Tài chính doanh nghiệp - Chapter 5: Option pricing

© 2004 South-Western Publishing1Chapter 5Option Pricing2OutlineIntroductionA brief history of options pricingArbitrage and option pricingIntuition into Black-Scholes3IntroductionOption pricing developments are among the most important in the field of finance during the last 30 yearsThe backbone of option pricing is the Black-Scholes model4Introduction (cont’d)The Black-Scholes model:5A Brief History of Options Pricing: The Early WorkCharles Castelli wrote The Theory of Options in Stocks and Shares (1877)Explained the hedging and speculation aspects of optionsLouis Bachelier wrote Theorie de la Speculation (1900)The first research that sought to value derivative assets6A Brief History of Options Pricing: The Middle YearsRebirth of option pricing in the 1950s and 1960sPaul Samuelson wrote Brownian Motion in the Stock Market (1955)Richard Kruizenga wrote Put and Call Options: A Theoretical and Market Analysis (1956)James Boness wrote A Theory and Measurement of Stock Option Value (1962)7A Brief History of Options Pricing: The PresentThe Black-Scholes option pricing model (BSOPM) was developed in 1973An improved version of the Boness modelMost other option pricing models are modest variations of the BSOPM8Arbitrage and Option PricingIntroductionFree lunchesThe theory of put/call parityThe binomial option pricing modelPut pricing in the presence of call optionsBinomial put pricingBinomial pricing with asymmetric branchesThe effect of time9Arbitrage and Option Pricing (cont’d)The effect of volatilityMultiperiod binomial option pricingOption pricing with continuous compoundingRisk neutrality and implied branch probabilitiesExtension to two periods10Arbitrage and Option Pricing (cont’d)Recombining binomial treesBinomial pricing with lognormal returnsMultiperiod binomial put pricingExploiting arbitrageAmerican versus European option pricingEuropean put pricing and time value11IntroductionFinance is sometimes called “the study of arbitrage”Arbitrage is the existence of a riskless profitFinance theory does not say that arbitrage will never appearArbitrage opportunities will be short-lived12Free LunchesThe apparent mispricing may be so small that it is not worth the effortE.g., pennies on the sidewalkArbitrage opportunities may be out of reach because of an impedimentE.g., trade restrictions13Free Lunches (cont’d)A University Example A few years ago, a bookstore at a university was having a sale and offered a particular book title for $10.00. Another bookstore at the same university had a buy-back offer for the same book for $10.50.14Free Lunches (cont’d)Modern option pricing techniques are based on arbitrage principlesIn a well-functioning marketplace, equivalent assets should sell for the same price (law of one price)Put/call parity15The Theory of Put/Call ParityIntroductionCovered call and short putCovered call and long putNo arbitrage relationshipsVariable definitionsThe put/call parity relationship16IntroductionFor a given underlying asset, the following factors form an interrelated complex:Call pricePut priceStock price andInterest rate17Covered Call and Short PutThe profit/loss diagram for a covered call and for a short put are essentially equalCovered callShort put18Covered Call and Long PutA riskless position results if you combine a covered call and a long putCovered callLong putRiskless position+=19Covered Call and Long PutRiskless investments should earn the riskless rate of interestIf an investor can own a stock, write a call, and buy a put and make a profit, arbitrage is present20No Arbitrage RelationshipsThe covered call and long put position has the following characteristics:One cash inflow from writing the call (C)Two cash outflows from paying for the put (P) and paying interest on the bank loan (Sr)The principal of the loan (S) comes in but is immediately spent to buy the stockThe interest on the bank loan is paid in the future21No Arbitrage Relationships (cont’d)If there is no arbitrage, then:22No Arbitrage Relationships (cont’d)If there is no arbitrage, then:The call premium should exceed the put premium by about the riskless rate of interestThe difference will be greater as:The stock price increasesInterest rates increaseThe time to expiration increases23Variable DefinitionsC = call premiumP = put premiumS0 = current stock priceS1 = stock price at option expirationK = option striking priceR = riskless interest ratet = time until option expiration24The Put/Call Parity RelationshipWe now know how the call prices, put prices, the stock price, and the riskless interest rate are related:25The Put/Call Parity Relationship (cont’d)Equilibrium Stock Price Example You have the following information:Call price = $3.5Put price = $1Striking price = $75Riskless interest rate = 5%Time until option expiration = 32 days If there are no arbitrage opportunities, what is the equilibrium stock price? 