Chapter 5

1. Chapter 5 Option Pricing

2. Outline • Introduction • A brief history of options pricing • Arbitrage and option pricing • Intuition into Black-Scholes

3. Introduction • Option pricing developments are among the most important in the field of finance during the last 30 years • The backbone of option pricing is the Black-Scholes model

4. Introduction (cont’d) • The Black-Scholes model:

5. A Brief History of Options Pricing: The Early Work • Charles Castelli wrote The Theory of Options in Stocks and Shares (1877) • Explained the hedging and speculation aspects of options • Louis Bachelier wrote Theorie de la Speculation (1900) • The first research that sought to value derivative assets

6. The Middle Years • Rebirth of option pricing in the 1950s and 1960s • Paul 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)

7. The Present • The Black-Scholes option pricing model (BSOPM) was developed in 1973 • An improved version of the Boness model • Most other option pricing models are modest variations of the BSOPM

8. Arbitrage and Option Pricing • Introduction • Free lunches • The theory of put/call parity • The binomial option pricing model • Binomial put pricing • Binomial pricing with asymmetric branches

9. Arbitrage and Option Pricing (cont’d) • The effect of time • The effect of volatility

10. Introduction • Finance is sometimes called “the study of arbitrage” • Arbitrage is the existence of a riskless profit • Finance theory does not say that arbitrage will never appear • Arbitrage opportunities will be short-lived

11. Free Lunches • The apparent mispricing may be so small that it is not worth the effort • E.g., pennies on the sidewalk • Arbitrage opportunities may be out of reach because of an impediment • E.g., trade restrictions

12. Free 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.

13. Free Lunches (cont’d) • Modern option pricing techniques are based on arbitrage principles • In a well-functioning marketplace, equivalent assets should sell for the same price • Put/call parity

14. The Theory of Put/Call Parity • Introduction • Covered call and short put • Covered call and long put • No arbitrage relationships • Variable definitions • The put/call parity relationship

15. Introduction • For a given underlying asset, the following factors form an interrelated complex: • Call price • Put price • Stock price and • Interest rate

16. Covered Call and Short Put • The profit/loss diagram for a covered call and for a short put are essentially equal Covered call Short put

17. Covered Call and Long Put • A riskless position results if you combine a covered call and a long put Long put Riskless position Covered call + =

18. Covered Call and Long Put • Riskless investments should earn the riskless rate of interest • If an investor can own a stock, write a call and buy a put and make a profit, arbitrage is present

19. No Arbitrage Relationships • The 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 stock • The interest on the bank loan is paid in the future

20. No Arbitrage Relationships (cont’d) • If there is no arbitrage, then:

21. No Arbitrage Relationships (cont’d) • If there is no arbitrage, then: • The call premium should exceed the put premium by about the riskless rate of interest • The difference will be greater as: • The stock price increases • Interest rates increase • The time to expiration increases

22. Variable Definitions C = call premium P = put premium S0 = current stock price S1 = stock price at option expiration K = option striking price R = riskless interest rate t = time until option expiration

23. The Put/Call Parity Relationship • We now know how the call prices, put prices, the stock price, and the riskless interest rate are related:

24. The Put/Call Parity Relationship (cont’d) Equilibrium Stock Price Example You have the following information: • Call price = $3.5 • Put price = $1 • Striking price = $75 • Riskless interest rate = 5% • Time until option expiration = 32 days If there are no arbitrage opportunities, what is the equilibrium stock price?

25. The Put/Call Parity Relationship (cont’d) Equilibrium Stock Price Example (cont’d) Using the put/call parity relationship to solve for the stock price:

26. The Binomial Option Pricing Model • Assume the following: • U.S. government securities yield 10% next year • Stock XYZ currently sells for $75 per share • There are no transaction costs or taxes • There are two possible stock prices in one year

27. The Binomial Option Pricing Model (cont’d) • Possible states of the world: $100 $75 $50 Today One Year Later

28. The 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 $75 • If the stock price is $100, the option will be worth $25 • If the stock price is $50, the option will be worth $0 • What should be the price of this option?

29. The 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 year • Buy the stock and write N call options

30. The Binomial Option Pricing Model (cont’d) • Possible portfolio values: $100 - $25N $75 – (N)($C) $50 Today One Year Later

31. The Binomial Option Pricing Model (cont’d) • We can solve for N such that the portfolio value in one year must be $50:

32. The 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 year • The future value is known and riskless and must earn the riskless rate of interest (10%) • The portfolio must be worth $45.45 today

33. The Binomial Option Pricing Model (cont’d) • Assuming no arbitrage exists: • The option must sell for $14.77!

34. The Binomial Option Pricing Model (cont’d) • The option value is independent of the probabilities associated with the future stock price • The price of an option is independent of the expected return on the stock

35. Binomial Put Pricing • Priced analogously to calls • You can combine puts with stock so that the future value of the portfolio is known • Assume a value of $100

36. Binomial Put Pricing (cont’d) • Possible portfolio values: $100 $75 + 2($P) $50 + N($75 - $50) Today One Year Later

37. Binomial Put Pricing (cont’d) • A portfolio composed of one share of stock and two puts will grow risklessly to $100 after one year

38. Binomial Pricing With Asymmetric Branches • The size of the up movement does not have to be equal to the size of the decline • E.g., the stock will either rise by $25 or fall by $15 • The logic remains the same: • First, determine the number of options • Second, solve for the option price

39. The Effect of Time • More time until expiration means a higher option value

40. The Effect of Volatility • Higher volatility means a higher option price for both call and put options • Explains why options on Internet stocks have a higher premium than those for retail firms

41. Intuition Into Black-Scholes • Continuous time and multiple periods

42. Continuous Time and Multiple Periods • Future security prices are not limited to only two values • There are theoretically an infinite number of future states of the world • Requires continuous time calculus (BSOPM) • The pricing logic remains: • A riskless investment should earn the riskless rate of interest