1、Primbs, MS&E3451The Return Form of Arbitrage PricingPrimbs, MS&E3452Pricing Theory:OptimizationReturn form(pdes)Linear function form(risk neutral)Primbs, MS&E3453Pricing Theory:Return form(pdes)Returns and Factor ModelsProfits and LossesAbsence of ArbitrageMarket Price of RiskFuturesRela

2、tionships between returns of assetsGeneralizes the Black-Scholes-Merton argumentMultiple FactorsPrimbs, MS&E3454Modeling Returns:1 periodYou put something in.P0You get something outP1001PPPrPrimbs, MS&E3455The time period can be:one yearone monthone weekone dayone secondone millionth of a se

3、condan instantaneous dt.Primbs, MS&E3456Assume we model a stock price as a geometric brownianmotion. SdzSdtdSdtStSt+dtWhat is the return?ttdttSSSrttSdSdzdtExample: an instantaneous dt:Primbs, MS&E3457This is an example of a factor model:dzdtSdSrbfaWhere:dtabknown at beginning of period dzf u

4、nknown factorPrimbs, MS&E3458bfadzdtSdSrNote that and can depend on S at time t (the beginningof the time period).dztSdttSSdSrtttt),(),(This is an example of a factor model:Primbs, MS&E3459The modeling paradigm:We describe the return of a security over a time period dt as a factor model:The

5、factors:Time: dt (this is for convenience)Random factors: dz1, dz2, . These random factors can be increments of Brownian Motion,Poisson Processes, or in general, whatever you want!I will write dt, but the time period could be of any length!Primbs, MS&E34510Pricing Theory:Return form(pdes)Returns

6、 and Factor ModelsProfits and LossesAbsence of ArbitrageMarket Price of RiskFuturesRelationships between returns of assetsGeneralizes the Black-Scholes-Merton argumentMultiple FactorsPrimbs, MS&E34511Profits and LossesThat is, I purchase sharestSxNow each share is worthSt+dtHow much money did I

7、make over dt?xSSxdtttInvest x dollars at initial timedtStSt+dtttdtttSSxSSxxSdSrxProfit:sharespriceinitial amountConsider an asset, S, with the following returndzdtrSPrimbs, MS&E34512Profit/Loss from a Portfoliodzdtr111dzdtr222dzdtr333We are given the returns on assets which all depend on a commo

8、n factor, dz.1S2S3SCost:321xxxProfit/Loss:332211xrxrxrLet xi be the dollar amount of money invested in asset i:1x2x3x111222333()()()dtdz xdtdz xdtdz x1 122331 12233()()xxx dtxxx dzxT1dzxdtxTT)()(where3213211111321xxxxPrimbs, MS&E34513Pricing Theory:Return form(pdes)Returns and Factor ModelsProfi

9、ts and LossesAbsence of ArbitrageMarket Price of RiskFuturesRelationships between returns of assetsGeneralizes the Black-Scholes-Merton argumentMultiple FactorsPrimbs, MS&E34514What would an arbitrage portfolio be?An arbitrage is a riskless profit which requires no investment.In the current sett

10、ing, a portfolio that (1) costs nothing(2) has no risk(3) but makes a profitPrimbs, MS&E34515Profit/Loss from a PortfolioCost:321xxxxT1Profit/Loss:332211xrxrxr33222111)()()(xdzdtxdzdtxdzdtdzxxxdtxxx)()(32211332211dzxdtxTT)()(Consider the following portfolio:No Cost:01xTNo Risk:0 xT0 xTNo Profit/

11、Loss:Otherwise, Arbitrage!Primbs, MS&E34516No Cost:01xTNo Risk:0 xTNo Profit/Loss:0 xTA necessary condition for No ArbitrageWhat are the implications of this?Primbs, MS&E34517Lets write the condition as follows:001xxTTTThis truthfulness of this implication is equivalent to saying that is a l

12、inear combination of 1 and .Why?1110Assume:01xxTTTTThen:Primbs, MS&E34518Lets write the condition as follows:001xxTTTThis truthfulness of this implication is equivalent to saying that is a linear combination of 1 and .Another Argument:then)(ANxand)(ANbut)()(TARANhence)(TARor1TATTA1LetPrimbs, MS&

