An Improved Black–Scholes Model to Determine the Optimal Boundary of Asset–Liability

The study addresses limitations of the Black–Scholes framework, specifically its reliance on a risk-neutral market and a self-financing hedging portfolio by proposing a generalized derivative pricing approach grounded in the efficient markets hypothesis. The research objective is to establish a valuation model in which a derivative’s fair value equals a conditional expectation discounted at the underlying asset’s drift, thereby explicitly retaining the asset’s drift rather than abstracting it away under risk neutrality. Methodologically, the paper develops a partial differential equation (PDE) that replaces the risk-free rate with an efficiency-consistent discount rate, derives a pricing formula for European call options that incorporates the underlying’s drift, and analyzes the optimal exercise boundary for American call options under varying parameters. Key findings show that the optimal exercise price increases with higher volatility and risk-free interest rates and decreases with higher dividend yield; moreover, it is never optimal to exercise an American call option early when the underlying pays no dividends. The study concludes that an efficiency-based discounting scheme offers a coherent alternative to risk-neutral valuation while preserving internal consistency with observed market dynamics. The contribution and implication are a drift-inclusive theoretical framework that refines PDE-based pricing, clarifies comparative statics for exercise policy, and provides practitioners with guidance for pricing and exercise decisions in settings where asset drift is informationally relevant.