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Unformatted text preview: Notes 6 Finance 5108 Learning Objectives 1. Understand the returns and distribution of stock returns and how they tie into the development of the Black-Scholes-Merton options pricing model. 2. Understand the basic assumptions and the derivation of the Black-Scholes- Merton Stochastic differential equation. 3. Learn to calculate the price of an option using the Black-Scholes formula. 4. Understand what implied volatility is and why it is used in the pricing of options. 5. Understand how the Black-Scholes could be modified to handle warrants and American Options with discrete dividends. 6. Apply the options pricing techniques to the a continuous payment of dividends Stock price are log normally distributed due to the fact that the price of stock cannot fall below zero as a price. From the results in chapters that we have not covered the distribution of the stock price is lognormal. The result follows that the 2 ln ln S + - T, 2 T S T This is just a mathematical representation that the stock price at any time has a mean value of the first term 2 ln S + - T 2 and has a standard deviation of T . For instance a stock that is currently trading at 100 per share and has an annual expected return of 10% and a volatility of 20%, the stock price should be [ ] 2 .2 ln ln 100 + .10 - 1, .2 1 2 or ln 4.685,.2 T T S S This means that assuming we wanted a 99% confidence interval the in of the price would be 4.685 plus or minus .2*2.33 or the range in value would be from 4.219 to 5.15. Taking the anti natural log (e) of these numbers, the range of the possible stock price is from 67.97 to 172.43. There is a 99% probability that the price in one year would be between these numbers. This implied that the expected value of the price is S e ut and ( 29 2 T 2 2 T T Var(S ) e 1 S e =- Using the stock above that would be S T = 100 e .1X1 = 110.52 with an expected variance of Var = 100 2 e 2x.1x1 (e .2x.2x1-1) = 498.46 or 22.33 for the standard deviation The return on the stock can be obtained by looking at the above equation and solving for the return we get T 2 1 S = ln with a distribution of T S x- , 2 x T Volatility The estimation of volatility using historical data can be accomplished using standard statistical techniques. Concepts underlying the Black-Scholes-Merton The Black-Scholes-Merton model is a stochastic differential equation that is based on a European call option on a non dividend paying stock. They establish a riskless portfolio through an instantaneous hedging system....
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