Option Premium = Intrinsic. Speculative Value. Value

Transcription

1 Chapters 4/ Part Options: Basic Concepts Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options An Option-Pricing Formula Investment in Real Projects and Options Summary and Conclusions Options Contracts: Preliminaries Option Definition. Calls versus Puts Call options Put options. Exercising the Option Strike Price or Exercise Price Expiration Date European versus American options Options Contracts: Preliminaries Value of an Option at Expiration Intrinsic Value Speculative Value Option Premium = Intrinsic Value + Speculative Value Impact of leverage Stock price is $. Buy shares Call strike is $, price is $. Buy contract. Put strike is $, price is $. Buy contract. ===================== C = S E P = E - S 3 Call Option Payoffs Put Option Payoffs Buy a put Write a call Write a put

2 Call Option Payoffs Call Option Payoffs Write a call Exercise price = $ 6 Exercise price = $ 7 Call Option Profits Put Option Payoffs Buy a put Option profits ($) Write a call Exercise price = $; option premium = $ 8 Exercise price = $ 9 Put Option Payoffs Put Option Profits Option profits ($) Write a put Buy a put -4 write a put Exercise price = $ Exercise price = $; option premium = $

3 Selling Options Writing Options The seller (or writer) of an option has an obligation. Option profits Option ($) profits ($) The purchaser of an option has an option. Write a put Buy a put Write a call 9 Call Option Payoffs at Expiration ( exercise) 6 E= E= Option Pricing Bounds at Expiration Reading The Wall Street Journal Upper bounds Call Options Put Options Lower Bounds Call option intrinsic value = max [, S - E] Put option intrinsic value = max [, E - S] In-the-money / Out-of-the-money Time premium/time decay At, an American call option is worth the same as a European option with the same characteristics. 4 --Call-- --Put-- Option/Strike Exp. Vol. Last Vol. Last IBM 3 Oct 364 ¼ 7 ¼ 38¼ 3 Jan 9½ 4 9¼ 38¼ 3 Jul 36 4¾ 43 3/6 38¼ 3 Aug 3 9¼ 94 ½ 38¼ 4 Jul 86 ¾ 47 ¾ 38¼ 4 Aug 93 6½ 8 7½ Valuing Options Option Value Determinants The last section concerned itself with the value of an option at. This section considers the value of an option prior to the date.. Exercise price. Stock price 3. Interest rate 4. Volatility in the stock price. Expiration date Call Put The value of a call option C must fall within max (S E, ) < C < S. The precise position will depend on these factors

4 Varying Option Input Values Varying Option Input Values Stock price: Call: as stock price increases call option price increases Put: as stock price increases put option price decreases Strike price: Call: as strike price increases call option price decreases Put: as strike price increases put option price increases Time until : Call & Put: as time to increases call and put option price increase Volatility: Call & Put: as volatility increases call & put value increase Risk-free rate: Call: as the risk-free rate increases call option price increases Put: as the risk-free rate increases put option price decreases 8 9 Figure.. Put and Call Option Prices 3 Figure.. Option Prices and Time to Expiration Option Price ($) Put Price Stock Price ($) Call Price Option Price ($) 3 Call Price Put Price Time to Expiration (months) Figure.3. Option Prices and Sigma Figure.4. Options Prices and Interest Rates Call Price Option Price ($) Call Price Put Price Option Price ($) Put Price Sigma (%) Interest Rate (%) 3 4

5 Option Value Determinants Market Value, Time Value and Intrinsic Value for an American Call Call Put. Exercise price +. Stock price + 3. Interest rate + 4. Volatility in the stock price + +. Expiration date + + The value of a call option C must fall within max (S E, ) < C < S. The precise position will depend on these factors. Profit loss The value of a call option C must fall within max (S E, ) < C < S. Time value C at > Max[S T - E, ] S T Market Value Intrinsic value E Out-of-the-money In-the-money S T - E S T 4 Combinations of Options Protective Put Strategy: Buy a Put and Buy the Underlying Stock: Payoffs at Expiration Puts and calls can serve as the building blocks for more complex option contracts. If you understand this, you can become a financial engineer, tailoring the risk-return profile to meet your client s needs. $ Buy the stock Protective Put strategy has downside protection and upside potential Buy a put with an exercise price of $ $ $ Value of stock at 6 7 Protective Put Strategy Profits Covered Call Strategy $4 Buy the stock at $4 $4 Buy the stock at $4 $ -$4 $4 $ Buy a put with $ for $ Protective Put strategy has downside protection and upside potential Value of stock at $ $ -$3 -$4 $3 $4 $ Sell a call with $ for $ Covered call Value of stock at 8 9

