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  • 1. Fundamentals of Investments 15 C h a p t e r Option Valuation second edition Valuation & Management Charles J. Corrado Bradford D. Jordan McGraw Hill / Irwin Slides by Yee-Tien (Ted) Fu
  • 2. Just What is an Option Worth?
  • 3. Option Valuation
    • Our goal in this chapter is to discuss stock option prices. We will look at the fundamental relationship between call and put option prices and stock prices. Then we will discuss the Black-Scholes-Merton option pricing model.
    Goal
  • 4. Put-Call Parity Put-call parity The difference between a call option price and a put option price for European-style options with the same strike price and expiration date is equal to the difference between the underlying stock price and the discounted strike price.
  • 5. Put-Call Parity C = call option price P = put option price S = current stock price K = option strike price r = risk-free interest rate T = time remaining until option expiration
  • 6. Put-Call Parity
    • Put-call parity is based on the fundamental principle of finance stating that two securities with the same riskless payoff on the same future date must have the same price.
    • Suppose we create the following portfolio:
      • Buy 100 shares of stock X.
      • Write one stock X call option contract.
      • Buy one stock X put option contract.
  • 7. Put-Call Parity
  • 8. Put-Call Parity
    • Since the payoff for the portfolio is always equal to the strike price, it is risk-free, and therefore comparable to a U.S. T-bill.
    • So, cost of portfolio = discounted strike price
      • S + P – C = Ke –rT
      •  C – P = S – Ke –rT
    • If the stock pays a dividend before option expiration, then C – P = S – Ke –rT – PV(D) , where PV(D) represents the present value of the dividend payment.
  • 9. Work the Web
    • To learn more about trading options, see:
      • http://www. ino .com
      • http://www. optionetics .com
  • 10. The Black-Scholes-Merton Option Pricing Model
    • Option pricing theory made a great leap forward in the early 1970s with the development of the Black-Scholes option pricing model by Fischer Black and Myron Scholes.
    • Recognizing the important theoretical contributions by Robert Merton, many finance professionals refer to an extended version of the model as the Black-Scholes-Merton option pricing model.
  • 11. The Black-Scholes-Merton Option Pricing Model
    • The Black-Scholes-Merton option pricing model states the value of a stock option as a function of six input factors:
    • S , the current price of the underlying stock
    • y , the dividend yield of the underlying stock
    • K , the strike price specified in the option contract
    • r , the risk-free interest rate over the life of the option contract
    • T , the time remaining until the option contract expires
    •  , the price volatility of the underlying stock
  • 12. The Black-Scholes-Merton Option Pricing Model
    • The price of a call option on a single share of common stock, C = Se –yT N(d 1 ) – Ke –rT N(d 2 )
    • The price of a put option on a share of common stock, P = Ke –rT N(–d 2 ) – Se –yT N(–d 1 )
    • where
    N(x) denotes the standard normal probability of the value of x
  • 13. The Black-Scholes-Merton Option Pricing Model
  • 14. Work the Web
    • To learn more about the Black-Scholes-Merton formula, see:
      • http://www. jeresearch .com
  • 15. Varying the Option Price Input Values
  • 16. Varying the Underlying Stock Price
  • 17. Varying the Time to Option Expiration
  • 18. Varying the Volatility of the Stock Price
  • 19. Varying the Interest Rate
  • 20. Work the Web
    • For option trading strategies and more, see:
      • http://www. numa .com
  • 21. Measuring the Impact of Input Changes
    • Delta measures the dollar impact of a change in the underlying stock price on the value of a stock option.
    • Call option delta = e –yT N(d 1 ) > 0
      • Put option delta = –e –yT N(–d 1 ) < 0
    • A $1 change in the stock price causes an option price to change by approximately delta dollars.
  • 22. Measuring the Impact of Input Changes
    • Eta measures the percentage impact of a change in the underlying stock price on the value of a stock option.
