Problem 1: Options Basics

Imagine AAPL stock closed at $520 on November 15th. At that time, the following options were traded for AAPL stock:

Call or put

Expiration date

Strike price

Option premium

Call

December 21st

$505

$26.50

Put

December 21st

$505

$13.30

Assume the interest rate is 0%.

(a)According to put-call parity, what is the present value of AAPL dividends betweenNovember 15th and December 21st?

Present value of dividends:

(b)Assuming AAPL's share price increases by 5% between now and December 21st, compute the payoff, profit, and return for each of the following positions:

(i)A long position in 100 call options, bought at $26.50 per call. (ii) A long position in 100 put options, bought at $13.30 per put.

As a reminder, the payoff from an options position is the value of the option at expiration (ignoring the premium paid), the profit is the payoff minus the premium, and the return equals the profit divided by the initial premium.

Position

Payoff

Profit

Return

Long 100 call options

Long 100 put options

(c)Assuming AAPL's share price decreases by 10% between now and December 21st, compute the payoff, profit, and return for each of the following positions:

(i)A long position in 100 call options, bought at $26.50 per call. (ii) A long position in 100 put options, bought at $13.30 per put.

As a reminder, the payoff from an options position is the value of the option at expiration (ignoring the premium paid), the profit is the payoff minus the premium, and the return equals the profit divided by the initial premium.

Position

Payoff

Profit

Return

Long 100 call options

Long 100 put options

Problem 2: Binomial Options Pricing (5 points)

Imagine a case where there were only two possible stock prices in one month: $100 and $80, occurring with probabilities 60% and 40% respectively, and the current stock price was $91:

$100

$91

40%$80

We will build up to finding the no-arbitrage price of a call option with strike K = 95 that expires exactly one month from now.

(a) What will the call payoff be in case that S_T = $100? What about S_T = $80? We'll call the first number O_u and the second O_d.

O_u =O_d =

(b) What portfolio of risk-free bonds with face value B and shares of the underlying stock will replicate the call option?

B = =

(c) What is the no-arbitrage price of the call option, assuming the continuously compounded risk-free rate is 2% annually?

C_t =

(d) Given the call option trades for the price you found in part (c), what is the expected return of the call option? What about the underlying stock? (Hint: you will need to use the probabilities 60% and 40%).

Call expected return:Stock expected return:

Problem 3: Two-stage Binomial Options Pricing (5 points)

A stock with current share price of $20 can either go up or down by $2 in each of the next two months, with up occurring 55% of the time and down occurring 45% of the time. To keep things simple, we will assume that the risk-free rate is 0%, and that you can both borrow and lend at this rate. There are call and put options traded with a strike price of $22 expiring in two months. Our goal will be to find the no-arbitrage price of both the call and the put today.

S_uu = $24 S_u = $22

S₀ = $20S_ud = $20

S_d = $18

S_dd = $16

(a) Find the no-arbitrage price of both the call and the put one month from now if the stock is worth $22.

No-arbitrage call price C_u:No-arbitrage put price P_u: (b) Find the no-arbitrage price of both the call and the put one month from now if the stock is worth $18.

No-arbitrage call price C_d:No-arbitrage put price P_d:

(c) Find the no-arbitrage price of both the call and the put today. (Hint: you'll need to use the values you found in parts (a) and (b) as your option payoffs O_u and O_d).

No-arbitrage call price C₀:No-arbitrage put price P₀:

Problem 4: Options Short Answers (5 points)

(a)Imagine AAPL stock closed at $520 on November 15th. At the same time, the followingoptions were traded for AAPL stock:

Call or put

Expiration date

Strike price

Option premium

Call

December 21st

505

$26.50

Put

December 21st

505

$13.30

Assuming AAPL's share price is $560 on December 21st, compute the payoff, profit, and return for a straddle (a portfolio of one call and one put) with strike price 505 purchased at market close on November 15th.

As a reminder, the payoff from an options portfolio is the total value of the underlying options at expiration (ignoring the premiums paid), the profit is the payoff minus the premiums paid, and the return equals the profit divided by the initial premium. Position

Payoff

Profit

Return

Long 1 straddle

(b)Say you were short Google stock in your portfolio but wanted to limit your losses to 25%of the amount your shorted. What option trade should you make (circle one of buy or sell, and one of put or call)?

Buy / Sell Google put / call options.

(c)Say you worked at Company A and found that Company B has stock returns moststrongly correlated with Company A's of all publicly traded firms. If you wanted to hedge the risk of being fired from Company A because the company peforms poorly, what option trade should you make in Company B's options?

Buy / Sell Company B put / call options.