ass4_f09 - STAT/ACTSC 446/846 Assignment#4(due 1 Let cfw_Wt...

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STAT/ACTSC 446/846 Assignment #4 (due November 23, 2009) 1. Let { W t } be defined by the SDE dW t = μdt + σdB t , where B t is a standard Brownian motion. Use Ito’s formula to write the following processes Y t in the form of a stochastic integral ( dY t = u ( Y t , t ) dt + v ( Y t , t ) dB t or an equivalent integral form): (a) Y t = W 3 t , t 0 (b) Y t = 10 + t 2 + e 4 W t , t 0 (c) Y t = exp ( W 2 t - σt ), t 0. 2. Let { B t } t 0 be standard Brownian motion. Generate three trajectories of the process X n ( i n ) , i = 1 , . . . , n , and show them on a graph, where { X n } is defined as follows: (a) X n ( i n ) = X n ( i - 1 n ) + 1 n Z i , Z i - i.i.d. binomial random variables taking values 1 and - 1 with probabilities p = 0 . 5 and q := 1 - p = 0 . 5, respectively, X n (0) = 0, and n = 150. (b) X n ( i n ) = X ( i n ), where X ( t ) satisfies the following SDE equations (use n = 150): (i) Cox, Ingersoll, and Ross model: dX t = 3(0 . 2 - X t ) dt + 0 . 06 p X t dB t , X (0) = 0 . 2 . (ii) Geometric Brownian motion: dX t = 0 . 03 X t dt + 0 . 25 X t dB t , X (0) = 100 . 3. Suppose a stock pays dividends at a rate δ , and its price under the real-world measure P can be
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