Financial Engineering Lecture 3 Option Valuation Methods Genentech

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Financial Engineering Lecture 3

Financial Engineering Lecture 3

Option Valuation Methods Genentech call options have an exercise price of $80 and expire

Option Valuation Methods Genentech call options have an exercise price of $80 and expire in one year. Case 1 Case 2 Stock price falls to $60 Stock price rises to $106. 67 Option value = $0 Option value = $26. 67

Option Valuation Methods If we are risk neutral, the expected return on Genentech call

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

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

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

Binomial Pricing Example Price = 36 Strike = 40 s =. 40 t =

Binomial Pricing Example Price = 36 Strike = 40 s =. 40 t = 90/365 D t = 30/365 r = 10% a = 1. 0083 u = 1. 1215 d =. 8917 Pu =. 5075 Pd =. 4925

Binomial Pricing 40. 37 36 32. 10

Binomial Pricing 40. 37 36 32. 10

Binomial Pricing 40. 37 36 32. 10

Binomial Pricing 40. 37 36 32. 10

Binomial Pricing 45. 28 50. 78 = price 36 40. 37 36 32. 10

Binomial Pricing 45. 28 50. 78 = price 36 40. 37 36 32. 10 28. 62 25. 52

Binomial Pricing 45. 28 50. 78 = price 10. 78 = intrinsic value 40.

Binomial Pricing 45. 28 50. 78 = price 10. 78 = intrinsic value 40. 37 36 32. 10 28. 62 0 25. 52 0

Binomial Pricing The greater of 45. 28 50. 78 = price 5. 60 10.

Binomial Pricing The greater of 45. 28 50. 78 = price 5. 60 10. 78 = intrinsic value 40. 37 36 36 . 37 32. 10 28. 62 0 25. 52 0

Binomial Pricing 45. 28 50. 78 = price 5. 60 10. 78 = intrinsic

Binomial Pricing 45. 28 50. 78 = price 5. 60 10. 78 = intrinsic value 40. 37 2. 91 36 . 37 . 19 36 1. 51 32. 10 28. 62 0 0 25. 52 0

Price Comparisons Black Scholes price= 1. 70 Binomial price = 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

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

Time Decay Implied Volatility is highest where time premium is highest…usually at the money

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

Volatility Surface Term Structure of Volatilities

Volatility Surface Term Structure of Volatilities

Volatility Smile Implied Volatility Asset Price Strike Price

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 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

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

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.

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

Implied Volatility Use Numa to calculate implied volatility. Example (same option) P = 41 r = 10% EX = 40 t = 30 days / 365 PRICE = 2. 67 v = ? ? Implied volatility = 42. 16%

Implied Volatility CBOE Example Use Actual option ◦ Calculate historical volatility ◦ Calculate implied

Implied Volatility CBOE Example Use Actual option ◦ Calculate historical volatility ◦ Calculate implied volatility http: //www. math. columbia. edu/~smirnov/options 13. html http: //www. cboe. com Bloomberg

Expected Returns Given a normal or lognormal distribution of returns, it is possible to

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 Steps for Infinite Distribution of Outcomes

Expected Return Example (same option) P = 41 r = 10% EX = 40

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

Expected Return Example (same option) P = 41 r = 10% EX = 40

Expected Return Example (same option) P = 41 r = 10% EX = 40 t = 30 days / 365 v =. 42 42% 58% $2. 67 63% 40 37% 42. 67

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

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