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課程名稱︰金融數學二
課程性質︰數學系/所選修
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）︰2013.06.17
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 :
Calculators allowed. Any results from lecture notes or homework answers may
be used without proof. Any other results used must be proved.
1.Suppose the yield curve at time t=0 is as in the table below:
T 0.5 1.0 1.5 2.0 2.5 3.0
r(0,T) 0.025 0.030 0.035 0.045 0.050 0.055
(a)Consider a coupon bond with maturity 3 years, face value 100 and coupon
rate 7%. Coupons are issued every 6 months.
Calculate the forward price at t=0 to buy this bond at time 1.5 (after
the third coupon has been issued).
(b)Suppose the market forward price at t=0 to buy the bond in (a) at time
1.5 is 98. Describe how you would arbitrage this mispricing, giving
full details of all transactions you make.
2.Consider a bond market with Ω={ω1,ω2,ω3,ω4,ω5,ω6}
and trading times t0=0, t1=1,t2=2, t3=3.
The zero cuopon bond price processes are as follows:
P(0,1)=0.9400, P(0,2)=x, P(0,3)=0.8400 for ω and
ω P(1,2) P(1,3) P(2,3)
ω1 0.9700 0.9360 0.9800
ω2 0.9700 0.9360 0.9500
ω3 0.9500 0.8902 0.9600
ω4 0.9500 0.8902 0.9300
ω5 0.9100 0.8367 0.9300
ω6 0.9100 0.8367 0.9000
(a)Find the range of x for which there is no arbitrage in this market. (Note
this range contais the value 0.89.)
(b)Set x=0.89. Consider a coupon bond which is issued at time t=0 with
face-value $100 and maturity T=3. Suppose the bond delivers a coupon of
$7 at the end of each period. Find the values of this coupon bond at time
t=1 (after the t=1 coupon has been delivered).
(c)Consider a European put option on the bond in (b) with strike price 102
and maturityT=1. What is its price at time t=0 ?
3.Consider the short-rate model which in the risk-neutral world has the form
dr = (α-β*r)dt + σ*exp(-γ*t)dΨ(t),
where dΨt=dΨ(t)= γdt +dWt and α>0, β>0, σ>0, γ>0. This is an affine
model so that we know the bond prices have the form
P(t,T)= exp( A(t,T)-B(t,T)*r(t) ).
Find B(t,T) and A(t,T).
4.Consider the short rate model dr = a(t,r)dt +b(t,r)dΨt in risk-neutral
world. Then when the short rate is r(t), the model gives the bond price
P(t,T) = V(t,r(t)), where V(t,r) solves the PDE:
Vt - r*V +a(t,r)*Vr +0.5 *b(t,r)^2 *Vrr =0
with V(T,r)=1.
(a) For the model dr = θ(t)dt +σdΨt, where σ>0 and θ(t) is a
deterministic functionm find the solution V(t,r) of the PDE with
V(T,r)=1. (Hint: try to find a solution of a particular form.)
(b) Suppose the prices P(0,T) given by the model (these prices depend on
r(0)) coincide with the market prices. Show it is 'necessary' that
θ(T)= partial T of f(0,T) plus σ^2;
where f(0,T)= partial T of (-ln P(0,T))
= d/dT ( -ln P(0,T) ).
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