2004 South-Western Publishing*Chapter 6The Black-Scholes Option Pricing Model

*OutlineIntroductionThe Black-Scholes option pricing modelCalculating Black-Scholes prices from historical dataImplied volatilityUsing Black-Scholes to solve for the put premiumProblems using the Black-Scholes model

*IntroductionThe 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 investors

*The Black-Scholes Option Pricing ModelThe modelDevelopment and assumptions of the modelDeterminants of the option premiumAssumptions of the Black-Scholes modelIntuition into the Black-Scholes model

*The Model

*The Model (contd)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 functions

*Development and Assumptions of the ModelDerivation from:PhysicsMathematical short cutsArbitrage arguments

Fischer Black and Myron Scholes utilized the physics heat transfer equation to develop the BSOPM

*Determinants of the Option PremiumStriking priceTime until expirationStock priceVolatilityDividends Risk-free interest rate

*Striking 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 price

*Time Until ExpirationThe longer the time until expiration, the more the option is worthThe option premium increases for more distant expirations for puts and calls

*Stock PriceThe higher the stock price, the more a given call option is worthA call option holder benefits from a rise in the stock price

*VolatilityThe 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 value

*DividendsA 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 date

*Risk-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 hold

*Assumptions of the Black-Scholes ModelThe stock pays no dividends during the options lifeEuropean exercise styleMarkets are efficientNo transaction costsInterest rates remain constantPrices are lognormally distributed

*The Stock Pays no Dividends During the Options 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 dividends

*The Stock Pays no Dividends During the Options Life (contd)The Robert Miller Option Pricing Model

*European 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 value

*Markets 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 bearish

*No Transaction CostsThere are no commissions and bid-ask spreadsNot trueCauses slightly different actual option prices for different market participants

*Interest 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 dynamic

*Prices Are Lognormally DistributedThe logarithms of the underlying security prices are normally distributedA reasonable assumption for most assets on which options are available

*Intuition 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)

*Intuition Into the Black-Scholes Model (contd)Cash InflowCash Outflow

*Intuition Into the Black-Scholes Model (contd)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 day

*Calculating 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 stock

*Calculating 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.

*Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (contd)

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.

*Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (contd)Using the BSOPM:

*Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (contd)

Using the BSOPM (contd):

*Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (contd)

Using normal probability tables, we find:

*Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (contd)

The value of the MSFT OCT 70 call is:

*Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (contd)

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 options life, which we cannot observe.

*Implied VolatilityIntroductionCalculating implied volatilityAn implied volatility heuristicHistorical versus implied volatilityPricing in volatility unitsVolatility smiles

*IntroductionInstead 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 volatility

*Calculating Implied VolatilitySigma cannot be conveniently isolated in the BSOPMWe must solve for sigma using trial and error

*Calculating Implied Volatility (contd)Valuing a Microsoft Call Example (contd)

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.

*An Implied Volatility HeuristicFor an exactly at-the-money call, the correct value of implied volatility is:

*Historical Versus Implied VolatilityThe volatility from a past series of prices is historical volatility

Implied volatility gives an estimate of what the market thinks about likely volatility in the future

*Historical Versus Implied Volatility (contd)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 markets forecast of future variance

*Pricing in Volatility UnitsYou cannot directly compare the dollar cost of two different options becauseOptions have different degrees of moneynessA more distant expiration means more time valueThe levels of the stock prices are different

*Volatility 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 smile

*Volatility Smiles (contd)

Chart2

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Current Stock Price

Striking Price

Implied Volatility (%)

Volatility SmileMicrosoft August 2000

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*Using Black-Scholes to Solve for the Put PremiumCan combine the BSOPM with put/call parity:

*Problems 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

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