推 t0444564 :已收錄 08/22 21:58
課程名稱︰金融數學二
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&數學所
考試日期（年月日）︰2013.04.15
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 :(In the following X1 means X_1, the subindex.)
This is an open book, open notes examination. Calculators are allowed.
No computers. All answers must be justified in full detail.
1.(a) Convert thelinear programming problemL minimize X1-X2 +X3 +2
subject to X1+2*X2-X3 = 3, X1+X2 ≦2, X1≧0, X3≧0 into a primal problem.
(b)Write down the dual of the primal problem found in (a)
(c)Does either of the problems in (a),(b) have a solution?
Justify your answer.
2.Solve the problem: maximize X1+2*X2-X3-1 subject to
2*X1-X2+X3≦1, X1+2*X2-X3≦-1, X1,X2,X3≧0.
3.Consider the one-period securities market with
S_0=┌10 ┐, D=┌ 8 10 12┐.
└15 ┘ └18 16 12┘
(a)Find all state price vectors ψ.
(b)Show the market is not complete and find all attainable contingent claims.
(c)Consider the call on the second security with strike 15.
Show this is not an attainable claim and find the interval [V_(C),V+(C)]
for it. V_(C) means the value of the max. lower bond of a contingent
claim C.
(d)If the price of the call is 0.50, give the details of an arbitrage
opportunity.
4.Consider a stock which in 9 months will give a dividend of $1. We consider
the stock price St as consisting of two parts
St=︴Zt + D*exp(-r(τ-r)) if 0≦t<τ
︴Zt ifτ≦t≦T, (St means S_t,subindex, similar for Zt)
where Zt is the "risy" part, D=1,τ=0.75 years, T=1 year and r=0.04.
We suppose Zt evolves as usual in the binomial model, that is,
Z_(t+δ*t) = Zt*u or Zt*d, where δ*t= T/2,2 being the number of periods.
ps. (t+δ*t) is the subindex Z.
If S_0=50, u=1.06, d=0.94, calculate the price of an American call with
strike 47 and maturity 1 year.
5. We suppose that with respect to the martingale measure Q the stock price
follows the stochastic differential equation(SDE):
dSt = r*St dt + σ*St dWt, where r is the interest rate.
Define M_T = sup Wt.
0≦t≦T
Then we know that the joint density function of (M_T, W_T) is
2(2a-x) -(2a-x)^2/(2T)
f(a,x)=─────*e , a≧0, x≦a.
T*√(2πT)
(a)Calculate Q{ max (St≧B) }.
0≦t≦T
(b)Consider a portfolio consisting of a long up-and-in call and a short
up-and-in put, both with barrier B, strike K and maturity T,
where S_0 <B.
(i) Write down the payoff C_T at time T to thes portfolio.
(ii)Calculate the t=0 price C_0 of thes portfolio.
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