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CHAPTER NINETEEN OPTIONS

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TYPES OF OPTION CONTRACTS n 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

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

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CALL OPTIONS n Role of Exchange 3 exchanges created the Options Clearing Corporation (CCC) to facilitate trading a standardized contract (100 shares/contract) 3 OCC helps buyers and writers to “close out” a position

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PUT OPTIONS n WHAT IS A PUT OPTION CONTRACT? DEFINITION: a legal contract that specifies four conditions 3 the company whose shares can be sold 3 the number of shares that can be sold 3 the selling price for those shares known as the exercise or strike price 3 the date the right expires

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OPTION TRADING n 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

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OPTION TRADING n TRADING ACTIVITY currently option trading takes place in the following locations: 3 the Chicago Board Options Exchange (CBOS) 3 the American Stock Exchange 3 the Pacific Stock Exchange 3 the Philadelphia Stock Exchange (especially currency options)

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OPTION TRADING n THE MECHANICS OF EXCHANGE TRADING Use of specialist Use of market makers

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THE VALUATION OF OPTIONS n VALUATION AT EXPIRATION FOR A CALL OPTION stock price value of option E 0

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THE VALUATION OF OPTIONS n 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

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THE VALUATION OF OPTIONS n VALUATION AT EXPIRATION In equation form IV c = max {0, P s, -E} where P s is the price of the stock E is the exercise price

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THE VALUATION OF OPTIONS n 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

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

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THE VALUATION OF OPTIONS n VALUATION AT EXPIRATION FOR A CALL OPTION 3 if the strike price is greater than $100, the option is worthless at expiration

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THE VALUATION OF OPTIONS in equation form IV c = max {0, - P s, E} where P s is the price of the stock E is the exercise price

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THE VALUATION OF OPTIONS n PROFITS AND LOSSES ON CALLS AND PUTS 100 pP PROFITS 0 0 CALLSPUTS LOSSES

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

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THE VALUATION OF OPTIONS n PROFIT EQUATIONS (CALLS) C = IV C - P C = max {0,P S - E} - P C = max {-P C, P S - E - P C } This means that the kinked profit line for the call is the intrinsic value equation less the call premium ( - P C )

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THE VALUATION OF OPTIONS n PROFIT EQUATIONS (CALLS) P = IV P - P P = max {0, E - P S } - P P = max {-P P, E - P S - P P } This means that the kinked profit line for the put is the intrinsic value equation less the put premium ( - P P )

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

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THE BINOMIAL OPTION PRICING MODEL (BOPM) n 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

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THE BINOMIAL OPTION PRICING MODEL (BOPM) n EXAMPLE: CALL OPTIONS ASSUMPTIONS: 3 price of Widget stock = $100 3 at current t: t=0 3 after one year: t=T 3 stock sells for either $125 (25% increase) $ 80 (20% decrease)

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THE BINOMIAL OPTION PRICING MODEL (BOPM) n EXAMPLE: CALL OPTIONS ASSUMPTIONS: 3 Annual riskfree rate = 8% compounded continuously 3 Investors cal lend or borrow through an 8% bond

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THE BINOMIAL OPTION PRICING MODEL (BOPM) n 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

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

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THE BINOMIAL OPTION PRICING MODEL (BOPM) n VALUATION What is a fair value for the call at time =0? 3 Two Possible Future States – The “Up State” when p = $125 – The “Down State” when p = $80

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THE BINOMIAL OPTION PRICING MODEL (BOPM) n Summary SecurityPayoff:Payoff: Current Up stateDown state Price Stock$125.00$ $ Bond $ Call ???

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BOPM: REPLICATING PORTFOLIOS n REPLICATING PORTFOLIOS The Widget call option can be replicated Using an appropriate combination of 3 Widget Stock and 3 the 8% bond The cost of replication equals the fair value of the option

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BOPM: REPLICATING PORTFOLIOS n REPLICATING PORTFOLIOS Why? 3 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

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BOPM: REPLICATING PORTFOLIOS COMPOSITION OF THE REPLICATING PORTFOLIO: 3 Consider a portfolio with N s shares of Widget 3 and N b risk free bonds In the up state 3 portfolio payoff = 125 N s N b = $25 In the down state 80 N s N b = 0

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BOPM: REPLICATING PORTFOLIOS COMPOSITION OF THE REPLICATING PORTFOLIO: 3 Solving the two equations simultaneously (125-80)N s = $25 N s =.5556 Substituting in either equation yields N b =

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BOPM: REPLICATING PORTFOLIOS n INTERPRETATION Investor replicates payoffs from the call by 3 Short selling the bonds: $ Purchasing.5556 shares of Widget

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

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BOPM: REPLICATING PORTFOLIOS n 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 $ = $14.53

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BOPM: REPLICATING PORTFOLIOS n MORE GENERALLY where V 0 = the value of the option P d = the stock price P b = the risk free bond price N d = the number of shares N b = the number of bonds

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THE HEDGE RATIO n 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

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

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THE HEDGE RATIO n THE HEDGE RATIO to replicate a call option 3 h shares must be purchased 3 B is the amount borrowed by short selling bonds B = PV(h P sd - P od )

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THE HEDGE RATIO the value of a call option V 0 = h P s - 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

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PUT-CALL PARITY n Relationship of hedge ratios: h p = h c - 1 where h p = the hedge ratio of a call h c = the hedge ratio of a put

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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

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PUT-CALL PARITY n FORMULA: P P + P S = P C + E / e RT where P P and P C denote the current market prices of the put and the call

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

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THE BLACK-SCHOLES MODEL n The Black-Scholes formula for valuing a call option where

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

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THE BLACK-SCHOLES MODEL n NOTES: E/e RT = the PV of the exercise price where continuous discount rate is used N(d 1 ), N(d 2 )= the probabilities that outcomes of less will occur in a normal distribution with mean = 0 and = 1

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THE BLACK-SCHOLES MODEL n 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

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THE BLACK-SCHOLES MODEL n 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

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THE BLACK-SCHOLES MODEL n LIMITATIONS OF B/S MODEL: It only applies to 3 European-style options 3 stocks that pay NO dividends

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END OF CHAPTER 19

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