Computing the right price of futures as a function of a strike price

Gabrielyan

2009-08-18

+1 818 640 0044

+1 818 676 6933

+41 21 550 0309

Let us have an operation price probability, a subject to normal Gaussian distribution (see normal distribution):

The following chart shows such distribution, where the mean value is equal to $4’000 and the variance is equal to $1’000:

[xls]

We seek to compute the ‘right’ price of futures (issued for our operation price) as a function of the strike price a. Such futures refund us each $1 of operation price exceeding the selected strike price a.

The statistical ‘margin-less’ cost of such futures must be equal to:

The cost computed in such a way contains neither a security nor a beneficiary margin of the issuer of futures.

Using the above formula the joined Excel file computes the cost of futures straight-forwardly by summing refund claims weighted by their probabilities. We use the Excel array formulas, as shown below. You must not type the curly brackets, they will appear when you press Ctrl-Shift-Enter, instead of Enter. The formula assumes that the x values of price are in the column B and the probabilities are in column C. For each price exceeding the strike price, the formula takes the difference between the price and the strike-price, multiplies it by the probability of the price, and adds all of them. The final sum is multiplied by the distance between the sample points on the x axis.

{=SUM((B5:B$500-B5)*C5:C$500)*(B$3-B$2)}

[xls]

Our objective is to find the analytical formula of the price of futures (and if not analytical then one using well known functions, such as the error function).

Due to the symmetric bell shape of the normal distribution:

Therefore:

CDF is the cumulative distribution function [see normal distribution]. It represents the probability that the price will fall below x (its argument).

[xls]

Considering the product rule (see derivative):

We can write that:

Therefore:

CDF can be expressed via error function as follows (see normal distribution):

ERF function is implemented in Excel so no approximations or simulations will be needed. By default ERF could be inactive in the Excel. It is necessary to enable the Analysis Tool Pak Add-In in order to use it.

It is known that (see list of integrals):

Taking into account the chain rule (see derivative):

The CDF integral is computed as follows:

Its Excel formula is shown below assuming that the B column contains the x values:

=1/2*(B2-mu)+1/2*sigma*2^0.5*(EXP(-(((B2-mu)/(sigma*2^0.5))^2))/PI()^0.5+ABS(B2-mu)/(sigma*2^0.5)*ERF(ABS(B2-mu)/(sigma*2^0.5)))

[xls]

The ‘margin-less’ price of futures can be therefore expressed as follows:

The excel formula of the futures is shown below, assuming that the column B contains the x values.

=-1/2*(B2-mu)+1/2*(sigma*(2/PI())^0.5*EXP(-(((B2-mu)/(sigma*2^0.5))^2))+ABS(B2-mu)*ERF(ABS(B2-mu)/(sigma*2^0.5)))

[xls]

The following chart shows an animation of the price of futures as a function of the strike price where the variance, i.e. the volatility, changes from $100 to $6500 over the time. The mean value is always the same and is equal to $4000: