1. Lecture 3

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

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

7. 40.37 32.10 36 Binomial Pricing

8. 40.37 32.10 36 Binomial Pricing

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

10. 50.78 = price 10.78 = intrinsic value 40.37.37 32.10 0 25.52 0 45.28 36 28.62 36 40.37 32.10 Binomial Pricing

11. 50.78 = price 10.78 = intrinsic value 40.37.37 32.10 0 25.52 0 45.28 5.60 36 28.62 40.37 32.10 36 The greater of Binomial Pricing

12. 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 Binomial Pricing

13.  Black Scholes price= 1.70  Binomial price = 1.51

14.  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 Option Price Stock Price Days to Expiration 90 60 30

16.  Term Structure of Volatilities

17. Strike Price Asset Price Implied Volatility

18. Strike Price Asset Price Implied Volatility

19. Strike Price Asset Price Implied Volatility

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

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

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

24.  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.  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. Steps for Infinite Distribution of Outcomes

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

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

29. See handout for specs Walk through sample project