Some important features of the underlying asset price process paths have been overlooked in classical derivative pricing models‎. ‎In these models‎, ‎three‎ ‎fundamental aspects of the stochastic movement of price namely speed‎, ‎scale‎, ‎and degree of continuity are not distinguishable‎. ‎These characteristics‎ ‎are inseparable in models where the source of randomness is defined by Brownian motion‎, ‎whereas these concepts are essential for a‎ ‎precise analysis of how asset prices move‎. To address this limitation‎, ‎we use discontinuous approximations that allow for the distinction‎ ‎between these three aspects‎. ‎In this regard‎, ‎we employ the generalized CGMY process where‎, ‎the parameter C represents speed‎ ‎and is modeled stochastically‎. ‎With a new approach‎, ‎we consider first-order variations as the driving force of the model‎, ‎which enables the modeling of‎ ‎stochastic volatility of volatility‎.
‎Subsequently‎, ‎we estimate the parameters of the proposed model using a two-step procedure‎. ‎In the first step‎, ‎the parameters of the CGMY process are‎ estimated using its characteristic function‎. Next, we estimate the parameters introduced during the construction of the generalized‎ CGMY process‎. ‎After determining all parameters‎, ‎we simulate sample paths of the new process and compare them with the original CGMY‎‎process‎.
‎We simulate the generalized CGMY process using the large jump approximation method‎. ‎In this approach‎, ‎the arrival rate of large jumps is modeled by a‎ ‎Poisson process‎, ‎and the sizes of these jumps follow an exponential distribution‎. ‎Finally‎, ‎using the ARCH test‎, ‎we confirm the presence of stochastic volatility of volatility in historical price data‎.