1. © 2004 South-Western Publishing 1 Chapter 6 The Black-Scholes Option Pricing Model

2. 2 Transition from discrete to continuous time T→lim0

3. 3 Natural Logarithm and e Continuous compounding/discounting

4. 4 The Model

5. 5 The Model (cont’d) Variable definitions: S=current stock price K=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

6. 6 Development and Assumptions of the Model Derivation from: – Physics – Mathematical short cuts – Arbitrage arguments Fischer Black and Myron Scholes utilized the physics heat transfer equation to develop the BSOPM

7. 7 Determinants of the Option Premium Striking price Time until expiration Stock price Volatility Risk-free interest rate

8. 8 Striking Price The lower the striking price for a given stock, the more the option should be worth

9. 9 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

10. 10 Stock Price Pricing based on discounted cash flows; Payoff occurs at expiration = S-K; best bet on S The role of two counterparties in the trade

11. 11 Volatility The greater the price volatility, the more the option is worth – The volatility estimate sigma cannot be directly observed and must be estimated – Volatility plays a major role in determining time value

12. 12 Risk-Free Interest Rate The higher the risk-free interest rate, the higher the option premium, everything else being equal – A higher “discount rate” means that the call premium must rise for the put/call parity equation to hold

13. 13 Assumptions of the Black- Scholes Model The stock pays no dividends during the option’s life European exercise style Markets are efficient No transaction costs Interest rates remain constant Prices are lognormally distributed

14. 14 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

15. 15 Markets Are Efficient The BSOPM assumes 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

16. 16 No Transaction Costs There are no commissions and bid-ask spreads – Not true – Causes slightly different actual option prices for different market participants

17. 17 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

18. 18 Prices Are Lognormally Distributed – the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributedprobability distributionrandom variable logarithmnormally distributed

19. 19 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)

20. 20 Time Value based on volatility Example Option at the money and 0 risk free rate

21. 21 Time Value based on volatility Example

22. 22 Time Value based on volatility Example

23. 23

24. 24

25. 25

26. 26

27. 27 Intrinsic Value plus Time Value 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

28. 28 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

29. 29 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.

30. 30 Calculating 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.

31. 31 Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using the BSOPM:

32. 32 Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using the BSOPM (cont’d):

33. 33 Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) Using normal probability tables, we find:

34. 34 Calculating Black-Scholes Prices from Historical Data Valuing a Microsoft Call Example (cont’d) The value of the MSFT OCT 70 call is:

35. 35 Calculating 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.

36. 36 Implied Volatility Introduction Calculating implied volatility An implied volatility heuristic Historical versus implied volatility Pricing in volatility units Volatility smiles

37. 37 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

38. 38 Calculating Implied Volatility Sigma cannot be conveniently isolated in the BSOPM – We must solve for sigma using trial and error

39. 39 Calculating 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.

40. 40 An Implied Volatility Heuristic For an exactly at-the-money call, the correct value of implied volatility is:

41. 41 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

42. 42 Historical Versus Implied Volatility (cont’d) Strong and Dickinson (1994) find – Clear evidence of a relation between the standard deviation of returns over the past month and the current level of implied volatility – That 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 variance

43. 43 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

44. 44 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

45. 45 Volatility Smiles (cont’d)

46. 46 Using Black-Scholes to Solve for the Put Premium Can combine the BSOPM with put/call parity:

47. 47 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