 Known dividend to be paid before option expiration ◦ Dividend has already been announced or stock pays regular dividends ◦ Option should be priced on S 0 – PV(dividends anticipated before t) = S 0 – Div * exp(-r*time dividend paid) ◦ Example: Coca-Cola pays relatively stable quarterly dividend around $0.31 per share on November 29 th each year. Assuming it is November 14, use S 0 – 0.31* exp(-r*15/365) for current stock price in Black-Scholes model for February option.

 Continuous Dividend Payments ◦ Index funds on basket of stocks (e.g. S&P 500 index): the many stocks pay out their dividends throughout the year ◦ Merton Model:  Assume continuous dividend yield k

 Strategies if anticipate that stock price will rise over period t ◦ Purchase calls ◦ Purchase stock ◦ Purchase stock and put to insure portfolio ◦ Write put  Strategies if anticipate that stock price will decline over period t ◦ Purchase puts ◦ Write calls  Covered call: purchase stock

 Want to guarantee that t periods from now you will have at least I*z ◦ z is a number generally between 0 and 1that guarantees a minimum value ◦ Want to invest in Stock with price S 0 and put on stock with exercise price X ◦ A package of share + put costs S 0 + P(S 0,X) ◦ Buy  packages where   =I/(S 0 + P(S 0,X))  Minimum $ return =  X which should be set to I*z  Pick X to guarantee that S 0 + P(S 0,X)=X/z

 Most financial models of stock prices assume that the stock’s price follows a lognormal distribution. (The logarithm of the stock’s price is normally distributed)  This implies the following relationship: P t = P 0 * exp[(μ-.5*σ 2 )*t + σ*Z*t.5 ]

◦ P 0 = Current price of stock ◦ t = Number of years in future ◦ P t = Price of stock at time t  Random Variable!! ◦ Z = A standard normal random variable with mean 0 and standard deviation 1  Random Variable!! ◦ μ = Mean percentage growth rate of stock per year expressed as a decimal ◦ σ = Standard deviation of the growth rate of stock per year expressed as a decimal. Also referred to as the annual volatility.

 Option price is the expected discounted value of the cash flows from an option on a stock having the same volatility as the stock on which the option is written and growing at the risk-free rate of interest.  The cash flows are discounted continuously at the risk-free rate  The option price does not depend on the growth rate of the stock!

 Simulate the stock price t years from now assuming that it grows at the risk-free rate r f. This implies the following relationship: P t = P 0 * exp[(r f -.5*σ 2 )*t + σ*Z*t.5 ]  Compute the cash flows from the option at expiration t years from now.  Discount the cash flow value back to time 0 by multiplying by e -rt to calculate the current value of the option.  Select the current value of the option as the output variable and perform many iterations to quantify the expected value and distribution for the option.

 Variance-Covariance Method ◦ Using an assumed distribution for the asset return (e.g. normally distributed), estimated mean, variances & covariance, compute the associated probability for the VaR  Historical Simulation ◦ Use sorted time series data to identify the percentile value associated with the desired VaR  Monte Carlo Simulation ◦ Specify probability distributions & correlations for relevant market risk factors and build a simulation model that describes the relationship between the market risk factors and the asset return. After performing iterations, identify the return that produces the desired percentile for the VaR.

 Use options to create other securities ◦ Bull spreads (written and purchased calls) ◦ Collars (stock, written call and purchased put) ◦ PPUP (Principal-Protected, upside potential: bond plus at-the-money call) ◦ Butterfly  Options can be replicated by a long or short position in the underlying stock and a long or short position in the risk-free asset (e.g. bond)