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CHAPTER TWENTY-FOUR

CHAPTER TWENTY-FOUR. OPTIONS. TYPES OF OPTION CONTRACTS. WHAT IS AN OPTION? Definition: a type of contract between two investors where one grants the other the right to buy or sell a specific asset in the future

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CHAPTER TWENTY-FOUR

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  1. CHAPTER TWENTY-FOUR OPTIONS

  2. TYPES OF OPTION CONTRACTS • WHAT IS AN OPTION? • Definition: a type of contract between two investors where one grants the other the right to buy or sell a specific asset in the future • the option buyer is buying the right to buy or sell the underlying asset at some future date • the option writer is selling the right to buy or sell the underlying asset at some future date

  3. CALL OPTIONS • WHAT IS A CALL OPTION CONTRACT? • DEFINITION: a legal contract that specifies four conditions • FOUR CONDITIONS • the company whose shares can be bought • the number of shares that can be bought • the purchase price for the shares known as the exercise or strike price • the date when the right expires

  4. CALL OPTIONS • Role of Exchange • exchanges created the Options Clearing Corporation (CCC) to facilitate trading a standardized contract (100 shares/contract) • OCC helps buyers and writers to “close out” a position

  5. PUT OPTIONS • WHAT IS A PUT OPTION CONTRACT? • DEFINITION: a legal contract that specifies four conditions • the company whose shares can be sold • the number of shares that can be sold • the selling price for those shares known as the exercise or strike price • the date the right expires

  6. OPTION TRADING • FEATURES OF OPTION TRADING • a new set of options is created every 3 months • new options expire in roughly 9 months • long term options (LEAPS) may expire in up to 2 years • some flexible options exist (FLEX) • once listed, the option remains until expiration date

  7. OPTION TRADING • TRADING ACTIVITY • currently option trading takes place in the following locations: • the Chicago Board Options Exchange (CBOS) • the American Stock Exchange • the Pacific Stock Exchange • the Philadelphia Stock Exchange (especially currency options)

  8. OPTION TRADING • THE MECHANICS OF EXCHANGE TRADING • Use of specialist • Use of market makers

  9. THE VALUATION OF OPTIONS • VALUATION AT EXPIRATION (E) • FOR A CALL OPTION E -100 value of option 0 200 100 stock price

  10. THE VALUATION OF OPTIONS • VALUATION AT EXPIRATION • ASSUME: the strike price = $100 • For a call if the stock price is less than $100, the option is worthless at expiration • The upward sloping line represents the intrinsic value of the option

  11. THE VALUATION OF OPTIONS • VALUATION AT EXPIRATION • In equation form IVc = max {0, Ps, -E} where Psis the price of the stock E is the exercise price

  12. THE VALUATION OF OPTIONS • VALUATION AT EXPIRATION • ASSUME: the strike price = $100 • For a put if the stock price is greater than $100, the option is worthless at expiration • The downward sloping line represents the intrinsic value of the option

  13. THE VALUATION OF OPTIONS • VALUATION AT EXPIRATION • FOR A PUT OPTION 100 value of the option E=100 0 stock price

  14. THE VALUATION OF OPTIONS • VALUATION AT EXPIRATION • FOR A CALL OPTION • if the strike price is greater than $100, the option is worthless at expiration

  15. THE VALUATION OF OPTIONS • in equation form IVc = max {0, - Ps, E} where Psis the price of the stock E is the exercise price

  16. THE VALUATION OF OPTIONS • PROFITS AND LOSSES ON CALLS AND PUTS PROFITS PROFITS CALLS PUTS 100 p P 0 0 100 LOSSES LOSSES

  17. THE VALUATION OF OPTIONS • PROFITS AND LOSSES • Assume the underlying stock sells at $100 at time of initial transaction • Two kinked lines = the intrinsic value of the options

  18. THE VALUATION OF OPTIONS • PROFIT EQUATIONS (CALLS) PC = IVC - PC = max {0,PS - E} - PC = max {-PC , PS - E - PC } This means that the kinked profit line for the call is the intrinsic value equation less the call premium (- PC)

  19. THE VALUATION OF OPTIONS • PROFIT EQUATIONS (CALLS) PP = IVP - PP = max {0, E - PS} - PP = max {-PP , E - PS - PP } This means that the kinked profit line for the put is the intrinsic value equation less the put premium (- PP)

  20. THE BINOMIAL OPTION PRICING MODEL (BOPM) • WHAT DOES BOPM DO? • it estimates the fair value of a call or a put option

  21. THE BINOMIAL OPTION PRICING MODEL (BOPM) • TYPES OF OPTIONS • EUROPEAN is an option that can be exercised only on its expiration date • AMERICAN is an option that can be exercised any time up until and including its expiration date

  22. THE BINOMIAL OPTION PRICING MODEL (BOPM) • EXAMPLE: CALL OPTIONS • ASSUMPTIONS: • price of Widget stock = $100 • at current t: t=0 • after one year: t=T • stock sells for either $125 (25% increase) $ 80 (20% decrease)

  23. THE BINOMIAL OPTION PRICING MODEL (BOPM) • EXAMPLE: CALL OPTIONS • ASSUMPTIONS: • Annual riskfree rate = 8% compounded continuously • Investors cal lend or borrow through an 8% bond

  24. THE BINOMIAL OPTION PRICING MODEL (BOPM) • Consider a call option on Widget Let the exercise price = $100 the exercise date = T and the exercise value: If Widget is at $125 = $25 or at $80 = 0

