Properties of Stock Options Chapter 10

Notation c : European call option price C : American Call option price p : European put option price S0 : Stock price today K : Strike price T : Life of option : Volatility of stock price C : American Call option price P : American Put option price ST :Stock price at option maturity D : Present value of dividends during option’s life r : Risk-free rate for maturity T with cont. comp.

Effect of Variables on Option Pricing K T  r D c p C P – – + + – – + + + + ? + + + + – – + + – – + +

American vs European Options An American option is worth at least as much as the corresponding European option C  c P  p

Calls: An Arbitrage Opportunity? Suppose that c = 3 S0 = 20 T = 1 r = 10% K = 18 D = 0 Is there an arbitrage opportunity? Yes!

Lower Bound for European Call Option Prices with No Dividends c  S0 –Ke –rT 3 < 3.713 Arbitrage! buy low, sell high Buy call, short stock, deposit $16.29 (PV of K)

Arbitrage the gap! (between the 2 sides of the inequality) T=0: buy call (-3), short stock (=20), deposit $16.29. NCF=0.71 (the gap) T=1 & S>=18: exercise call (-18), use stock obtained to cover short, close out bank account (+18). NCF=0 T=1 & S<18: allow call to lapse unexercised, buy stock to cover short (-S), close out bank account (+18). NCF=18-S>0

@ Expiry: Call vs. Portfolio (long stock & borrow debt service = K)

c  (S0 - D )–Ke –rT c  (S0 e –dT)–Ke –rT Lower Bound for European Call Option Prices with Dividends D=PV of Dividends, d=cc dividend yield c  (S0 - D )–Ke –rT c  (S0 e –dT)–Ke –rT

Puts: An Arbitrage Opportunity? Suppose that p = 1 S0 = 37 T = 0.5 r =5% K = 40 D = 0 Is there an arbitrage opportunity? Yes!

Lower Bound for European Put Prices with No Dividends p  Ke -rT–S0 1 < 2.01 Arbitrage! Buy low, sell high Buy put, buy stock, borrow $39.01

Arbitrage the gap! (between the 2 sides of the inequality) T=0: buy put (-1), borrow 39.01, buy stock (-37). NCF= 1.01 (this is the gap) T=0.5 & S<=40: exercise put (+40) delivering stock owned, payoff loan (-40). NCF=0. T=0.5 & S>40: Allow put to lapse unexercised, sell stock (+S), payoff loan (-40). NCF=(S-40)>0.

@ Expiry: Put vs. Portfolio (short stock & deposit account value = K)

p  Ke –rT – (S0 - D ) p  Ke –rT – (S0 e –dT) Lower Bound for European Put Option Prices with Dividends D=PV of Dividends, d=cc dividend yield p  Ke –rT – (S0 - D ) p  Ke –rT – (S0 e –dT)

Put-Call Parity with No Dividends Consider the following 2 portfolios: Portfolio A: European call on a stock + PV of the strike price in cash Portfolio B: European put on the stock + the stock Both are worth max(ST , K ) at the maturity of the options They must therefore be worth the same today. This means that c + Ke -rT = p + S0

Arbitrage Opportunities (when put-call parity is violated) Suppose that c = 3 S0 = 31 T = 0.25 r = 10% K =30 D = 0 p = 1.26: arbitrage possibilities when p = 2.25 ? p too high p = 1 ? p too low

Arbitrage when p not = 1.26 Buy low, sell high! p = 2.25 (too high) T=0: sell put (2.25), buy call (-3), short stock (31), deposit 29.26. NCF=0.99 (the gap) p = 1 (too low) T= 0: buy put (-1), sell call (3), buy stock (-31), borrow 29.26. NCF=0.26 (the gap)

Early Exercise Usually there is some chance that an American option will be exercised early Exception is an American call on a non-dividend paying stock; it should never be exercised early But if stock price is sufficiently low, American put on non-dividend paying stock should be exercised early

An Extreme Situation For an American call option: S0 = 100; T = 0.25; K = 60; D = 0 Should you exercise immediately? What should you do if You want to hold the stock for the next 3 months? You do not feel that the stock is worth holding for the next 3 months?

Reasons For Not Exercising a Call Early (No Dividends) No income is sacrificed We delay paying the strike price Holding the call provides insurance against stock price falling below strike price

Should Puts Be Exercised Early ? Yes when stock price is very low. Note: stock price can’t be lower than 0. Are there any advantages to exercising an American put when S0 = 10; T = 0.25; r=10% K = 100; D = 0 or > 0

The Impact of Dividends on Lower Bounds to Option Prices

Extensions of Put-Call Parity American options; D = 0 S0 - K < C - P < S0 - Ke -rT European options; D > 0 c + D + Ke -rT = p + S0 American options; D > 0 S0 - D - K < C - P < S0 - Ke -rT