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Chapter 13 – Boot Strap Method

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Boot Strapping It is a computer simulation to generate random numbers from a sample. In Excel, it can simulate 5000 different scenarios of possible closing prices of the stock when the option expires. The average present value of these 5000 scenarios determines the fair price for the option.

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Example of a computer simulation Suppose you are flipping a coin. The probability of getting heads is 50%. Simulate this process 20 times using Excel. Use Excel and follow the steps on page 215-216.

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Example of a Random Selection Make a list of 10 names. Use Randbetween and Index commands to randomly choose one person from the list. Use Excel and follow the steps on page 217.

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Fair Price It is a price to charge so that the expected value for both the buyer and seller is zero. Do #6. Find the fair price in playing the game?

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Focus on the Project: Fair Price of a stock option E.g. Suppose the strike price for a stock option is $32. A random process simulated the closing prices as follows: $30, $29, $33,$32, $34, $27, $37, $39, $29, $30 What would be a fair price for the stock option?

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Focus on the Project: Fair Price of a stock option The value of the option for each simulated closing prices: $0, $0, $3, $0, $2, $0, $5, $4, $0, $0 The mean of these values is 1.4 Hence, a fair price for this option is $1.40. A fair price of a stock option is the mean of the value of all options.

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Focus on the Project: Future value of a stock The future value of a stock is as random as the weekly ratio is. We can also simulate this as well. For instance, a stock’s current price is $20.87 and we randomly draw the weekly ratio to be 1.04. We can simulate a 4% increase next week. So the price of the stock next week is (20.87)(1.04)=21.7

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Focus on the Project: Future Value of a stock Suppose the option expires in 20 weeks, we can continue this process by randomly choosing 20 ratios and multiplying them to the starting price to simulate a possible price 20 weeks from now.

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Focus on the Project: If the closing price is greater than the strike price, the stock option is in-the- money, and the future value of the option is C-S. If the closing price is less than or equal to the strike price, it is out-of-the- money, and the future value of the option is $0.

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Example: #8 Suppose the current price of a stock is $45.90. You hold a 4-week stock option with a strike price of $48.00 and ratios of 1.01, 0.99, 0.976, 1.08 Find the future value and determine if the option is in-the-money or out-of-the- money.

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Future value = $45.9(1.01)(0.99)(0.976)(1.08)=48.38 Since this is more than the strike price of $48, then it is in-the-money and the value of the stock option is $0.38

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Boot Strap: Bootstrap is a statistical method for estimating the distribution of a sample by resampling with replacement from the original sample. In our project, our original sample is our normalized weekly ratios. We use Excel to randomly select these normalized weekly ratios with replacement from original sample. These randomly selected normalized ratios will represent the weekly ratios for the next n weeks until our stock option expires.

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Focus on the Project: after bootstrapping…. We are going to collect 5000 samples of size n. For each sample, we can compute the closing price by using the formula closing price=starting price*r1*r2*r3…*rn We can find the future value of our stock option based on the future closing price computed above. We can find the present value of our stock option. The fair price of the option is the average of these 5000 present values of our stock option.

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Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 14.1 Value at Risk Chapter 14.

Options, Futures, and Other Derivatives, 4th edition © 1999 by John C. Hull 14.1 Value at Risk Chapter 14.

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