Finding the optimal strike price and volume of futures for hedging against the volatility of expenses

Aram Gabrielyan

Emin Gabrielyan

2009-08-19

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We analyze a model where specific expenses of an enterprise are hedged by futures. In this model we assume that futures are issued directly on these expenses. This means that we deal with future contracts that can refund a fraction of expenses of enterprise exceeding a pre-selected strike price. The choice of the refunded fraction (of the excess with respect to the strike price) is made by the choice of the amount of purchased futures. This fraction can be less than, more than, or equal to 1. This model is applicable to a more general case, where instead of futures issued directly on the enterprise expenses, the futures are issued on another underlying instrument highly correlated to the enterprise expenses. Examples are futures of HDD versus the expenses of heating companies.

Our objective is to find the optimal strike price and volume of futures, purchased for hedging the enterprise expenses against volatility. For this purpose we introduce the volatility cost of enterprise. On the other hand we shall introduce the beneficiary margin of the issuer of futures. In our example model the average amount of expenses (mu) is equal to $12m and the variance (sigma) is equal to $3.2m. The cost of the volatility is introduced via two parameters. A debt cost (debtc) of the case when the enterprise expenses end up at their mean (mu), and a multiplication factor (debtk) by which the debt cost of enterprise would increase, if the company ends up with an expense higher from the average by sigma. (i.e. mu+sigma). The higher is debtk the higher is the interest to stay closer to mu (the average being anyway equal to mu).

In [090817 ii] we developed price probability density transformation achieved by the purchase of futures. In [090819a ii] we developed a formula for computing the neutrally right price of futures. In [090819b ii] we showed that the mean of insured expenses does not change, if the futures are bought at neutrally right prices. This means that if the market price of futures is always right, we are definitively interested in purchasing of futures all the time, even if the volatility cost is low. Futures are narrowing the probability density of expenses and are therefore minimizing the volatility related debt costs.

However the pleasure of having a future based insurance shall often have a market cost exceeding its neutrally right price [090819a ii]. The difference between the market price of futures and the right price computed by our formulas [090819a ii] is precisely the cost of this pleasure. This cost must be counterbalanced with savings achieved on the volatility side. The margin added on the right price of futures is represented in percentages. We analyze a range from 0.5% to 90%.

In the following chart we assume that the default debt cost (debtc) is equal to $300’000 (the default price applicable for all scenarios where expenses end up at the average of $12m). The multiplication factor (debtk) is a parameter changing from 1.5 to 5. For instance if this factor is 2, it means that if the enterprise ends up with an expense of $15.2m (i.e. one variance away from the mean), it will cost extra $600’000 to the company (e.g. due to short-term debt interests).

The surface shown in the figure below represents the overall cost (the volatility debt cost together with the beneficiary margins paid to issuers of futures) as a function of the choice of the strike price (in the range from $4m to $20m) and the insurance ratio (in the range from 10% to 200%). The animation shows the changes in the shape when changing the beneficiary margin of issuers of futures (fairness) and the debt cost factor (dependence). For each frame (i.e. for each pair of fairness of futures and dependence of enterprise from the volatility), the text data panel shows the best recommended strike price and insurance ratio.

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This animation demonstrates the concave shaped form of the surface for all pairs of fairness and of dependence showing that there is always an optimal choice to make depending on the market prices of futures and the extra cost of the volatility of your expenses.

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