# Black scholes model

PART A Q1 Use Black scholes model to price the following a. A call option on a stock whose current price is 50 with an exercise price X = 50, T=0.5, r=10% and standard deviation= 25%. b. A put option with the same parameters. Q2 Use the data in Q1 and DATA/TABLE to produce graphs to show: a. The sensitivity of the Black Scholes put price to changes in standard deviation. Discuss your results in relation to option theory. b. The sensitivity of the Black Scholes call price to changes in the time to maturity T. Discuss your results in relation to option theory. c. The sensitivity of the Black-Scholes call price to changes in the interest rate T. Discuss your results in relation to option theory. Q3. Produce a graph comparing a call’s intrinsic value [defined as max (S-X, 0)] and its Black Scholes price. From the graph you should be able to deduce that it is never optimal to exercise early a call priced by the Black-Scholes formula. Discuss your results on relation to option theory. Q4. Produce a graph comparing a put’s intrinsic value[defined as (max(X-S, 0)] and its Black-Scholes price. From the graph you should be able to deduce that it may be optimal to exercise early a put priced by the Black-Scholes formula. Discuss your results in relation to option theory. PART B Using data from YAHOO finance calculate the volatility of the S&P500 for the period Jaunary 2006 to December 2015. Present your results in graphical and tabular form. Discuss your findings in a clear manner. Compare your volatility results with an alternative measure of volatility. Discuss how your estimated results compared to your chosen alternative measure. Reference: John Hull, Options, Futures and Other Derivatives, 8th edition Pearson