Option Pricing under Randomised GBM Models

By employing a randomisation procedure on the variance parameter of standard geometric Brownian motion (GBM) model, we construct new families analytically tractable asset pricing models. In particular, develop two explicit processes that are respectively referred to as randomised gamma (G) and inverse (IG) models, both characterised by shape scale parameter. Both models admit relatively simple closed-form analytical expressions for transition density no-arbitrage prices European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in log-forward moneyness. Surprisingly, integer-valued arbitrary positive real parameter, option formulas involve only elementary functions even more straightforward than (constant volatility) formulas. Moreover, show some interesting characteristics risk-neutral densities G IG exhibiting fat tails. fact, has finite moments order less or equal one. contrast, first moment with higher depending time-to-maturity its We how efficiently accurately calibrated market equity data, having pronounced volatility across several strikes maturities. also calibrate same data wellknown SABR (Stochastic Alpha Beta Rho) model.