Volatility Estimation in a Jump Diusion Geometric Brownian Motion with Reinvested Dividends and Transaction Cost.
This paper derives a local volatility model for an asset whose dynamics follow a jump di usion geometric Brownian motion, incorporating reinvested divi- dend yields and proportional transaction costs. Building upon the seminal works of Black and Scholes (1973), Merton (1976), and the more recent de- velopments by Opondo et al. (2021, 2025), the study extends Dupire’s local volatility framework to accommodate discontinuities and market frictions in asset price behavior. By integrating tools from stochastic calculus, jump pro- cess theory and transaction cost modeling, we derive a modi ed Dupire-type volatility equation tailored for complex nancial environments. This model enhances the accuracy of derivative pricing and risk assessment under more realistic market conditions.
Keywords: Dupire Volatility, Jump Di usion, Geometric Brownian Motion, Reinvested Dividends, Transaction Costs and Fokker-Planck equation.
2000 MSC: Primary 91GXX, 91G50, 62 P05, 97M30.
