In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance10(1), 39–52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method.

This research examines the predictability of a profit maximization model, an expected value-variance utility maximization (E-V) model, and two versions of the target-MOTAD model for modeling risky agricultural production decisions. Model solutions were translated into expected value and variance of farm income for analysis. Direct comparison and chi-square analysis of actual and predicted expected income distributions were used in the analyses. It was concluded that the utility maximization and cash-cost target-MOTAD models predicted distributions of farm income better than the variable-cost target-MOTAD and profit maximization models.