04-newpde(f)

04-newpde(f) - Primbs MS&E 345 1 Applications of the...

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Unformatted text preview: Primbs, MS&E 345 1 Applications of the Return Form of Arbitrage Pricing: Equity Derivatives Primbs, MS&E 345 2 Deriving Equations for Derivative Assets: Three step algorithm: (1) Derive factor models for returns of tradable assets. (often involves Ito’s lemma.) (2) Apply absence of arbitrage condition. (μ= [1 σ]λ29 (3 29 Apply appropriate boundary conditions and solve. Primbs, MS&E 345 3 Black-Scholes Poisson (Cox and Ross) Jump diffusion model (Merton) Dividends Options on futures (Black) Multiple factors Stochastic Volatility (Hull and White) Exchange one asset for another (Margrabe) Option on Max, Min Option on an average (Asian options) Path Dependent Primbs, MS&E 345 4 Black-Scholes (Again) Step 1: Derive factor models for returns of tradable assets. Risk Free Asset: rdt B dB = Underlying Stock: dz dt S dS σ μ + = Ito’s Lemma: dz Sc dt c S Sc c dc S SS S t σ σ μ + + + = ) ( 2 2 2 1 ) , ( t S c t What is a factor model for the return on ) , ( t S c t Let be a security derivative to the stock. ? dz c Sc dt c c S Sc c c dc S SS S t σ σ μ + + + = ) ( 2 2 2 1 Return: Primbs, MS&E 345 5 + = + + c Sc c c S Sc c r S SS S t σ σ λ λ σ μ μ 1 1 1 ) ( 1 2 2 2 1 Black-Scholes (Again) Derivative: dz c Sc dt c c S Sc c c dc S SS S t σ σ μ + + + = ) ( 2 2 2 1 Step 1: Derive factor models for returns of tradable assets. Risk Free Asset: rdt B dB = Underlying Stock: dz dt S dS σ μ + = [ ] λ μ K 1 = Step 2: Apply Primbs, MS&E 345 6 r = λ First Equation Second Equation σ μ λ ) ( 1 r- = Third Equation σ μ σ σ μ ) ( ) ( 2 2 2 1 r c Sc r c c S Sc c S SS S t- + = + + + = + + c Sc c c S Sc c r S SS S t σ σ λ λ σ μ μ 1 1 1 ) ( 1 2 2 2 1 [ ] λ μ K 1 = Step 2: Apply Black-Scholes (Again) Primbs, MS&E 345 7 Third Equation rc c S rSc c SS SS t = + + 2 2 2 1 σ The Black-Scholes Equation This is for an option on a non-dividend paying asset which follows a geometric Brownian motion. σ μ σ σ μ ) ( ) ( 2 2 2 1 r c Sc r c c S Sc c S SS S t- + = + + Step 3: Apply appropriate boundary condition and solve! Black-Scholes (Again) Primbs, MS&E 345 8 Step 3: European Calls and Puts rc c S rSc c SS S t = + + 2 2 2 1 σ +- = ) ( ) , ( K S T S c ) , ( = t c rp p S rSp p SS S t = + + 2 2 2 1 σ +- = ) ( ) , ( S K T S p ) ( ) , ( t T r Ke t p-- = Solution: ) ( ) ( ) , ( 2 ) ( 1 d N Ke d SN t S c t T r--- = t T t T r K S d-- + + = σ σ ) )( ( ) / ln( 2 2 1 1 t T d d-- = σ 1 2 where: ) ( ⋅ N distribution function for a standard Normal (i.e. N(0,1) ) ) ( ) ( ) , ( 1 2 ) ( d SN d N Ke t S p t T r--- =-- These formulas are basic...know them!!! Primbs, MS&E 345 9 European Calls and Puts ) ( ) ( ) , ( 2 ) ( 1 d...
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04-newpde(f) - Primbs MS&E 345 1 Applications of the...

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