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Properties of Stock Options

Properties of Stock Options. Chapter 10. c : European call option price p : European put option price S 0 : Stock price today K : Strike price T : Life of option  : Volatility of stock price. C : American Call option price P : American Put option price

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Properties of Stock Options

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  1. Properties of Stock Options Chapter 10

  2. c : European 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 Twith cont. comp. Notation

  3. Variable S0 K ? ? T  r D Effect of Variables on Option Pricing c p C P – – + + – – + + + + + + + + – – + + – – + +

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

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

  6. 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)

  7. 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

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

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

  10. 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!

  11. 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

  12. 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.

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

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

  15. 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 thatc + Ke -rT = p + S0

  16. 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

  17. 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)

  18. 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

  19. 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?

  20. 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

  21. 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

  22. The Impact of Dividends on Lower Bounds to Option Prices

  23. 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

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