Chapter1 FoundationsofFinancialRisk Management
1.1Introductoryconceptsofthesecuritiesmarket.Subjectoffinancialmathematics
Thenotionofan asset (anythingofvalue)isoneofthefundamentalnotionsinthe financialmathematics.Assetscanbe risky and non-risky.Here risk isunderstood asanuncertaintythatcancauselosses(e.g.,ofwealth).Themosttypical representativesofsuchassetsarethefollowing basicsecurities: stocks S and bonds (bank accounts) B .Thesesecuritiesconstitutethebasisofa financialmarket thatcanbe understoodasaspaceequippedwithastructurefortradingtheassets.
Stocks aresharesecuritiesissuedforaccumulatingcapitalofacompanyforits successfuloperation.Thestockholdergetstherighttoparticipateinthecontrolof thecompanyandtoreceivedividends.Bothdependonthenumberofsharesowned bythestockholder.
Bonds (debentures)aredebtsecuritiesissuedbyagovernmentoracompanyfor accumulatingcapital,restructuringdebts,etc.Incontrasttostocks,bondsareissued foraspecifiedperiodoftime.Theessentialcharacteristicsofabondincludethe exercise(redemption)time, facevalue (redemptioncost), coupons (paymentsupto redemption)and yield (returnuptotheredemptiontime).Thezero-couponbondis similartoabankaccountanditsyieldcorrespondstoabankinterestrate.
An interestrate r ≥ 0 istypicallyquotedbybanksasanannualpercentage. Supposethataclientopensanaccountwithadepositof B0 ,thenattheendofa 1-yearperiodtheclient’snon-riskyprofitis ∆B1 = B1 B0 = rB0 .After n years thebalanceofthisaccountwillbe Bn = Bn 1 + rB0 ,giventhatonlytheinitial deposit B0 isreinvestedeveryyear.Inthiscase r isreferredtoasa simpleinterest Alternatively,theearnedinterestcanbealsoreinvested(compounded), thenatthe endof n yearsthebalancewillbe Bn = Bn 1 (1+ r )= B0 (1+ r )n .Notethathere theratio ∆Bn /Bn 1 reflectstheprofitabilityoftheinvestmentasitisequalto r ,the compoundinterest.
Nowsupposethatinterestiscompounded m timesperyear,then
Such rate r (m) is quoted as a nominal (annual) interest rate and the equivalent effective (annual) interest rate is equal to r = 1 + r (m) m m 1. Let t ≥ 0, and consider the ratio
where r (m) is a nominal annual rate of interest compounded m times per year Then
is called the nominal annual rate of interest compounded continuously. Clearly, Bt = B0 er t .
Thus, the concept of interest is one of the essential components in the description of time evolution of ‘value of money’. Now consider a series of periodic payments (deposits) f0 , f1 ,..., fn (annuity). It follows from the formula for compound interest that the present value of k -th payment is equal to fk 1+ r k ,andthereforethe presentvalueoftheannuityis
WORKED EXAMPLE 1.1
Letaninitialdepositintoabankaccountbe $10, 000.Giventhat r (m) =0 1, findtheaccountbalanceattheendof2yearsfor m =1, 3 and 6.Alsofind thebalanceattheendofeachofyears 1 and 2 iftheinterestiscompounded continuouslyattherate r =0.1.
SOLUTION
Usingthenotionofcompoundinterest,wehave
forinterestcompoundedonceperyear;
forinterestcompoundedthreetimesperyear;
forinterestcompoundedsixtimesperyear. Forinterestcompoundedcontinuouslyweobtain
Stocks aresignificantlymorevolatilethanbonds,andthereforetheyarecharacterizedas riskyassets.Similarlytobonds,onecandefinetheir profitability ρn =∆Sn /Sn 1 ,n =1, 2,...,where Sn isthepriceofastockattime n.Thenwe havethefollowingdiscreteequation Sn = Sn 1 (1+ ρn ),S0 > 0.
Themathematicalmodelofafinancialmarketformedbyabankaccount B (with aninterestrate r )andastock S (withprofitabilities ρn )isreferredtoasa (B,S )market.
Thevolatilityofprices Sn iscausedbyagreatvarietyofsources,someofwhich maynotbeeasilyobserved.Inthiscase,thenotionof randomness appearstobe appropriate,sothat Sn ,andtherefore ρn ,canbeconsideredas randomvariables. Sinceateverytimestep n thepriceofastockgoeseitherupordown,thenitisnatural toassumethatprofitabilities ρn formasequenceofindependentrandomvariables (ρn )∞ n=1 thattakevalues b and a (b>a)withprobabilities p and q respectively (p + q =1).Next,wecanwrite ρn asasumofitsmean µ = bp + aq andarandom variable wn = ρn µ whoseexpectationisequaltozero.Thus,profitability ρn canbedescribedintermsofan‘independentrandomdeviation’ wn fromthemean profitability µ.