6 Long Straddle: Buy a Call and a Put Short Straddle: Sell a Call and a Put $4 $3 $ -$ -$ $3 $4 $ $6 $7 A Long Straddle only makes money if the stock price moves $ away from $. with an $ for $ Buy a put with an $ for $ Value of stock at 3 $ $ $ -$3 -$4 A Short Straddle only loses money if the stock price moves $ away from $. Sell a put with $ for $ $3 $4 $ $6 $7 Sell a call with an $ for $ Value of stock at 3 Put-Call Parity S + P = C + Ee C = Call option price S = Current stock price r = Risk-free rate S + P C = Ee P = Put option price E = Option strike price T = Time until option Buy the stock, buy a put, and write a call; the sum of which equals the strike price discounted at the risk-free rate Put-Call Parity Buy Stock & Buy Put Position Value Share Price Combination: Long Stock & Long Put Long Stock Long Put 3 33 Put-Call Parity Buy Call & Buy Zero Coupon Risk-Free Exercise Price Position Value Combination: Long Stock & Long Bond Long Bond Position Value Put-Call Parity Share Price Long Stock Combination: Long Stock & Long Put Long Put Position Value Share Price Long Call Combination: Long Stock & Long Bond Long Bond Share Price Long Call 34 In market equilibrium, it must be the case that option prices are set such that: S + P = C + Ee Otherwise, riskless portfolios with positive payoffs exist. 3 6

7 The Black-Scholes Model Black-Scholes Model Value of a stock option is a function of 6 input factors:. Current price of underlying stock.. Strike price specified in the option contract. 3. Risk-free interest rate over the life of the contract. 4. Time remaining until the option contract expires.. Price volatility of the underlying stock. The price of a call option equals: C = S N ( d) E e N ( d) C = S N ( d) E e N ( d) Where the inputs are: S = Current stock price E = Option strike price r = Risk-free interest rate T = Time remaining until option σ = Sigma, representing stock price volatility, standard deviation Black-Scholes Model Black-Scholes Models C = S N ( d) E e N ( d) Where d and d equal: d = ln ( S ) σ + r T E + σ T d = d σ T 38 Remembering put-call parity, the value of a put, given the value of a call equals: S + P = C + Ee P = C S + Ee Also, remember at : C = S E P = E S 39 The Black-Scholes Model Find the value of a six-month call option on the Microsoft with an $ The current value of a share of Microsoft is $6 The interest rate available in the U.S. is r = %. The option maturity is 6 months (half of a year). The standard deviation of the underlying asset is 3% per annum. Before we start, note that the intrinsic value of the option is $ our answer must be at least that amount. The Black-Scholes Model Assume S = $6, X = $, T = 6 months, r = %, and σ = 3%, calculate the value of a call. First calculate d and d ln( S / E) + ( r +. ) T d = σ T ln(6 /) + (. +.(.3) ). d =.3. Then d, d = d σ T = = d =.8.3. =

8 The Black-Scholes Model C = S N( d) Ee rt N( d) d =.8 d =. 36 C = $6.73 e C = $.9 N(d ) = N(.8) =.73 N(d ) = N(.36) = Another Black-Scholes Example Assume S = $, X = $4, T = 6 months, r = %, and σ = 8%, calculate the value of a call and a put. d ln ( ) = =. d = = 686. From a standard normal probability table, look up N(d ) =.8 and N(d ) =.74 (or use Excel s normsdist function) C = (. 8) 4 e. (. ) (. 74) = $ P = $ 8. 3 $ + $ 4e. (. ) = $. 43 Real Options Collar: Buy a Put, Buy the Stock, Sell the Call Real estate developer buys 7 acres in a rural area. He plans on building a subdivision when the population from the city expands this direction. If growth is less than anticipated, the developer thinks he can sell the land to a country club to build a golf course on the property. The development option is a option. The golf course option is a option. How would these real options change the standard NPV analysis? $49.33 $.76 $ $.67 -$7.9 -$8 $ Buy a put with exercise price of $ for $.67 $8 Buy the stock at $8 $ Sell a call with $ for $.76 Collar $4. Value of stock at 44 NTS 4 8