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Within the optimal production and hedging decision framework, Lien compares the exponential utility function with its second order approximation under the normality distribution assumption. In this paper, we first extend the result further by comparing the exponential utility function with a 2
*n*-order approximation for any integer
*n*. We then propose an approach with illustration to find the smallest n that provides a good approximation.

Using polynomials to approximate the expected utility function is one of the important issues in finance (see, for example, Feldstein [

Suppose that, at time 0, a producer intends to produce q units of a commodity that are planned to be sold at time 1. The production cost is c(q) and there is no production risk. we assume that the price,

We further assume that the hedger has an exponential utility function u(.) such that

where k is the Arrow-Pratt risk aversion coefficient. Consequently,

where

It is well known in the literature that the firm’s optimal production decision

Following Tsiang [

where

Let

While it is common to use second-order approximation (see, for example, Pulley [

Consequently,

where

Let

For

From the above equation,

Define

porate Equation (4.1) into the formula of M(h), we get

Thus,

Proposition 4.1. Consider a one-period production and futures hedging framework. Given that the producer is endowed with an the exponential utility function and the spot price in the future is normally distributed,

1) if

2)if

We now turn to the general case. Consider the 2n-th order approximation of the exponential utility function u in (2.2):

Upon taking the expectation, we get

where

Let

For

From the above equation, we obtain

After substituting this equation into the formula of

Thus,

which in turn leads to

Proposition 4.2. Consider a one-period production and futures hedging framework. Given that the producer is endowed with an the exponential utility function and the spot price in the future is normally distributed,

1) if

2) if

In this section we compare the optimal futures position under the 2n-order approximation with the true optimal position under the true expected utility function:

Let

From the previous section, we rewrite

Thus,

implying the sign of

Proposition 5.1. Consider a one-period production and futures hedging framework. Given that the producer is endowed with an the exponential utility function and the spot price in the future is normally distributed, we have

1) if

2) if

We now propose an approach to find the smallest n that will provide a good approximation. Since it is well known that

Let

Thus,

Theorem 6.1. Let

1) if

2) if

3)

4) for any

Thus, to obtain a good approximation for

choose the level of tolerance,

Below we present an example to illustrate Theorem 6.1. Consider

assume

is the solution to the following equation:

which can be rewritten as:

Solving the above quadratic equation, we have

Now, we let

The second order condition requires

Thus,

and

If we assume

According to the second order condition, we obtain

Thus,

By using the “solve” function in MATLAB, we find

In this paper, we analyze a one-period production and hedging decision problem where the producer is endowed with an exponential utility function. Our findings are summarized as follows. First, it is well-known that a normal distribution coupled with an exponential expected utility produces a mean-variance (MV) approach. Meanwhile, a quadratic approximation also leads to a mean-variance approach. Our first finding is that the two approaches lead to different results (see Lien [

Lastly, Hlawitschka [

This research is partially supported by grants from Beijing Normal University, Nanjing University of Aeronautics and Astronautics, University of Texas at San Antonio, Tsinghua University, Asia University, Lingnan University, Hong Kong Baptist University, and Research Grants Council of Hong Kong.

Xu Guo,Donald Lien,Wing-Keung Wong, (2016) Good Approximation of Exponential Utility Function for Optimal Futures Hedging. Journal of Mathematical Finance,06,457-436. doi: 10.4236/jmf.2016.63036