Whenthetimestepsbecomesmaller,theoscillationsofprofitabilitybecomemore chaotic.Formallythe‘limit’continuousmodelcanbewrittenas
where µ isthemeanprofitability, σ isthevolatilityofthemarketand ˙ wt isthe Gaussianwhitenoise.
Theformulaeforcompoundandcontinuousratesofinteresttogetherwiththe correspondingequationforstockprices,definethebinomial(Cox-Ross-Rubinstein) andthediffusion(Black-Scholes)modelsofthemarket,respectively.
Aparticipantinafinancialmarketusuallyinvestsfreecapitalinvarious available assetsthatthenforman investmentportfolio.Theprocessofbuildingandmanaging suchaportfolioisindeedthemanagementofthecapital.Theredistributionofa portfoliowiththegoaloflimitingorminimizingtheriskinvariousfinancialtransactionisusuallyreferredtoas hedging.Thecorrespondingportfolioisthencalled a hedgingportfolio.Aninvestmentstrategy(portfolio)thatmaygiveaprofiteven withzeroinitialinvestmentiscalledan arbitrage strategy.Thepresenceofarbitrage reflectstheinstabilityofafinancialmarket.
Thedevelopmentofafinancialmarketofferstheparticipantsthe derivativesecurities,i.e.,securitiesthatareformedonthebasisofthebasicsecurities–stocks andbonds.Thederivativesecurities(forwards,futures,optionsetc.)requiresmaller initialinvestmentandplaytheroleofinsuranceagainstpossiblelosses.Also,they increasetheliquidityofthemarket.
Forexample,supposecompanyAplanstopurchasesharesofcompanyBatthe endoftheyear.Toprotectitselffromapossibleincreaseinsharesprices,company AreachesanagreementwithcompanyBtobuythesharesattheendoftheyearfor afixed(forward)price F .Suchanagreementbetweenthetwocompaniesiscalleda forwardcontract (orsimply, forward).
NowsupposethatcompanyAplanstosellsomesharestocompanyBattheend oftheyear.Toprotectitselffromapossiblefallinpriceofthoseshares, company Abuysa putoption (seller’soption),whichconferstherighttosellthesharesatthe endoftheyearatthefixed strikeprice K .Notethatincontrasttotheforwardscase, aholderofanoptionmustpaya premium toitsissuer.
Futurescontract isanagreementsimilartotheforwardcontractbutthetrading takesplaceona stockexchange,aspecialorganizationthatmanagesthetradingof variousgoods,financialinstrumentsandservices.
Finally,wereiterateherethatmathematicalmodelsoffinancialmarkets, methodologiesforpricingvariousfinancialinstrumentsandforconstructingoptimal(minimizingrisk)investmentstrategiesareallsubjecttomodernfinancialmathematics.
1.2Probabilisticfoundationsof financialmodellingand pricingofcontingentclaims
Supposethatanon-riskyasset B andariskyasset S arecompletelydescribedat anytime n =0, 1, 2,... bytheirprices.Therefore,itisnaturaltoassumethatthe pricedynamicsofthesesecuritiesistheessentialcomponentofafinancialmarket. Thesedynamicsarerepresentedbythefollowingequations
∆Bn = rBn 1 ,B0 =1 ,
where ∆Bn = Bn Bn 1 ,
Sn = Sn Sn 1 ,n =1, 2,... ; r ≥ 0 isaconstant rateofinterestand ρn willbespecifiedlaterinthissection. Anotherimportantcomponentofafinancialmarketisthesetofadmissibleactionsorstrategiesthatareallowedindealingwithassets B and S .Asequence π =(πn )∞ n=1 ≡ (βn ,γn )∞ n=1 iscalledan investmentstrategy (portfolio)ifforany n =1, 2,... thequantities βn and γn aredeterminedbyprices S1 ,...Sn 1 .In otherwords, βn = βn (S1 ,...Sn 1 ) and γn = γn (S1 ,...Sn 1 ) arefunctionsof S1 ,...Sn 1 andtheyareinterpretedastheamountsofassets B and S ,respectively, attime n.The value ofaportfolio π is
where βn Bn representsthepartofthecapitaldepositedinabankaccountand γn Sn representstheinvestmentinshares.Ifthevalueofaportfoliocanchange onlydue tochangesinassetsprices: ∆X π n = X π n X π n 1 = βn ∆Bn + γn ∆Sn ,then π is saidtobea self-financing portfolio.Theclassofallsuchportfoliosisdenoted SF . Acommonfeatureofallderivativesecuritiesina (B,S )-marketistheirpotentialliability(payoff) fN atafuturetime N .Forexample,forforwardswehave fN = SN F andforcalloptions fN =(SN K )+ ≡ max{SN K, 0}.Such
liabilities inherent in derivative securities are called contingent claims One of the most important problems in the theory of contingent claims is their pricing at any time before the expiry date N . This problem is related to the problem of hedging contingent claims. A self-financing portfolio is called a hedge for a contingent claim fN if X π n ≥ fN for any behavior of the market. If a hedging portfolio is not unique, then it is important to find a hedge π ∗ with the minimum value: X π ∗ n ≤ X π n for any other hedge π . Hedge π ∗ is called the minimal hedge. The minimal hedge gives an obvious solution to the problem of pricing a contingent claim: the fair price of the claim is equal to the value of the minimal hedging portfolio. Furthermore, the minimal hedge manages the risk inherent in a contingent claim.
