Useful Formulas and Reference. cumulative standard normal prob. function
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1 Useful Formulas and Reference Black-Scholes Value of a European Call Option VCt, N d1 PN d2 PV EX Black-Scholes Value of a European Put Option VPt, N d2 PV EX N d1 P Notes ln d1 T P PV ( EX ) T 2 d d T 2 1 N d cumulative standard normal prob. function GBM Risk-Neutral Parameters dt r f 1 2 dt 2 dt Excel Functions Goal seek: Solver Sum Average Product Mmult Minverse Transpose Data What if Analysis 2 nd choice Data Solver Adds a column or row of numbers Averages a column or row of numbers Multiplies a column or row of numbers Multiplies two matrices and/or vectors Inverses a square matrix Transposes a matrix or vector 3
2 1. Short Answer Questions (35 min; 35 points in total) A. (10 min; 10 points) Below, we give you the riskless forward rate curve (all 1-yr forward rates). Please give us the riskless spot rate curve by filling-in the spaces below. Write answers as percentages (%) with two decimal places. Data is located in the associated worksheet. We have filled-in some spot rates for you. Maturity (Years from Today) 1yr Fwd Rate Spot Rate % 1.00% % % 2.00% % % 2.35% % % 2.36% % % 2.25% % B. (5 min; 5 points) Plot the spot rate curve from above. We have already plotted the forward rate curve for your reference. 4.00% 3.00% Rate 2.00% 1.00% 0.00% Maturity 4
3 C. (10 min; 10 points) Using the spot rate curve from above, solve for the coupon rate of a 4-year bond that is priced at par. This is a standard Eurobond that makes annual payments. Please write your answer as a percentage (%) with two decimal places. Coupon Rate D. (10 min; 10 points) Find the all the solutions (or roots ) of following equation: Y X X X X By solutions we want you to find all values of X such that Y = 0. Since this is a 4 th degree polynomial, there are a maximum of four solutions. Please write your answers as numbers with two decimal places. Solution #1 Solution #2 (if necessary) Solution #3 (if necessary) Solution #4 (if necessary) 5
4 2. Options and Volatilities (35 min; 35 points in total) A. (5 min; 5 points) What are the three ways of pricing options that we covered in this course? i. ii. iii. Consider the following information for pricing options on XYZ stock. Information about exercise price(s) and volatilities will be given later. Current XYZ stock price Riskfree rate (CCR) 5.00% Time to maturity 9 months Dividend rate 0.00% Consider a given type of option (call or put) and given exercise price. Suppose you can see that option s price in the market. Given the information above, and the option s price, you should be able to back-solve for the volatility. This number is called the implied volatility of the option. B. (20 min; 20 points) The following table lists call and put option prices at different strike prices. Read one row at a time going across. Fill-in the missing implied volatility numbers. Write your answers as percentages (%) with two decimal places. Exercise Price Call Price Implied Volatility from Call Price Put Price Implied Volatility from Put Price % % % % % % % % 6
5 C. (5 min; 5 points) Please plot the implied volatilities of the call options. Draw a line connecting the points and mark the line clearly. Please plot the implied volatilities for the puts. Mark the line clearly. 35.0% 32.5% Implied Volatility 30.0% 27.5% 25.0% 22.5% 20.0% Exercise Price D. (5 min; 5 points) Please suggest two economic reasons to explain the shapes you observed in the plot above? i) ii) 7
6 3. Securitized Mortgages (40 min; 40 points in total) Below is the 1-yr spot rate tree. It shows today s 1-yr rate (3.00%) and possible rates in yr-1 and yr-2. The up-down probabilities are 50/50 throughout the tree % 5.46% 8.10% 4.56% 7.00% 6.05% A. (5 min; 5 points) Show the possible prices of a 3-yr zero coupon bond that pays $1 at maturity. We have started the tree off for you. Please write prices as numbers with four (4) decimals B. (3 min; 3 points) Show the possible prices of a 2-yr zero coupon bond that pays $1 at maturity. We have started the tree off for you. Please write prices as numbers with four (4) decimals C. (2 min; 2 points) Solve for today s price of a 1-yr zero coupon bond that pays $1 at maturity. We have started the tree off for you. Please write prices as numbers with four (4) decimals
