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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