Introduction to the Mathematics of Finance. HOMEWORK 2. Due March 15, XXXXXXXXXX59pm Please write a pledge that homework solutions represent your own work and that you did not copy solutions from the...

Please complete 1-6 and 9. 7 in total


Introduction to the Mathematics of Finance. HOMEWORK 2. Due March 15, 2021 11.59pm 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) Describe in one sentence each of the smiles in file HW2smiles.pdf (Which is symmetric, more upward sloped on the left or on the right.) 2.(10pt) Describe in one sentence volatility term structure for SPY in file HW2smiles.pdf 3.(20pt) Suppose the stock price is 50, the risk-free rate is 2% continuously compounded. What is the price of a 1 year call struck at 50 if the volatility is 0. How would you hedge the call. Check your answer with the option calculator making volatility smaller and smaller. 4.(20pt) 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.) 5.(10pt) Explain the European call-put parity argument. Why it can not be used for American options 6.(20pt) Calculate the implied volatility of Microsoft stock using calls expiring March 19, 2021 with strike 235 and with strike 240. Get the quotes from any data provider, for example, finance.yahoo.com and use all other necessary data. 7.(20pt) 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.(20pt) Download Excel Brownian Motion model from the courseworks. Save as a source. Open in Excel. Modify it to Geometric Brownian motion with growth rate µ = 0.10, volatility σ = 0.21 and 250 trajectories. Submit excel formulas printout. Geometric Brownian motion starts with positive Xo so you must change a starting value from 0 to a positive number that you may choose (you can choose 1, 100 or other positive number). 9.(20pt) Suppose that the value X of a variable that follows a Standard Brownian Motion is initially 20. The time is measured in years. Write the probability density function of the distribution of X after 0.5 year, 1 year, 2 years, 4years and 5 years? What is the probability of variable after 1 year to be higher than 20. We usually defined Standard Brownian motion to start at Xo=0 but you can start at any Xo , it is equivalent to adding constant Xo to all trajectories of the zero start Standard Brownian Motion. SPYEquitySKEW(Smileforop5onsonSP500ETFSPY) QQQEquitySKEW(Smileforop4onsonNasdaqETFQQQ) Atthemoneyop4onsonSPYvola4litytermstructure (Smilesforop4onsonAAPLstockon3differentdatesinthepast) NGJ21COMDTYSKEW(Smileforop4onsonApril2021NaturalGasfuture) CLJ1COMDTYSKEW(Smileforop4onsonApril2021CrudeOilfuture) NQH1INDEXSKEW(Smileforop4onsonMarch2021e-miniNasdaqfuture) GCJ1COMDTYSKEW(Smileforop4onsonApril2021Goldfuture)