This question is about fixed income securities and stocks $[50$ marks$]$

Use discrete compounding of interest rates and assume that bond coupons are annual throughout this question.

a$)[5$ marks$]$ Consider two bonds, both with face value $100.$ The coupons are paid annually. Bond $1$ is a three$-$year $10\%$ coupon bond and bond $2$ is a two$-$year zero. Bond $1$ is traded at a price of $99\mathrm{.}5$ and bond $2$ at a price of $80.$ In addition to these bonds, a two$-$year annuity with annual payments of $10$ is traded at a price of $17.$ What are the discount factors d$1,$ d$2,$ and d$3?$

b$)[10$ marks$]$ Consider a three$-$year annuity with an annual payment of $55,$ which is traded at a price of $132\mathrm{.}75.$ Construct a portfolio of the bonds and the annuity in part a$)$ which has the same payoffs as this three$-$year annuity. Is there an arbitrage opportunity in the market? If your answer is yes, how many units of each asset you need to buy or sell to earn an arbitrage profit of $1000?$

c$)[13$ marks$]$ An insurance company needs to pay out liabilities of $10$ million in each of the years $2,3,$ and $4$ from now. The term structure is flat at r$=5\%.$ The assets of the company are in cash, and the value of the assets is equal to the value of liabilities. The company decides to invest all its cash in T$-$ year zero coupon bonds with face value $100$ to immunize its liabilities so that the company$$s net worth is approximately insensitive to parallel shifts of the term structure. What should be the maturity T of these bonds? Round up T to the nearest integer, and find how many units of bonds with this maturity the company need to buy.

d$)[10$ marks$]$ XYZ corporation has net income NI in year t$=1,$ cost of capital r$,$ and payout ratio p$.$ Moreover, its ROE is equal to the cost of capital r$.$ Derive the value of the firm as a function of NI and r$.$ Is the price affected by the payout ratio? Provide intuition for your result.

e$)[12$ marks$]$ Consider a perpetuity with annual payments starting in year $5.$ The first payment in year $5$ is $10,$ and then the payment grows at a rate of $1\%$ per year forever. Consider a forward contract that delivers this perpetuity in year $3($that is$,$ the forward price is paid in year $3$ and the first perpetuity payment is in year $5).$ What is the forward price at date $0$ if the term structure is flat at r$=2\%?$ What is the value of the long position in this forward contract one year later if in year $1$ the term structure shifts to $1\mathrm{.}8\%$ and remains flat?

$2.$

This question is about options, futures, and forwards $[50$ marks$]$

Use discrete compounding of interest rates throughout this question.

$[12$ marks$]$ The current price of stock XYZ is $100.$ In each period it either goes up by $10\%$ or down by $10\%.$ Assume that the stock does not pay dividends. The riskless interest rate per period is $2\%.$ Construct a binomial tree with two periods and three dates $($t$=0,1,2)$ for the stock price movements. Calculate the risk neutral probabilities of the up and down moves. Calculate the time$-0$ price of an

American put option written on the square root of the stock price. This option pays $$ #$!$ when it is exercised and $0$ otherwise. The strike price is $10\mathrm{.}5.[$Hint: the method of option pricing is the same as for standard options, only option payoffs are different.$]$

$[11$ marks$]$ A European at the money straddle $($that is$,$ the strike of the options needed to construct this straddle is equal to the current stock price S$0)$ with maturity T$=1$ year on EFG stock costs $$10.$ The current stock price is $$100.$ The annualized riskless interest rate is r$=5\%.$ What are the prices of at the money call and put options on EFG stock with maturity T$=1$ year implied by the straddle price? Suppose at the money call and put options on EFG stock with maturity T$=1$ year are traded at prices C and P$,$ respectively, such that C$=$P$+$$$4.$ Is there an arbitrage? If so$,$ construct an arbitrage strategy by trading stocks, bonds, call and put options. $[$Hint: straddle is a portfolio of one call and one put with the same strike and maturity, and written on the same stock.$]$

$[8$ marks$]$ Consider a security with payoff D$($ST$)$ depicted below. The time to maturity is T$=1$ year, the riskless rate is r$=5\%,$ and the prices of call options with strikes $10,30,$ and $50$ are given by $5,3\mathrm{.}5,$ and $2,$ respectively. What is the price of this security at date $0?$ Note that D$(0)=20,$ D$(10)=20,$ D$(30)=0,$ D$(50)=20,$ and D$(60)=20.$ What kind of bet does this security represent?

$[7$ marks$]$ Consider two European$-$type options that have payoffs $()=$ max $(1,0)$ and $"$##

$()=$ max $(1$ #$,0),$ where stock price $$ can take any value greater than zero. Both options $###

have the same maturity date and are written on the same stock. Which of these options costs more at date $0?$ Explain your answer.

$[12$ marks$]$ Consider a forward contract to deliver one unit of stock of ABC corporation. The maturity is $1$ year from now, and the forward price at the current date $($date $0)$ is F$0=100.$ The stock price is $110$ and it will pay $10$ divi