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）︰101.01.09
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, calculators allowed, no computers.
Table of normal distribution is provided.
Show all working: points will be deducted if sufficient working is not shown.
1. Let Wt be Brownian motion.
(a) Using the independence of Wt - Ws and Ws or otherwise, find
P(Wt≦y∣Ws = x) if t>s>0 and so write down the conditional density
function of Wt fiven Ws = x.
(b) Write down the density function of Ws if s>0.
(c) Using (a)and (b) or otherwise, write down the joint density function of
Wt and Ws when t>s>0.
(d) Write down the density function of Wt if t>0.
(e) Using (c) and (d) or otherwise, write doown the conditional density
function os Ws, given that Wt = y, when t>s>0.
2. (a) Using the independence of Wt-Ws and Ws or otherwise,for x≧0 and t>s>0,
find the conditional probability P( Wt^2 - Ws^2 ≦0 ∣Ws^2 = x).
(b) t s
Show that ∫ Wu dWu and ∫ Wu dWu are not independent when
s 0
t>s>0. (Hint:apply Ito's formula to Wt^2 and use (a).)
3. Find the solution of the stochastic differential equation
dZt = (a*log Zt + b)*Zt dt +σZt dWt
with value Zo at t=0, where a, b and σ are constants with a≠0.
(Hint: define Yt = log Zt and use Ito's formula; then make a further
transformation.)
4. Consider the Black-Scholes market consisting of a sotck whose price St
follows geometric Brownian motion dSt = μSt dt +σSt dWt
and a risk-free asset Bt= e^rt wgere So= 50, σ= 0.25, and r=0.06.
Calculate the price at time t=0 of the options with payoffs at time T=0.25:
(註:ST 中的T應為下標，但打不出來。)
(a) 55-ST if ST<55 and 0 otherwise;
(b) 45-ST if ST<45 and 0 otherwise;
(c) 10 if ST≦45, 55-ST if 45<ST<55 and 0 if ST≧55.
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