© 2002 South-Western Publishing 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 $ 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 callShort put

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

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 S 0 = current stock price S 1 = 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: $75 $50 $100 TodayOne 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: $75 – (N)($C) $50 $100 - $25N TodayOne 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: $75 + 2($P) $50 + N($75 - $50) $100 TodayOne 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

43 Probabilities and Risk Neutrality (P up U) + (P down D) = e rt (P up U) + (1- P up ) D = e rt P up U + (D – P up D) = e rt P up = (e rt – D)/(U – D) P down = 1 - P up