金融工程課件07linear2(f)

1、Primbs, MS&E 3451The Linear Functional Form of ArbitragePrimbs, MS&E 3452Linear PricingThe basic argumentWhat linear functionals look likeA first step toward risk neutralityGirsanovs TheoremThe Big PictureSummaryInterpretation as state pricesPrimbs, MS&E 3453Linear PricingThe basic argum

2、entWhat linear functionals look likeA first step toward risk neutralityGirsanovs TheoremThe Big PictureSummaryInterpretation as state pricesPrimbs, MS&E 3454Derivative pricing is nothing more than fitting data pointswith a linear function.PayoffPricexxBondStockThe stock and bond are our “data po

3、ints”Price=L(Payoff)The Big PicturePrimbs, MS&E 3455Derivative pricing is nothing more than fitting data pointswith a linear function.PayoffPricexxBondStockPrice=L(Payoff)The Big PictureTo price a new payoff, we just evaluate our linear function.Call optionPayoffxxxCallPricePrimbs, MS&E 3456

4、Linear PricingThe basic argumentWhat linear functionals look likeA first step toward risk neutralityGirsanovs TheoremThe Big PictureSummaryInterpretation as state pricesPrimbs, MS&E 3457Background:XL:We will consider a pricing functional:PricePayoff:L0)(SSLTPricing the stock0)(BBLTPricing the bo

5、ndWe need to define the space of payoffs. Consider the following example.Example:Primbs, MS&E 3458t = 0t = 1t = 2t = 3t = 4t = 5t = 6t = 0t = 1t = 2t = 3t = 4t = 5t = 6Stock LatticeProbability Space),(PThink of each w as being a pathfor the stock (we want the entire path for path dependent deriv

6、atives.)The payoff can be a functionof the path of the stock. )(wTfwConsiderPrimbs, MS&E 3459Background:XL:We will consider a pricing functional:PricePayoff:LOur space of payoffs should be:Payoff:)(wTf(That is, functions of the path of the stock)(2PL(Technical condition)Primbs, MS&E 34510t =

7、 0t = 1t = 2t = 3t = 4t = 5t = 6An arbitrage is:)()(,(20PLffTwa (price, payoff) pair:with:0)(,(0wTffThat is, a positive payoff that has a negative price!You just receive cash flows!00 fand0)0)(wTfPorPrimbs, MS&E 34511t = 0t = 1t = 2t = 3t = 4t = 5t = 6The Basic Argument:Linearity:)()()(TTTTgLfLg

8、fLIf there is a continuous strictly positive linear pricing functional, then there is no arbitrage.Strict Positivity:0)0)(wTfP0)(wTfL0)(wTfPrimbs, MS&E 34512t = 0t = 1t = 2t = 3t = 4t = 5t = 6Assume there is a strictly positive linear pricing functional,Consider0)0)(wTfP0)(wTfwith0)(0)0)(0ffLfPT

9、Twwby positivityno arbitrage!The Basic Argument:If there is a continuous strictly positive linear pricing functional, then there is no arbitrage.Primbs, MS&E 34513t = 0t = 1t = 2t = 3t = 4t = 5t = 6Consider0)(wTf0)(0ffLTwby linearityThe Basic Argument:Assume there is a strictly positive linear p

10、ricing functional,no arbitrage!If there is a continuous strictly positive linear pricing functional, then there is no arbitrage.Primbs, MS&E 34514Notes:In a discrete setting, there is an equivalence betweenthe existence of a continuous strictly positive linear pricing functional, and absence of

11、arbitrage. In a continuous setting, technical conditions need to beadded before an equivalence can be stated. For practical purposes, you can take them to be equivalent.Primbs, MS&E 34515Linear PricingThe basic argumentWhat linear functionals look likeA first step toward risk neutralityGirsanovs

