ACT 370 Final Exam Page 1 of 7

UNIVERSITY OF TORONTO

Faculty of Arts & Science

AUGUST 2021 EXAMINATIONS

ACT 370H1S

Financial Principles for Actuarial Science II

RESTRICTED EXAM

Duration: 3 hours

Student Name and Number:

Exam Reminders:

Non-programmable calculators are permitted.

State any assumptions you feel you are required to make to answer the question.

Answer all questions – there is no choice; part marks can only be awarded if you write something.

Do not begin writing the actual exam until the announcements have ended and the Exam Facilitator has

started the exam.

As a student, you help create a fair and inclusive writing environment.

ANSWER ALL 8 QUESTIONS

180 MARKS IN TOTAL

SUBMIT YOUR SOLUTIONS AND ID VERIFICATION BY UPLOADING TO QUERCUS

ACT 370 Final Exam Page 2 of 7

Question 1 (Options Markets and Properties of Options) – (15 Marks)

Part A (5 Marks)

What happens when futures options are exercised? Why would a trader prefer futures options?

Part B (5 Marks)

What makes warrants, employee stock options and convertible bonds different from other exchange-traded

options?

Part C (5 Marks)

Should American options be exercised early? Explain why and/or why not.

Question 2 (Trading Strategies Involving Options) – (35 Marks)

Part A (30 Marks)

You are considering a Butterfly strategy in which you believe the underlying stock price will stay at $60 for a

certain period of time.

You have the following call options available to you:

Strike Prices of $50, $60 and $70 with each option priced at $22, $15 and $10 respectively

You have the following put options available to you:

Strike Prices of $50, $60 and $70 with each option priced at $11.50, $14.40 and $19.30 respectively

All options are European with 1 year to maturity. The risk free rate is 1% (continuous compounding).

1. Create your strategy using only call options. What option position(s) would you make? Draw the payoff and

profit diagrams. Clearly label your charts. What range of stock prices would this strategy be profitable?

2. Create your strategy using only put options. What option position(s) would you make? Draw the payoff and

profit diagrams. Clearly label your charts. What range of stock prices would this strategy be profitable?

3. Create an Iron Butterfly (i.e. using calls and puts). What option position(s) would you make? Draw the payoff

and profit diagrams. Clearly label your charts. What range of stock prices would this strategy be profitable?

4. Given these three methods to create a Butterfly, explain why you would choose one method over the other.

Pick a call and a put option with the same strike price. Compare these prices using put-call parity. Do these

prices make sense? Please show your calculations.

Part B (5 Marks)

A trader is considering a Condor strategy instead of a Butterfly. Identify one advantage and one disadvantage a

Condor strategy has compared to a Butterfly. Draw a payoff diagram for the Condor and illustrate how you

would create this. (Note: Part B is separate from Part A)

ACT 370 Final Exam Page 3 of 7

Question 3 (Binomial Option Pricing Model) – (25 Marks)

An American put futures option has a strike price of $0.55 and a time to maturity of 1 year. The current futures

price is $0.60. The volatility of the futures price is 25% and the interest rate (with continuous compounding) is

6% per annum. Use a four step tree to value the option.

Question 4 (Black-Scholes-Merton Model) – (15 Marks)

Consider an option on a non-dividend-paying stock when the stock price is $19, the exercise price is $20, the

risk-free interest rate is 1.5% per annum (continuous compounding), the volatility is 20% per annum, and the

time to maturity is one year.

a) What is the price of the option if it is a European call?

b) What is the price if it is a European put (hint: use put-call parity)?

c) Is there another way to calculate the put price? Explain.

d) Explain the concept and the assumptions underlying the Black-Scholes-Merton Pricing formula.

Question 5 (The Greeks) – (25 Marks)

For this question please refer to the following call options:

Call Options C1 C2 C3 C4

Strike Price X = $90 X = $100 X = $110 X = $100

Time to

Maturity

T = 180

days T = 180 days T = 180 days T = 90 days

Option Price 16.33 10.3 6.06 6.91

Delta 0.786 0.6151 0.4365 0.582

Gamma 0.0138 0.0181 0.0187 0.0262

Theta -11.2054 -12.2607 -11.4208 -15.8989

Vega 20.4619 26.8416 27.6602 19.3905

Rho 30.7085 25.2515 18.5394 12.6464

Part A (15 Marks)

A trader can create a delta neutral portfolio from a stock and a call option written on that stock or from a stock

and a put option written on that stock. In general, a stock plus one option can be made neutral to one of the

Greek parameters.

a) A portfolio composed of a stock and a single option written on that stock can only be made both delta

and gamma neutral if what special condition is true? (5 Marks)

b) Create a portfolio that is both Delta and Gamma neutral using stock, call C2 and call C3. Also calculate

and comment on this portfolio’s theta, vega and rho. (10 Marks)

Part B (10 Marks)

Consider a calendar spread formed of call C2 and call C4. What is the cost of establishing this position

now? Calculate the Greek sensitivities for this option position now. Comment on these values.