Next we introduce some basic notions from probability theory and stochastic analysis that are helpful in studying risky assets. We start with the fundamental notion of an ‘experiment’ when the set of possible outcomes of the experiment is known but it is not known a priori which of those outcomes will take place (this constitutes the randomness of the experiment).
Example 1.1 (Trading on a stock exchange)
A set of p ossible exchange rates b etween the dollar and the euro is always known b efore the b eginning of trading, but not the exact value.
Let Ω be the set of all elementary outcomes ω and let F be the set of all events (non-elementary outcomes), which contains the impossible event ∅ and the certain event Ω.
Next, suppose that after repeating an experiment n times, an event A ∈ F occurred nA times. Let us consider experiments whose ‘randomness’ possesses the following property of statistical stability: for any event A there is a number P (A) ∈ [0, 1] such that nA /n → P (A) as n → ∞ This number P (A) is called the probability of event A. Probability P : F → [0, 1] is a function with the following properties:
1. P (Ω) = 1 and P (∅)= 0;
2. P ∪k Ak = k P (Ak ) for Ai ∩ Aj = ∅.
The triple (Ω, F , P ) is called a probability space. Every event A ∈ F can be associated with its indicator:
IA (ω )= 1 , if ω ∈ A 0 , if ω ∈ Ω \ A .
Anymeasurablefunction X :Ω → R iscalleda randomvariable.Anindicatoris animportantsimplestexampleofarandomvariable.Arandomvariable X iscalled discrete iftherangeoffunction X (·) iscountable: (xk )∞ k =1 .Inthiscasewehavethe followingrepresentation
X (ω )= ∞ k =1 xk IAk (ω ) ,
where Ak ∈ F and ∪k Ak = Ω A discrete random variable X is called simple if the corresponding sum is finite. The function
FX (x) := P ({ω : X ≤ x}) , x ∈ R is called the distribution function of X For a discrete X we have
FX (x)= k :xk ≤x P ({ω : X = xk }) ≡ k :xk ≤x pk
The sequence (pk )∞ k =1 is called the probability distribution of a discrete random variable X . If function FX (·) is continuous on R , then the corresponding random variable X is said to be continuous If there exists a non-negative function p(·) such that
FX (x)= x ∞ p(y )dy , then X is called an absolutely continuous random variable and p is its density The expectation (or mean value) of X in these cases is
E (X )= k ≥1 xk pk and
E (X )= R xp(x)dx ,
respectively. Given a random variable X , for most functions g : R → R it is possible to define a random variable Y = g (X ) with expectation
E (Y )= k ≥1 g (xk )pk in the discrete case and E (Y )= R g (x)p(x)dx foracontinuous Y .Inparticular,thequantity
V (X )= E X E (X ) 2 iscalledthe variance of X .
Example 1.2 (Examples of discrete probability distributions)
1.Bernoulli:
p0 = P ({ω : X = a})= p,p1 = P ({ω : X = b})=1 p, where p ∈ [0, 1]and a,b ∈ R.
2.Binomial: pm = P ({ω : X = m})= n k pm (1 p)n m ,
where p ∈ [0, 1],n ≥ 1and m =0, 1,...,n.
3.Poisson(withparameter λ> 0):
for m =0, 1,...
Oneofthemostimportantexamplesofanabsolutelycontinuousrandomvariable isaGaussian(ornormal)randomvariablewiththedensity
)=
,x,m ∈ R ,σ> 0 ,
where m = E (X ) isitsmeanvalueand σ 2 = V (X ) isitsvariance.Inthiscaseone usuallywrites X = N (m,σ 2 ).
Considerapositiverandomvariable Z onaprobabilityspace (Ω, F ,P ).Suppose that E (Z )=1,thenforanyevent A ∈F defineitsnewprobability
(A)= E (ZIA ) (1.1)
Theexpectationofarandomvariable X withrespecttothisnewprobabilityis
Theproofofthisformulaisbasedonthefollowingsimpleobservation
forrealconstants ci .Randomvariable Z iscalledthe density oftheprobability P withrespectto P .
