Numerical methods to value an option including risk aversion with a constant relative risk aversion function

Abstract

This study develops a comprehensive discrete numerical model for option valuation that explicitly incorporates risk preferences, which may deviate from risk neutrality. Unlike the traditional binomial tree models - strictly under the risk-neutral paradigm - our framework embeds a constant relative risk aversion (CRRA) utility specification, capturing heterogeneous attitudes toward risk while preserving the arbitrage-free pricing rule.The model extends the multiplicative binomial recombination tree (MBRT) by adjusting key parameters - transition probabilities, growth factors, discount rates and drift/diffusion terms - to reflect the investor's degree of risk aversion. The classical Cox-Ross-Rubinstein binomial tree (CRR) emerges as a special case when risk aversion is set to zero. The methodology remains consistent with geometric Brownian motion (GBM) dynamics and is benchmarked against a modified Monte Carlo simulation to ensure robustness.Results show that option values can be consistently derived under both traditional risk-neutral settings and preference-driven settings. Sensitivity analysis highlights the impact of time to maturity, volatility, strike price and the risk-free rate under varying levels of risk aversion.While this research offers significant theoretical and practical contributions, certain limitations warrant further study. Computational complexity: the CRRA-based valuation method introduces additional numerical challenges, requiring precise calibration and advanced optimization techniques. Dependence on risk aversion estimates: the model assumes that investor risk preferences can be accurately measured and remain stable, which may not always reflect dynamic market conditions. Absence of a closed-form solution: our proposed approach lacks an analytical closed-form solution. Therefore, it is crucial to dedicate efforts to its development.The integration of CRRA utility functions into derivative valuation represents a key innovation, as it explicitly accounts for investor risk preferences beyond the traditional risk-neutral paradigm. This framework advances the literature on utility-based and nonlinear risk-adjusted pricing by demonstrating how variations in the relative risk aversion (RRA) coefficient shape option values. From a practical perspective, the model offers a flexible tool for portfolio managers, traders and policymakers by aligning valuations with observed market behavior while preserving consistency with classical models under specific conditions. Accurate calibration of risk preferences thus becomes essential for reliable pricing and policy design.The novelty of this research lies in bridging utility-based preferences with recombining lattice valuation: while prior studies focused exclusively on risk-neutral or arbitrage-based approaches, our model incorporates explicit risk aversion into the numerical structure. By deriving general algebraic expressions and validating the framework through numerical experiments, this study offers a tractable and versatile tool for analyzing option prices under heterogeneous risk attitudes, without losing the analytical clarity of traditional methods.