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Chapter 5

Chapter 5. Option Pricing. Outline. Introduction A brief history of options pricing Arbitrage and option pricing Intuition into Black-Scholes. Introduction. Option pricing developments are among the most important in the field of finance during the last 30 years

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Chapter 5

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  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 $10.00. 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 call Short put

  18. 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 + =

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

  21. 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?

  22. 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:

  23. 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 S0 = $50 K = $50 R = 6.00% t = 6 months

  24. The Put/Call Parity Relationship (cont’d) • Based on the provided information, the put value should be: P = $4.75 - $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

  25. 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

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

  27. 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?

  28. 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

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

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

  31. 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

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

  33. 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

  34. 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

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

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

  37. 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

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

  39. 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

  40. 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

  41. 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

  42. 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 • Pup = probability that the stock goes up • Pdown = probability that the stock goes down • ert = continuously compounded interest rate factor

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

  44. 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:

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

  46. 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?

  47. Extension to Two Periods (cont’d) $133.33 (UU) $100 $66.67 (UD = DU) $75 $50 $33.33 (DD) Today One Year Later Two Years Later

  48. 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

  49. 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:

  50. Extension to Two Periods (cont’d) $58.33 (UU) $34.72 $0 (UD = DU) $20.66 $0 $0 (DD) Today One Year Later Two Years Later

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