Forthesakeofsimplicity,inthefollowingdiscussionwerestrictourselvesto thecaseofdiscreterandomvariables X and Y withvalues (xi )∞ i=1 and (yi )∞ i=1 respectively.Theprobabilities
form the joint distribution of X and Y
Denote pi = j pij and pj = i pij , then random variables X and Y are called independent if pij = pi pj , which implies that E (X Y )= E (X )E (Y ).
The quantity
E (X |Y = yi ) := i xi pij pj
is called the conditional expectation of X with respect to {Y = yi }. The random variable E (X |Y ) is called the conditional expectation of X with respect to Y if E (X |Y ) is equal to E (X |Y = yi ) on every set {ω : Y = yi }. In particular, for indicators X = IA and Y = IB we obtain
E (X |Y )= P (A|B ) = P (AB ) P (B ) .
We mention some properties of conditional expectations:
1. E (X )= E E (X |Y ) , in particular, for X = IA and Y = IB we have P (A)= P (B )P (A|B ) + P (Ω \ B )P (A|Ω \ B );
2. if X and Y are independent, then E (X |Y )= E (X );
3. since by the definition E (X |Y ) is a function of Y , then conditional expectation can be interpreted as a prediction of X given the information from the ‘observed’ random variable Y
Finally, for a random variable X with values in {0, 1, 2,.. .} we introduce the notion of a generating function φX (x)= E (z X )= i z i pi
Wehave
(1)=1 ,
and
X1 +···
forindependentrandomvariables X1 ,...,Xk .
i (
Example 1.3 (Trading on a stock exchange: Revisited) Considerthefollowingtimescale: n =0(presenttime), ...,n = N (canbe onemonth,quarter,yearetc.).
Anelementaryoutcomecanbewrittenintheformofasequence ω = (ω1 ,...,ωN ),where ωi isanelementaryoutcomerepresentingtheresultsof tradingattimestep i =1,...,N .Nowweconsideraprobabilityspace
(Ω, FN ,P )thatcontainsalltradingresultsuptotime N .Forany n ≤ N we alsointroducethecorrespondingprobabilityspace(Ω, Fn ,P )withelementary outcomes(ω1 ,...,ωn ) ∈Fn ⊆FN .
Thus,todescribeevolutionoftradingonastockexchangeweneedafiltered probabilityspace(Ω, FN , F,P )calleda stochasticbasis,where F =(Fn )n≤N iscalleda filtration (or informationflow):
F0 = {∅, Ω}⊆F1 ⊆ ... ⊆FN .
Fortechnicalreasons,itisconvenienttoassumethatif A ∈Fn ∈ F,then Fn alsocontainsthecomplementof A andisclosedundertakingcountable unionsandintersections,thatis Fn isa σ -algebra.
Nowconsidera (B,S )-market.Sinceasset B isnon-risky,wecanassumethat B (ω ) ≡ Bn forall ω ∈ Ω.Forariskyasset S itisnaturaltoassumethatprices S1 ,...,SN arerandomvariablesonthestochasticbasis (Ω, FN , F,P ).Eachof Sn iscompletelydeterminedbythetradingresultsuptotime n ≤ N orinother words,bythe σ -algebraofevents Fn .Wealsoassumethatthesourcesoftrading randomnessareexhaustedbythestockprices,i.e. Fn = σ (S1 ,...,Sn ) isa σalgebrageneratedbyrandomvariables S1 ,...,Sn .
Letusconsideraspecificexampleofa (B,S )-market.Let ρ1 ,...,ρN beindependentrandomvariablestakingvalues a and b (a<b)withprobabilities P ({ω : ρk = b})= p and P ({ω : ρk = a})=1 p ≡ q .Definetheprobabilitybasis: Ω= {a,b}N isthespaceofsequencesoflength N whoseelements areequaltoeither a or b; F =2Ω isthesetofallsubsetsof Ω.Thefiltration F is generatedbytheprices (Sn ) orequivalentlybythesequence (ρn ):
Fn = σ (S1 ,...,Sn )= σ (ρ1 ,...,ρn ) , whichmeansthateveryrandomvariableontheprobabilityspace (Ω, Fn ,P ) isa functionof S1 ,...,Sn or,equivalently,of ρ1 ,...,ρn duetorelations
∆Sk Sk 1 1= ρk ,k =0, 1,....
Afinancial (B,S )-marketdefinedonthisstochasticbasisiscalled binomial. Consideracontingentclaim fN .Sinceitsrepaymentdayis N ,theningeneral, fN = f (S1 ,...,SN ) isafunctionofall‘history’ S1 ,...,SN .Thekeyproblem nowistoestimate(orpredict) fN atanytime n ≤ N giventheavailablemarket information Fn .Wewouldlikethesepredictions E (fN |Fn ) ,n =0, 1,...,N ,to havethefollowingintuitivelynaturalproperties:
1. E (fN |Fn ) isafunctionof S1 ,...,Sn ,butnotoffutureprices Sn+1 ,...,SN .
2.Apredictionbasedonthe trivial information F0 = {∅, Ω} shouldcoincide withthemeanvalueofacontingentclaim: E (fN |F0 )= E (fN ).
