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Financial Engineering

Financial Engineering. Lecture 3. Option Valuation Methods. Case 1 Stock price falls to $60 Option value = $0. Case 2 Stock price rises to $106.67 Option value = $26.67. Genentech call options have an exercise price of $80 and expire in one year. . Option Valuation Methods.

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Financial Engineering

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  1. Financial Engineering Lecture 3

  2. Option Valuation Methods Case 1 Stock price falls to $60 Option value = $0 Case 2 Stock price rises to $106.67 Option value = $26.67 Genentech call options have an exercise price of $80 and expire in one year.

  3. Option Valuation Methods If we are risk neutral, the expected return on Genentech call options is 2.5%. Accordingly, we can determine the price of the option as follows, given equal probabilities of each outcome.

  4. Binomial Model The price of an option, using the Binomial method, is significantly impacted by the time intervals selected. The Genentech example illustrates this fact.

  5. Binomial Pricing The prior example can be generalized as the binomial model and shown as follows.

  6. Binomial Pricing a = 1.0083 u = 1.1215 d = .8917 Pu = .5075 Pd = .4925 Example Price = 36 s = .40 t = 90/365 D t = 30/365 Strike = 40 r = 10%

  7. Binomial Pricing 40.37 32.10 36

  8. Binomial Pricing 40.37 32.10 36

  9. Binomial Pricing 50.78 = price 40.37 32.10 25.52 45.28 36 28.62 40.37 32.10 36

  10. Binomial Pricing 50.78 = price 10.78 = intrinsic value 40.37 .37 32.10 0 25.52 0 45.28 36 28.62 40.37 32.10 36

  11. Binomial Pricing 50.78 = price 10.78 = intrinsic value 40.37 .37 32.10 0 25.52 0 45.28 5.60 36 28.62 The greater of 40.37 32.10 36

  12. Binomial Pricing 50.78 = price 10.78 = intrinsic value 40.37 .37 32.10 0 25.52 0 45.28 5.60 36 .19 28.62 0 40.37 2.91 32.10 .10 36 1.51

  13. Price Comparisons • Black Scholes price= 1.70 • Binomial price = 1.51

  14. Volatility • Only non-observable variable • Historical volatility • Predictive models • ARCH (Robert Engel) • GARCH • Weighted Average Historical Volatility • Implied Volatility • VIX – Exchange traded volatility option • 1993 • S&P 500 Implied Volatility

  15. Implied Volatility is highest where time premium is highest…usually at the money Time Decay Days to Expiration 90 60 30 Option Price Stock Price

  16. Volatility Surface • Term Structure of Volatilities

  17. Volatility Smile Implied Volatility Asset Price Strike Price

  18. Volatility Smirk Implied Volatility Asset Price Strike Price

  19. Volatility Smirk Implied Volatility Asset Price Strike Price

  20. Volatility • Calculate the Annualized variance of the daily relative price change • Square root to arrive at standard deviation • Standard deviation is the volatility

  21. Volatility • Develop Spreadsheet • Download data from internet http://finance.yahoo.com

  22. Implied Volatility • All variables in the option price can be observed, other than volatility. • Even the price of the option can be observed in the secondary markets. • Volatility cannot be observed, it can only be calculated. • Given the market price of the option, the volatility can be “reverse engineered.”

  23. Implied Volatility Use Numa to calculate implied volatility. Example (same option) P = 41 r = 10% PRICE = 2.67 EX = 40 t = 30 days / 365 v = ???? Implied volatility = 42.16%

  24. Implied Volatility • CBOE Example • Use Actual option • Calculate historical volatility • Calculate implied volatility http://www.math.columbia.edu/~smirnov/options13.html http://www.cboe.com http://www.numa.com

  25. Expected Returns • Given a normal or lognormal distribution of returns, it is possible to calculate the probability of having an stock price above or below a target price. • Wouldn’t it be nice to know the probability of making a profit or the probability of being “in the money?”

  26. Expected Return Steps for Infinite Distribution of Outcomes

  27. Expected Return Example (same option) P = 41 r = 10% v = .42 EX = 40 t = 30 days / 365 Example

  28. Expected Return Example (same option) P = 41 r = 10% v = .42 EX = 40 t = 30 days / 365 37% 58% $2.67 63% 40 42.67

  29. Option Pricing Project • See handout for specs • Walk through sample project

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