Properties of Stock Options

Question 1

What are the factors affecting option prices?

Answer 1

1. Stock Price 2. Exercise Price 3. Time to Maturity 4. Risk-free Rate 5. Volatility 6. Dividends So, Before Maturity: Price = f(S,K,T,r,δ,D)

Question 2

How do we value call and put options at maturity?

Answer 2

On the expiry date, the value of an in-the-money call option is simply the difference between the stock’s price and the exercise price; St-K On the expiry date, the value of an in-the-money put option is simply the difference between the exercise price and the stock’s price; K-St

Question 3

As Stock price increases, European & American call option value…?

Answer 3

Increases. European & American Call option value increases with stock price because an increase in the stock price today increases the expected stock price at maturity. If the expected stock price, E(St), at maturity increases the payoff, the value of the option at maturity also increases. A higher expected value at maturity implies that the option’s value should be higher today (take the present value).

Question 4

As Dividends are paid, European & American call option value…?

Answer 4

Decreases. Dividends reduce stock prices, decreasing expected stock price at maturity, lowering the expected option value at maturity, lowering the call option price today.

Question 5

As Strike price increases, European & American call option value?

Answer 5

Decreases. The higher the strike price, the more you need to pay at maturity to buy the underlying asset. The payoff becomes smaller and this leads to a lower option value at maturity. The current value of the option is lower as a result.

Question 6

As the risk-free rate increases, European & American call option value?

Answer 6

Increases. The higher the risk-free rate, the lower the strike price in present value terms. If we simply think that the current value of a call option today is given by max (0, S-PV(K)), then increasing the risk-free rate leads to a higher call price today.

Question 7

As Term to Maturity increases, European and American Call value …?

Answer 7

As TTM increases, we’re unsure of the effect on a European call/put, as while an increase in T decreases the value, it also increases the probability that the price will change more, making the option more valuable.

Question 8

What is put-call parity?

Answer 8

Put-call parity can be derived from arbitrage arguments. It states that the value of a European call with a certain exercise price and date can be derived from the value of a European put with the same exercise price and date. This is not the case for American options.

Question 9

c + Ke^(-rt) = ?

Answer 9

p + S Where: c = Price of European Call Option K = Strike price e = Euler’s e Ke^(-rt) = PV of strike price (risk-free investment for strike price). r = risk-free rate p = Price of European Put Option s = Price of Share

Question 10

p + s = ?

Answer 10

c + Ke^(-rt) Where: c = Price of European Call Option K = Strike price e = Euler’s e Ke^(-rt) = PV of strike price (risk-free investment for strike price). r = risk-free rate p = Price of European Put Option s = Price of Share

Question 11

What is the formula for American option put-call parity?

Answer 11

It is not close form. It is: S - K ≤ C - P≤ S - Ke^(-rt)

Question 12

What is the upper bound of a european call option?

Answer 12

The maximum price of a european call option cannot exceed the stock price. 
The best that can happen with a call is that you end up owning the stock (where strike k = 0). Otherwise we could sell a call-option, and use the premium to buy a share, with some cash left over. At expiry we have a share to satisfy the call, if exercised, together with cash Upper Bound: c ≤ S

Question 13

What is the lower bound of a European Call option?

Answer 13

The minimum price of a European call option must be at least as great as the price implied by the put-call parity with a zero put value. Lower Bound: c ≥ max⁡(0, S - Ke^(-rt)) (Expected to be in the money (as it has a value), therefore the corresponding put is not ITM, therefore p = 0). The price is =S-Ke^(-rt)+TV, will converge (=) by t=0.

Question 14

What is the upper bound of a european put option?

Answer 14

The Upper Bound of a European put option cannot be more than 
the strike price. (Share price goes to zero, buy & sell at strike) The strike price is the greatest payoff a put can have (St=0): p ≤ Ke^(-rt)

Question 15

What is the lower bound of a european put option?

Answer 15

The Lower Bound of a European put option must be at least as great as the price implied by put-call parity with a zero call value b/c it’s not worth anything if it’s expected to be out of the money: p ≥ max⁡(0, Ke^(-rt) - S) The value will converge (= zero call value) by t = 0).

Question 16

What are the upper and lower bounds for an American call option?

Answer 16

Upper Bound: c ≤ S Lower Bound: c≥max⁡(0,S - Ke^(-rt)) – We have the PV here because there is no benefit w/ activating early. You give up flexibility upon exercise.

Question 17

What are the upper and lower bounds for an American put option?

Answer 17

Upper Bound: p ≤ K - Differs as we can exercise the American put option early to get K, so no compounding required. Lower Bound: p ≥ max⁡(0, K - S) Don’t have the PV b/c we can exercise early.

Question 18

Consider a one-year European call option on a non-dividend paying stock with an exercise price of $25 and a call premium of $4. The current stock price is $32 and the continuously compounded risk-free rate is 10% per annum. Use bounds to check for an arbitrage opportunity and show and arbitrage opportunity.

Answer 18

Check the bounds for an arbitrage opportunity: We can check the bounds for this arbitrage opportunity b/c if the price is out of the bound then there’s an AO. Call Max: c = c ≤ S = 32 - No problem here Call Min: S - Ke^(-rt) = 32 - 25e^(-0.10(1)) = $9.38 4 = c

Question 19

When should an American Option be exercised prior to maturity?

Answer 19

The owner of an option would exercise an American option early if doing so brings net benefits, including dividend payments over the option’s life.

Question 20

When should an American Call Option be exercised prior to maturity?

Answer 20

Unless there is a net benefit, the rule for an American call option on a non-dividend paying stock is: It is never optimal to exercise the option early. This implies C = c, meaning we can value an American call on a 
non-dividend paying stock as if it were a European call option. If you exercise your American call now rather than at maturity, you: - Pay exercise price now (miss out on time value of money). - You give up the future flexibility of not exercising it at maturity (what if the stock price is much lower by maturity). If you think an American call option is overvalued, you want to sell it on the market b/c if you exercise you’ll get S - K, while the minimum bound is c ≥ S - Ke^(-rt), which should be more than S - K as the K has time value.

Question 21

When should an American Put option be exercised prior to maturity?

Answer 21

It may be optimal to exercise an American put option early if the option is sufficiently deep in the money. When StT = $0 & K = $30, K - S = $30 This is because the return won’t be more than $30, so it should then be exercised. If not 0, just sell the option, otherwise you’re foregoing the time value on the option.

Question 22

When should American Options on dividend paying stocks be exercised?

Answer 22

Rule for call options: it may be optimal to exercise the option just prior to an ex-dividend date. Rule for put options: it may be optimal to exercise the option except just prior to an ex-dividend date; a put option will be more valuable when the price drops just after the ex-dividend date.

Question 23

How does one show an arbitrage profit?

Answer 23

(1) Calculate theoretical value (2) Compare this to the market value (3) Take advantage by constructing the AO.

Question 24

What is the put-call parity equation with dividends?

Answer 24

European: c + Ke^(-rt) + D = p + S American: S - D - K ≤ C - P ≤ S - Ke^(-rt)

Question 25

What are the lower bounds of european options with dividends?

Answer 25

The lower bound of a European call option becomes c ≥ max⁡(0, S - D - Ke^(-rt)). The lower bound of a European put option becomes p ≥ max⁡(0, D + Ke^(-rt) - S).