3.Predictionsmustbecompatible:
inparticular
4.Apredictionbasedonallpossibleinformation FN shouldcoincidewiththe contingentclaim: E (fN |FN )= fN
5.Linearity:
(φfN + ψgN |Fn )= φE (fN |Fn )+ ψE (gN |Fn ) for φ and ψ definedbytheinformationin Fn .
6.If fN doesnotdependontheinformationin Fn ,thenapredictionbasedon thisinformationshouldcoincidewiththemeanvalue
(fN |Fn )= E (fN ) .
7.Denote fn = E (fN |Fn ),thenfromproperty3weobtain
forall n ≤ N .Suchstochasticsequencesarecalled martingales.
Howtocalculatepredictions?Comparingthenotionsofaconditionalexpectation andaprediction,weseethatapredictionof fN basedon Fn = σ (S1 ,...,Sn ) is equaltotheconditionalexpectationofarandomvariable fN withrespecttorandom variables S1 ,...,Sn .
WORKEDEXAMPLE1.2
Supposethatthemonthlypriceevolutionofstock S isgivenby Sn = Sn 1 (1+ ρn ) ,n =1, 2,..., whereprofitabilities ρn areindependentrandomvariablestakingvalues 0.2 and 0.1 withprobabilities 0.4 and 0.6 respectively.Giventhatthecurrent price S0 =200($), findthepredictedmeanpriceof S forthenexttwomonths.
SOLUTION Since E (ρ1 )= E (ρ2 )=0.2 · 0.4 0.1 · 0.6=0.02 ,
Wefinishthissectionwithsomefurthernotionsandfactsfromstochasticanalysis. Let (Ω, F , F,P ) beastochasticbasis.Forsimplicityweassumethat Ω isfinite. Considerastochasticsequence X =(Xn , Fn )n≥0 adoptedtofiltration F andsuch that E (|Xn |) < ∞ forall n.If
forall n ≥ 1,then X iscalleda martingale.If
forall n ≥ 1,then X iscalleda submartingale ora supermartingale,respectively. Letapositiverandomvariable Z bethedensityoftheprobability P (see (1.1))withrespectto P .Considerboththeseprobabilitiesonmeasurablespaces (Ω, Fn ),n ≥ 0,anddenotethecorrespondingdensities Zn .Then Zn = E (Z |Fn ) givesanimportantexampleofamartingale.
Anysupermartingale X admitstheDoobdecomposition Xn = Mn An ,
where M isamartingaleand A isanon-decreasing(∆An = An An 1 ≥ 0) (predictable)stochasticsequencesuchthat A0 =0 and An iscompletelydetermined by Fn 1 .Thisfollowsfromthefollowingobservation
Since M 2 isasubmartingale,thenusingDoobdecompositionwehave
2 n = mn + M,M n , where m isamartingaleand M,M isapredictableincreasingsequencecalledthe quadraticvariation of M .Weclearlyhave
and
Forsquare-integrablemartingales M and N onecandefinetheircovariance
M,N n = 1 4 M + N,M + N n − M N,M N n .
Martingales M and N aresaidtobe orthogonal if M,N n =0 or,equivalently,if theirproduct MN isamartingale.
Let M beamartingaleand H beapredictablestochasticsequence.Thenthe quantity
iscalleda discretestochasticintegral.Notethat
.
Considerastochasticsequence U =(Un )n≥0 with U0 =0.Definenewstochastic sequence X by
Thissimplelinearstochasticdifferenceequationhasanobvioussolution
whichiscalleda stochasticexponential. If X isdefinedbyanon-homogeneousequation
thenithastheform
Stochasticexponentialshavethefollowingusefulproperties:
2. ε(U ) isamartingaleifandonlyif U isamartingale;
3. εn (U )=0 forall n ≥ τ0 :=inf {k : εk (U )=0} ;
isthemultiplicationrule.
1.3Thebinomialmodelofa financialmarket.Absence ofarbitrage,uniquenessofarisk-neutralprobability measure,martingalerepresentation.
