Reference no: EM132384289

COMM 477 - Risk Management and Financial Engineering

Assignment

1. You can buy or sell the following default-free bonds. All bonds mature at Year 3. Their prices today and cash flows at the end of each year are the following:

Bond Price CF Year 1 CF Year 2 CF Year 3

A 94 4 4 104

B 100 6 6 106

C 112 12 12 112

Given these prices, you realize that yoiu can take a position to lock in an arbitrage profit of $6 today with no future net cash flows. What is your investment strategy that guarantees this arbitrage profit? (Feel free to use a computer to solve this question).

2. There are four default-free bonds on the market. All bonds mature at or before the end of Year 3. Their prices today and cash flows at the end of each year are the following:

Bond Price CF Year 1 CF Year 2 CF Year 3

A 124.67 15 20 125

B 99.89 5 10 115

C 100.12 7 107

D 95.23 100

Is there an arbitrage opportunity? If yes, state your investment strategy. (Feel free to use a computer to solve this question).

3. NOTE: this question needs to be solved numerically (Excel, Python,. . . ) Consider the following information about five default-free bond positions. Assume that the pricing of these five positions admits no arbitrage strategies.

Position Price CF Year 1 CF Year 2 CF Year 3 CF Year 4 CF Year 5

A 309 75 85 48 66 110

B 1918 374 562 400 492 532

C 341 37 111 104 114 46

D 682 74 222 208 228 92

E 325 56 98 76 90 78

In addition, consider two positions with the following cash flows.

Position CF Year 1 CF Year 2 CF Year 3 CF Year 4 CF Year 5
F 169 157 68 108 252
G 18 124 129 138 14

(a) Attempt to price position F using the information given about positions A through E. If you cannot price this position by no arbitrage, compute the upper and lower bound for this position’s price, that is,

i. Find the minimum cost of constructing a portfolio of bonds A–E with cash flow at least as good as the cash flows of bond F.

ii. Find the maximum cost cost of constructing a portfolio of bonds A–E with cash flow at most as good as the cash flows of bond F.

(b) Attempt to price position G using the information given about positions A through E. If you cannot price this position by no arbitrage, compute the upper and lower bound for this position’s price, that is,

i. Find the minimum cost of constructing a portfolio of bonds A–E with cash flow at least as good as the cash flows of bond G.

ii. Find the maximum cost cost of constructing a portfolio of bonds A–E with cash flow at most as good as the cash flows of bond G.

4. Suppose that firms face a 40% income tax rate on positive profits and that net losses receive no credit. That is, if profits are positive, after tax income is (1 − 0.4) × Profit, while if there is a loss, after-tax income is the amount lost. Firm A has a 50% probability of a $1,000 profit and a 50% probability of a $600 loss each year.

(a) What is the expected pre-tax profit for firm A?

(b) What is the expected after-tax profit for firm A?

(c) Discuss whether risk management can help firm A to increase its aftertax profit. Provide a numerical/graphical example if you can.

5. The SPX index spot price is 1100 and the continuously compounded risk-free rate is 5%. You observe a 9-month forward price of 1129.257.

(a) What dividend yield is implied by this forward price?

(b) Suppose you believe the dividend yield over the next 9 months will be only 0.5%. What arbitrage would you undertake?

(c) Suppose you believe the dividend yield over the next 9 months will be 3%. What arbitrage would you undertake?

6. Suppose the S&P500 currently has a level of 875. The continuously compounded return on a 1-year T-bill is 4.75%. You wish to hedge an $800,000 portfolio that has a market beta of 1.1 and a correlation of 1 with the S&P.

(a) What is the 1-year futures price for the S&P500 assuming no dividends?

(b) Assume that the payoff of one long future contract with future price F0,T is $250×

(S˜T − F0,T ), where S˜

T denotes the S&P index level at time T. How many S&P futures contract should you short to hedge your portfolio? What return do you expect on the hedged portfolio?

7. The current price of oil is $32.00 per barrel. Forward prices for 3, 6, 9, 12 months are $31.37, $30.75, $30.14, $29.54. Assume a 2% continuously compounded annual risk free rate.

(a) What is the annualized convenience yield (or lease rate) for each maturity?

(b) What is the Swap price for a 1-year swap with payments in month 6 and 12?

8. The following table contains information on current prices of zero-coupon bonds, oil futures and euro futures:

Quarter 1 2 3 4 5 6 7 8

T-bill prices 0.9852 0.9701 0.9546 0.9388 0.9231 0.9075 0.8919 0.8763

Oil fwd prices 21 21.1 20.8 20.5 20.2 20 19.9 19.8

Euro fwd prices($/e) 0.9056 0.9115 0.9178 0.9244 0.9312 0.9381 0.9452 0.9524

(a) What is the price of an 8-period swap for which two barrels of oil are delivered in even-numbered quarters and one barrel of oil in odd-numbered quarters?

(b) What is the fixed rate in a 8-quarter interest rate swap with the first settlement in quarter 1?

(c) What is the fixed rate in a 4-quarter interest rate swap with the first settlement in quarter 1?

(d) What is the fixed rate in a 5-quarter interest rate swap with the first settlement in quarter 2?

(e) What is the dollar value of an 8-quarter annuity paying e1 each quarter?

9. Payoff Diagrams

The stock of Klaatu Inc. is currently trading at S0 dollars per share. Assuming K1 < K2 < K3 < K4 < K5, consider a portfolio of European put and call options which pay the following on expiration in 6 months:

Stock Price (S6) Portfolio Payoff

S6 < K1 3 × (K1 − S6)

K1 ≤ S6 ≤ K2 0

K2 < S6 ≤ K3 K2 − S6

K3 < S6 < K4 S6 − K4

K4 ≤ S6 ≤ K5 0

K5 < S6 3 × (K5 − S6)

In addition, make the following assumptions:

• K3 = 1 2

(K2 + K4),

• 6 month European calls and puts are available with strike prices K1 and K5,

• only 6 month European puts are available with strike prices K2, K3, and K4,

• S0 = Present value of K5.

(a) Construct a final payoff diagram for this strategy.

(b) Identify the relevant options needed to create this portfolio. Indicate the position taken in each option.

10. In each of the following cases, identify whether there is an arbitrage and, if so, demonstrate how you would make money by creating a table showing your payoff.

(a) Consider two European options on the same stock with the same time to expiration. The 50-strike put costs $7 and the 55-strike put costs $14.

(b) Consider two European option on the same stock with the same time to expiration. The 50-strike call costs $16 and the 55-strike call costs $10. Please assume that the risk-free interest rate is positive.

11. Suppose the interest rate is 0 and the stock of ABC has a positive dividend yield.

(a) Is there any circumstance in which you would early exercise an American ABC call?

(b) Is there any circumstance in which you would early exercise an American ABC put?

12. A stock currently sells for $32.00. A 6-month call option with a strike of $35.00 has a premium of $2.27. Assuming a 4% continuously compounded risk-free rate and a 6% continuous dividend yield, what is the price of the associated put option?