课程名称︰金融数学一 课程性质︰数学系选修 课程教师︰彭栢坚 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2011/01/08 考试时限（分钟）：180min. 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Open book, open notes. Calculatiors are allowed. No computers or mobile phones. No points for answers without full justification.] 1. Let Wt be suandard Brownian motion. (a) Find P(Wt≧1 for some t, 0≦t≦2). (b) Find P(W1≦1, Wt≧3 for some t belongs to [0,1]). (c) Find P(W1>W2, W4<W1). t 2. Show that Yt = Wt^3 - 3* ∫ Ws ds is a martingale. Also find Var(Yt). 0 3. By applying Ito's formula to Xt = ln Zt or otherwise, solve the stochastic differential equation dZt = Zt* (-ln Zt + 0.5*t^2) dt + t*Zt dWt. 4. (a) Suppose the current stock prise is 90, the interest rate 0.03 and the volatility 0.25. Use the Black-Scholes formula to calculate the price of a call with expiry t0 = three months and exercise price c=95. (b) Same as (a) but now the option is a put. (c) Deduce the price of an option which pays S下标t0 -95 if S下标t0 >95 and 95-S下标t0 if S下标t0 <95. (d) Deduce the price of an option which pays S下标t0 if S下标t0 <95 and 0 otherwise. 5. Gvie a formula for the Black-Scholes price of the option with payoff (ln S下标t0)^3. Also give the stock holding in the replicating portfolio. --

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