1. Dr. Hassan Mounir El-SadyChapter 6 1 Black-Scholes Option Pricing Model (BSOPM)

2. Chapter 6 Dr. Hassan Mounir El-Sady 2 Introduction The Black-Scholes option pricing model (BSOPM) has been one of the most important developments in finance in the last 50 years – Has provided a good understanding of what options should sell for – Has made options more attractive to individual and institutional investors

3. Chapter 6 Dr. Hassan Mounir El-Sady 3 The Model

4. Chapter 6 Dr. Hassan Mounir El-Sady 4 The Model (cont’d) Variable definitions: S= Current Stock Price X= Option Strike Price e= Base of Natural Logarithms R= Riskless Interest Rate T= Time Until Option Expiration  = Standard Deviation (Sigma) of Returns on the Underlying Security ln= Natural Logarithm N(d 1 ) and N(d 2 ) = Cumulative Standard Normal Distribution Functions

5. Chapter 6 Dr. Hassan Mounir El-Sady 5 Determinants of the Option Premium 1. Striking Price – The lower the striking price for a given stock, the more the option should be worth, because a call option lets you buy at a predetermined striking price 2. Time Until Expiration – The longer the time until expiration, the more the option is worth – The option premium increases for more distant expirations for puts and calls 3. Stock Price – The higher the stock price, the more a given call option is worth

6. Chapter 6 Dr. Hassan Mounir El-Sady 6 4. Volatility – The greater the price volatility, the more the option is worth. – The volatility estimate sigma cannot be directly observed and must be estimated. 5. Dividends – A company that pays a large dividend will have a smaller option premium than a company with a lower dividend, everything else being equal (Capital Gain Effect). – Listed options do not adjust for cash dividends. 6. Risk-Free Interest Rate – The higher the risk-free interest rate, the higher the option premium, everything else being equal. Determinants of the Option Premium (cont’d)

7. Chapter 6 Dr. Hassan Mounir El-Sady 7 Assumptions of the Black-Scholes Model 1. The stock pays no dividends during the option’s life 2. European exercise style 3. Markets are efficient 4. No transaction costs 5. Interest rates remain constant 6. Prices are lognormally distributed

8. Chapter 6 Dr. Hassan Mounir El-Sady 8 Assumptions of the Black-Scholes Model: 1. The stock pays no dividends during the option’s life. – If 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 premium – Developed BSOPM account for the payment of dividends.

9. Chapter 6 Dr. Hassan Mounir El-Sady 9 The BSOPM if Stock Pays Dividends During the Option’s Life The Robert Miller Option Pricing Model

10. Chapter 6 Dr. Hassan Mounir El-Sady 10 Assumptions of the Black-Scholes Model (cont’d) : 2. European Exercise Style: – A European option can only be exercised on the expiration date – American options are more valuable than European options – Few options are exercised early due to time value 3. Informational Efficiency – People cannot predict the direction of the market or of an individual stock – Put/call parity implies that you and everyone else will agree on the option premium, regardless of whether you are bullish or bearish 4. No Transaction Costs – There are no commissions and bid-ask spreads – This is not true, because transaction costs causes slightly different actual option prices for different market participants

11. Chapter 6 Dr. Hassan Mounir El-Sady 11 Assumptions of the Black-Scholes Model (cont’d): 5. Interest Rates Remain Constant 5. Interest Rates Remain Constant : – There is no real “riskfree” interest rate – Often the 30-day T-bill rate is used – Must look for ways to value options when the parameters of the traditional BSOPM are unknown or dynamic 6. Prices Are Lognormally Distributed: – The logarithms of the underlying security prices are normally distributed – A reasonable assumption for most assets on which options are available

12. Chapter 6 Dr. Hassan Mounir El-Sady 12 Intuition Into the Black-Scholes Model The valuation equation has two parts – One gives a “pseudo-probability” weighted expected stock price (an inflow) – One gives the time-value of money adjusted expected payment at exercise (an outflow) Cash InflowCash Outflow

13. Chapter 6 Dr. Hassan Mounir El-Sady 13 Intuition 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 day.

14. Chapter 6 Dr. Hassan Mounir El-Sady 14 Calculating Black-Scholes Prices from Historical Data To calculate the theoretical value of a call option using the BSOPM, we need: – The stock price – The option striking price – The time until expiration – The riskless interest rate – The volatility of the stock

15. Chapter 6 Dr. Hassan Mounir El-Sady 15 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. 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.

16. Chapter 6 Dr. Hassan Mounir El-Sady 16 Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using the BSOPM:

17. Chapter 6 Dr. Hassan Mounir El-Sady 17 Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using the BSOPM (cont’d): Using normal probability tables, we find: The value of the MSFT OCT 70 call is:

18. Chapter 6 Dr. Hassan Mounir El-Sady 18 Using Black-Scholes to Solve for the Put Premium Can combine the BSOPM with put/call parity:

19. Chapter 6 Dr. Hassan Mounir El-Sady 19 Implied Volatility Introduction Calculating implied volatility An implied volatility heuristic Historical versus implied volatility Pricing in volatility units Volatility smiles

20. Chapter 6 Dr. Hassan Mounir El-Sady 20 Introduction Instead of solving for the call premium, assume the market-determined call premium is correct – Then solve for the volatility that makes the equation hold – This value is called the implied volatility

21. Chapter 6 Dr. Hassan Mounir El-Sady 21 Calculating Implied Volatility Sigma cannot be conveniently isolated in the BSOPM – We must solve for sigma using trial and error Valuing a Microsoft Call Example 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.

22. Chapter 6 Dr. Hassan Mounir El-Sady 22 An Implied Volatility Heuristic For an exactly at-the-money call, the correct value of implied volatility is:

23. Chapter 6 Dr. Hassan Mounir El-Sady 23 Historical Versus Implied Volatility The 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

24. Chapter 6 Dr. Hassan Mounir El-Sady 24 Pricing in Volatility Units You cannot directly compare the dollar cost of two different options because – Options have different degrees of “moneyness” – A more distant expiration means more time value – The levels of the stock prices are different

25. Chapter 6 Dr. Hassan Mounir El-Sady 25 Volatility Smiles Volatility smiles are in contradiction to the BSOPM, which assumes constant volatility across all strike prices – When you plot implied volatility against striking prices, the resulting graph often looks like a smile

26. Chapter 6 Dr. Hassan Mounir El-Sady 26 Volatility Smiles (cont’d)

27. Chapter 6 Dr. Hassan Mounir El-Sady 27 Problems Using the Black-Scholes Model Does not work well with options that are deep-in- the-money or substantially out-of-the-money Produces biased values for very low or very high volatility stocks – Increases as the time until expiration increases May yield unreasonable values when an option has only a few days of life remaining