INSTRUCTIONS:

❼ The total marks for this assignment are 100.

❼ Please work on this assignment in a group of at most 4 people, photocopy your work and

submit one pdf version on eclass by 11:30 pm, April 12.

❼ Please write down the names and student ID’s of the group members.

❼ Please write your answers clearly, since 0 marks will be assigned to ambiguous answers.

❼ Partial credit will be given for showing logically correct reasoning, even if the final answer is

not correct.

❼ If not otherwise specified, put 2 decimal places in the final result for dollar amount (eg, $1.99)

and rates (eg, 1.99%); therefore, you need to retain more decimal places in your intermediate calculation steps (6 decimal places recommended).

❼ If not otherwise mentioned, 1 unit of derivative is on 1 unit of underlying.

❼ If not otherwise mentioned, rates are continuously compounding, except for interest rate

swaps and forward rate agreements.

1.  Interest Rate Swaps (12 points)

Bank of Pokemonland entered an interest rate swap several months ago.  According to the terms and conditions of the IRS, the swap rate is 3.2%, the notional principal is $100 million, and it in- volves semiannual payments. Bank of Pokemonland is receiving the swap rate and paying LIBOR. Currently, the IRS still has 18 months to maturity.  As of today, the continuously compounded risk-free rates per annum are 2.2%, 2.4% and 2.8% for the future 6 months, 12 months and 18 months.  The 6-month LIBOR forward rates per annum are 3.0%, 3.5% and 4.0% for the periods ending in 6 months, 12 months and 18 months. To Match the payment frequency of the IRS, the LIBOR forward rates and swap rates are quoted as semiannual APR.

(a) [6 points] Assuming that the actual rates for LIBOR happen to be the same as the forward rates given above, what are the net cash flow received/paid by Bank of Pokemonland in 6 months, 12 months and 18 months respectively?

(b) [4 points] What is the value of IRS today to Bank of Pokemonland?

(c) [2 points] What is the value of IRS today to the counterparty of Bank of Pokemonland?

2.  Interest Rate Swaps (12 points)

Wigglytuff Co. has just entered into an IRS (notional principal $1,000,000 with quarterly payment) where the reference rate is the 3-month SOFR. The continuously compounded risk-free spot rates per annum based on SOFR are the following:

. for the future 3 months: 5.00%

. for the future 6 months: 4.75%

. for the future 9 months: 5.00%

. for the future 12 months: 5.25%

The following are the corresponding forward rates per annum in continuous compounding based on SOFR and implied from the above spot rates:

❼ 0 months → 3 months: 5.00%

❼ 3 months → 6 months: 4.50%

❼ 6 months → 9 months: 5.50%

❼ 9 months → 12 months: 6.00%

Note: the compounding frequency of rates in the contract should match payment frequency of the IRS.

(a) [6 points] The company is the fixed-rate payer in this swap. What is the “fair”swap rate rK per annum that the company should pay?

(b) [4 points] If the company is willing to pay only 5% per annum as the fixed rate, how much should the dealer charge the company as a “fair”one-time fee today, in addition to the regular fees?

(c) [2 points] Suppose the company has an existing 1-year debt with quarterly payment of interest amount, where the interest rate per annum is SOFR + 4%.  Illustrate with a chart on how the company could use the swap in (b) to convert such a floating-rate debt to a fixed-rate debt.

3.  Forward Rate Agreements (22 points)

Six months ago, Charmander Capital Management entered an FRA to pay 4.5% per annum (quar- terly APR), and receive SOFR (quarterly APR) with notional principal of $1,000,000 and quarterly payment. At that time, the FRA still had a life of 15 months.

(a) [2 points] As of six months ago, what was the 3-month forward rate per annum (quarterly APR) ending at the maturity date of the contract, assuming the FRA has been priced fairly?

(b) [5 points] Today, the zero rates per annum for SOFR (continuously compounding) are given as below, calculate 3-month forward rates per annum (continuously compounding) ending in 3 months, 6 months, 9 months, 12 months and 15 months.

(c) [5 points] Based on the forward rates per annum (continuously compounding) in (b), calculate the corresponding forward rates per annum (quarterly APR).

