03-ReturnFormP(f)

03-ReturnFormP(f) - Primbs MS&E345 1 The Return Form of...

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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 Black-Scholes-Merton 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 Black-Scholes-Merton 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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03-ReturnFormP(f) - Primbs MS&E345 1 The Return Form of...

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