26The Put/Call Parity Relationship (cont’d)Equilibrium Stock Price Example (cont’d) Using the put/call parity relationship to solve for the stock price:27The Put/Call Parity Relationship (cont’d)To understand why the law of one price must hold, consider the following information:C = $4.75P = $3S0 = $50K = $50R = 6.00%t = 6 months28The Put/Call Parity Relationship (cont’d)Based on the provided information, the put value should be:P = $4.75 - $50 + $50/(1.06)0.5 = $3.31The actual call price ($4.75) is too high or the put price ($3) is too low29The Put/Call Parity Relationship (cont’d)To exploit the arbitrage, arbitrageurs would:Write 1 call @ $4.75Buy 1 put @ $3Buy a share of stock at $50Borrow $48.56 at 6.00% for 6 monthsThese actions result in a profit of $0.31 at option expiration irrespective of the stock price at option expiration30The Put/Call Parity Relationship (cont’d)Stock Price at Option Expiration$0$50$100From call4.754.75(45.25)From put47.00(3.00)(3.00)From loan(1.44)(1.44)(1.44)From stock(50.00)0.0050.00Total$0.31$0.31$0.3131The Binomial Option Pricing ModelAssume the following:U.S. government securities yield 10% next yearStock XYZ currently sells for $75 per shareThere are no transaction costs or taxesThere are two possible stock prices in one year32The Binomial Option Pricing Model (cont’d)Possible states of the world:$75$50$100TodayOne Year Later33The Binomial Option Pricing Model (cont’d)A call option on XYZ stock is available that gives its owner the right to purchase XYZ stock in one year for $75If the stock price is $100, the option will be worth $25If the stock price is $50, the option will be worth $0What should be the price of this option?34The Binomial Option Pricing Model (cont’d)We can construct a portfolio of stock and options such that the portfolio has the same value regardless of the stock price after one yearBuy the stock and write N call options35The Binomial Option Pricing Model (cont’d)Possible portfolio values:$75 – (N)($C)$50$100 - $25NTodayOne Year Later36The Binomial Option Pricing Model (cont’d)We can solve for N such that the portfolio value in one year must be $50:37The Binomial Option Pricing Model (cont’d)If we buy one share of stock today and write two calls, we know the portfolio will be worth $50 in one yearThe future value is known and riskless and must earn the riskless rate of interest (10%)The portfolio must be worth $45.45 today38The Binomial Option Pricing Model (cont’d)Assuming no arbitrage exists:The option must sell for $14.77!39The Binomial Option Pricing Model (cont’d)The option value is independent of the probabilities associated with the future stock priceThe price of an option is independent of the expected return on the stock40Put Pricing in the Presence of Call OptionsIn an arbitrage-free world, the put option cannot also sell for $14.77; If it did, an astute arbitrageur would:Buy a 75 callWrite a 75 putSell the stock shortInvest $68.18 in T-billsThese actions result in a cash flow of $6.82 today and a cash flow of $0 at option expiration41Put Pricing in the Presence of Call OptionsActivityCash Flow TodayPortfolio Value at Option ExpirationPrice = $100Price = $50Buy 75 call-$14.77 $250Write 75 put+14.770-$25Sell stock short+75.00-100-50Invest $68.18 in T-bills-68.1875.0075.00Total$6.82$0.00$0.0042Binomial Put PricingPriced analogously to callsYou can combine puts with stock so that the future value of the portfolio is knownAssume a value of $10043Binomial Put Pricing (cont’d)Possible portfolio values:$75 $50 + N($75 - $50)$100TodayOne Year Later44Binomial Put Pricing (cont’d)A portfolio composed of one share of stock and two puts will grow risklessly to $100 after one year45Binomial Pricing With Asymmetric BranchesThe size of the up movement does not have to be equal to the size of the declineE.g., the stock will either rise by $25 or fall by $15The logic remains the same:First, determine the number of optionsSecond, solve for the option price46The Effect of TimeMore time until expiration means a higher option value47The Effect of VolatilityHigher volatility means a higher option price for both call and put optionsExplains why options on Internet stocks have a higher premium than those for retail firms48Multiperiod Binomial Option PricingIn reality, prices change in the marketplace minute by minute and option values change accordinglyThe logic of binomial pricing can be easily extended to a multiperiod setting using the recursive methods of solving for the option value49Option Pricing With Continuous CompoundingContinuous compounding is an assumption of the Black-Scholes modelUsing continuous compounding to revalue the call option from the previous example:50Risk Neutrality and Implied Branch ProbabilitiesRisk neutrality is an assumption of the Black-Scholes modelFor binomial pricing, this implies that the option premium contains an implied probability of the stock rising51Risk Neutrality and Implied Branch Probabilities (cont’d)Assume the following:An investor is risk-neutralHe