課程名稱︰金融數學一
課程性質︰數學系選修
課程教師︰彭栢堅
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/01/08
考試時限（分鐘）：180min.
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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下標t0 -95 if S下標t0 >95 and 95-S下標t0 if S下標t0 <95.
(d) Deduce the price of an option which pays S下標t0 if S下標t0 <95 and 0
otherwise.
5. Gvie a formula for the Black-Scholes price of the option with payoff
(ln S下標t0)^3. Also give the stock holding in the replicating portfolio.
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