Thebinomialmodelofa (B,S )-marketwasintroducedintheprevioussection. SometimesthismodelisalsoreferredtoastheCox-Ross-Rubinsteinmodel.Recall thatthedynamicsofthemarketarerepresentedbyequations
∆Bn = rBn 1 ,B0 =1 ,
∆Sn = ρn Sn 1 ,S0 > 0 , where r ≥ 0 isaconstantrateofinterestwith 1 <a<r<b,andprofitabilities
ρn = b withprobability p ∈ [0, 1] a withprobability q =1 p ,n =1,...,N,
formasequenceofindependentidenticallydistributedrandomvariables.The stochasticbasisinthismodelconsistsof Ω= {a,b}N ,thespaceofsequences x =(x1 ,...,xN ) oflength N whoseelementsareequaltoeither a or b; F =2Ω , thesetofallsubsetsof Ω.Theprobability P hasBernoulliprobabilitydistribution with p ∈ [0, 1],sothat
Thefiltration F isgeneratedbythesequence (ρn )n≤N : Fn = σ (ρ1 ,...,ρn ). Intheframeworkofthismodelwecanspecifythefollowingnotions.Apredictablesequence π =(πn )n≤N ≡ (βn ,γn )n≤N isan investmentstrategy (portfolio).A contingentclaim fN isarandomvariableonthestochasticbasis (Ω, F , F,P ). Hedge foracontingentclaim fN isaself-financingportfoliowiththeterminalvalue X π n ≥ fN .Ahedge π ∗ withthevalue X π ∗ n ≤ X π n foranyotherhedge π ,iscalledthe minimalhedge.Aself-financingportfolio π ∈ SF iscalledan arbitrage portfolioif X π 0 =0 ,X π N ≥ 0 and P {ω : X π N > 0} > 0 , whichcanbeinterpretedasanopportunityofmakingaprofitwithoutrisk.
Notethattheriskynatureofa (B,S )-marketisassociatedwithrandomnessof prices Sn .Aparticularchoiceofprobability P (intermsofBernoulliparameter p) allowsonetonumericallyexpressthisrandomness.Ingeneral,theinitialchoiceof P cangiveprobabilisticpropertiesof S suchthatthebehaviorof S isverydifferent fromthebehaviorofanon-riskyasset B .Ontheotherhand,itisclearthatpricing ofcontingentclaimsshouldbe neutraltorisk.Thiscanbeachievedbyintroducing anewprobability P ∗ suchthatthebehaviorsof S and B aresimilarunderthis probability: S and B areonaveragethesameunder P ∗ .Inotherwords,thesequence ofdiscountedprices (Sn /Bn )n≤N mustbe,onaverage,constantwithrespectto probability P ∗ :
For n =1 thisimplies
where p∗ isaBernoulliparameterthatdefines P ∗ .Wehave
andtherefore
is unique, and
Notethatinthiscasewecanfind density Z ∗ N ofprobability P ∗ withrespectto probability P ,i.e.anon-negativerandomvariablesuchthat
Since Ω isdiscrete,weonlyneedtocomputevaluesof Z ∗ N foreveryelementary event {x}.Wehave
andhence
Todescribethebehaviorofdiscountedprices Sn /Bn undertherisk-neutralprobability P ∗ ,wecomputethefollowingconditionalexpectationsforall n ≤ N :
Thismeansthatthesequence (Sn /Bn )n≤N isamartingalewithrespecttotheriskneutralprobability P ∗ .Thisisthereasonthat P ∗ isalsoreferredtoasa martingale probability (martingalemeasure).
Thenextimportantpropertyofabinomialmarketistheabsenceofarbitragestrategies.Suchamarketisreferredtoasa no-arbitragemarket.Consideraself-financing strategy π =(πn )n≤N ≡ (βn ,γn )n≤N ∈ SF withdiscountedvalues X π n /Bn .Usingpropertiesofmartingaleprobability,wehavethatforall n ≤ N E ∗ X π n
whichimpliesthatthediscountedvalueofaself-financingstrategyisamartingale withrespecttotherisk-neutralprobability P ∗ .Thispropertyisusuallyreferredto asthe martingalecharacterizationofself-financingstrategies SF
Further,supposethereexistsanarbitragestrategy π .Fromitsdefinitionwehave
Ontheotherhand,themartingalepropertyof X π n /Bn implies
Now,forprobabilities P and P ∗ thereisapositivedensity Z ∗ sothat P ∗ (A)= E (Z ∗ N IA ) foranyevent A ∈FN .Therefore
, whichcontradictstheassumptionofarbitrage.
Nowweprovethat,inthebinomialmarketframework,anymartingalecanberepresentedintheformofadiscretestochasticintegralwithrespecttosome basicmartingale.Let (ρn )n≤N beasequenceofindependentrandomvariableson (Ω, F ,P ∗ ) definedby
withprobability
withprobability
where 1 <a<r<b.Considerfiltration F generatedbythesequence (ρn ): Fn = σ (ρ1 ,...,ρn ) . Anymartingale (Mn )n≤N ,M0 =0,canbewritteninthe form
where (φn )n≤N ispredictablesequence,and
isa(‘Bernoulli’)martingale.