12、 TheoremThe Big PictureSummaryInterpretation as state pricesPrimbs, MS&E 34516Riesz Representation Theorem:Let )(: )(2PLLbe a continuous linear functional, then there exists a unique )(2PLsuch that:dPffL)(LetHL: )(be a continuous linear functional ona Hilbert space H, then there exists a unique

13、Hsuch that:,)(ffLor, (a little) more generally:Primbs, MS&E 34517A bit of intuition:How do you create linear functions?),(21x1x2221121),(xxxxL),(),(2121xxk12xx122 dimensionsPrimbs, MS&E 34518A bit of intuition:How do you create linear functions?),(21x1x2nnkxxxnkxL.).1,(2211).1,(),.1,(nknkxkk

14、n dimensionsThis is hard to visualize!k12.xx12nxnnkkkx1Primbs, MS&E 34519A bit of intuition:How do you create linear functions?),(21x1x2)()()()(wwwwdPxxL)(),(wwx dimensionsThis is hard to visualize!w)(wPrimbs, MS&E 34520Proof of the Riesz Representation theorem for Hilbert Space:Let M=N(L).

15、M is a closed linear subspace of H (since L is continuous). Assume M is not equal to H, (otherwise, 0, then there is a vector f in M with L(f)=1. Now, if h is a vector in H and L(h)=a, then L(h-af)=L(h)-a=0. So h-L(h)f is in M. Thus:ffhLfhffhLh,)(,)(0fff,So, ifthen,)(hhLUniqueness is left to you.(Re

16、ference: Conway, “A course in functional analysis”)Primbs, MS&E 34521Terminology:Warning: It also goes by the names:State price deflatorStochastic discount factorAnd is closely related to:State price densitiesGreens functionsRisk neutral probabilitiesEquivalent martingale measuresAs far as I can

17、 tell, the main reason for all the different names is to add confusion to the subject. is known as the “Pricing Kernel”:IfdPffL)(is our pricing functional,Primbs, MS&E 34522Linear PricingThe basic argumentWhat linear functionals look likeA first step toward risk neutralityGirsanovs TheoremThe Bi

18、g PictureSummaryInterpretation as state pricesPrimbs, MS&E 34523Interpretation as state prices:Let = R. To each state (path) w we assign a security which pays off $1 if that state (path) occurs.t = 0t = 1t = 2t = 3t = 4t = 5t = 6$1w1I can write the payoff of this security as)(1wwCall its price)(

19、 1wthen)( )(11wwwLPrimbs, MS&E 34524Interpretation as state prices:Let = R. To each state (path) w we assign a security which pays off $1 if that state (path) occurs.t = 0t = 1t = 2t = 3t = 4t = 5t = 6I can write the payoff of this security as)(2wwCall its price)( 2wthen)( )(22wwwL$1w2Primbs, MS

20、&E 34525t = 0t = 1t = 2t = 3t = 4t = 5t = 6For a general payoff, we can decompose it into a linear combination of delta functions.)()()(zzffRzTTwwBy linearity, the price is:)()()()()(zLzfzzfLfLRzTRzTTwww)( )(zzfRzTInterpretation as state prices:In the continuous limit:RzTRzTdzzzfzzf)( )()( )(Pri

21、mbs, MS&E 34526Connections with Riesz Representation:)(wTfL)( )(zzfRzTRzTdzzzf)( )(is a state price density.State prices:Riesz Representation Theorem:dPffLTTw)(RzTdzzpzzf)()()(win terms of a density.RzTRzTdzzzfdzzpzzf)( )()()()(w)( )()(zzpzwThe state price density incorporatesthe underlying prob

22、abilities into it. TogetherPrimbs, MS&E 34527Linear PricingThe basic argumentWhat linear functionals look likeA first step toward risk neutralityGirsanovs TheoremThe Big PictureSummaryInterpretation as state pricesPrimbs, MS&E 34528A Trick:Since is positive, if it integrated to 1, I could ac

