Probability density function of insured prices and futures with right prices

Gabrielyan

2009-08-19

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In [090817 ii] we developed the probability density function of an operation price insured by futures issued on this price. This density is also a function of selected strike price a and of insurance ratio k (representing the refund ratio of price units exceeding the strike price a, or otherwise the amount of purchased futures). In the document we assumed the cost of futures c as an input parameter. No formula was provided for computing the cost of futures.

In [090819 ii] we developed a formula for computing the ‘right’ cost of futures for a given strike price a. Under term ‘right’ or ‘neutral’ we understand the price, such futures would cost to an insurance company taking into account the volatility of your cost. This shall be an integral of excess of operation prices exceeding the strike price a weighted by the normal distribution of the operation price (see normal distribution).

The formula of futures price that we developed in [090819 ii], is fully analytical, with the exception of error function ERF implemented in Excel (see error function).

Because the price of futures is right, if computed as follows:

The following must hold for all a and k

In order to validate both of our formulas, we use an Excel model. The mean of our operation cost at $12m and the variance is equal to $3.2m. For a two-dimensional set of parameters, (a) strike prices varying from $4m to $20m and (b) the insurance ratios varying from 10% to 200%, we compute ‘right’ prices of futures using our formula . The blow chart shows this function for the two dimensional set of input parameters:

[xls]

Then, sub-sequentially, based on the right price, we compute probability density values with our formula by loading results into an array with prices ranging from $0 to $23’500’000.

The following animation shows that the choice of a and k can change the shape of price distribution significantly:

[slow] [fast] [xls]

By having, for each shape (dictated by a pair of strike price a and insurance ratio k), the probability density array, we compute the mean of the insured operation cost:

Irrespectively how wildly the probability density shape is changed (by the choice of two parameters of futures), the Excel simulation shows that the computed mean of insured operation price is always equal to the mean of the uninsured price, i.e. to (with exceptions of precision errors). This suggests no hidden costs in prices of futures.

[xls]

The results of this simulation validate our two formulas for calculation of the ‘right’ price of futures and for subsequent calculation of the insured price’s probability density .

References:

Validation of the formula of the ‘right’ price of futures and of the function of the insured price’s probability density (this page):

http://switzernet.com/people/aram-gabrielyan/public/090819-average-of-hedged-cost/

http://unappel.ch/people/aram-gabrielyan/public/090819-average-of-hedged-cost/

Right price of futures as a function of the strike price:

http://switzernet.com/people/aram-gabrielyan/public/090819-neutral-price-vs-Black-Scholes/

http://unappel.ch/people/aram-gabrielyan/public/090819-neutral-price-vs-Black-Scholes/

Probability density of a hedged price:

http://unappel.ch/people/aram-gabrielyan/public/090817-hedging-cost-with-futures/

http://switzernet.com/people/aram-gabrielyan/public/090817-hedging-cost-with-futures/

Normal Distribution and Cumulative distribution function:

http://en.wikipedia.org/wiki/Normal_distribution

Error function:

http://en.wikipedia.org/wiki/Error_function

Black-Scholes model

http://en.wikipedia.org/wiki/Black-Scholes

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