ACT 370 Final Exam Page 4 of 7

Question 6 (Forward and Futures Prices) – (15 Marks)

Part A (10 Marks)

Consider a stock with a current price of $50. The stock will pay quarterly dividends of $0.50 starting 3 months

from now. The risk-free rate of interest (continuously compounded) is 1.75% for all maturities.

a) What should the forward price of an 8-month contract be?

b) Now suppose you found a contract offering a forward price of $55. Is there an arbitrage opportunity? If

yes, explain in detail what you would do to exploit the arbitrage opportunity and calculate the ultimate

profit you would realize. If no, explain.

Part B (5 Marks)

We discussed about the forward curve (i.e. a series of consecutive month’s prices for future delivery of an asset)

for NYMEX HO (i.e. heating oil) and NYMEX Natural Gas. What is the significance of the forward curve? What

should the forward curve look like for each of these two commodities?

Question 7 (Interest Rate Futures) – (20 Marks)

Part A (10 Marks)

The cheapest-to-deliver bond in a Treasury bond futures contract is an 8% coupon bond, and delivery is

expected in 296 days. Coupon payments on the bond will be made in 175 days and 357 days from now. The last

coupon date was 9 days ago. The rate of interest with continuous compounding is 5% per annum for all

maturities (i.e. term structure is flat). The conversion factor for the bond is 1.2191. The current quoted bond

price is $137. Calculate the quoted futures price for the contract (i.e. settlement price of futures contract).

Part B (5 Marks)

A Eurodollar futures quote for the period between 5.1 and 5.35 year in the future is 97.1. The standard deviation

of the change in the short-term interest rate in one year is 1.4%. Estimate the forward interest rate in an FRA.

Part C (5 Marks)

What do you know about the conversion factor for a bond?

ACT 370 Final Exam Page 5 of 7

Question 8 (Swaps) – (30 Marks)

Part A (15 Marks)

Interest rates of all maturities (i.e. short term to long term) are 1% with continuous compounding in the U.S. and

2% with continuous compounding in the UK (note: assume LIBOR is 2% – i.e. flat yield curve). Assume that rates

have remained unchanged for the last year. The current exchange rate is 1.3000 (USD/£). Consider a company

that has entered into a swap where it will receive a fixed rate of 4 percent in US dollars on a principal of $13

million, and pay LIBOR in pounds on a principal of 10 million British pounds. Payments are annual. The swap has

a remaining life of 18 months.

a) Calculate the value today in US dollars of the payments that will be received in US dollars.

b) Calculate the value today in British pounds of the payments that will be made in pounds (Hint: The

LIBOR rate given above is expressed with continuous compounding. There’s an extra step you need to

consider.).

c) What is the value of the swap today in USD?

d) Part of a bank’s bid-offer spread in the swap market is compensation for credit risk. Explain why the

credit risk component of the bid-offer spread is usually higher for currency swaps than for interest rate

swaps.

Part B (15 Marks)

Firm AAA issued a 5-year bond for a principal of $100 million at a fixed yearly rate of 3.8% (semi-annual

compounding), while Firm BBB issued a 5-year loan based on the same principal at the floating rate of LIBOR +

1% (semi-annual compounding). The two companies also entered at the same time a bilateral 5-year swap

agreement in which AAA pays LIBOR to BBB and receives 3.6% from BBB. The total gain from the swap is 0.8%

that is shared equally between AAA and BBB. Payments are semi-annual, and there are 13 months left before

the bond and the swap expires. The LIBOR rate was 4.2% (semi-annual compounding) 5 months ago. Interest

rates for all maturities are 5% (continuous compounding).

a) Calculate the overall (net) rate paid by AAA and BBB

b) Calculate the value of the swap for AAA and BBB

c) BBB is doing extremely poorly and has been downgraded. As a result, BBB’s floating loan rate has

increased to LIBOR + 10%. Answer b) in this new scenario.

d) Given the value of the swap to AAA, is AAA facing a higher credit risk due to the downgrade of company

BBB?

e) Considering what has happened to BBB, did BBB made the right decision in entering the swap?

Reference: Hull

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