7 D. (10 min; 10 points) You are considering buying a pool of existing mortgages. The pool currently has a balance (principal) of $15,000 and 3 yrs until maturity. You can treat the pool as one big mortgage-style loan for this problem. 1 If the pool is paying a rate of 3.75%, fill-out the loan amortization schedule below. Write numbers as dollars with zero (0) decimal places. Initial Total Ending Year Balance Interest Principal PMT Balance 0 15, , Any payments from the mortgage pool are divided into three tranches that pay sequentially. Each tranche currently has a balance of $5,000. Tranche A received the first $5,000 of principal, Tranche B receives the next $5,000 of principal, Tranche C receives the last $5,000 of principal, and all three tranches receive interest on their outstanding principal each period (at the same rate used above). E. (10 min; 10 points) Fill in the scheduled payments to Tranche B (there are no prepayments) Initial Total Ending Year Balance Interest Principal PMT Balance 0 5, , F. (5 min; 5 points) Given the spot rate tree from above, what is the value of the whole mortgage pool? Write the value as dollars with zero (0) decimals. Value of pool G. (5 min; 5 points) What are the values of tranches A, B, and C? Tranche A: Tranche B: Tranche C: 1 In real life, pools typically have values of $15,000,000 or more. We are using less numbers to save you writing extra digits during the exam. 9
8 4. Delta Hedging (55 min; 40 points in total) Consider an at-the-money European call option and stock as described below: Current stock price = Exercise price = Riskfree rate (CCR) = 5.00% Time to maturity (yrs) = 1.00 Volatility = 15.00% Dividend rate = 0.00% A. (10 min; 10 points) What is the Black-Scholes price for this option? Call value B. (5 min; 5 points) The seller of the option is exposed to the risk. Ultimately, the stock price may end above the strike price. We know the seller can hedge the option by holding a certain portfolio comprised of stock shares ( delta ) and riskless bonds. The hedging strategy must be dynamic in the sense that the seller must rebalance the portfolio over time. According to the Black-Scholes formula, N(d1) is the delta of a European call option. What is the initial delta for a call option shown above? How much does the seller need to initially borrow to finance his portfolio? Number of shares (Delta) Value of stock position ($) Value of bond position ($) 10
9 C. (10 min; 10 points) Suppose the seller rebalances the portfolio quarterly. We have given you stock prices representing a possible stock price path that might happen over the next nine months. How many shares (Delta) does the seller need on each rebalancing date? How much trading does the seller need to do over the nine month interval? Define share volume traded as the absolute value of changes in shares held between any two dates. Time Stock Price Shares Held (Delta) Share Volume Traded 0.00 (today) N/A D. (5 min; 5 points) What was the total volume of trading generated by the option s seller in the example above? Total volume is given in shares and equals the initial position plus the sum of the quarterly share volumes. Total volume (shares) E. (15 min; 5 points; Save until the end) Use the Monte Carlo Simulation to estimate the amount of trading an option seller can expect to do over nine months. Expected trading (shares) F. (10 min; 5 points; Save until the end) If the seller rebalances weekly (39 times over the nine months) what is the expected amount of trading? Expected trading (shares) 11
10 5. LAD Regressions (30 min; 15 points in total; Save until the end) A. (10 min; 5 points; Save until the end) Estimate a fund s alpha and beta using a simple market model (CAPM). For this estimation, we would like you to use Solver and minimize sum of squared errors (SSE). Data is in the associated worksheet. r r r r i, t f, t m, t f, t i, t Alpha Beta B. (5 min; 2 points; Save until the end) Plot your best fit regression line from above. We have already plotted the X-Y data for you. 20% 16% 12% 8% 4% 0% -20% -16% -12% -8% -4% 0% -4% 4% 8% 12% 16% 20% -8% -12% -16% -20% C. (5 min; 2 points; Save until the end) Define the following averages: Y bar = average(y t ) X bar = average(x t ) Please calculate the following two quantities using your coefficient estimates from above. Y bar Alpha + Beta X bar 12
11 C. (5 min; 3 points; Save until the end) Re-estimate a fund s alpha and beta using a simple market model (CAPM). For this estimation, we would like you to use Solver again. However, for this part, we would like you minimize LAD or Least Absolute Deviation. The absolute deviation of one observation is defined as: Y t - Y t (fitted) Alpha(LAD) Beta(LAD) D. (5 min; 3 points; Save until the end) Please give an economic reason why Alpha(LAD) and Beta(LAD) coefficients are different from the typical/sse regression coefficients: 13
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