can invest funds risk free over one year at a continuously compounded rate of 10%The stock either rises by 33.33% or falls 33.33% in one yearAfter one year, one dollar will be worth $1.00 x e.10 = $1.1052 for an effective annual return of 10.52%52Risk Neutrality and Implied Branch Probabilities (cont’d)A risk-neutral investor would be indifferent between investing in the riskless rate and investing in the stock if it also had an expected return of 10.52%We can determine the branch probabilities that make the stock have a return of 10.52%53Risk Neutrality and Implied Branch Probabilities (cont’d)Define the following:U = 1 + percentage increase if the stock goes upD = 1 – percentage decrease if the stock goes downPup = probability that the stock goes upPdown = probability that the stock goes downert = continuously compounded interest rate factor54Risk Neutrality and Implied Branch Probabilities (cont’d)The average stock return is the weighted average of the two possible price movements:55Risk Neutrality and Implied Branch Probabilities (cont’d)If the stock goes up, the call will have an intrinsic value of $100 - $75 = $25If the stock goes down, the call will be worthlessThe expected value of the call in one year is:56Risk Neutrality and Implied Branch Probabilities (cont’d)Discounted back to today, the value of the call today is:57Extension to Two PeriodsAssume two periods, each one year long, with the stock either rising or falling by 33.33% in each periodWhat is the equilibrium value of a two-year European call shown on the next slide?58Extension to Two Periods (cont’d)$75$50$100TodayOne Year Later$133.33 (UU)$66.67 (UD = DU)$33.33 (DD)Two Years Later59Extension to Two Periods (cont’d)The option only winds up in the money when the stock advances twice (UU)There is a 65.78% probability that the call is worth $58.33 and a 34.22% probability that the call is worthless60Extension to Two Periods (cont’d)There is a 65.78% probability that the call is worth $34.72 in one year and a 34.22% probability that the call is worthless in one yearThe expected value of the call in one year is:61Extension to Two Periods (cont’d)$20.66$0$34.72TodayOne Year Later$58.33 (UU)$0 (UD = DU)$0 (DD)Two Years Later62Recombining Binomial TreesIf trees are recombining, this means that the up-down path and the down-up path both lead to the same point, but not necessarily the starting pointTo return to the initial price, the size of the up jump must be the reciprocal of the size of the down jump63Binomial Pricing with Lognormal ReturnsBlack-Scholes assumes that security prices follow a lognormal distributionWith lognormal returns, the size of the upward movement U equals:The probability of an up movement is:64Multiperiod Binomial Put PricingTo solve for the value of a put using binomial logic, just change the terminal intrinsic values and work backward just as with call pricingThe branch probabilities do not change65Exploiting ArbitrageArbitrage Example Binomial pricing results in a call price of $28.11 and a put price of $2.23. The interest rate is 10%, the stock price is $75, and the striking price of the call and the put is $60. The expiration date is in two years. What actions could an arbitrageur take to make a riskless profit if the call is actually selling for $29.00?66Exploiting Arbitrage (cont’d)Arbitrage Example (cont’d) Since the call is overvalued, and arbitrageur would want to write the call, buy the put, buy the stock, and borrow the present value of the striking price, resulting in the following cash flow today: Write 1 call $29.00 Buy 1 put ($2.23) Buy 1 share ($75.00) Borrow $60e-(.10)(2) $49.12 $0.8967Exploiting Arbitrage (cont’d)Arbitrage Example (cont’d) The value of the portfolio in two years will be worthless, regardless of the path the stock takes over the two-year period.68American Versus European Option PricingWith an American option, the intrinsic value is a sure thingWith a European option, the intrinsic value is currently unattainable and may disappear before you can get at itAn American option should be worth more than a European option69European Put Pricing and Time ValueWith a European put, the longer the option’s life, the longer you must wait to see sales proceedsMore time means greater potential dispersion in underlying asset values, and this pushes up the put valueA European put’s value with respect to time until expiration is indeterminate70European Put Pricing and Time Value (cont’d)Often, an out-of-the-money put will increase in value with more timeOften, an in-the-money put decreases in value for more distant expirations71Intuition Into Black-ScholesContinuous time and multiple periods72Continuous Time and Multiple PeriodsFuture security prices are not limited to only two valuesThere are theoretically an infinite number of future states of the worldRequires continuous time calculus (BSOPM)The pricing logic remains:A risk less investment should earn the riskless rate of interest