Tài chính doanh nghiệp - Chapter 6: The black - Scholes option pricing model

© 2004 South-Western Publishing1Chapter 6The Black-Scholes Option Pricing Model2OutlineIntroductionThe Black-Scholes option pricing modelCalculating Black-Scholes prices from historical dataImplied volatilityUsing Black-Scholes to solve for the put premiumProblems using the Black-Scholes model3IntroductionThe Black-Scholes option pricing model (BSOPM) has been one of the most important developments in finance in the last 50 yearsHas provided a good understanding of what options should sell forHas made options more attractive to individual and institutional investors4The Black-Scholes Option Pricing ModelThe modelDevelopment and assumptions of the modelDeterminants of the option premiumAssumptions of the Black-Scholes modelIntuition into the Black-Scholes model5The Model6The Model (cont’d)Variable definitions:S = current stock priceK = option strike pricee = base of natural logarithmsR = riskless interest rateT = time until option expiration = standard deviation (sigma) of returns on the underlying securityln = natural logarithmN(d1) and N(d2) = cumulative standard normal distribution functions7Development and Assumptions of the ModelDerivation from:PhysicsMathematical short cutsArbitrage argumentsFischer Black and Myron Scholes utilized the physics heat transfer equation to develop the BSOPM8Determinants of the Option PremiumStriking priceTime until expirationStock priceVolatilityDividends Risk-free interest rate9Striking PriceThe lower the striking price for a given stock, the more the option should be worthBecause a call option lets you buy at a predetermined striking price10Time Until ExpirationThe longer the time until expiration, the more the option is worthThe option premium increases for more distant expirations for puts and calls11Stock PriceThe higher the stock price, the more a given call option is worthA call option holder benefits from a rise in the stock price12VolatilityThe greater the price volatility, the more the option is worthThe volatility estimate sigma cannot be directly observed and must be estimatedVolatility plays a major role in determining time value13DividendsA company that pays a large dividend will have a smaller option premium than a company with a lower dividend, everything else being equalListed options do not adjust for cash dividendsThe stock price falls on the ex-dividend date14Risk-Free Interest RateThe higher the risk-free interest rate, the higher the option premium, everything else being equalA higher “discount rate” means that the call premium must rise for the put/call parity equation to hold15Assumptions of the Black-Scholes ModelThe stock pays no dividends during the option’s lifeEuropean exercise styleMarkets are efficientNo transaction costsInterest rates remain constantPrices are lognormally distributed16The Stock Pays no Dividends During the Option’s LifeIf you apply the BSOPM to two securities, one with no dividends and the other with a dividend yield, the model will predict the same call premiumRobert Merton developed a simple extension to the BSOPM to account for the payment of dividends17The Stock Pays no Dividends During the Option’s Life (cont’d)The Robert Miller Option Pricing Model 18European Exercise StyleA European option can only be exercised on the expiration dateAmerican options are more valuable than European optionsFew options are exercised early due to time value19Markets Are EfficientThe BSOPM assumes informational efficiencyPeople cannot predict the direction of the market or of an individual stockPut/call parity implies that you and everyone else will agree on the option premium, regardless of whether you are bullish or bearish20No Transaction CostsThere are no commissions and bid-ask spreadsNot trueCauses slightly different actual option prices for different market participants21Interest Rates Remain ConstantThere is no real “riskfree” interest rateOften the 30-day T-bill rate is usedMust look for ways to value options when the parameters of the traditional BSOPM are unknown or dynamic22Prices Are Lognormally DistributedThe logarithms of the underlying security prices are normally distributedA reasonable assumption for most assets on which options are available23Intuition Into the Black-Scholes ModelThe valuation equation has two partsOne gives a “pseudo-probability” weighted expected stock price (an inflow)One gives the time-value of money adjusted expected payment at exercise (an outflow)24Intuition Into the Black-Scholes Model (cont’d)Cash InflowCash Outflow25Intuition Into the Black-Scholes Model (cont’d)The value of a call option is the difference between the expected benefit from acquiring the stock outright and paying the exercise price on expiration day26Calculating Black-Scholes Prices from Historical DataTo calculate the theoretical value of a call option using the BSOPM, we need:The stock priceThe option striking priceThe time until expirationThe riskless interest rateThe volatility of the stock27Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example We would like to value a MSFT OCT 70 call in the year 2000. Microsoft closed at $70.75 on August 23 (58 days before option expiration). Microsoft pays no dividends. We need the interest rate and the stock volatility to value the call. 28Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Consulting the “Money Rate” section of the Wall Street Journal, we find a T-bill rate with about 58 days to maturity to be 6.10%. To determine the volatility of returns, we need to take the logarithm of returns and determine their volatility. Assume we find the annual standard deviation of MSFT returns to be 0.5671. 29Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using the BSOPM:30Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using the BSOPM (cont’d):31Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using normal probability tables, we find:32Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) The value of the MSFT OCT 70 call is:33Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) The call actually sold for $4.88. The only thing that could be wrong in our calculation is the volatility estimate. This is because we need the volatility estimate over the option’s life, which we cannot observe.34Implied VolatilityIntroductionCalculating implied volatilityAn implied volatility heuristicHistorical versus implied volatilityPricing in volatility unitsVolatility smiles35IntroductionInstead of solving for the call premium, assume the market-determined call premium is correctThen solve for the volatility that makes the equation holdThis value is called the implied volatility36Calculating Implied VolatilitySigma cannot be conveniently isolated in the BSOPMWe must solve for sigma using trial and error37Calculating Implied Volatility (cont’d)Valuing a Microsoft Call Example (cont’d) The implied volatility for the MSFT OCT 70 call is 35.75%, which is much lower than the 57% value calculated from the monthly returns over the last two years. 38An Implied Volatility HeuristicFor an exactly at-the-money call, the correct value of implied volatility is:39Historical Versus Implied VolatilityThe volatility from a past series of prices is historical volatilityImplied volatility gives an estimate of what the market thinks about likely volatility in the future40Historical Versus Implied Volatility (cont’d)Strong and Dickinson (1994) findClear evidence of a relation between the standard deviation of returns over the past month and the current level of implied volatilityThat the current level of implied volatility contains both an ex post component based on actual past volatility and an ex ante component based on the market’s forecast of future variance41Pricing in Volatility UnitsYou cannot directly compare the dollar cost of two different options becauseOptions have different degrees of “moneyness”A more distant expiration means more time valueThe levels of the stock prices are different42Volatility SmilesVolatility smiles are in contradiction to the BSOPM, which assumes constant volatility across all strike pricesWhen you plot implied volatility against striking prices, the resulting graph often looks like a smile43Volatility Smiles (cont’d)44Using Black-Scholes to Solve for the Put PremiumCan combine the BSOPM with put/call parity:45Problems Using the Black-Scholes ModelDoes not work well with options that are deep-in-the-money or substantially out-of-the-moneyProduces biased values for very low or very high volatility stocksIncreases as the time until expiration increasesMay yield unreasonable values when an option has only a few days of life remaining