Solution_7 - IEOR 4701 Stochastic Models for Financial...

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IEOR 4701 - Stochastic Models for Financial Engineering Solution to Homework 7 Exercise 1: Consider a model of interest rate dynamics which follows the so-called Cox- Ingersoll-Ross model (also known as Feller diffusion) given by dX ( t ) = κ ( µ - X ( t )) dt + σ X ( t ) dB ( t ) X (0) = x, with κ, σ, µ > 0. In addition let is consider an asset which follows the Black-Scholes dynamics given by dY ( t ) = βY ( t ) dt + vY ( t ) dW ( t ) , Y (0) = 1 , where for β and v given. Assume that W ( · ) and B ( · ) are independent standard Brownian motions. a) Consider the discounted price process Z ( t ) = Y ( t ) exp ( - X ( t ) ds ) . If we wish to make sure that Z ( · ) is a martingale (i.e. E ( dZ ( t )) = 0) which relations must be satisfied by the various parameters of these models (in particular κ , µ , and β ). ( Hint : Apply Ito’s lemma).
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into the right hand side, and collecting terms, we have dZ ( t ) = e X ( t ) Y ( t )[( κ + σ 2 / 2) X ( t ) - κµ + β ] dt - σe X ( t ) X ( t ) Y ( t ) dB ( t ) + νe X ( t ) Y ( t ) dW ( t ) . In order to make the dt term vanish, we enforce β = κµ, and σ 2 2 + κ = 0 . b) What if W ( · ) and B ( · ) were the same Brownian motion? Which relations must be satisfied?