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Example 12.8 Simulating Stock Price and Options

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17 Background Information n A share of AnTech stock currently sells for $42. n A European call option with an expiration date of 6 months and an exercise price of $40 is available. n The stock has an annual standard deviation of 20%. n The stock price has tended to increase at a mean rate of 15% per year. The risk-free rate is 10% per year. n What is a fair price for this option?

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17 European Options n A European option on a stock gives the owner of the option the right to buy (if the option is a call option) or sell (if the option is a put option) one share of a stock on a particular date for a particular price. n The date on which the option must be used is called the expiration date. n Cox et al. derived a method for pricing options. Their model states that the price of an option must be the expected discounted value of the cash flows from an option on a stock having the same standard as the stock on which the option is written and growing at the risk-free rate of interest.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17 ANTECH1.XLS n According to Cox et al. we need to know the mean of the cash flow from this option, discounted to the present time (time 0), assuming that the stock price increases at the risk-free rate. n Therefore, we will simulate many 6-moth periods, each time finding the discounted cash flow of the option. n The average of these discounted cash flows represents an estimate of the true mean. n The spreadsheet model is quite simple. This file contains the setup for the model.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17 Developing the Spreadsheet Model n The model can be formed with the following steps: –Inputs. Enter the inputs in the shaded cells. Note that the expiration date is expressed in years. Also note that we enter the mean growth rate of the stock in cell B6. However, this value is not used in the model. –Simulated stock price at exercise date. To simulate the stock price in 6 months we enter the formula =B4*EXP((B8-.5*B7^2)*B9+B7*RISKNORMAL(0,1)*SQRT(B9)) in cell B12.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17 Developing the Spreadsheet Model -- continued –Cash flow from option. Calculate the cash flow from the option by entering the formula =MAX(FutPrice-ExerPrice,0) in cell B13. This says that if the value in B12 is greater than the value in cell B5, we make the difference; otherwise, we make nothing. –Discount the cash flow. Discount the cash flow in cell B14 with the formula =RISKOUTPUT ( ) + EXP(- Duration*RFRate)*OptCFlow This represents the net present value of cash flow (if any) realized at the expiration date. Because the price of the option will be the average of this discounted value, it must be designated as an @Risk output cell.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17 Developing the Spreadsheet Model -- continued –Average of output cell. We might as well take advantage of @Risks RISKMEAN function to get the eventual price of the option on the spreadsheet itself. To do this, enter the formula =RISKMEAN(DiscVal) in cell B16.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17 Using @Risk n Set the number of iterations to 10,000, and set the number of simulations to 1. n After running @Risk, the value of $4.76 appears in cell B16. It turns out that this is the exact price of the option (using the formula) so the simulation got it exactly right! n We recognize, however, that the simulated mean might not be exactly equal to the true mean. Therefore, we calculate a 95% confidence interval for the true mean in row 19.

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12.112.1 | 12.2 | 12.3 | 12.4 | 12.5 | 12.6 |12.7 | 12.9 | 12.10 | 12.11 | 12.12 | 12.13 | 12.14 | 12.15 | 12.16 | 12.1712.212.312.412.512.612.712.912.10 12.1112.1212.1312.1412.1512.1612.17 Using @Risk -- continued n To do this, we first enter the formula in cell B18. n This standard deviation indicates the variability of the discounters cash flow in the 10,000 iterations. n Then we go out 1.96 standard errors on each side of the mean to form the confidence interval in row 19, where the standard error is the standard deviation in cell B18 divided by the square root of 10,000. n Based on the simulation, we cannot be absolutely sure of the option price, but we are 95% confident that it is between $4.66 and $5.06.

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Learning Objectives for Section 3.2

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