Introduction to the Mathematics of Finance. LAST HOMEWORK. Due December 9 1. Suppose that the spot interest rates with continuous compounding are 6.0 % For 1 yr 6.7 % For 2 yr 7.0 % For 3 yr 7.5 % For 4 yr Calculate forward interest rates for second, third, and fourth year, and for a 2 year period between years 2 and 4. (Hull p. 79) 2. Using bootstrap method (Hull p.82) calculate zero coupon yield curve from coupon bearing bonds. Principal Maturity Yr Coupon (paid semiannually) Bond Price 100 0.25 0 98 100 0.50 0 95.25 100 1.00 0 90.5 100 1.50 6 % 94 100 2.00 8.5 % 96 3. Suppose we add another bond Principal Maturity Yr Coupon (paid semiannually) Bond Price 100 2.25 0 99 Using bootstrap method and linear interpolation calculate zero coupon yield. (*)If you know other methods of approximation that are better then linear (for example splines) calculate zero coupon yield. 4. Consider a 30 year 7 % coupon bond with a face value 100. Suppose that the yield on a bond is 6 % per annum. Coupon payments are of 3.5$ per 100$ face value are made every 6 months. Calculate the bond price. 5. Calculate the above bond Price risk (sometimes called dollar duration), Value of 01 and Modified duration in years. If the yield increases to 6.01% how much the bond price goes down. 6. Calculate the convexity for the bond in problem 4. 7. Using the second order approximation calculate the approximate price for that bond at a Yield of 6.5 %. Compare it to exact price. 8. What happens to call and put greeks (Delta, Gamma, Vega, Theta, Rho) when we increase Stock price, Strike price, Time to expiration, Volatility, Interest rate, Dividends (i.e. one of these factors increases and all the rest reamain the same). Make 5 tables for put greeks and 5 tables for call greeks similar to the table on page 157 of the Hull's book. Mathematics of Finance. Another PRACTICE Final For In Class Exam This practice final is a part of the Homework and should be submitted with it. \item{\bf 1.} What is a call option. What is a put option. \item{\bf 2.} Suppose you are short call with strike 30 and long put with strike 25. What is the payoff from your position if the stock price at expiration is 24? If the stock price is 35? \item{\bf 3.} What is a butterfly spread? \item{\bf 4.} What are the parameters affecting foreign currency option price? What happens to option prices when one of these parameters changes with all the others remaining the same? Make the table. \item{\bf 5.} Write the Black-Scholes equation for dividend paying stocks. \item{\bf 6.} Write the Black-Scholes formula for call options on futures. \item{\bf 7*.} What are the parameters affecting prices European and American calls and puts on Futures. How do the prices change when one of the parameters changes with all the others remaining the same? Make a table.(Be careful!) \item{\bf 8.} Suppose that the stock price follows geometric Brownian motion $dX_t=0.1 X_t dt +0.3 X_t dW_t$. What is the distribution of the stock prices in 1 year if the current stock price is 11. \item{\bf 9.} Write the probability density function for the stock price in 1 year for the stock described in the problem 8. \item{\bf 10.} Write the probability density function for a normal distribution with mean $\mu$ and the standard deviation $\sigma$. \item{\bf 11.} What is the forward price of the dividend paying stock? \ \item{\bf 12.} What is the price of the forward contract on the dividend paying stock? \item{\bf 13.} What is the put-call parity for European options on dividend paying stocks? \item{\bf 14.} Does put-call parity holds for American options? \item{\bf 15.} Does put-call parity holds for European options but when the stock price distribution is different from log normal? \item{\bf 16.} Define $\Delta$, $\Gamma$, vega, $\rho$, $\Theta$. \item{\bf 17.} Describe delta hedging. \item{\bf 18.} Write the formula for delta of a European call on a nondividend paying stock. \item{\bf 19.} 6 months spot rate is $5\% $, 1 year spot rate is $5.5\% $. Calculate the forward rate between 6 months and 1 year. \item{\bf 20.} What is the difference between implied and historical volatilities. \item{\bf 21.} What is the difference between stocks, bonds, futures, and options. \item{\bf 22.} Why is the log-normal model for stock prices better then normal? \item{\bf 23.} What is a yield curve? \item{\bf 24.} What is a duration of a bond portfolio? \item{\bf 25.} What is a convexity of a bond portfolio \end