Hi, tutor, please complete this financial math worksheet with step by step solutions. You must have essential knowledge of options and futures, arbitrage,Black-Scholes model , and Brownian motion, and access to Bloomberg terminal is needed. Please answer as many questions as you can. A good tip will be offered if you can provide satisfying answer. (*Skip Question 7 and 8)

http://www.math.columbia.edu/~smirnov/options13.html,

here is the link of an option price calculator, You will need it when you do questions.

Introduction to the Mathematics of Finance. HOMEWORK 2. Due October 10, 2016 Please write a pledge that homework solutions represent your own work and that you did not copy solutions from the work of other students. 1.(10pt) European call and put on a stock both have a strike price 100 and an expiration date of 3 months. Both trade for 15$. The risk free rate is 10% per year, the current stock price is 95$ and a 5$ dividend is expected in 1 month. Is it possible to create an arbitrage using call, put, stock and cash. If yes, explain how to do the arbitrage. 2.(5pt) Explain why an American option on a stock paying continuous dividend yield is always worth as much as its intrinsic value. Give a numerical example of a situation when European option is worth less than intrinsic value. (Give the numerical value of stock price, strike price, time to expiration, etc.) 3.(5pt) Explain the European call-put parity argument. Why it can not be used for American options 4.(5pt) What is the difference between historic and implied volatilities? Which one is higher for Apple stock AAPL? Go to the Bloomberg terminal and type: AAPL HIVG . Print the output. 5.(5pt) What is the volatility smile? Using Bloomberg terminal function SKEW (CLZ6 Comdty SKEW ) plot the implied volatility smiles for TYZ6 Comdty, CLF7 Comdty, NGZ6 Comdty, ESZ6 Index, SPY Equity. What are these securities? For description type CLF7 Comdty DES. Compare their smiles. Submit printouts. (Bloomberg terminals upgrade functions from time to time. If these functions do not work ask TAs for help.) 6.(5pt) Calculate the implied volatility of Microsoft stock using November 2016 calls expiring November 18, 2016 with strike 60 and November 2016 calls expiring November 18, 2016 with strike with strike 57.5. Get the quotes from any data provider, for example, finance.yahoo.com and use all other necessary data. On Bloomberg type MSFT Equity CALL (page down as needed). 7.(10pt) Download Excel options model with VBA code from the courseworks . Save as a source. Open in Excel. Modify the Black model for futures to Black-Scholes model for stocks paying dividends at rate q (like in the Hull's book). Make the necessary changes in Visual Basic code. Use Excel help or consult TA's if you do not know what to do. Check that code works and submit the code printout. 8.(10pt) Download Excel Brownian Motion model from the courseworks. Save as a source. Open in Excel. Modify it to Geometric Brownian motion with growth rate = 0.15, volatility = 0.38 and 250 trajectories. Submit excel formulas printout. 9.(5pt) Suppose that the value X of a variable that follows a Standard Brownian Motion is initially 60. The time is measured in years. Write the probability density function of the distribution of X after 0.5 year, 1 year, 2 years, 3 years? 10.(10pt) Suppose the stock price is 60, the riskless rate is 2%. What is the price of a 1 year call struck at 60 if the volatility is 0. How would you hedge the call. Check your answer with the option calculator making volatility smaller and smaller