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Chapter 11

Chapter 11. Put and Call Options. A call option is the right to buy an underlying security at an exercise (strike) price during a stated time interval. C = Market value of the call option. P = Market value of the underlying asset. E = Exercise price (strike price). Call Options. 0.

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Chapter 11

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  1. Chapter 11 Put and Call Options

  2. A call option is the right to buy an underlying security at an exercise (strike) price during a stated time interval. C = Market value of the call option. P = Market value of the underlying asset. E = Exercise price (strike price). Call Options

  3. 0 Expiration Would like to find value here But first need to determine value here

  4. Value of call option at expiration, E = $100 P < E P = E P > E e.g., P = 90 P = 100 P = 110 C = 0 C = 0 C = P – E e.g., C = 10 out-of-the-money at-the-money in-the-money

  5. Arbitrage Guarantees That C = P – E If C < P - E at expiration Suppose P = 110, E = 100, C* = 6. Arbitrage: Buy Call -6 Exercise -100 Sell Underlying +110 Arbitrage Profit +4

  6. Case of C > P – E: Suppose P = 110, E = 100, C** = 17. Arbitrage: Write Call +17 Exercised +100 Buy Underlying -110 Arbitrage Profit +7

  7. Value of call option before expiration, E = $100 P < E P = E P > E e.g., P = 90 P = 100 P = 110 C > 0 C > 0 C > P – E e.g., C > 10

  8. Call Option Bounds C P P – E Arbitrage Feasible call prices Arbitrage P E

  9. Arbitrage if C > P Time 0 Write Call +C Buy Underlying -P C – P > 0 Expiration P < EP = EP > E Sell Underlying +P +P = E Call Exercised +E Net +P +E +E

  10. Profit Profiles 0 Expiration Determine Profit or Loss Overlooking Dividends and Interest Take a position Close entire position

  11. 0 Expiration Buy or Call-4 Close: Exercise if in-money. Let expire if out-of-money.

  12. Profits or Losses for Call Buyer Price of underlying at expiration 98 100 102 104 Buy call -4 Exercise call at expiration Sell underlying acquiredfrom exercise Net profit = - C -4 Net profit = – C – E + P -2 0 -4 - 4 -4 -100 +102 -4 -100 +104

  13. Payoff Function: Buy Call + – c – c + [Pexp – E] E 0 Pexp E + c – c Breakeven –

  14. Profit Profile for Buying a Call $ Call out-of-money Call in-money +Profit -C+ [P–E] -C Buy Call 100 P at Expiration 0 E 104 -Loss -4

  15. Profits or Losses for Call Writer Price of underlying at expiration 98 100 102 104 Write call +4 Sell underlying Buy underlying Net profit = + C Net profit = + C + E – P +2 0 +4 +4 +4 +100 -102 +4 +100 -104

  16. Profit Profile for Writing a Call $ $ Call out-of-money Call in-money +Profit +Profit +C +C – [P-E] +4 100 P at Expiration P at Expiration 0 0 E 104 104 -Loss -Loss Write Call

  17. Payoff Function: Write Call + + c + c – [Pexp – E] + c E 0 Pexp E + c Breakeven –

  18. Profits or Losses from Writing a Covered Call Price of underlying at expiration 98 100 102 104 Buy underlying -100 Write call +4 Sell underlying at exerciseprice when call is exercised Sell underlying at market +100price Net profit +4 -100 +4 +98 +2 -100 +4 +100 +4 -100 +4 +100 +4

  19. Profit Profile for Writing a Covered Call Option $ Buy underlying security Call out-of-money Call in-money +Profit C  Write covered call 100 Underlying asset at expiration 0 E -Loss

  20. Profit Profile for Call Option Profit Buy underlying security Call out-of-money Call in-money Write call +4 Buy call 100 Underlying asset at expiration 0 E 104 -4 Loss

  21. Put Option A put option is the right to sell the underlying security at an exercise price during a stated time interval. 0 Expiration First, find value at expiration

  22. Put Options Value of put option at expiration, E = $100 P < E P = E P > E e.g., P = 90 P = 100 P = 110 Put = E – P Put = 0 Put = 0 e.g., Put = 10 in-the-money at-the-money out-of-money

