Enhanced Valuation of European Options Under Jump Processes and Innovative Characterization of Implied Volatility Smile

An enhanced option pricing framework that makes use of both continuous and discontinuous time paths based on a geometric Brownian motion and Poisson-driven jump processes respectively is performed in order to better fit with real-observed stock price paths while maintaining the analytical trackability of the Black-Scholes model. The main advantage of this model is to lie on a consistent framework that does not make stringent assumptions in order to derive a closed-form option pricing formula and to capture rare events such as major political changes or catastrophic events through the use of discontinuous stochastic processes. Moreover, this model does not much face any calibration issue given that little dependence with non-observable parameters is introduced. Moreover, an innovative quantification of pricing differences is proposed between the in-house (i.e. the own made pricing formula including jumps) and the classic Black-Scholes model through implied volatility and its curve resembles a smile, meaning that the introduction of jumps is quantified via a smile according to implied volatility. In order to derive such an implied volatility smile, an iterative search procedure referred to as the Newton-Raphson algorithm is proposed. Numerical experiments of both the in-house pricing formula and its implied volatility recursive algorithm are presented and results show that both the two are fast computing via computers through the use of coding languages