an increasing function ofvolatility. We have also shown that
restofthis section.
thesameholds inthejump-diffusionmodel as afunction ofeachofvolatility and
We will show that for European options with convex pay-offs, price is an in jumpintensity.
creasing function of uncertainty.Recall that a function, f, on an inteIVal U is
convex iffor all 叽Y E Uand () E [0,1],wehave that
1.3. Convexityand optionality
f(()x 十 (1 - ())y) ζ ()f(x) + (1- ())f(y).
Intheprevious section,by invokingJensen's inequality we wereabletoshow
Geometrically,thechordbetweenany twopointson thegraphliesonorabovethe
that the prices ofEuropean derivatives with convex pay-offs increase with uncer
graph.
tainty. Inthis section,weexplorethefinance behind thisTesult. Forwhatfollows,
YτT、挝h1览er…r1…ω创Ostimpor此t凯antresult台ffrom严pr灿a汕bility出the∞orη')γy吨a灿
itis importantthatwearecarefultomakethedistinction betweenan option anda
is Jensenγ's inequali让ty.
derivative. Anoptionis achoicebetweenoneoftwo,orpossiblymore,portfolios,
THEOREM 1.1. 扩fisaconvexβmction， andX归αrandomν'ariablesuchthat whereas a derivative is the right and obligation to receive a function ofthe stock
JE(/X/) isfinite, andJE(lf(X)I) isfinite then price at maturity. All options can be viewed as derivatives simply by taking the
function to bethe maximum ofthe portfolio values.
JE(f(X)) 注 f(JE(X)).
Forexample,a call option is the rightto choosebetween the empty portfolio
and a portfolio consisting ofa unit ofstock and - K riskless bonds. A put is the
Fora proofsee,forexample,[197].
righttochoosebetweentheemptyportfolioandaportfolioofK bondsandminus
Now suppose we have a derivative C paying a convex function f at time T.
onestock.Notethatwerequirethecompositionofeachportfoliotobefixed atthe
Supposewehavetworisk---neutral models A andB suchthatthemndom variable,
start.