1、Primbs, MS&E 3451From the Return Form to the Linear Functional Form of ArbitragePrimbs, MS&E 3452Pricing Theory:OptimizationReturn form(pdes)Linear function form(risk neutral)Primbs, MS&E 3453Pricing Theory:Return form(pdes)Linear function form(risk neutral)OptimizationPrimbs, MS&E 3454The Linear Fu

2、nctional Form of Arbitrage:What is the return form lacking?rrdtBdBBB(1+r)ddtSdSS)1 (S)1 (SThere is a clear arbitrage opportunity!The return form doesnt capture it!Consider the following example:Primbs, MS&E 3455In our example, does there exist a l such that:lrYes! of course!Something is missing from

3、 the return form.The return form is a necessary condition for no arbitrage, butit is not sufficient.Lets return to the basics.The return form doesnt capture it!Primbs, MS&E 3456What is an arbitrage opportunity?tvudttvddttvpricepayoffIf we plot all the cash flows associated with the purchase of a por

4、tfolio v, there must be some positive and some negativecash flows. If all cash flows were positive (actually non-negative), thenthis would represent an arbitrage opportunity.Primbs, MS&E 3457Portfolio: x shares of stock and y bonds:Price:ttyBxS Payoff:up:down:dttudttyBxSdttddttyBxSLets take a simple

5、 setting:bond and stock.tSudttSddttStBdttBdttBWhat is an arbitrage opportunity?000negative pricepositive payoffArbitrageAt least one strictpricepayoffIn Pictures:You just receivecash flows!Primbs, MS&E 3458pricepayoffttyBxS dttudttyBxSdttddttyBxSArbitrage000At least one strictArbitrage is the existe

6、nce of a solution to the above set of inequalities.Absence of arbitrage is that there exists no solution to the above set of inequalities.Therefore, we need to find conditions which guarantee thatno solutions to the above inequalities exist.Just to make things easier, lets flip the first inequalitie

7、s sothat they all look the same.Primbs, MS&E 3459pricepayoffttyBxS dttudttyBxSdttddttyBxSArbitrage000At least one strictNow, consider the following:If they are all non-negative (one strict), then if I multiply them by positive numbers and add them together, that mustbe positive.ttyBxS dttudttyBxSdtt

8、ddttyBxS000000u0d0)( )( )(0dttddttddttudttuttyBxSyBxSyBxSPrimbs, MS&E 34510pricepayoffttyBxS dttudttyBxSdttddttyBxSArbitrage000At least one strictBut what if for some 0 0, u 0, d 0, and all x and y: 0)()()(0dttddttddttudttuttyBxSyBxSyBxSThis would rule out arbitrage!ttyBxS dttudttyBxSdttddttyBxS0000

9、00u0d0)( )( )(0dttddttddttudttuttyBxSyBxSyBxSThen clearly the above inequalities are never satisfied!Primbs, MS&E 34511pricepayoffttyBxS dttudttyBxSdttddttyBxSArbitrage000At least one strict0)()()(dttddttddttudttuttyBxSyBxSyBxSWithout loss of generality, we can choose:10Since this condition must be

10、true for all (x,y) not identically zero.0)(ddttdudttutSSS0)(dttddttutBBBNoArbitrage0)()(dttddttutddttdudttutBBBySSSxRearranging:Primbs, MS&E 345120)(ddttdudttutSSS0)(dttddttutBBBddttdudttudttddttddttudttudttddttuddttdudttutttvvyBxSyBxSBBySSxyBxSv )()( )()( ddttdudttutSSSdttddttutBBBThese are absence

11、 of arbitrage conditions, and pricing formulas!For a genericportfolioPrimbs, MS&E 34513ddttdudttutSSSdttddttutBBBddttdudttutvvvududududttSddttSdttBdttBudttvddttvIn Pictures:tStBtvPrimbs, MS&E 34514ddttdudttutvvvududttvddttvddutv10ud10Interpretation of as state prices:tvPayoff 1 in thedown state.ud01udutv01Payoff 1 in theup state.d is the price of the down state.u is the price of the up state.Primbs, MS&E 34515ddttdudttutvvvududttvddttvtvLinear pricing functional and inner products:We ca