Topic A – Suggested Time Allocation: 20 Minutes

a. Hey, this is a question about derivatives.

b. This is a related question about derivatives, but you can answer it even if you couldn’t answer a.

c. A third related question, but you can answer it even if you couldn’t answer a or b.

Problem 2; 40 marks (Mark breakdown: 20-10-10)

An Example – Suggested Time Allocation: 10 Minutes

We calculated N(d1) for lots of things. Suppose a stock has a current price of 9, an expected return of 15% and volatility of 12%, and the risk-free rate is 5%. You have a call option with a strike price of 10 and maturity of 5 years.

a. Calculate d1.

b. Ignore your answer for a. Suppose you found d1 = 0.10. What is N(d1)?

c. Ignore your answer for a. Suppose you found d1 = 0.10. What is N(d2)?

1. You have been asked to calculate VaR for a bank. The bank’s assets are currently 1 Billion and liabilities are 750 Million.

a. What is the bank’s equity?

b. The variance of equity values is 250 Million. You are told the expected gain in assets is 10% and the expected gain in liabilities is 12%. What is the 10% VaR?

c. What is the 1% VaR?

2. As income increases, tax rates also increase (this is called a “progressive tax rate”). Suppose taxes are 0% for income < 20,000, 25% for income between 20,000 and 35,000, and 35% for income above 35,000. You can invest in a new technology with current market value of 25,000 and volatility of 20%. The risk-free rate is 2%, and you’ll learn about the true value of the new technology in one year.

a. What portfolio mimics this payoff (Hint: Payoff Diagram/Table!)?

b. What is the price of that portfolio?

c. Is c. the “no-arbitrage” price?

d. If the volatility of the technology is 25%, is the value higher or lower?

3. The forward price is 100 for a one-year forward contract, the risk-free rate is 2%, and the current spot price is 100. The underlying asset is onions.

a. What is the convenience yield/cost of carry?

b. What is the Forward price if the convenience yield/cost of carry is zero?

c. You see two Forward contracts available on a share of stock. There is no cost of carry or convenience yield. One forward is a six-month contract with forward price of 100.45. The other is a one-year contract with forward price of 101. You may take either side of either contract. Interest rates are 1% for borrowing or lending. Is there an arbitrage involving buying or selling 1 of the six-month contract?

4. The “Harry Potter” (I made it up) is short a put option with a low strike price, short a call option with the same strike price and long two call options with a medium-low strike price, short three calls with a medium-high strike price, and long four calls with a high strike price. Draw the payoff table and payoff diagram. Note: It’s better if the medium-low strike is closer to the low strike than the medium-high strike.

5. A stock’s price is currently 20, and the stock has a volatility of 15%. The risk-free rate is 2%. Draw a two-step tree that covers the next two months’ time. Suppose buying or selling stock requires $0.10, but buying or selling derivatives is free. You can also borrow or lend at 0.1%. What prices do not permit arbitrage for a European call option with strike price equal to $21 and 2-month maturity?

6. A “something” sets two price levels. If the spot price at maturity is lower than the lowest price level but the average price over the contract’s life is higher, the payoff is the average price over the contract’s life. If the spot price ends up higher than the highest price level but the average price over the contract’s life is lower, the payoff is the highest price level. Otherwise, the payoff is the average of the two price levels.

Price a three-month “something” using the following information: The current spot is $10, the risk-free rate is 0%, and the “something” price levels are $7.50 and $15. You’ve simulated the following 5 series of returns (each series has three one-month returns):

0.0441

-0.3154

-0.3218

-0.0018

0.3168

-0.3589

-0.0145

0.3063

0.3647

-0.2974

-0.151

0.2949

0.0441

-0.3154

-0.3218

7. Short answer:

a. You have a contract that allows you to choose to purchase an underlying asset for a specified price at any time over the next year. What is this?

b. You are a farmer growing wheat. What derivatives might you use to hedge your risks?

c. “You can never Delta and Gamma hedge an option with one other option.” Explain why this is incorrect in a few sentences.

d. Suppose you make jewelry. You use gold as an input, which you’ll need in two months’ time. You short a gold futures contract. Are you hedged? What are your payoffs at maturity?

SPACE FOR ROUGH WORK

Examiners will NOT take into account anything written on this page.

SPACE FOR ROUGH WORK

Examiners will NOT take into account anything written on this page.

SPACE FOR ROUGH WORK

Examiners will NOT take into account anything written on this page.

SPACE FOR ROUGH WORK

Examiners will NOT take into account anything written on this page.

FORMULA SHEET

𝑈=𝑒𝜎√𝑇

𝜎2=[𝑥1−𝑥]2+[𝑥2−𝑥]2+⋯+[𝑥𝑇−𝑥]2𝑇

𝑑1=ln(𝑆𝐾)+(𝑟+𝜎22)𝑇𝜎√𝑇

𝑑2=ln(𝑆𝐾)+(𝑟−𝜎22)𝑇𝜎√𝑇

𝑐=𝑆 𝑁(𝑑1)− 𝐾 𝑒−𝑟𝑇𝑁(𝑑2)

𝑁′(𝑥)=1√2𝜋𝑒−𝑥22

𝐺𝑎𝑚𝑚𝑎=1𝑆𝜎√𝑇𝑁′(𝑑1)