finance question blackscholes model

Assignment 1 (10% of final result) Recall that in the Black-Scholes model, the stock price is assumed to satisfy dS, = S, dt+oS, dw, (1) where W is a standard Brownian motion. The initial time zero value of the stock price is So. 4. Suppose we measure time in years and S, actually denotes the price of a well- diversified market index. What would be typical values of u and a compatible with empirical evidence? Note: you are not asked to perform a full-fledged estimation of these parameters. 5. For fixed h > 0, what is the distribution of Seth conditional on S.? 6. Use the answer to item 5. to simulate T = 5 years of stock prices according to the Black-Scholes SDE with a frequency of h = 1/12 (i.e., a monthly frequency). For this simulation you use the parameter values from item 4. above. Use this simulation to calculate the expectation of St given So. How much does the simulated value differ from the exact (analytically calculated) value? 7. Using the same code, calculate the (approximate) variation Sin - S019h| and (approximate) quadratic variation In Sin - Sa-13h | Make a graph/table of both for h = 1, h=1/12, h = 1/252 and h = 1/2520. Describe and explain what you observe. Assignment 1 (10% of final result) Recall that in the Black-Scholes model, the stock price is assumed to satisfy dS, = S, dt+oS, dw, (1) where W is a standard Brownian motion. The initial time zero value of the stock price is So. 4. Suppose we measure time in years and S, actually denotes the price of a well- diversified market index. What would be typical values of u and a compatible with empirical evidence? Note: you are not asked to perform a full-fledged estimation of these parameters. 5. For fixed h > 0, what is the distribution of Seth conditional on S.? 6. Use the answer to item 5. to simulate T = 5 years of stock prices according to the Black-Scholes SDE with a frequency of h = 1/12 (i.e., a monthly frequency). For this simulation you use the parameter values from item 4. above. Use this simulation to calculate the expectation of St given So. How much does the simulated value differ from the exact (analytically calculated) value? 7. Using the same code, calculate the (approximate) variation Sin - S019h| and (approximate) quadratic variation In Sin - Sa-13h | Make a graph/table of both for h = 1, h=1/12, h = 1/252 and h = 1/2520. Describe and explain what you observe