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Construct a three-stage tree to model the growth of the asset over a threemonth period. in .NET Receive Code 3 of 9 in .NET Construct a three-stage tree to model the growth of the asset over a threemonth period.




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Construct a three-stage tree to model the growth of the asset over a threemonth period. using vs .net toembed 3 of 9 barcode with asp.net web,windows application interleaved 2 of 5 Solution Consider Figur e 7.16. Observe: (1) We could use different u and d values in each time period.

In this way, we could model changing market conditions. However, in this book we will keep the same u, d values throughout the tree. (2) In this model, an up move followed by a down move gives the same asset value as a down move followed by an up move.

When this happens, we have what is known as a recombining tree. The recombining property is more for computational ef ciency than for realistic asset modelling..

Example 6 Today, the ass et is priced at $10. Let u = 1.03, d = 0.

99. Suppose that, in the real world, up and down moves are equally likely. So prob(up) = 0.

5,. Option pricing 5x1.1x1.1x1.

1= 6.66 5x1.1x1.

1=6.05 5x1.1=5.

5 5 5x0.95=4.75 5x0.

95x0.95=4.51 5x0.

95x0.95x0.95=4.

29 5x1.1x0.95=5.

23 5x1.1x0.95x0.

95=4.96 5x1.1x1.

1x0.95=5.75.

prob(down) = 0 ANSI/AIM Code 39 for .NET .5.

Consider a model with 100 time periods. We simulate one possible route through the multi-stage tree. In Figure 7.

17 we show the Excel spreadsheet and the graph of the simulated asset values. This is beginning to look like the history of a share price. By varying u and d, the reader could generate different asset price histories.

. Financial Products To calculate o 39 barcode for .NET ption values from a multi-stage tree The method is an extension of that described above. Procedure (1) Decide how many time periods (changes in asset value) are to occur before maturity.

(2) Starting with today s value for the asset, sweep out a tree with the number of time periods agreed in (1). For example: three periods will give the tree in Figure 7.18.

In this way, we sweep forward from today through to maturity, calculating, at each vertex, a value of the asset. (3) Calculate the interest rate multiplying factor for each time period. This will be of the form 1 + r for discrete compounding and e Rt (where R is the annual rate and t is the time period) for continuous compounding.

(4) Use the interest rate multiplying factor together with u and d to calculate, for each one-stage tree, the risk neutral probabilities {q, 1 q}. (5) At maturity, calculate the pay-off from the option for each possible nal value of the asset. The pay-off is the value of the option at that vertex of the tree.

(6) Let V be a vertex one time period back from maturity. Originating from V is a one-stage tree terminating at maturity. Use the known option values at maturity (X, Y), the risk neutral probabilities and the interest rate multiplying factor for this tree to calculate the value of the option at V (Figure 7.

19). Repeat for all vertices one time period back from maturity. Now, we know the value of the option both at maturity and one time period back from maturity.

(7) Let V be a vertex two time periods back from maturity. Originating at V is a one-stage tree where, at both end points, the option value (X and Y) is known. The interest rate multiplying factor and the risk neutral probabilities are known.

Use Figure 7.19 to calculate the option value at V. In this way, calculate the option value at all vertices two time periods back from maturity.

(8) Repeat this method, working backwards through the tree, moving back one time period at each stage, until the option value for the rst vertex (today) has been calculated. Each vertex of the tree will now have both an asset value and an option value. The option value today is the price we require.

Now we illustrate the method. We proceed as before, rst numerically and then, to nd out what is actually happening in the process, algebraically..

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