binomial model (in the context of stocks): doesn't work well

can someone kindly help me with the binomial model (in the context of stocks): on the page 36 [here]

(https://www.mathworks.com/help/releases/R2021a/pdf_doc/finance/fintbx.pdf)

in the

OptionPrice matrix

there are always 3 numbers non-zero in the 4,5 and 6th column ?

I mean, why the column 4,say, has the same number (three 3) of non-zero elements and not more, like

StockPrice matrix [it has 4] *above* it:

up-up-up,up-up-down,up-down-down,down-down-down

PS. **when are 2 paths exactly considered as the *same/different* in the binomial model** ?

Answer 1

[Govind KM]

Hi Jan,

The “OptionPrice” matrix gives the option value for each possible path in the binomial model, which is the price paid by the buyer to obtain the right to buy or sell a stock at a certain price. A negative option price would indicate getting paid to buy an option, creating arbitrage opportunities that can destabilize the market. Option prices are hence defined as the maximum of zero or the difference between the asset price and the exercise price.

In the provided example, some option values in the given periods (columns 4,5 and 6) were set as zero as the predicted option prices were negative. Considering another example, if the exercise price is changed from 95 to 70, there are four nonzero options in the 4th column of the “OptionPrice” matrix as all the predicted option values for that period were positive.

[StockPrice, OptionPrice] = binprice(100, 70, 0.10, 0.25, 0.05,  0.50, 1, 0, 5.0, 3)

StockPrice = 6×6

100.0000 111.2713 123.8732 137.9629 148.6915 166.2807 0 89.9677 100.0495 111.3211 118.8981 132.9629 0 0 80.9994 90.0175 95.0744 106.3211 0 0 0 72.9825 76.0243 85.0175 0 0 0 0 60.7913 67.9825 0 0 0 0 0 54.3608

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OptionPrice = 6×6

31.1252 41.9678 54.2223 67.9629 79.0406 96.2807 0 20.8293 30.3986 41.3211 49.2472 62.9629 0 0 11.6767 20.0175 25.4236 36.3211 0 0 0 3.6349 7.3883 15.0175 0 0 0 0 0 0 0 0 0 0 0 0

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In the binomial model, it is assumed that stock prices will either move up or down, with certain constant probabilities. The order of these movements does not affect the final stock price, so the paths of “up-down-up” and “down-up-up” are seen as equivalent. Two paths are considered the same if they have an equal number of upward and downward steps, irrespective of their order.

Hope this clarifies the issue!