Econ 136: Financial Economics Problem Set #2 – Solutions Due Date: September 13, 2018 General Instructions: • Please upload a PDF of your problem set to Gradescope by 11:00 pm. • Late homework will...

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Econ 136: Financial Economics Problem Set #2 – Solutions Due Date: September 13, 2018 General Instructions: • Please upload a PDF of your problem set to Gradescope by 11:00 pm. • Late homework will not be accepted. • Please put your name, student ID & your GSI’s name at the upper right corner of the front page. 1. A 6-month at-the-money European call option on a non-dividend-paying stock is cur- rently selling for GBP 3.79 while a put with the same terms is selling for GBP 2.33. The underlying stock price is GBP 55.00, and the risk-free rate is 5% per year. Is there an opportunity for you as an arbitrageur? If so describe the trades you would undertake and the profit you would make; if not explain why not. Solution: We begin by considering the relative prices of the portfolios that are on each side of our expression for put-call parity C + Ke−rt = S + P (1) The portfolio on the left-hand side of this expression is currently priced at C + Ke−rt = 3.79 GBP + 55e−0.05×6/12 GBP (2) = 3.79 GBP + 53.64 GBP = 57.43 GBP (3) and the portfolio on the right-hand side of this expression is currently priced at S + P = 55 GBP + 2.33 GBP = 57.33 GBP (4) So we will sell (go short) the left-hand-side portfolio, buy (go long) the right-hand- side portfolio, and make a profit as illustrated on the next page. 1 • Today: – Sell the call for 3.79 GBP – Buy the stock for 55.00 GBP – Buy the put for 2.33 GBP – Borrow the 53.54 GBP (= 55.00 + 2.33 - 3.79) needed for the purchases at the rate of 5% per year. • At expiration: – Sell the stock for 55 GBP. If the price of the stock is: ∗ Equal to 55 GBP: Sell the stock. Both options expire worthless. ∗ Less than 55 GBP: Exercise your long put to sell the stock at 55. The call expires worthless. ∗ More than 55 GBP: Your short call will be exercised against you by your counterparty who is long the call, and you will be required to sell the stock to them at 55.00 GBP. – Pay back the loan you took out to make your purchases. The loan has grown to 54.90 GBP (=53.54e+0.05×6/12: of which 53.54 is the principal that you borrowed and 1.36 is the interest accrued over the 6-months). You make this payment using a portion of the 55.00 you made on the stock sale. – You keep the difference of 0.10 GBP (= 55.00 GBP - 54.90 GBP). 2. Develop your own Black-Scholes option pricer and reproduce the Black-Scholes Excel example shown in class to demonstrate that your pricer is working (Note: reproducing a known result is called “validation”). Then (a) Generate a screen-shot of your reproduction. Solution: The screen-shot of the example from class is shown in the figure below. 2 (b) Calculate the call and put prices if the stock price increases to 110 and all other variables remain unchanged. Briefly explain the change in the call and put price. Solution: If the stock price increases to 110 the call price rises to 12.97 and the put price drops to 4.87. The increase in the call price is due to it now being in the money. Similarly, the drop in the put price follows because it is now out of the money. (c) Calculate the call price if the stock volatility decreases to 20% and all other variables remain unchanged. Briefly explain the change in the call price. Solution: The call price has decreased to 4.79. This is due to the narrowing of the probability density represented by the decrease in the volatility (the standard deviation of the return density). This narrowing has put less proba- bility mass in the positive regions of the call payoff diagram which results in an decrease in the expected payoff. Since the discount factor did not change and the call price is the present value of (or discount factor times) the expected payoff, the decrease in volatility decreases the price of the call. 3. Create an extension of the option calculator you built in part 2 of this problem set to reproduce the call diagram shown in slide 2 of lecture 5. The parameters for the options shown are a strike of 50, time to expiration of 1 year, risk-free rate of 10%/year, and volatility of 39.115%1. The red dashed curve corresponds to a time-to-expiration of one year. The solid blue line corresponds to a time-to-expiration of zero years. However, the Black-Scholes formula does not react well to zero time to expiration (you’ll get “divide by zero” errors if you try this), so use a very small time to expiration instead. I find that 1/1,000,000 years works well as a proxy for zero years. Your answer to this question is (i) a copy of your reproduction of the call diagram shown in slide 2 of lecture 5 and (ii) a screenshot of your calculation: part of your spreadsheet if you calculated this in a spreadsheet or part of your code if you did this using something like MatLab or Python. Solution: See figure on next page. 1Strictly speaking volatility should be quoted as 39.115%/year 1 2 The reason for this strange time unit is that the product σ √ t needs to be dimensionless (i.e., have no units). Since the units of √ t are the square root of time the units of σ must include the inverse of the square root of time so that these square roots cancel in the product. However, in actual practice nobody speaks of the time aspect and simply refers to the volatility as a percent. 3 Documentation of my Excel/VBA implementation is shown below. The reproduc- tion of the call diagram is shown in the graph. The option price with a year to expiration at an underlying level of 50 should be 10, and we see that the red curve has a value of 10 when the x-axis is 50 which confirms this aspect of the valuation. The highlighted cell shows that the function “mybsm” was used to generate the option price in the cell and the panel in lower right of the screen shows the code for this function. 4 4. Review the derivation of digital calls and puts in the study guide and use these results to derive a general expression for an option position that at expiration pays (i) nothing below K1, (ii) one (1) unit of currency (e.g., 1 USD) between K1 and K2, and (iii) nothing above K2. Note: digital calls have come up in job interviews and financial economics exams. It is a good idea to be familiar with these derivations. Solution: A digital call is an option that at expiration pays nothing (0) below the strike and one (1) above the strike. Similarly, a digital put is an option that at expiration pays nothing (0) above the strike and one (1) below the strike. The expressions for these option prices are: C(K) = e−rtN (d2,K) (5) and P (K) = e−rtN (−d2,K) (6) where we’ve slightly modified the notation to include the strikes. As long as your strikes are indicated in some way, any notation to this end is fine. There are two ways one can create the payoff described above. Either of the following ways is an acceptable; you didn’t need to do both. (a) Long one K1-strike digital call and short one K2-strike digital call. For this portfolio the long call provides the “zero below K1 and 1 above K1” features of the payoff and the short digital call sets the payoff function to zero above K2 by adding -1 to the +1 payoff of the K1-strike call. The expression for this option position is Position = e−rtN (d2,K1)− e−rtN (d2,K2) (7) or Position = e−rt [ N (d2,K1)−N (d2,K2) ] (8) (b) Long one K2-strike digital put and short one K1-strike digital put. For this portfolio the long put provides the “zero above K2 and 1 below K2” features of the payoff and the short digital put sets the payoff function to zero below K1 by adding -1 to the +1 payoff of the K2-strike put. Position = e−rtN (−d2,K2)− e−rtN (−d2,K1) (9) or Position = e−rt [ N (−d2,K2)−N (−d2,K1) ] (10) 5
Answered Same DaySep 13, 2021

Answer To: Econ 136: Financial Economics Problem Set #2 – Solutions Due Date: September 13, 2018 General...

Sumit answered on Sep 16 2021
129 Votes
1.
The first step of the solution is to consider the relative price of the portfolio on the either
side of the expression for put-call parity:
C + Ke-rt = S + P
The Portfolio on the left-hand side of the expression is currently priced at:
C + Ke-rt = 14.63 NOK + 125e-004 x 7/12 NOK
= 14.63 NOK + 116.65 NOK
The portfolio on the right-hand of the expression is currently priced at:
125 + 12.35
= 137.35 NOK
Hence, we will go long (buy) on the left-hand side...
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