PLEASE DON'T USE CHAT GPT TO ANSWER.

PRICING AND DERIVATIVES QUESTIONS

1. Explain carefully the difference between hedging, speculation, and arbitrage

2. Choose the correct answer. In the binomial tree model, the price of a European call option is computed by:

a) Multiplying the option payoff at maturity with the actual probabilities and discounting at the risk-adjusted rate

b) Multiplying the option payoff at maturity with the risk-adjusted probabilities and discounting at the risk-free rate

c) Multiplying the option payoff at maturity with the risk-adjusted probabilities and discounting at the risk-adjusted rate

d) Multiplying the option payoff at maturity with the actual probabilities and discounting at the risk-free rat

3. The risk-free rate is 0%. Consider two derivative contracts written on the same stock:

A European call option with an exercise price of K=50 and maturity T=1 year with a current price of $5.

A European put with the same exercise price and maturity and a current price of $15. Answer to the following questions:

i) What is the current stock price assuming a complete market and no arbitrage opportunities?

ii) Consider another derivative contract worth $45 today that is written on the same stock and with the same maturity, but a payoff at maturity equal to twice the distance between the underlying stock price and $50. Is it possibly to benefit from an arbitrage opportunity if one trades this security together with the other two derivative contracts?

iii) Consider yet another derivative contract in the form of a forward contract with a delivery date in one year and the same underlying security. The forward price of this contracts is $ 50 at the outset. Can one build an arbitrage portfolio by trading this new security along with the call and put option contracts?

4. A trader enters into a short cotton futures contract when the futures price is 50 cents per pound. The contract is for the delivery of 50,000 pounds. How much does the trader gain or lose if the cotton price at the end of the contract is (i) 48.20 cents per pound and (ii) 51.30 cents per pound?

5. Briefly explain put-call parity.

6. EMG is currently trading at $90 per share. An American call on EMGs stock with a strike price of $80 and 6 months to maturity is currently trading at $11. The risk-free (continuously compounded) interest rate is 8% p.a. Intel will pay out a dividend in exactly a month, but will not make no other dividend payment until the contract matures. It has not yet been announced what the amount of the dividend will be, but analysts speculate it will be between $1 and $5 per share. Answer the following questions.

i) Can a hedge fund manager ascertain she will achieve an arbitrage profit by trading the above securities?

ii) Could the manager earn an arbitrage profit had the firm announced that it will pay a dividend of $1 in one month? Explain carefully the trading positions to generate the profit if there is an arbitrage opportunity.

7. Choose the correct answer: The IBM stock is trading at $52. It is known that IBM will pay a dividend in 2 months time; this will be the only dividend paid in the next three months. Assume the annualized interest rate is 6%. You have a long position in an American call option with a strike price of $50 and a maturity of 3 months. Should you exercise the option in 1 month if the stock price has increased by 20% next month?

a) True

b) False

8. As a hedge fund manager, you are in charge of devising a dynamic hedging strategy to offset the risk of a portfolio made up of a long call and a short put. The options are European and written on the same non-dividend-paying stock, with the same exercise price and the same time to maturity. Compute the delta, gamma, theta, rho and vega of that particular portfolio (or synthetic/redundant security)

9. Consider a securities market comprising (a) a money-market instrument with a continuously compounded interest rate of 1% per annum, (b) a risky stock currently trading at $20 which can increase to $22 or decrease to $18 and (c) a European call option written on that risky stock with a strike price of $21 which matures in 3 months. Answer the following questions.

i) Can you create a portfolio that mimics the return on the money-market instrument? If yes, please indicate the exact composition of this portfolio. What is the present value of this portfolio? Can you infer the fair price of the call option?

ii) Can you interpret this call option value as the present value of an expected option payoff? If yes, what is the probability of a decrease over the 3-month period that you should use in your computation? Is the call option price then consistent with the result in part 1 of the question?

10. Explain why an American option is always worth at least as much as a European option on the same asset with the same strike price and exercise date.

11. Choose the correct answer: Both European call and put options decline in value when volatility decreases

a) True

b) False

12. At dinner, one of your parents announces they believe Apple stock (currently at $54 with a volatility of 54%) will increase and want to speculate massively by investing in a European call option written on that stock (with a strike price of $56, a maturity of 2 years). As a student educated at emlyon, you are used to adopt an analytic and down-to-earth approach to security trading and argue that your parent should beware that they acquire the call option at a fair price. Calculate the fair price of this option using the Cox-Ross-Rubinstein binomial model with a step of one year. You also obtained an option value using the Black-Scholes-Merton formula, which differs from the one you just computed. Suggest reasons why these option values do not perfectly match. N.B. A government bonds currently yields an interest of 3% per annum. Apple stock does not pay dividends.

13. EMGs capital structure is made of 500,000 shares of common stock and of 50,000 bonds. Currently, one share trades at $40. The debt instrument is zero-coupon bond maturing in exactly 2 years, each with a face value of $1,000. Historically, the assets in this industry have achieved a return with a volatility of 25%. The interest rate for a Treasury bond maturing in 2 years time is 4% per year. Explain how to determine the current value of the firms assets.

14. Choose the correct answer: Which of the following parties has the right to sell an asset at a predetermined price?

a) A put writer

b) A put buyer

c) A call buyer

d) A call write

15. As a hedge fund manager, you own a portfolio composed of (1) a long position in a European call and (2) short position in a put written on the same stock, with the same strike and the same maturity. Answer the following questions.

i) You are interested in generating arbitrage profits. Is there a standard traded derivative instrument you should trade if you observe a differential between the value of your portfolio and the price of that derivative instrument?

ii) What is the value of your option portfolio at time as a function of the option contracts primitives (stock price, drift parameter, diffusion parameter, strike price, maturity date, risk-free rate).