Option Pricing under Deterministically Time-Varying Volatility: A Practical Analytical Approximation to the Black-Scholes Framework

Abstract

: This study expands the Black – Scholes option pricing model by assuming volatility is deterministically evolving over time. The traditional formulation assumes constant volatility, which is inconsistent with empirical studies showing predictable changes during periods of market uncertainty (e.g., macro announcements, earnings reports and changing market conditions). To address this limitation while preserving the simplicity of analytical derivations, this paper models volatility as a known function of time. By integrating the instantaneous variance to cumulative variance and converting it into an effective volatility parameter, the pricing formula of the European option is given in a closed-form expression and preserves the conventional form of the Black-Scholes model. The paper also shows how the DTVV model could be used to illustrate option value adjustment when uncertainty is higher or lower during the option lifecycle. Besides, the paper discusses several economic implications of the time-varying model, which firms and financial institutions could apply when making option pricing, hedging, calibration, and risk management decisions. Although the DTVV extension improves the realism of the Black – Scholes model, it fails to capture unforeseen volatility shocks. This limitation has led to the potential extensions of the segmented volatility model or hybrid deterministic and stochastic formulas.