© 2004 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 A Brief History of Options Pricing: 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 A Brief History of Options Pricing: 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 Put pricing in the presence of call options Binomial put pricing Binomial pricing with asymmetric branches The effect of time

9 Arbitrage and Option Pricing (cont’d) The effect of volatility Multiperiod binomial option pricing Option pricing with continuous compounding Risk neutrality and implied branch probabilities Extension to two periods

10 Arbitrage and Option Pricing (cont’d) Recombining binomial trees Binomial pricing with lognormal returns Multiperiod binomial put pricing Exploiting arbitrage American versus European option pricing European put pricing and time value

11 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

12 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

13 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.

14 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 (law of one price) – Put/call parity

15 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

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

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

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

19 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

20 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

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

22 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

23 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

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

25 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?

26 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:

27 The Put/Call Parity Relationship (cont’d) To understand why the law of one price must hold, consider the following information: C = $4.75 P = $3 S 0 = $50 K = $50 R = 6.00% t = 6 months

28 The Put/Call Parity Relationship (cont’d) Based on the provided information, the put value should be: P = $ $50 + $50/(1.06) 0.5 = $3.31 – The actual call price ($4.75) is too high or the put price ($3) is too low

29 The Put/Call Parity Relationship (cont’d) To exploit the arbitrage, arbitrageurs would: – Write 1 $4.75 – Buy 1 $3 – Buy a share of stock at $50 – Borrow $48.56 at 6.00% for 6 months These actions result in a profit of $0.31 at option expiration irrespective of the stock price at option expiration

30 The Put/Call Parity Relationship (cont’d) Stock Price at Option Expiration $0$50$100 From call4.75 (45.25) From put47.00(3.00) From loan(1.44) From stock(50.00) Total$0.31

31 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

32 The Binomial Option Pricing Model (cont’d) Possible states of the world: $75 $50 $100 TodayOne Year Later

33 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?

34 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

35 The Binomial Option Pricing Model (cont’d) Possible portfolio values: $75 – (N)($C) $50 $100 - $25N TodayOne Year Later

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

37 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

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

39 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

40 Put Pricing in the Presence of Call Options In an arbitrage-free world, the put option cannot also sell for $14.77; If it did, an astute arbitrageur would: Buy a 75 call Write a 75 put Sell the stock short Invest $68.18 in T-bills These actions result in a cash flow of $6.82 today and a cash flow of $0 at option expiration

41 Put Pricing in the Presence of Call Options ActivityCash Flow Today Portfolio Value at Option Expiration Price = $100Price = $50 Buy 75 call-$14.77$250 Write 75 put $25 Sell stock short Invest $68.18 in T-bills Total$6.82$0.00

42 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

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

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

45 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

46 The Effect of Time More time until expiration means a higher option value

47 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

48 Multiperiod Binomial Option Pricing In reality, prices change in the marketplace minute by minute and option values change accordingly The logic of binomial pricing can be easily extended to a multiperiod setting using the recursive methods of solving for the option value

49 Option Pricing With Continuous Compounding Continuous compounding is an assumption of the Black-Scholes model Using continuous compounding to revalue the call option from the previous example:

50 Risk Neutrality and Implied Branch Probabilities Risk neutrality is an assumption of the Black-Scholes model For binomial pricing, this implies that the option premium contains an implied probability of the stock rising

51 Risk Neutrality and Implied Branch Probabilities (cont’d) Assume the following: – An investor is risk-neutral – He can invest funds risk free over one year at a continuously compounded rate of 10% – The stock either rises by 33.33% or falls 33.33% in one year After one year, one dollar will be worth $1.00 x e.10 = $ for an effective annual return of 10.52%

52 Risk Neutrality and Implied Branch Probabilities (cont’d) A risk-neutral investor would be indifferent between investing in the riskless rate and investing in the stock if it also had an expected return of 10.52% We can determine the branch probabilities that make the stock have a return of 10.52%

53 Risk Neutrality and Implied Branch Probabilities (cont’d) Define the following: – U = 1 + percentage increase if the stock goes up – D = 1 – percentage decrease if the stock goes down – P up = probability that the stock goes up – P down = probability that the stock goes down – e rt = continuously compounded interest rate factor

54 Risk Neutrality and Implied Branch Probabilities (cont’d) The average stock return is the weighted average of the two possible price movements:

55 Risk Neutrality and Implied Branch Probabilities (cont’d) If the stock goes up, the call will have an intrinsic value of $100 - $75 = $25 If the stock goes down, the call will be worthless The expected value of the call in one year is:

56 Risk Neutrality and Implied Branch Probabilities (cont’d) Discounted back to today, the value of the call today is:

57 Extension to Two Periods Assume two periods, each one year long, with the stock either rising or falling by 33.33% in each period What is the equilibrium value of a two- year European call shown on the next slide?

58 Extension to Two Periods (cont’d) $75 $50 $100 TodayOne Year Later $ (UU) $66.67 (UD = DU) $33.33 (DD) Two Years Later

59 Extension to Two Periods (cont’d) The option only winds up in the money when the stock advances twice (UU) – There is a 65.78% probability that the call is worth $58.33 and a 34.22% probability that the call is worthless

60 Extension to Two Periods (cont’d) There is a 65.78% probability that the call is worth $34.72 in one year and a 34.22% probability that the call is worthless in one year – The expected value of the call in one year is:

61 Extension to Two Periods (cont’d) $20.66 $0 $34.72 TodayOne Year Later $58.33 (UU) $0 (UD = DU) $0 (DD) Two Years Later

62 Recombining Binomial Trees If trees are recombining, this means that the up-down path and the down-up path both lead to the same point, but not necessarily the starting point To return to the initial price, the size of the up jump must be the reciprocal of the size of the down jump

63 Binomial Pricing with Lognormal Returns Black-Scholes assumes that security prices follow a lognormal distribution – With lognormal returns, the size of the upward movement U equals: – The probability of an up movement is:

64 Multiperiod Binomial Put Pricing To solve for the value of a put using binomial logic, just change the terminal intrinsic values and work backward just as with call pricing The branch probabilities do not change

65 Exploiting Arbitrage Arbitrage Example Binomial pricing results in a call price of $28.11 and a put price of $2.23. The interest rate is 10%, the stock price is $75, and the striking price of the call and the put is $60. The expiration date is in two years. What actions could an arbitrageur take to make a riskless profit if the call is actually selling for $29.00?

66 Exploiting Arbitrage (cont’d) Arbitrage Example (cont’d) Since the call is overvalued, and arbitrageur would want to write the call, buy the put, buy the stock, and borrow the present value of the striking price, resulting in the following cash flow today: Write 1 call$29.00 Buy 1 put($2.23) Buy 1 share($75.00) Borrow $60e -(.10)(2) $49.12 $0.89

67 Exploiting Arbitrage (cont’d) Arbitrage Example (cont’d) The value of the portfolio in two years will be worthless, regardless of the path the stock takes over the two-year period.

68 American Versus European Option Pricing With an American option, the intrinsic value is a sure thing With a European option, the intrinsic value is currently unattainable and may disappear before you can get at it An American option should be worth more than a European option

69 European Put Pricing and Time Value With a European put, the longer the option’s life, the longer you must wait to see sales proceeds More time means greater potential dispersion in underlying asset values, and this pushes up the put value A European put’s value with respect to time until expiration is indeterminate

70 European Put Pricing and Time Value (cont’d) Often, an out-of-the-money put will increase in value with more time Often, an in-the-money put decreases in value for more distant expirations

71 Intuition Into Black-Scholes Continuous time and multiple periods

72 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 risk less investment should earn the riskless rate of interest