432 An Introduction to Stochastic Modeling 8.4.5 Change a Brownian motion with drift X(t) into an absorbed Brownian motion with drift X A (t) by defining X A (t) = where = min{t 0; X(t) = 0}. (We suppose that X(0) = x > 0 and that µ < 0, so that absorption is sure to occur eventually.) What is the probability that the absorbed Brownian motion ever reaches the height b > x? What is the probability that a geometric Brownian motion with drift parameter = 0 ever rises to more than twice its initial value? (You buy stock whose fluctuations are described by a geometric Brownian motion with = 0. What are your chances to double your money?) A call option is said to be "in the money" if the market price of the stock is higher than the striking price. Suppose that the stock follows a geometric Brownian motion with drift , variance 2 , and has a current market price of z What is the probability that the option is in the money at the expiration time ? The striking price is a. Verify the Hewlett-Packard option valuation of $6.03 stated in the text when 1 = 2 , z = $59, a = 60, r = 0.05, and = 0.35. What is the Black-Scholes valuation if = 0.30? Let be the first time that a standard Brownian motion B(t) starting from B(0) = x > 0 reaches zero. Let be a positive constant. Show that w(x) = E e - |B(0) = x = e - 2x X(t), 0, for t < , for t , 8.4.6 8.4.7 8.4.8 8.4.9 . Hint: Develop an appropriate differential equation by instituting an infinites- imal first step analysis according to w(x) = E E e - |B( t) |B(0) = x = E e - t w(x + B) . 8.4.10 Let t 0 = 0 < t 1 < t 2 < · · · be time points, and define X n = Z(t n ) exp(-rt n ), where Z(t) is geometric Brownian motion with drift parameters r and vari- ance parameter 2 (see the geometric Brownian motion in the Black-Scholes formula (8.53)). Show that {X n } is a martingale. 8.5 The Ornstein­Uhlenbeck Process The Ornstein­Uhlenbeck process {V(t); t 0} has two parameters, a drift coefficient > 0 and a diffusion parameter 2 . The process, starting from V(0) = v, is defined in This section contains material of a more specialized nature.