Hedging the operation price with hypothetical futures issued directly on your operation price
2009-08-17
We analyze transformations of a probability density of an operation price, caused by purchases of futures (issued on that price). We assume two free variables in the purchasing of futures, the strike price, and the quantity of futures. We assume that all futures are purchased with the same strike price. When your operation cost exceeds the strike price, the futures refund you a given percentage of the excess of your operation cost. That percentage depends on the amount of futures purchased. It can vary from 0% to 100% (underinsurance) or can be more than 100% (over-insurance). The choice of the quantity of purchased futures determines the insurance ratio.
The following example shows in an animation, how the probability curve of price changes when the ratio of insurance changes from 10% to 250%. The strike price is the same in all samples. The cost of futures is computed with a very simplified empiric formula (it is sufficiently good for this demonstration). As expected, this animation shows that the more you increase the insurance ratio toward 100%, the narrower the volatility of your prices becomes. We also see that the price, when over insured, never exceeds the strike price plus the cost of insurance.
The probability density of the hedged price is computed as follows:
Where:
c is the cost of purchased futures
a is the strike price
k is the insurance ratio (1 corresponds to 100%)
Obviously:
The corresponding Excel formula looks as follows:
=IF(B2-cost<a,1/(sig*SQRT(2*PI()))*EXP(-((B2-cost-mu)^2/(2*sig^2))),0)+
IF((B2-cost-a*k)/(1-k)>a,1/(sig*SQRT(2*PI()))*EXP(-(((B2-cost-a*k)/(1-k)-mu)^2/(2*sig^2)))/ABS(1-k),0)
Files:
Excel file with formulas and charts [xls]. Values in blue are inputs and can be changed.
Excel file with a simulation of the insured price probability demonstrating that the analytical function is correct [xls]
Samples of simulated version of the hedged price probability [htm]
References:
http://en.wikipedia.org/wiki/Normal_distribution
http://en.wikipedia.org/wiki/Kronecker_delta
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