23、t like it is a probability density!To get it to integrate to one, I can just normalize it by a constant, K.(Note, is in L1(P) since it is in L2(P), so we can integrate it.)Since nowKz)(integrates to one, I can act like it is the density for a newmeasure, Q.zTTzdPzzffL)()()()(wWe have)()()()()()()(zd

24、PKzzfKzdPzzffLzTzTTwwherezzdPzK)()(So)()()()()()(TQzTzTTfKEzdQzfKzdPKzzfKfLwdQdPKz)(That is:Primbs, MS&E 34529A Trick:Since is positive, if it integrated to 1, I could act like it is a probability density!)()()()(TQzTTfKEzdPzzffLwWe haveAll we have done is to rewrite our integral to make it look

25、 like an expectation.Note that Q is not the real probability measure, it is just normalized so that it integrates to one.Lets verify, and translate our restrictions of linearity and strict positivity to our probability setting:Primbs, MS&E 34530A Trick:Since is positive, if it integrated to 1, I

26、 could act like it is a probability density!)()()()(TQzTTfKEzdPzzffLwWe haveLinearity: this comes from the property of expectations.)()()(TTTQTQTTQTTgLfLgKEfKEgfKEgfLPrimbs, MS&E 34531A Trick:Since is positive, if it integrated to 1, I could act like it is a probability density!)()()()(TQzTTfKEz

27、dPzzffLwWe haveStrict positivity requires that Q and P be equivalent measures.That is Q(A)=0 iff P(A)=0, for all measurable sets A in .On the other hand, assume Q(A)=0, but P(A)0. Then consider thepayoff function:ATf1then0)(1 )(AKQKEfKEfLAQTQTThis payoff costs nothing, but has a positive probability

28、 of occurring (P(A)0).This is an arbitrage.So we must have P(A)=0, and Q and P are equivalent.SincedPKzdQ)(it is clear that P(A)=0 Q(A)=0.Primbs, MS&E 34532The Radon-Nikodym Theorem:(Reference: Royden, “Real Analysis”, Chapter 11)Let be a s-finite measure space, and let Q be a measure defined on

29、 which is absolutely continuous with respect to P. Then there is a nonnegative measurable function such that for each A in we haveThe function is unique modulo sets of P measure zero. AdPAQ)(),(P)(1PLNotation: is called the Radon-Nikodym derivative and is often denoted: dPdQFor us, the Radon-Nikodym

30、 theorem just cements the fact that we can indeedinterchangeably talk in terms of a pricing kernel or an equivalent measure Q. Primbs, MS&E 34533The probabilities Q are known as the “risk neutral” probabilitiesor as an “equivalent martingale measure”. We now know where the “equivalent” comes fro

31、m in the term “equivalent martingale measure”. We will see where the “martingale” term comes from later.Now, lets see why people refer to Q as the “risk neutral”probability measure.Primbs, MS&E 34534Derivative pricing is nothing more than fitting data pointswith a linear function.PayoffPricexxBo

32、ndStockThe stock and bond are our “data points”Price=L(Payoff)The Big PicturePayoffKEQPrimbs, MS&E 34535Calibration: Lets fit our linear pricing functional to the data:In the Black-Scholes setting, we have two data points: the bond and stock) 1 ()(0LBLBeTrTSo: 1 QKEKrTeK0)(ffEefKEfLTQrTTQTUnder

33、Q, all portfolios earn the risk free rate!Hence, the name “risk neutral probabilities”.rTTQeffE0Lets start with the bond:Bond:Price:Payoff:rTeB01TBTherefore: given a generic payoff fTPrimbs, MS&E 34536Calibration: Lets fit our linear pricing functional to the data:In the Black-Scholes setting, w

34、e have two data points: the bond and stock)(0TQrTTSEeSLSSo:Now for the stock:Stock:Price:Payoff:0STzTTeSSss)(0221S has a mean return equal to the risk free rate under Q.Under the real probabilities, (P), S looks like:SdzSdtdSsso maybe under Q, it looks like:SdzrSdtdSsThis is a good guess, and it is

