Intuitively, we can think of an option's value as C=E[Cr]e i.e., the option's expected payoff E[C]...
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Intuitively, we can think of an option's value as C=E[Cr]e¹¹ i.e., the option's expected payoff E[C] discounted back by the option's required rate of return (rc). To implement this equation, we need to address two issues: 1) how do we compute the option's expected payoff, and 2) what is the option's required rate of return? The purpose of this assignment is to get you thinking about the first of these issues. An option's expected payoff depends on the distribution of possible asset prices at the time the option expires. For example, consider a twomonth call option on a stock. Suppose the current stock price is $50 and the option's exercise price is $60. If there is no chance that the stock price will be greater than $60 in two months, then the option's expected payoff is zero. On the other hand, if the probability of realizing a stock price greater than $60 is large, then the option's expected payoff is large. In order to model the distribution of stock prices at expiration, we will use the following approach. (Warning: don't spend too much time studying the details until we get to the example that follows.) $ u²S S d's The time from now until the option's expiration is T. We divide this time period into N time steps so that the length of each time step is dt = T/N. We assume that the asset price follows a binomial process which means that at each time step the asset price either increases or decreases. If the price increases, the return is u = explo(dt)"} where o is the annualized volatility of the asset's returns. And if the price decreases, the return is d1/u. We will assume that the probability that the asset price increases at each step is q(b* d)/(ud) where be is the asset's cost of carry rate per time step, and we will define * = e(*) as the riskless interest rate per time step. Let's consider a specific example to see how this works. Suppose we want to use a twostep binomial tree to model the S&P 500 index during the next three months. The current index level is 3790.00, the riskless interest rate is 3%, the index dividend yield is 2% (i.e., the cost of carry rate is 3%  2% = 1%), and the volatility of S&P 500 returns is 20%. Using this information, we can now define: dt = (3/12)/2 = 0.125 U d r* = exp{0.03(0.125)} = 1.003757 b* = exp{0.01(0.125)} = 1.001251 q = (1.001251 0.931731)/(1.073271 0.931731) = 0.491167 exp{0.20(0.125¹)} = 1.073271 1/u 0.931731 = And we build the binomial tree as follows: 3790.00 3790.00 4067.70 3531.26 4067.70 3531.26 At the first step in the tree, the stock price can either increase to 3790.00(1.073271) = 4067.70 or decrease to 3790.00(0.931731) = 3531.26. If the stock price increases, then at the next step the price can either increase to 4067.70(1.073271) = 4365.74 or decrease to 4067.70(0.931731) = 3790.00. 4365.74 3790.00 3290.19 4365.74 At each step in the tree, the probability that the stock price increases is 49.1167% (this was the 9 that we calculated earlier). Therefore, the probability that the stock price will be 4365.74 in three months is 24.1245%, the probability that the stock price will be 3790.00 is 49.9844%, and the probability that the stock price will be 3290.19 is 25.8911%. We could calculate these probabilities in Excel using the function =binom.dist(#up, N, q, 0) where "#up" is the number of upsteps needed to reach that point in the tree. In this case, we need two upsteps to reach 4365.74, one upstep to reach 3790.00 (i.e., the price can go up and then down or it can go down and then up), and zero upsteps to reach 3290.19. 3790.00 Obviously, this is a very simplistic view of what can happen to the S&P 500 index level over the next three months (i.e., we are assuming that it will be either 4365.74, 3790.00, or 3290.19). But we can get a more realistic distribution if we divide the threemonth period into more time steps we just increase the N in our model. Fortunately, this is easy to do in Excel. The key is to construct the tree in the form of a right triangle and then it is easy to copy the formulas throughout the tree: 3290.19  2 At this point, you should try this on your own. Open the Binomial.xlsx file that you downloaded from Canvas. All of the input parameters for the example have been entered into the spreadsheet as well as the formulas needed to compute the parameters for the tree. All you need to do is input the number of time steps you want and then multiply the current stock price by the u's and the d's to build the tree. Note that the "Input Parameters" and "Binomial Settings" have been named (press F3 and click on "paste list" to see all the names). So, to calculate the asset price in cell C13, just type "=B13*u". Try to build the twostep tree described above to make sure you can verify my results. The Assignment Build a 10step binomial tree to model the S&P 500 index during the next three months. The current index level is 3790.00, the riskless interest rate is 3%, the index dividend yield is 2% and the index volatility rate is 20%. You can start with the same spreadsheet as before but remember to change the N under the columns labeled "Binomial Settings." When you are finished with the tree, you should have 11 possible asset prices at the end of the three month period. Now, in the column next to each asset price, compute the corresponding threemonth continuously compounded return, i.e., =ln(S7/So). In the next column, use the binom. dist() function described above to compute the probability of realizing each of these returns. You might want to add up the probabilities to make sure that they sum to one. Use your completed tree to answer the following questions: 1. Compute the expected threemonth continuously compounded return. To do this, multiply each of the returns (i.e., =ln(S7/So)) by their corresponding probabilities and then add them all up. 2. Compute the expected stock price in three months. Do this the same way as in question 1 above, except now multiply the stock prices (i.e., Sr) by the probabilities. Note: You don't need to do anything yet with the cells below row 25 (i.e., the option prices). We will complete this part of the spreadsheet after we discuss option valuation in class. The Deliverable Print your completed spreadsheet on a single page of paper. Write your name in the top corner. Write your answer for the expected threemonth return (question 1 above) below your name and then below that write your answer for the expected stock price (question 2). Upload this printout to Canvas. Finally, download "BlackScholes.xlsx" from the course page in Canvas. You don't need to do any thing with this spreadsheet; just make sure that it's on your laptop and ready to use during class.  3 INPUT PARAMETERS S cxobt X T ASSET PRICES 0 3790.00 3790.00 3825.00 0.0300 0.0100 0.2000 0.2500 EUROPEAN PUT PRICES 1 0 AMERICAN PUT PRICES 1 0 N 2 2 BINOMIAL SETTINGS zo sot bo N dt b* 3 10 0.0250 1.0321 0.9689 1.0008 1.0003 0.4960 4 4 LO 5 LO 5 LO (o (0) (0) 7 7 7 00 8 8 9 9 10 10 10 INPUT PARAMETERS S cxobt X T ASSET PRICES 0 3790.00 3790.00 3825.00 0.0300 0.0100 0.2000 0.2500 EUROPEAN CALL PRICES 1 0 AMERICAN CALL PRICES 1 0 N 2 2 BINOMIAL SETTINGS zo sot bo N dt b* 3 10 0.0250 1.0321 0.9689 1.0008 1.0003 0.4960 4 4 LO 5 LO 5 LO (o (0) (0) 7 7 7 00 8 8 9 9 10 10 10
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Sure here are the answers to your questions 1 Compute the expected three month continuously compounded return To do this we multiply each of the returns i e ln S by their corresponding probabilities a View the full answer
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Management Accounting Information for DecisionMaking and Strategy Execution
ISBN: 9780137024971
6th Edition
Authors: Anthony A. Atkinson, Robert S. Kaplan, Ella Mae Matsumura, S. Mark Young
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