Problem 1 (30 pts) Xn' + (Xn-2) - In Suppose {Xn}, n = 0, 1,2, ... is a symmetric random walk, 2 xx - 4 xx+4 - Th -1 ) " + ('n - 1 - 1 1/2 PXn+1 = Xn + 1|In] =?' P Xn+1 = Xn - 1| Fn] = (19 - 2Xn+ 2 )- with Xo = 0. The filtration { Fn } is defined by the history of the process {X, }: F, = (Xo, X1,..., Xn). (a) Is the process In = (X, - 1)2, n = 0, 1, 2, ..., a martingale, submartingale, or supermartingale? x X - 2 (b) Derive the Doob decomposition of the process {Y,}, that is, find a predictable nondecreasing - In process {An} such that {Yn + An} is a martingale (decide, which of the signs "+" or "- should be used in this formula). Problem 2 (30 pts) Suppose the stock prize today is So. Its evolution in time steps n = 0, 1, 2, ... is modeled by the following tree process: Sn+1 = Snelotovo where u > 0 is the drift and o > 0 the volatility. The constant 8 > 0 is a small time step and we assume that ov8 > us. The interest rate is r > 0 and a bond with nominal value By = exp(nro), n = 0, 1,2, ..., is available at each time period. Both stock and bond can be bought or sold short in any quantities at any time. (a) What probabilities qn, n = 0, 1, 2, ..., of the "up moves" make arbitrage impossible? D is wanted (b) Find the present price of a European call option with strike price Soelo and expiration time n = 2 (a call option is the right to buy one share for the strike price at expiration). Problem 3 (20 pts) Suppose {W/ } 20 and { Br} 20 are two independent standard Brownian motions. (a) Prove that for every a the process X, = W, sin(a) + B, cos(a), t 2 0, is a standard Brownian motion. (b) Derive the covariation process of the processes {W, } and {B, }? Problem 4 (20 pts) P ( W LI ) 7 / 2a ) Suppose {W/} 20 is a standard Brownian motion, and let Ta be the hitting time of a level a > 0. Calculate the probability that Ta _ Land W(1)