© 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 Basic Option Pricing Models The theory of put/call parity The binomial option pricing model Binomial put pricing Binomial pricing with asymmetric branches The effect of time The effect of volatility

9 Arbitrage and Option Pricing 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 Option pricing techniques are based on basic arbitrage principles Efficient market - equivalent assets should sell for the same price – ‘the law of one price’

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

11 Put-Call Parity Theory For a given underlying asset, the following factors form an interrelated complex: – Call price – Put price – Stock price and – Interest rate - risk free rate – time to expiration.....the price of European style puts and calls on the same stock with the identical strike price and expiration dates have a special relationship

12 Put – Call Parity Theory ‘When European options are at the money and the stock pays no dividends, relative call prices should exceed relative put prices by an amount approximately equal to the riskless rate of interest for the option term times the stock price.’ ….a theory about how to price options relative to other options

13 Put-Call Parity theory - Covered Call and Short Put The profit/loss diagram for a covered call and for a short put are essentially equal Covered callShort put

14 Put Call Parity theory - Covered Call and Long Put A riskless position results if you combine a covered call and a long put Covered call Long putRiskless position + =....stock price at option expiration has no impact on the profit/loss position - ‘riskless’

15 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....and will be quickly taken advantage of.

16 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

17 No Arbitrage Relationships - State of Equilibrium 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 (S)(r) – 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

18 No Arbitrage Relationships If there is no arbitrage and assuming European at the money options, then:

19 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 for the option term times the strike price – The difference will be greater as: The stock price increases Interest rates increase The time to expiration increases

20 The Put/Call Parity Relationship We now know how the call prices, put prices, the stock price, and the riskless interest rate are related What about when the strike price is different from the stock price ie. In/out of the money options ?

21 The Put-Call Parity Relationship …where the strike price is less than the stock price S 0 (in the money call) the call price will reflect this intrinsic Value

22 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?

23 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:

24 Put-Call Parity – another look Another way to illustrate the theory Instead of the covered call and long put with borrowing – an equivalent situation is: – A share of stock plus a put is equivalent to a call plus an investment in risk free bonds

25 The Binomial Option Pricing Model A theory about establishing the option price from the factors that influence it: – Another simplified pricing model - used to illustrate various influences on option prices: Stock price Strike price Interest rates Stock price volatility – The simplifying assumption is: a single period binomial model – there are only two possible outcomes and one time period – The stock can go up or down in price over the period

26 Binomial 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 ….we want to construct a riskless portfolio of stock and options – one where the future value of the portfolio is independent of the future value of the stock.

27 The Binomial Option Pricing Model (cont’d) Possible states of the world: $75 $50 $100 Stock Price Today Stock Price 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 (no risk) – 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 Total Investment Today Total Investment 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%) Discounting at 10% for one year 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 otherwise there would exist an arbitrage situation: – If the option is selling for more than $14.77 sellers would step in or if the price was less than $14.77 buyers of the option would find it attractive

34 The Binomial Option Pricing Model (cont’d) The option value is independent of the probabilities associated with the future stock price and hence: The price of an option is independent of the expected return on the stock! The price of an option tells you nothing about the future path of the stock price!

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 Option Pricing Model Establish a riskless or hedge portfolio where no arbitrage opportunity exists - state of equilibrium

37 Binomial Put Pricing (cont’d) Possible portfolio values: $75 + 2($P) $50 + N($75 - $50) $100 TodayOne Year Later

38 Binomial Put Pricing Solve for N – how many put options to purchase – $50 + N(25) = $100 – 25 N = $50 – N = 2

39 Binomial Put Pricing (cont’d) A portfolio composed of one share of stock and two puts will grow risklessly to $100 after one year (discounted to $90.91 in today’s dollars)

40 Back to Put-Call Parity Let’s reconcile: Using binomial theory we arrived at a call price of $14.77 and a put price of $7.95 Does this reconcile with the put-call parity theory?

41 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

42 The Effect of Time More time until expiration means a higher option value – ….remember an option can have time value and or intrinsic value

43 The Effect of Volatility Higher volatility means a higher option price for both call and put options – E.g. high tech vs utility type stocks ….work through the same call option example with smaller range of possible outcomes (lower volatility)

44 Black-Scholes Model - the next evolution Pricing logic remains the same – riskless investments should earn riskless rates of return Input factors are the same – Stock price – Strike price – Time until expiration – Interest rates – Stock volatility Using developing computer power in the early 70’s, the model moves to continuous time calculus – outcomes now not limited to only two, many different time intervals - in theory there are an infinite number of future outcomes

45 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