  25. THE BINOMIAL OPTION PRICING MODEL (Price Tree) Annual Analysis: $125 P0=25 $100 $80 P0=$0 Semiannual Analysis: $125 P0=65 $111.80 $100 $100 P0=0 $89.44 $80 P0=0 t=0 t=.5T t=T

  26. THE BINOMIAL OPTION PRICING MODEL (BOPM) • VALUATION • What is a fair value for the call at time =0? • Two Possible Future States • The “Up State” when p = $125 • The “Down State” when p = $80

  27. THE BINOMIAL OPTION PRICING MODEL (BOPM) • Summary Security Payoff: Payoff: Current Up state Down state Price Stock $125.00 $ 80.00 $100.00 Bond 108.33 108.33 $100.00 Call 25.00 0.00 ???

  28. BOPM: REPLICATING PORTFOLIOS • REPLICATING PORTFOLIOS • The Widget call option can be replicated • Using an appropriate combination of • Widget Stock and • the 8% bond • The cost of replication equals the fair value of the option

  29. BOPM: REPLICATING PORTFOLIOS • REPLICATING PORTFOLIOS • Why? • if otherwise, there would be an arbitrage opportunity • that is, the investor could buy the cheaper of the two alternatives and sell the more expensive one

  30. BOPM: REPLICATING PORTFOLIOS • COMPOSITION OF THE REPLICATING PORTFOLIO: • Consider a portfolio with Ns shares of Widget • and Nb risk free bonds • In the up state • portfolio payoff = 125 Ns + 108.33 Nb = $25 • In the down state 80 Ns + 108.33 Nb = 0

  31. BOPM: REPLICATING PORTFOLIOS • COMPOSITION OF THE REPLICATING PORTFOLIO: • Solving the two equations simultaneously (125-80)Ns= $25 Ns = .5556 Substituting in either equation yields Nb = -.4103

  32. BOPM: REPLICATING PORTFOLIOS • INTERPRETATION • Investor replicates payoffs from the call by • Short selling the bonds: $41.03 • Purchasing .5556 shares of Widget

  33. BOPM: REPLICATING PORTFOLIOS Portfolio Component Payoff In Down State Payoff In Up State .5556 x $125 = $6 9.45 .5556 x $80 = $ 44.45 Stock -$41.03 x 1.0833 = -$44.45 -$41.03 x 1.0833 = -$ 44.45 Loan Net Payoff $25.00 $0.00

  34. BOPM: REPLICATING PORTFOLIOS • TO OBTAIN THE PORTFOLIO • $55.56 must be spent to purchase .5556 shares at $100 per share • but $41.03 income is provided by the bonds such that $55.56 - 41.03 = $14.53

  35. BOPM: REPLICATING PORTFOLIOS • MORE GENERALLY where V0 = the value of the option Pd = the stock price Pb = the risk free bond price Nd = the number of shares Nb = the number of bonds

  36. THE HEDGE RATIO • THE HEDGE RATIO • DEFINITION: the expected change in the value of an option per dollar change in the market price of an underlying asset • The price of the call should change by $.5556 for every $1 change in stock price

  37. THE HEDGE RATIO • THE HEDGE RATIO where P = the end-of-period price o = the option s = the stock u = up d = down

  38. THE HEDGE RATIO • THE HEDGE RATIO • to replicate a call option • h shares must be purchased • B is the amount borrowed by short selling bonds B = PV(h Psd - Pod )

  39. THE HEDGE RATIO • the value of a call option V0 = h Ps - B where h = the hedge ratio B = the current value of a short bond position in a portfolio that replicates the payoffs of the call

  40. PUT-CALL PARITY • Relationship of hedge ratios: hp = hc - 1 where hp = the hedge ratio of a call hc = the hedge ratio of a put

  41. PUT-CALL PARITY • DEFINITION: the relationship between the market price of a put and a call that have the same exercise price, expiration date, and underlying stock

  42. PUT-CALL PARITY • FORMULA: PP + PS = PC + E / eRT where PP and PC denote the current market prices of the put and the call

  43. THE BLACK-SCHOLES MODEL • What if the number of periods before expiration were allowed to increase infinitely?

  44. THE BLACK-SCHOLES MODEL • The Black-Scholes formula for valuing a call option where

  45. THE BLACK-SCHOLES MODEL and where Ps = the stock’s current market price E = the exercise price R = continuously compounded risk free rate T = the time remaining to expire s = risk (standard deviation of the stock’s annual return)

  46. THE BLACK-SCHOLES MODEL • NOTES: • E/eRT = the PV of the exercise price where continuous discount rate is used • N(d1 ), N(d2 )= the probabilities that outcomes of less will occur in a normal distribution with mean = 0 and s = 1

  47. THE BLACK-SCHOLES MODEL • What happens to the fair value of an option when one input is changed while holding the other four constant? • The higher the stock price, the higher the option’s value • The higher the exercise price, the lower the option’s value • The longer the time to expiration, the higher the option’s value

  48. THE BLACK-SCHOLES MODEL • What happens to the fair value of an option when one input is changed while holding the other four constant? • The higher the risk free rate, the higher the option’s value • The greater the risk, the higher the option’s value

  49. THE BLACK-SCHOLES MODEL • LIMITATIONS OF B/S MODEL: • It only applies to • European-style options • stocks that pay NO dividends

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