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hw3(1) - Math 623 W 2007 Homework 3 For full credit your...

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Math 623, W 2007: Homework 3. For full credit, your solutions must be clearly presented and all code included. All the problems in this homework set deal with the following (Heston) stochastic volatility model for the price S t of a stock dY t = 2(0 . 04 - Y t ) dt + 0 . 2 Y t dW t dS t S t = 0 . 01 dt + Y t ( - 0 . 7 dW t + 0 . 51 dZ t ) (1) for 0 t 0 . 25. Here W t and Z t are independent (uncorrelated) Brownian motions under the risk neutral measure Q . Time is counted in years unless otherwise stated. By convention, one year is 360 days. Today is t = 0 and Y 0 = 0 . 04. The stock pays no dividends. (1) First some general questions. (a) What is the risk-free interest rate r ? (b) The current stock price S 0 is such that the E Q [ S 0 . 25 ] = 100. What is S 0 ? (c) What is (formally) the expected value and variance of the relative stock return dS t /S t (conditioned on S t and Y t )? (d) What is (formally) the expected value and variance of dY t (conditioned on S t and Y t )? (e) Why is it reasonable to view Y t as the square of the volatility level at time t ?
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