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05-Interest(f)

# 05-Interest(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: Interest Rate Derivatives Primbs, MS&E 345 2 Interest Rate Derivatives Single Factor Short Rate Models Multi Factor Models Heath-Jarrow-Morton Basics Defaultable Bonds Primbs, MS&E 345 3 Basic Quantities: The short rate: r(t) applies to an infinitesimal time period dt . Instantaneous forward rates: f(t,s) the time t forward interest rate between times s and s+ds. Zero coupon bonds: B(t;T) is the price at time t with maturity at time T (and payoff of \$1). The term structure of interest rates: T t r(t) The short rate Primbs, MS&E 345 4 Basic Relationships: The term structure of interest rates: T t r(t) The short rate ) ( ) , ( t r t t f = ∫ =- T t ds s t f e T t B ) , ( ) ; ( )) ; ( log( ) , ( T t B T t f T ∂ ∂- = Primbs, MS&E 345 5 Interest Rate Derivatives Single Factor Short Rate Models Multi Factor Models Heath-Jarrow-Morton Basics Defaultable Bonds Primbs, MS&E 345 6 The General Single Factor Model: Underlying Factor: (The short rate) dz t r b dt t r a dr ) , ( ) , ( + = dt rB dB = Risk Free Asset dz T B T bB dt T B T B b T aB T B T B T dB r rr r t ) ( ) ( ) ( )) ( ) ( ) ( ( ) ( ) ( 2 2 1 + + + = Bond of maturity T: notation) for ) ( ( ) ; , ( T B T t r B = Step 1: Primbs, MS&E 345 7 The General Single Factor Model: 1 2 2 1 ) ( ) ( 1 1 ) ( )) ( ) ( ) ( ( λ λ + = + + T B T bB T B T B b T aB T B r r rr r t Step 2: r = λ First Equation 1 2 2 1 ) ( ) ( ) ( )) ( ) ( ) ( ( λ T B T bB r T B T B b T aB T B r rr r t + = + + Second Equation Primbs, MS&E 345 8 is the market price of risk. 1 λ 1 2 2 1 ) ( ) ( ) ( )) ( ) ( ) ( ( λ T B T bB r T B T B b T aB T B r rr r t + = + + ) ( ) ( ) ( ) ( ) ( 2 2 1 1 T rB T B b T B b a T B rr r t = +- + λ Generally it is determined from the market. That is, it is chosen to calibrate the model to the market prices. (Recall that λ 1 = λ 1 (r,t) ) Primbs, MS&E 345 9 Different models for the short rate: Rendleman and Bartter: rdz rdt dr σ μ + = Vasicek: dz dt r b a dr σ +- = ) ( Cox, Ingersoll, Ross: dz r dt r b a dr σ +- = ) ( etc......
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05-Interest(f) - Primbs MS&E 345 1 Applications of the...

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