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Unformatted text preview: Primbs, MS&E345 1 The Return Form of Arbitrage Pricing Primbs, MS&E345 2 Pricing Theory: Optimization Return form (pdes) Linear function form (risk neutral) Primbs, MS&E345 3 Pricing Theory: Return form (pdes) Returns and Factor Models Profits and Losses Absence of Arbitrage Market Price of Risk Futures Relationships between returns of assets Generalizes the BlackScholesMerton argument Multiple Factors Primbs, MS&E345 4 Modeling Returns: 1 period You put something in. P You get something out P 1 1 P P P r = Primbs, MS&E345 5 The time period can be: one year one month one week one day one second one millionth of a second an instantaneous dt . Primbs, MS&E345 6 Assume we model a stock price as a geometric brownian motion. Sdz Sdt dS σ μ + = dt S t S t+dt What is the return? t t dt t S S S r = + t t S dS = dz dt σ μ + = Example: an instantaneous dt : Primbs, MS&E345 7 This is an example of a factor model: dz dt S dS r σ μ + = = bf a + = Where: dt a μ = σ = b known at beginning of period dz f = unknown factor Primbs, MS&E345 8 bf a + = dz dt S dS r σ μ + = = Note that μ and σ can depend on S at time t (the beginning of the time period). dz t S dt t S S dS r t t t t ) , ( ) , ( σ μ + = = This is an example of a factor model: Primbs, MS&E345 9 The modeling paradigm: We describe the return of a security over a time period dtr as a factor model: The factors: • Time: dt (this is for convenience) •Random factors: dz 1 , dz 2 , ... 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&E345 10 Pricing Theory: Return form (pdes) Returns and Factor Models Profits and Losses Absence of Arbitrage Market Price of Risk Futures Relationships between returns of assets Generalizes the BlackScholesMerton argument Multiple Factors Primbs, MS&E345 11 Profits and Losses That is, I purchase shares t S x Now each share is worth S t+dt How much money did I make over dt ? x S S x dt t t + Invest x dollars at initial time dt S t S t+dt t t dt t t S S x S S x = + x S dS = rx = Profit: shares price initial amount Consider an asset, S , with the following return dz dt r σ μ + = S Primbs, MS&E345 12 Profit/Loss from a Portfolio dz dt r 1 1 1 σ μ + = dz dt r 2 2 2 σ μ + = dz dt r 3 3 3 σ μ + = We are given the returns on assets which all depend on a common factor, dz....
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This note was uploaded on 06/17/2010 for the course MS&E 345 taught by Professor Jimprimbs during the Winter '10 term at Stanford.
 Winter '10
 JIMPRIMBS

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