A family of jump process models is derived by applying Gauss-Hermite quadrature to the recursive integration problem presented by a compound option model. The result is jump processes of any order with known efficiency properties in valuing options. In addition, these processes arise in the replication of options over finite periods of time with two or more assets where they again have known efficiency properties. A “sharpened” trinomial process is designed that accounts for the first-derivative discontinuity in option valuation functions at critical exercise points. It is shown to have accuracy superior to that of conventional binomial and trinomial processes and is nearly identical to the trinomial process optimized by Boyle (1988) through trial and error.

ABSTRACT It has been noted that a certain continuous‐time trading strategy, termed the “doubling strategy”, generates a positive net return on borrowed funds, with probability one and within a finite period of time. Since the doubling strategy seems to represent a “free lunch” or arbitrage opportunity, a variety of constraints to render it infeasible have been proposed. In this paper, we show that the doubling strategy generates infinite disutility for a large class of utility functions, and we can think of no utility function for a risk‐averse agent which is a counterexample.

It has been noted that a certain continuous-time trading strategy, termed the “doubling strategy”, generates a positive net return on borrowed funds, with probability one and within a finite period of time. Since the doubling strategy seems to represent a “free lunch” or arbitrage opportunity, a variety of constraints to render it infeasible have been proposed. In this paper, we show that the doubling strategy generates infinite disutility for a large class of utility functions, and we can think of no utility function for a risk-averse agent which is a counterexample.

The dynamic nonmyopic portfolio behavior of an investor who trades a risk-free and risky asset is derived for all HARA utility functions and a stochastic risk premium. Conditions are found for when the investor holds more or less than the myopic amount of the risky asset; hedges against or speculates the risk-premium uncertainty; is long or short on the risky asset; and holds more or less of the risky asset at longer horizons. The analytical solutions derived take multiple mathematical forms and include extreme cases in which investors with long but finite horizons can attain nirvana.

[The dynamic nonmyopic portfolio behavior of an investor who trades a risk-free and risky asset is derived for all HARA utility functions and a stochastic risk premium. Conditions are found for when the investor holds more or less than the myopic amount of the risky asset; hedges against or speculates the risk-premium uncertainty; is long or short on the risky asset; and holds more or less of the risky asset at longer horizons. The analytical solutions derived take multiple mathematical forms and include extreme cases in which investors with long but finite horizons can attain nirvana.]