13、amp;E34519No Cost:01xTNo Risk:0 xTNo Profit/Loss:0 xTA necessary condition for No ArbitrageThere exists a such that: 1Primbs, MS&E34520Pricing Theory:Return form(pdes)Returns and Factor ModelsProfits and LossesAbsence of ArbitrageMarket Price of RiskFuturesRelationships between returns of assets

14、Generalizes the Black-Scholes-Merton argumentMultiple FactorsPrimbs, MS&E34521We have derived a relationship between:1Expected Returnsand Risk(Actually, this is nothing more that Ross 1976 Arbitrage Pricing Theory.)1110The s tell you how much expected return you are rewarded for taking risk.Our

15、return was given by:dzdtrThey are generally referred to as the “Market Price of Risk”.Lets look at them in a little more depth.Primbs, MS&E34522Consequences of the Return Form of AOA.dtrrf0the risk free rateThen, from AOA:We can think of 0 as the“market price of time”.001fr10110 is the risk free

16、 rate!A Risk Free Asset:Primbs, MS&E34523Consequences of the Return Form of AOA.A Risk Free Asset:dtrrf0Two other assets:dzdtr111dzdtr222Then, from AOA:0fr(1)(2)111101fr121202frSolve for 1:22111ffrrWe refer to 1 as the “market price of risk” for dz.Primbs, MS&E34524The Market Price of Risk22

17、111ffrr(1) It looks like the Sharpe ratio.(2) The market price of risk is associated with the factor dz.All securities that only depend on the random factor dz will have the same market price of risk (instantaneous Sharpe ratio).Primbs, MS&E34525The Market Price of Risk22111ffrrAn important Note

18、:dtStSt+dtThe market price of risk is often a function of St and t because it depends on and which are often functions of St and t.The market price of risk is known here, just like and . I wont always make this dependence explicit, but you should keep it in mind!Primbs, MS&E34526Connections with

19、 CAPM:CAPM says:)(fMfrrrWe just derived that:frHence, we should have that:)(fMrr If returns are uncorrelated with the market:00Sometimes used to determine market price of risk.Example:Primbs, MS&E34527Pricing Theory:Return form(pdes)Returns and Factor ModelsProfits and LossesAbsence of Arbitrage

20、Market Price of RiskFuturesRelationships between returns of assetsGeneralizes the Black-Scholes-Merton argumentMultiple FactorsPrimbs, MS&E34528An extension to multiple factorsVector notation:01xTNo Cost0 xKTNo Risk0 xTNo ProfitNo Arbitragemmdzdzdtr111111.mmdzdzdtr212122.mnmnnndzdzdtr.111x2xnxKd

21、zdtrwherenmnm1111mdzdzdz1n1Primbs, MS&E3452901xT0 xKT0 xTNo CostNo RiskNo ProfitNo ArbitrageK1There exists such that:Primbs, MS&E34530Does any of this make sense?3132121110132322212102Necessary condition for no arbitrage:In general, these equations do not have a unique solution for the s!Tha

22、t is, there could be many s that satisfy the equations. Nevertheless, it is still a necessary condition for no arbitrage.31311111.dzdzdtr32312122.dzdzdtrConsider:Primbs, MS&E34531Pricing Theory:Return form(pdes)Returns and Factor ModelsProfits and LossesAbsence of ArbitrageMarket Price of RiskFu

23、turesRelationships between returns of assetsGeneralizes the Black-Scholes-Merton argumentMultiple FactorsPrimbs, MS&E34532An interesting special case: Futures ContractsWhen you enter into a futures contract, no money exchanges hands. So, initially, it costs nothing. Thismakes it a special case.

24、But, when I enter into this contract, I pay nothingHow does this fit into our factor model framework?Let, f be the futures price, and assume f follows:dfdtdzfPrimbs, MS&E34533Profit/Loss from a Portfolio that contains Futuresdzdtr111dzdtr222dzdtr333We are given the returns on assets which all de

25、pend on a common factor, dz.1S2SfCost:21xx x011Profit/Loss:332211xrxrxr33222111)()()(xdzdtxdzdtxdzdtdzxxxdtxxx)()(32211332211dzxdtxTT)()(Future PriceLet x1, x2 be the amount of money invested in assets 1 and 2:1x2x3xBut, x3 is the total future price. You agree to pay this at the delivery date, not now!Primbs, MS&E34534No Cost:021 xxNo Risk:0332211xxxNo Profit/Loss:0332211xxxA nec