Lecture 3

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 $ Option value = $26.67

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.

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.

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

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

Binomial Pricing

Binomial Pricing

50.78 = price Binomial Pricing

50.78 = price = intrinsic value Binomial Pricing

50.78 = price = intrinsic value The greater of Binomial Pricing

50.78 = price = intrinsic value Binomial Pricing

 Black Scholes price= 1.70  Binomial price = 1.51

 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

 Implied Volatility is highest where time premium is highest…usually at the money Time Decay Option Price Stock Price Days to Expiration

 Term Structure of Volatilities

Strike Price Asset Price Implied Volatility

Strike Price Asset Price Implied Volatility

Strike Price Asset Price Implied Volatility

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

 Develop Spreadsheet  Download data from internet

 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.”

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

 CBOE Example  Use Actual option ◦ Calculate historical volatility ◦ Calculate implied volatility

 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?”

Steps for Infinite Distribution of Outcomes

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

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

See handout for specs Walk through sample project