The Binomial-tree Option Calculator



The Option Strategy Calculator

This is a short documentation of how to use the php-program for using the binomial method and Black-Scholes for calculations on strategies with options. The binomial methods used in the calculations are the well-known Cox-Ross-Rubinstein's binomial model and a few others. The value of Vega and Rho are scaled to show the change of the option value when the value of the volatility and the risk-free interest rate will change by one percent. The value of Theta is scaled to show the change of the option value if the time to maturity is changed by one day.

Input data

First of all you have to fill in the form with the needed input data. This is asset data:

• The underlying asset price,

• The risk free interest rate,

• The volatility and

• The number of stocks

and option data:

• The option strike price,

• Time to maturity,

• The number of options

• The exercise type (American or European)

• Option type (Call or Put)

• The number of binomial steps (if American type).

If you want to use Black-Scholes for American Call options, click the checkbox at the top.

When all the input data is given you must select (if your strategy includes American options):

• The type of binomial tree to use.

Calculate

Now, you can start the calculation by pressing Calculate.

The button Clear, clears all the input data and let you start from the beginning.

Numerical Methods

A great advantages of this program, is that you easily can change the type of tree to use in the calculations. This gives you the possibility to investigate the different results given by the models. You can also create plots to examine and compare how the tree-models converge. The graphical possibilities are described below.

The trees

You can select between the following trees:

• CRR – Cox, Ross Rubenstein [1979] tree,

• CRR 2 – Cox, Ross Rubenstein tree, (a modified CRR model)

• JR – Jarrow Rudd [1983] tree,

• TIAN – Tian [1993] tree,

• TRG – Trigeorgis [1991] tree,

• LR – Leisen Reimer[1995] tree and

Since this is NOT a document with the aim to give a full description of the binomial method, I refer the reader to the literature. I give some of my references in the end of this document. But, I will give a shot description of the parameters used in the trees.

All trees are built from four parameters (u, d, p and q). The parameters u and d tell how much the underlying will go up or down in each discrete time, and the parameters p and q is the probabilities for the price to go up and down respectively. Therefore p + q = 1.

CRR model

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CRR-2 model

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JR model

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TIAN model

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TRG model

In the TRG model the logarithm of the price (instead of the price itself) is used to build the tree. Therefore, when the tree is build we add u = dx and subtract by d = -dx.

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LR model

This model differs from the other methods in a common sense. The model converges much faster than the other ones and do not oscillate. For the reader I suggest the article by Leisen and Reimer (see references). But a short explanation is given here:

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Where B is the inverse of the binomial distribution and N the number of refinements. We use the Peizer-Pratt method [case: j + ½ = n – (j + ½), n = 2j + 1]:

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Then we have

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Graphics

The most important functions in the program are the graphical possibilities. Below the input/output form there are a number of buttons that will give you a number of graphs.

• Value

• Delta

• Gamma

• Theta

• Vega

• Rho

• Multi Value

Each of these buttons creates different graphs, the given property as function of the underlying value, as function of time and as function of the volatility. If the checkbox “Print input data in the plots”, is checked, a box with the input values is presented in the upper right corner. With Multi Value you get a plot of all options and the stock and the total value in the same graph.

A printable result

With the button Printable you get a printable version of the calculation result. This can be printed or copied and pasted into a document etc.

Hedging

Two types of hedging can be calculated with the strategy analyzer: Delta-Hedging and Delta-Gamma-Hedging.

Delta Hedging

With Delta-Hedging the program calculates the number of options needed to be delta neutral. To use this function, first input all data for the underlying stock (price, volatility and the number of stocks you want to hedge) and the risk free interest rate. Then you investigate the market of options and input one, two, three or four different options. You need to fill in the strike price, the time to maturity, the exercise and option type. Then press the Delta-Hedge button. The results are displayed as Number: which means the number of Option 1, 2, 3 or 4 needed to be delta neutral. Observe, the numbers don’t need to be integers.

Delta-Gamma Hedging

With Delta-Gamma-Hedging the program calculates the number of options needed to be both delta and gamma neutral. To be neutral in both delta and gamma you need to buy two different options. To use this function, first input all data for the underlying stock (price, volatility and the number of stocks you want to hedge) and the risk free interest rate. Then you investigate the market of options and input two different options. You need to fill in the strike price, the time to maturity, the exercise and option type. Then press the Delta-Gamma-Hedge button. The results are displayed as Number: which means the number of Option 1and 2 needed to be delta and gamma neutral. Observe, the numbers don’t need to be integers.

Example of delta-gamma hedging:

You need to hedge 1000 shares of a stock at the price of $35. The risk free interest rate is 4.5% and the volatility 37.5 %. You find two options on the market, one with strike $37 and another with strike $30, both with 102 days to maturity. They are American type. Therefore try to use put options with the strike $37 and call options with strike $30. With this data, press the Delta-Gamma-Hedge button. The result given is to buy 551 put options and to sell 843 call options.

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Press the button Value and you see the curve:

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As you can see, the total value of your portfolio is constant for Asset Prices in the interval $30 - $40. You are HEDGED!!

The cost of this hedge is -$3084 (an income) + courtage. The risk is that the call options will be exercised. Instead you can switch the option types and recalculate.

This gives:

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and

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As you can see, the total value of your portfolio is constant for Asset Prices in the interval $30 - $40. You are HEDGED!!

The cost of this hedge is -$1514 (an income) + courtage. The risk is that the call options will be exercised.

References

[1] Cox, J., Ross, S.A., Rubenstein M. (1979): “Option Pricing: A simplified

Approach”, Journal of Financial Economics 7, 1979, pp. 145-166.

[2] Hull J. “Option Futures and other Derivatives” Prentice-Hall, New Jersey.

[3] Jarrow, R. Rudd A.(1983): “Option Pricing”. Homewood, Illinois 1983,

pp. 183-188.

[4] Pratt, J. W. (1968): “A Normal Approximation for Binomial, F, Beta, and

Other Common, Related Tail Probabilities, II”, The Journal of the American

Statistical Association, Bd. 63. 1968, pp. 1457-1483.

[5] Tian, Y. (1993): “A Modified Lattice Approach to Option Pricing”, Journal

of Futures Markets, Vol 13, No. 5, pp. 564-577.

[6] Trigeorgis, Lenos (1991): “A Log-transformed Binomial Numerical Analysis

for Valuing Complex Multi-Option Investments”, Journal of Financial and

Quantitative Analysis 26, No. 3, September 1991, pp. 309-326.

[7] Leisen, D., Reimer, M. (1996): ”Binomial Models for Option Valuation –

Examine and Improving Convergence”. Applied Mathematical Finance, vol. 3

1996, pp. 319-346.

[8] Leisen, D., (1998): “Pricing the American put option: A detailed convergence

analysis for binomial models”. Journal of Economic Dynamics and Control.

22 (1998), pp. 1419-1444.

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