35、correct!Lets see why.Primbs, MS&E 34537Linear PricingThe basic argumentWhat linear functionals look likeA first step toward risk neutralityGirsanovs TheoremThe Big PictureSummaryInterpretation as state pricesPrimbs, MS&E 34538Gaussians beget Gaussians:)()(exp)2(1)(1212xxxpXTXn1211121exp)(exp

36、)2(12TTTxxxn)2(exp)2(1111212TTTxxxnGaussian with mean zero)(0 xpXXSome multiplying factorSo: )()(0 xpxpXXMultiplication by changes the mean!The variance stays the same!Primbs, MS&E 34539So: )()(0 xpxpXXMultiplication by changes the mean!In this case, our randomness comes from the driving Brownia

37、n motion.), 0(tNztzzzpTtZn210exp)2(1)(2For any time t, we can easily shift the mean of this to t using:tztTT)(exp)(21dPSedQSeSEeTrTTrTTQrTWe want to do a similar thing, but for Ito processes:We need to change measure (find ) so that all portfolios have meanreturn equal to the risk free rate. Q)(ttZz

38、ptzt)(0tZzptzPPrimbs, MS&E 34540In fact, since each increment of Brownian motion is normally distributed, we can change the mean over each increment.We do this over every dt using:dttttdzdtTTt),(),(),(exp)(21)(www)()(dtt)()(dtt)()(dtt)()(dtt).()(dtttTtTdtttdzt0210),(),(),(expwww), 0(dtNdztdzdzdz

39、pTdtdZn210exp)2(1)(2time)(tdtttdztdttTTtt),(),(),(exp)()(21)()(wwwMultiplying all the together gives:Primbs, MS&E 34541tTtTtdsssdzst0210)(),(),(),(exp)(wwwUsing:tzPWe can shift: under:ttdssz0),(wQ to: under:tdztdw),(Notes:Our change of measure (Radon-Nikodym derivative)satisfies the Itos equatio

40、n:TdssE0221),(expwIn general, we need to guarantee that our proposed change of measure is a martingale. A common sufficient conditionfor this is the Novikov condition:Primbs, MS&E 34542“Quasi” Statement of the Girsanov TheoremLetntRx be an Ito process of the form:ttdztdttdx),(),(2/1wwwheremtRz n

41、tRmnR2/1Suppose there exist processes ),(tw),(trwsatisfying certain conditions (measurable, adapted, etc.I am being vague here)with),(),(),(),(2/1tttrtwwwwAssume that satisfies the Novikov condition: ),(twTdssE0221),(expwSetTsTtdzsdss00221),(),(expwwanddPdQThentttzdssz0),(wis a Brownian motion with

42、respect to Q. Furthermore, the process x satisfies:tzdtdttrdx),(),(2/1ww(See, for instance, Oksendal, Chapter 8, Section 6, for a real statement)Primbs, MS&E 34543How do we use Girsanov in finance?zSddtSS)(sszSdrSdtsrBdtdB )(dtzdSSdtdSsQrBdtdB SdzSdtdSsPTradableAssetsThis is an equivalent change

43、 of measures. We just need to be able to set the mean return of all tradable assets to r under Q:s rWhich means we need to be able to solve:We are back at essentially the same characterization as we found whenusing returns!)(dtzdSSdtdSsPrimbs, MS&E 34544rBdtdB )(,(),(dtzdtSKdttSdSQNote: If we we

44、re to write the Ito equations as instantaneous returns, we would essentially recover our vector return form conditions. The vector case:rBdtdB dztSKdttSdS),(),(PTradableAssetsntRS mtRz mnRKntRHere we will assume that zt consists of m independent Brownian motions.KrS To convert the mean returns to r, we need to be able to solve:Primbs, MS&E 34545Poisson ProcessesWe can change the intensity parameter