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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 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 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.
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%
Binomial Pricing 40.37 32.10 36
Binomial Pricing 40.37 32.10 36
Binomial Pricing 50.78 = price 40.37 32.10 25.52 45.28 36 28.62 40.37 32.10 36
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
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
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
Price Comparisons • Black Scholes price= 1.70 • Binomial price = 1.51
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
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
Volatility Surface • Term Structure of Volatilities
Volatility Smile Implied Volatility Asset Price Strike Price
Volatility Smirk Implied Volatility Asset Price Strike Price
Volatility Smirk Implied Volatility Asset Price Strike Price
Volatility • Calculate the Annualized variance of the daily relative price change • Square root to arrive at standard deviation • Standard deviation is the volatility
Volatility • Develop Spreadsheet • Download data from internet http://finance.yahoo.com
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.”
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%
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
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?”
Expected Return Steps for Infinite Distribution of Outcomes
Expected Return Example (same option) P = 41 r = 10% v = .42 EX = 40 t = 30 days / 365 Example
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
Option Pricing Project • See handout for specs • Walk through sample project