(d) [4 points] What is the value today of the FRA to Charmander Capital Management ? What is the value today of the FRA to the counterparty of Charmander Capital Management?

(e) [3 points] Today, Pika Securities has an existing debt (principal is $500,000, the interest rate is SOFR, the only interest payment is in 15 months, and the interest amount is given by the formula in (f)).  The company is worried about an increase in SOFR that leads to an increase in interest amount.  Use a chart to help explain how Pika Securities may use an FRA to hedge against the risk of increase in interest rate. What is the fair contract rate rK  per annum in this FRA?

(f) [3 points] Continuing with (e), the interest amount of the existing debt in 15 months is

500, 000 ×

SOFR

尸         使

× 

quarterly compounding

What is the effective total cash flow to Pika Securities in 15 months, combining interest amount of the existing debt position and the cash flow of the FRA position?

4.  Futures Options:  Mechnism (6 points)

(a) [3 points] Suppose you have a long position in a put option on October gold futures with a strike price of $1,500 per ounce. Each contract is for the delivery of 100 ounces. What happens if you exercise when the October futures price is $1,350?

(b) [3 points] Suppose you have a short position in a call option on October wheat futures with a strike price of $6.2 per bushel. Each contract is for the delivery of 2,000 bushels. What happens if the contract is exercised when the futures price is $8?

5.  Futures Options:  Black Model (12 points)

(a) [4 points] A futures price is currently $50, the volatility of the returns of the futures price is 20% per annum, and the risk-free rate is 5% per annum. What is the value of a 9-month European call on the futures with a strike price of $50, according to the Black model?

(b) [4 points] Using the same parameters in (a), what is the value of a 9-month European put on the futures, according to the Black model? Compare the result with that in (a), what do you find?

(c) [4 points] Use the Black model to prove mathematically that:  “The price of an at-the-money European futures call option always equals the price of a similar at-the-money European futures put option”.

6.  Exotic Options (12 points)

The price of stock follows geometric Brownian motion with the volatility of returns σ = 25% per annum and continuous dividend yield q = 2%. The risk-free rate is 4% and the current stock price is $25. A chooser option is on the stock where the option style to be chosen is either a European call or a European put. The decision date is in 6 month while the maturity date is in 1 year. We’d like to price this chooser option with the help of the Black-Scholes model.

(a) [3 points] Applying the decomposition in class to price the chooser, what is the price of the European call with strike price of $20 and time to maturity of 1 year?

(b) [3 points] Applying the decomposition in class to price the chooser, what is the strike price of the European put with time to maturity of 0.5 years? What is the number of units for the put?

(c) [3 points] Based on (b), what is the put price?

(d) [3 points] Based on the results above, what is the price of the chooser?

7.  Exotic Options (20 points)

The current level of the S&P 500 index is 4,000, the risk-free rate is 4% per annum, the continuous dividend yield on the index is 4% per annum, and the volatility of the returns of the index is 20%. Assume the S&P 500 index level follows the geometric Brownian motion. You are betting against your friend Alex.

Your friend Alex offers you the following alternatives

A1: he will pay you $4000 if the index level is above 4000 in 6 months; 0 otherwise.

A2: he will pay you the value of the index level in 6 months if the index level is above 4000 in 6 months; 0 otherwise.

In return, you offer him the following alternatives

B1: you will pay him $4000 if the index level is below 4000 in 6 months; 0 otherwise.

B2: you will pay him the value of the index level in 6 months if the index level is below 4000 in 6 months; 0 otherwise.

(a) [4 point] What is the name of the derivative that has the payoff structure as in A1? What is the present value of the payoff for the betting in A1?

(b) [4 point] What is the name of the derivative that has the payoff structure as in A2? What is the present value of the payoff for the betting in A2?

(c) [2 points] A European call option is on the index with time to maturity of 6 months and strike price of $4000.  What is the no-arbitrage price of the call today, without using the Black-Scholes model.

(d) [4 point] What is the name of the derivative that has the payoff structure as in B1? What is the present value of the payoff for the betting in B1?

(e) [4 point] What is the name of the derivative that has the payoff structure as in B2?  What is the present value of the payoff for the betting in B2?