  23. If P < E, There is Arbitrage unless Put = E – P Case of Put < E – P: Suppose P = 90, E = 100, Put* = 6. Arbitrage: Buy Put -6 Exercise +100 Buy Underlying -90 Arbitrage Profit +4

  24. Case of Put > E – P: Suppose P = 90, E = 100, Put** = 17. Arbitrage: Write Put +17 Exercised -100 Sell Underlying +90 Arbitrage Profit +7

  25. Value of put option before expiration, E = $100 P < E P = E P > E e.g., P = 90 P = 100 P = 110 Put > E – P Put > 0 Put > 0 e.g., Put = 10 in-the-money at-the-money out-of-money

  26. Profits or Losses for Buying a Put Option Price of underlying at expiration 94 97 100 104 Buy put -3 Buy underlying Exercise put Net profit = – Put -3 -3 Net profit = – Put + E – P +3 0 -3 -3 -94 -97 +100 +100 -3

  27. Payoff Function: Buy Put + – put + [E – Pexp] – put E 0 Pexp E – Put – put Breakeven –

  28. Profit Profile for Put Option Profit Put in-money Put out-of-money -PUT -PUT + [E-P] Buy put 100 Underlying asset at expiration 0 97 E -3  Shortsell Loss

  29. Writing a Put Price of underlying at expiration 94 97 100 104 Write put +3 Sell underlying Put exercised Net profit = + Put +3 +3 Net profit = + Put – E + P +3 +94 -100 -3 +3 +97 -100 0 +3

  30. Payoff Function: Write Put + + put – [E – Pexp] + put + put E 0 Pexp E – put Breakeven –

  31. Profit Profile for Writing a Put Profit Put in-money Put out-of-money +PUT - [E-P] +PUT +3 100 Underlying asset at expiration 0 97 E Loss

  32. Put-Call Parity C = Put + P – E D 0 Expiration Presentvalue = .98 = D = E D = 98 $1 E 100

  33. Profit of Put-Call Parity 0 Expiration Buy call Cash flows from call Cash flows from portfolio Buy portfolio If cash flows at Expiration are the same for call as for portfolio, then the Time 0 value must be the same.

  34. Put-Call Parity P < E P = E P > E Cash flows at expiration from buying call Call 0 0 P – E Cash flows at expiration from buying put, buying underlying and borrowing present value of exercise price Put E – P 0 0 Underlying +P +P +P Loan –E –E –E

  35. Implications of Put-Call Parity C = Put + [P – E D] = Put + [Levered position in underlying] 5 = 3 + [100 – 98].

  36. Leverage RORon levered Levered Unlevered Higher return from levered i Lower return from levered i ROR unlevered i = Interest rate

  37. If C = Put + P – E D, then C  P – E D C P – E D P – E Call P E

  38. Arbitrage if C < P – ED • Suppose P = 110, E = 100, D = .98. • P – E = 10. • P – ED = 12.

  39. Suppose C* = 11. Arbitrage: Time 0 Buy Call -11 Short Underlying +110 Lend ED -98 Arbitrage Profit +1

  40. Expiration P < EP = EP > E Call 0 0 P – E Buy Underlying -P -P -P Receive E +E +E +E Net E – P > 0 E – P = 0 0

  41. Factors Affecting the Value of a Call Option Call--greater value of call 1. P--greater value of underlying 2. E--lower value of exercise price 3. Greater time to expiration 4. Higher volatility of underlying

  42. Impact of Longer Remaining Life on the Value of a Call Option C C2 has a longer life than C1 P  ED is higher for C2 because D2 < D1 P  ED2 P  ED1 P  E C2 C1 P E

  43. Value of a Call Call Option of Security 1 Prices of underlying 90 100 110 Value of call option 0 0 10 Probability 1/3 1/3 1/3 Mean value = (0)(1/3) + (0)(1/3) + (10)(1/3) = 3.33 Call Option of Security 2 Prices of underlying 80 100 120 Value of call option 0 0 20 Probability 1/3 1/3 1/3 Mean value = (0)(1/3) + (0)(1/3) + (20)(1/3) = 6.67

  44. Impact of the Volatility of the Underlying Asset on the Value of a Call Option C P  ED C2 has greater volatility of underlying asset P  E C2 C1 P E

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