Analytic solving of asset pricing models: The by force of habit case
Analytic methods for solving asset pricing models are developed to solve asset pricing models. Campbell and Cochrane's [1999. By force of habit, a consumption-based explanation of aggregate stock market behavior. Journal of Political Economy 107, 205-251] habit persistence model provides a prototypical example to illustrate this method. When the parameters involved satisfy certain conditions, the integral equation of this model has a solution in the space of continuous functions that grows exponentially at infinity. However, the parameters advocated by Campbell and Cochrane do not satisfy one of these conditions. The existence problem is removed by restricting the price-dividend function to avoid values of dividend growth that are extreme. Thus, existence and uniqueness of the solution in the space of continuous and bounded functions is proved. Using complex analysis the price-dividend function is also shown to be analytic in a region large enough to cover all relevant values of dividend growth. Next, a numerical method is presented for computing higher order polynomial approximations of the solution. Finally, a uniform upper bound on the error of these approximations is derived. An intensive search of the parameter space results in no parameter values for which the solution matches the historic equity premium and Sharpe ratio within Campbell and Cochrane's model.
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