    • Call option eta = e –yT N(d 1 )S / C > 1
      • Put option eta = –e –yT N(–d 1 )S / P < –1
    • A 1% change in the stock price causes an option price to change by approximately eta%.
  • 23. Measuring the Impact of Input Changes
    • Vega measures the impact of a change in stock price volatility on the value of a stock option.
    • Vega is the same for both call and put options.
    • Vega = Se –yT n(d 1 )  T > 0
      • where n(x) represents a standard normal density
    • A 1% change in sigma changes an option price by approximately the amount vega.
  • 24. Measuring the Impact of Input Changes
    • Gamma measures delta sensitivity to a stock price change. A $1 stock price change causes delta to change by approximately the amount gamma.
    • Theta measures option price sensitivity to a change in time remaining until option expiration. A one-day change causes the option price to change by approximately the amount theta.
  • 25. Measuring the Impact of Input Changes
    • Rho measures option price sensitivity to a change in the interest rate. A 1% interest rate change causes the option price to change by approximately the amount rho.
  • 26. Implied Standard Deviations
    • Of the six input factors for the Black-Scholes-Merton stock option pricing model, only the stock price volatility is not directly observable.
    • A stock price volatility estimated from an option price is called an implied standard deviation (ISD) or implied volatility (IVOL) .
    • Calculating an implied volatility requires that all input factors (except sigma) and either a call or put option price be known.
  • 27. Implied Standard Deviations
    • Sigma can be found by trial and error, or by using the following formula, which yields accurate implied volatility values as long as the stock price is not too far from the strike price of the option contract.
  • 28. Work the Web
    • For applications of implied volatility, see:
      • http://www. ivolatility .com
  • 29. Hedging a Portfolio with Index Options
    • Many institutional money managers make some use of stock index options to hedge the equity portfolios they manage.
    • To form an effective hedge, the number of option contracts needed =
      • Portfolio beta  Portfolio value .
      • Option delta  Option contract value
    • Note that regular rebalancing is needed to maintain an effective hedge over time.
  • 30. Work the Web
    • For stock option reports, see:
      • http://www. aantix .com
  • 31. Implied Volatility Skews
    • Volatility skews (or volatility smiles ) describe the relationship between implied volatilities and strike prices for options.
      • Recall that implied volatility is often used to estimate a stock’s price volatility over the period remaining until option expiration.
  • 32. Implied Volatility Skews
  • 33. Implied Volatility Skews
  • 34. Implied Volatility Skews
    • Logically, there can be only one stock price volatility, since price volatility is a property of the underlying stock.
    • However, volatility skews do exist. There is widespread agreement that the major cause factor is stochastic volatility.
    • Stochastic volatility is the phenomenon of stock price volatility changing randomly over time.
  • 35. Implied Volatility Skews
    • The Black-Scholes-Merton option pricing model assumes that stock price volatility is constant over the life of the option.
    • Nevertheless, the simplicity of the model makes it an excellent tool. Furthermore, the model yields accurate option prices for options with strike prices close to the current stock price.
  • 36. Work the Web
    • For volatility summaries, see:
      • http://www. pmpublishing . com
  • 37. Chapter Review
    • Put-Call Parity
    • The Black-Scholes-Merton Option Pricing Model
  • 38. Chapter Review
    • Varying the Option Price Input Values
      • Varying the Underlying Stock Price
      • Varying the Option’s Strike Price
      • Varying the Time Remaining until Option Expiration
      • Varying the Volatility of the Stock Price
      • Varying the Interest Rate
      • Varying the Dividend Yield
  • 39. Chapter Review
    • Measuring the Impact of Input Changes on Option Prices
      • Interpreting Option Deltas
      • Interpreting Option Etas
      • Interpreting Option Vegas
      • Interpreting an Option’s Gamma, Theta, and Rho
    • Implied Standard Deviations
    • Hedging a Stock Portfolio with Stock Index Options
  • 40. Chapter Review
    • Implied Volatility Skews