Since σ -algebras Fn aregeneratedby ρ1 ,...,ρn ,and Mn arecompletelydeterminedby Fn ,thenthereexistfunctions fn = fn (x1 ,...,xn ) with xk equaltoeither a or b,suchthat Mn (ω )= fn (ρ1 (ω ),...,ρn (ω )) ,n ≤ N. Therequiredrepresentation(1.2)canberewrittenintheform ∆Mn (ω )= φk (ω )∆mk or fn (ρ1 (ω ),...,ρn 1 (ω ),b) fn 1 (ρ1 (ω ),...,ρn 1 (ω ))= φn (ω )(b r ) , fn (ρ1 (ω ),...,ρn 1 (ω ),a) fn 1 (ρ1 (ω ),...,ρn 1 (ω ))= φn (ω )(a r ) ,
φn (ω )= fn (ρ1 (ω ),...,ρn 1 (ω ),b) fn 1 (ρ1 (ω ),...,ρn 1 (ω )) (b r )
whichwenowestablish.Themartingalepropertyimplies
or p ∗ fn (ρ1 ,...,ρn 1 ,b) (1 p ∗ )fn (ρ1 ,...,ρn 1 ,a)= fn 1 (ρ1 ,...,ρn 1 )
whichinviewofthechoice p∗ =(r a)/(b a) provestheresult.
Usingtheestablishedmartingalerepresentationwenowcanprovethefollowing representationfordensity Z ∗ N ofthemartingaleprobability P ∗ withrespectto P : Z ∗ N = N k =1 1 µ r σ 2 (ρk µ) = εN µ r σ 2 N k =1 (ρ
, where µ = E (ρk ) ,σ 2 = V (ρk ) ,k =1,...,N
Indeed,consider Z ∗ n = E Z ∗ N Fn ,n =0, 1,...,N .Fromthepropertiesof conditionalexpectationswehavethat (Z ∗ n )n≤N isamartingalewithrespecttoprobability P andfiltration Fn = σ (ρ1 ,...,ρn ).Therefore, Z ∗ n canbewritteninthe form Z ∗ n =1+ n k =1 (ρk µ)φk ,
where φk isapredictablesequence.Since Z ∗ n > 0,wehavethatitsatisfiesthe followingstochasticequation Z ∗ n =1+ n k =1 Z ∗ k 1 φk Z ∗ k 1 (ρk µ) =1+ n k =1 Z ∗ k 1 ψk (ρk µ) ,
andhence
Z ∗ n = n k =1 1+ ψk (ρk µ) .
Takingintoaccountthat Z ∗ N isthedensityofamartingaleprobability,wecancomputethecoefficients ψk = φk /Z ∗ k 1 .For N =1 wehave
0= E ∗ (ρ1 r ) F0 = E ∗ (ρ1 r )= E Z ∗ 1 (ρ1 r ) = E 1+ ψ1 (ρ1 µ) (ρ1 r )
=(µ r )+ ψ1 σ 2 , thus ψ1 = (µ r )/σ 2 .
Nowsupposethat ψk = (µ r )/σ 2 forall k =1,...,N 1,thenusing independenceof ρ1 ,...,ρN weobtain
0= E ∗ (ρN r ) FN 1 = E Z ∗ N (ρN r ) FN 1 Z ∗ N 1 = E 1+ ψN (ρN µ) (ρN r ) FN 1 = E (ρN r )+ ψN (ρN µ)(ρN r ) FN 1 = E (ρN r )+ ψN E (ρN µ)(ρN r ) FN 1
=(µ r )+ ψN σ 2 , whichgives ψN = (µ r )/σ 2 andprovestheclaim.
1.4Hedgingcontingentclaimsinthebinomialmarket model.TheCox-Ross-Rubinsteinformula.Forwardsandfutures.
Intheframeworkofabinomial (B,S )-marketweconsiderafinancialcontract associatedwithacontingentclaim fN withthefuturerepaymentdate N .
If fN isdeterministic,thenitsmarketriskcanbetriviallycomputedsince E (fN |FN ) ≡ fN .Infact,thereisnoriskassociatedwiththerepaymentofthis claimasonecaneasilyfindthepresentvalueofthediscountedclaim fN /BN
If fN dependsonthebehaviorofthemarketduringthecontractperiod [0,N ], thenitisarandomvariable.Theintrinsicriskinthiscaseisrelatedtotheabilityto repay fN .Toestimateandmanagethisrisk,oneshouldbeabletopredict fN given thecurrentmarketinformation Fn ,n ≤ N
Westartthediscussionofamethodologyofpricingcontingentclaimswith two simpleexamplesthatillustratetheessenceof hedging.
