4. a. There are two 3year bonds with face values equaling $1000. The coupon rate of bond A is 0.05 and 0.08 for bond B. A third bond C also exists. wit-h a maturity of 2 years. 30nd C also has a face value of $1000; it has a coupon rate of 11%. rT'he prices of the three bonds are $878.9172. $955.4787. and 351055.419. respectively. Find a portfolio of bonds 14.3. and C that would replicate the cash ow structure of bond D. which has a face value of $1000. a maturity of 3 years. and a coupon rate of 3%. b. Security A will pay 5 in out-come 1. 7 in outcome 2. and 9 in outcome 3. Security B will pay 2 in outcome 1. 4 in outcome 2. and 8 in out-come 3. Security C will pay 9 in outcome 1. 1 in outcome 2. and 3 in outcome 3. Both securities A and C currently sell for $5 and security 13 currently sells for $3. \"7 hat would be the value of security D. which will pay 1 in each of the three outcomes? a. Verify that in a onetimeperiod binomial economy with a riskfree return rate equal to '3'" and a stock currently selling for So that the synthetic (riskneutral) probability for upward price movement in this economy is characterized by [1+ 7' d)/ (u _ d) where u and d are multiplicative upward and downward stock price movements. respectively. meaning that the stock price will equal either 51 = USO or 51 = ng. b. Harper Company stock currently sells for $14 per share and is expected to be worth either $10. $16. or $25 in 1 year. The current riskfree rate is 0.125. A 1year call with an exercise price of $15 currently sells for $3. i. \"That would be the value of a 1year call with an exercise price of $9? ii. \"'hat are the synthetic probabilities in this economy? a. Find the value of a bond that has a face value of $1000 that matures in 1 year. The monthly rate at which interest accrues at time t (in months) after origination is determined by the following simple model: Tl?) = 0-007 000003132. b. Solve the following initial value problem: dz; 3%.: yt and yo: 100 a. Suppose that a statistical time series analysis revealed that an investment grows at a rate proportional to the square root of its current value with constant of proportionality k 1 011." x/ _ _ 71'? = k l" and 1/ (0) = 1/0 Find the value of this investment as a function of time. b. A Tyear zerocoupon bond with face value BI = F has a price pat-h given by the following differential equation: dB ' = rndt T t Provide a general and a nancially appropriate particular solution to this differen tial equation