Finance homework. Swaps. File is attached.

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Problem Set 5 FIN 424, Winter 2017 Robert Novy-Marx This problem set is due before the start of class on Wednesday, February 22. You should work as a study group, but every one should be involved in answering each question (this is really practice for the exams, which you will have to do solo, so please do not just divide the problems up). Each group should hand in one solution. If (and only if) you must all miss class that day, then please email your group's solutions before the start of class the day they are due. Please think of your product/deliverable as something you might give your employer. If you email solutions, I will open one and only one electronic file, and push the print button exactly once (i.e., I won't print multiple Excel sheets). If you do the problems in an Excel, please do your \"scratch work\" on back worksheets and make sure that your answers are all organized onto the first sheet. Include your names. No question is deliberately designed to trick or confuse you, or be overly complex, so assume the obvious if you feel that sufficient detail is missing. 1. (Constructing a simple implied volatility trees.) The current price of a non-dividend-paying stock is S0 D 50. The annual interest rate (simple) is r D 2%. We also observe the market option prices, c.K; T /, for three European calls on the stock (T in years): c.50; 1/ D 12:67 c.30; 2/ D 25:92 c.80; 2/ D 6:91: (a) Construct a two-year, two-step Derman-Kani (implied) tree consistent with the observed stock price, interest rate, and option prices. 1 \u000f You can solve the equations analytically if you want, but feel free to use the Excel Solver tool to solve equations if you prefer (you'll need to use the \"subject to constraint\" function within the Solver routine to deal with more than one equations). q0 q11 S22 q10 S0 S11 S0 S10 S20 (b) What is the tree-implied volatility at each node (i.e., the \"local volatilities\"), assuming the simple, annual expected return of the stock is 8%? \u000f Remember, the tree-implied volatility and annualized (continuously compounded) expected rate of return for the underlying security are given by p p.1 p/ ln SSdu \u001bloc D p t ln.p SSu C .1 p/ SSd / D t where S denotes the underlying price at the node, Su and Sd denote next period's up and down prices, respectively, and p denotes the objective probability (i.e., not the risk-neutral probability) of the up-move at the node. Solving the first of those for p yields \u0016c:c: loc c:c: e \u0016loc t D p SSu C .1 c:c: ) p D Sd S e \u0016loc t Su S c:c: p/ SSd Sd S D 1 C \u0016loc Su S Sd S Sd S where \u0016loc D e \u0016loc t is the simple one period expected stock return. 2 (c) Use the constructed tree to price a two-year ATM (i.e., K D 50) American put option. 2. (The volatility \"smile\" for Cell Therapeutics (CTIC:Nasdaq) options.) Sometimes the lognormal assumption is worse than others, and sometimes it's really bad. At the close on Friday March 4, 2005, CTIC was at $10 (its realized volatility over the previous year was 68%), the one, three and six month t-bill yields were 2.56%, 2.76% and 3.01%, respectively, and puts and calls with one, two, five and eight months to maturity were prices according to the following matrix: Strike 5 7.5 10 12.5 15 17.5 20 19-Mar Call Put 5.35 0.35 3.85 1.2 2.775 2.7 2.05 4.5 1.475 6.35 1.05 8.45 0.65 10.6 16-Apr Call Put 5.65 0.55 4.25 1.7 3.3 3.25 2.6 5 2.1 6.95 1.325 8.75 0.95 10.8 18-Jun Call Put 5.75 0.7 4.4 1.925 3.6 3.5 2.8 5.2 2.25 7.05 1.575 8.95 1.225 11.1 17-Sep-05 Call Put 5.95 0.825 4.7 2.125 3.75 3.65 3 5.35 2.45 7.25 1.875 9.15 1.475 11.2 Figure 1: Cell Therapeutics Option Prices (a) Estimate: i. The number of trading days to maturity on each of the option series. ii. The approximate annualized continuously compounded yield-tomaturity for each expiration date. (b) Calculate the implied volatilities for each put series, and plot these volatility \"smiles\" on one graph. 3 (c) How do the implied volatility vary across strike and time to maturity? i. Does it look like a smile, or a smirk? Or something else? ii. How can you reconcile these implied volatilities with the historic vol.? iii. What sort of stock price dynamics might support the volatility surface you've just produced? (d) In the \"smile\" lecture we made an ad-hoc adjustment for earnings announcements: we added a few days to the contract life (the exact number depends on the size of the firm's historical earnings announcement surprises). This was intended to adjust for the impact of the earnings announcement volatility, which added uncertainty to the returns to the underlying over the life of the contract. If the expected surprise it months) will also have a something we would like followed p return variance D D big, this adjustment (which could then be material impact on the time discounting, to avoid. The holding period adjustment q q 2 2 n\u001bNEAD C \u001bEA .n q .n v u u D t.n D D \u001bNEAD 2 2 2 1/\u001bNEAD C \u001bNEAD C \u001bEA 2 2 1/\u001bNEAD C \u001bEAD 2 1/\u001bNEAD C v u u t.n 1/ C 2 \u001bEAD 2 \u001bNEAD ! 2 \u001bEAD 2 \u001bNEAD ! \u0001 2 \u001bNEAD p That is, the \"\u001b T \" that goes in Black-Scholes used the standard deviation of non-earnings announcement day returns, but added (You must annualize these numbers, of course, before you plug them into Black-Scholes.) 4 We could, instead, have done the adjustment on the volatility: letting \u001b be the annualized normal (everyday) variance, q p 2 return variance D \u001b 2 T C \u001bEA 0s 1 2 p p \u001bEA A T D \u001bT T D @ \u001b2 C T q \u001b2 . The disadvantage of this procedure is that where \u001bT \u0011 \u001b 2 C EA T the volatility adjustment depends on the time-to-maturity; the advantage is that the adjustment doesn't throw off the time discounting implicit in Black-Scholes. The advantages outweigh the cost when the expected surprises are large. Please plot the \"pricing errors\" (model prices minus market prices) for each put series, first assuming \"normal\" volatility is 90% (i.e., \u001b D 0:9), and second assuming the 90% volatility, but that the market also expects a normally distributed log-price jump with a standard deviation of 70% (i.e., \u001bEA D 0:7). 5 3. (Forward rates and swaps) Suppose you observe the following term structure of interest rates (zero coupon bond prices, per $100 dollars of face): Maturity (years) 0.5 1 1.5 2 Zero Coupon Bond Prices 99.01 97.547 95.75 93.795 (a) What are the six-, twelve-, and 18-month ahead forward prices of the six-month rate? What about the six- and twelve-month ahead forward prices of the one-year rate? (b) What is the two year swap rate for swaps making semi-annual payments? What about the two year swap rate for swaps making annual payments? (c) What is the price of a two year receive-fixed swap making semi-annual payments on a notional of $1 million at a fixed rate of 3.5%? What about the two year pay-fixed swap making semi-annual payments on the same notional a fixed rate of 3.0%? (d) If you owned both the swaps from the previous part, what would be the net cash flows? What would the position be worth? 6