WORKEDEXAMPLE1.3
Let Ω= {ω1 ,ω2 } and F0 = {∅, Ω} , F1 = ∅, {ω1 }, {ω2 }, Ω .Consider asingle-periodbinomial (B,S )-marketwith B0 =1($),S0 =100($),B1 = B0 (1+ r )=1+ r =1.2($) assumingthattheannualrateofinterestis r =0.2, and S1 = 150($) withprobability p =0 4 70($) withprobability 1 p =0.6 .
FindthepriceforaEuropeancalloption f1 =(S1 K )+ ≡ max{0,S1 K } ($) withstrikeprice K =100($)
SOLUTION Clearly f1 =(S1 100)+ ≡ max{0,S1 100} = 50($)withprobability0.4 0($)withprobability0.6 .
Theintuitivepriceforthisoptionis
Now,usingtheminimalhedgingapproachtopricing,weconstructaselffinancingstrategy π0 =(β0 ,γ0 )thatreplicatesthefinalvalueoftheoption: X π 1 = f1 .Since X π 1 = β0 (1+ r )+ γ0 S1 ,thenwehave β0 1.2+ γ0 150=50 ,
.
γ
70=0 , whichgives β0 = 36.5and γ0 =5/8.Therefore,the‘minimalhedging’price is X π 0 = β0 + γ0 S0 = 36.5+100 × 5/8 ≈ 26 .
Notethatthisstrategyofmanagingrisk(ofrepayment)assumesthatthe writeroftheoptionattime0sellsthisoptionfor26dollars,borrows36.5 dollars(as β0 isnegative)andinveststheobtained62.5dollarsin5/8(= 62.5/100)sharesofthestock S . Alternatively,wecanfindarisk-neutralprobability p∗ fromtheequation 100= S0 = E ∗ S1 1+ r = 150 p∗ +70(1 p∗ ) 1 2
So p∗ =5/8andthe‘risk-neutral’priceis E ∗ f1 1+ r = 50 × 5/8 1.2 ≈ 26 .
Onthesamemarket, findthepriceofanoptionwiththe finalrepayment f1 =max{S0 ,S1 }− S1 .
SOLUTION Notethat f1 = 30($)withprobability0.6 0($)withprobability0.4 .
Theintuitivepriceforthisoptionis
Usingaminimalhedgingself-financingstrategy π0 =(β0 ,γ0 )wehave
3
8and
8.Therefore,the ‘minimalhedging’priceis
Finally,the‘risk-neutral’priceis
Incontrasttothepreviousexample,thisstrategyassumesthatthewriter of theoptionattime0sellsthisoptionfor9 3dollars,borrows3/8sharesofthe stock S (worthof37.5dollars)andinveststheobtained46.8dollarsinabank account.
Notethatinbothexamplesthe‘minimalhedging’pricecoincideswiththe‘riskneutral’priceandtheydifferfromtheintuitivepricefortheoption.Thisobservation leadsustoamoregeneralstatement: thepriceofacontingentclaimisequaltothe expectationofitsdiscountedvaluewithrespecttoarisk-neutralprobability.
Toverifythis,weconsideracontingentclaim fN onabinomial (B,S )-market. Theconditionalexpectation(withrespecttoarisk-neutralprobability)ofitsdiscountedvalue
isamartingalewiththeboundaryvalues
Itadmitsthefollowingrepresentation
where
Inparticular,
whichmeansthat π ∗ isahedgefor fN .Foranyotherhedge π ,frompropertiesof conditionalexpectationswehave
Thus π ∗ istheminimalhedgeforacontingentclaim fN .
Theinitialvalue CN (f ):= X π ∗ 0 ofthisminimalhedgeiscalledthe price acontingentclaim fN .Asweobservedbefore,itisequalto E ∗ (fN /BN ). NowwecomputethepriceofanarbitraryEuropeancalloptiononabinomial (B,S )-market.Inthiscase fN =(SN K )+ ≡ max{0,SN K }.Recallthata Europeancalloptiongivesitsholdertherighttobuysharesofthestock S atafixed strikeprice K (whichcanbedistinctfromthemarketprice SN )attime N .The writerofsuchanoptionisobligedtosellsharesatthisprice K . Usingthedescribedabovemethodologywehave
Tocomputethelatterexpectationwewrite
Denote
where [[x]] istheintegerpartofarealnumber x.Nowsince
E εN µ r σ 2 N k =1 (
k µ) KI{ω : SN ≥K } = K N k =k0
= K N k =k0
Next,usingpropertiesofstochasticexponentialsandtherepresentation
0 εN N k
,weobtain
N
0 N k =k0
= S0 N k =k0
Introducingthenotation
n
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Title: The minister had to wait
Author: Roger D. Aycock
Release date: December 27, 2023 [eBook #72518]
Language: English
Original publication: New York, NY: King-Size Publications, Inc, 1953
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