Convergence of optimal expected utility for a sequence of binomial models

Area of Science:

• Quantitative Finance
• Financial Mathematics
• Stochastic Calculus

Background:

• The Black-Scholes-Merton model is a cornerstone of modern option pricing.
• Discrete-time models, such as binomial models, are often used as approximations.
• Previous research established counter-examples for positively skewed binomial models.

Purpose of the Study:

• To investigate the convergence of discrete-time utility maximization problems in binomial models to the continuous-time Black-Scholes-Merton model.
• To address the open problem regarding convergence for symmetric binomial models.
• To extend convergence results to negatively skewed binomial models.

Main Methods:

• Analysis of discrete-time utility maximization problems.
• Application of fine estimates for the tail behavior of binomial random variables.
• Review of general convergence results for discrete models to the Black-Scholes-Merton framework.

Main Results:

• Convergence is demonstrated for symmetric binomial models.
• Convergence is proven for negatively skewed binomial models.
• The proofs rely on detailed analysis of binomial random variable tail distributions.

Conclusions:

• The convergence of discrete utility maximization problems to the Black-Scholes-Merton model holds for symmetric and negatively skewed binomial cases.
• This resolves a previously open problem in financial mathematics.
• The findings enhance the theoretical understanding of discrete approximations to continuous financial models.