Option Pricing under Randomised GBM Models
https://doi.org/10.26794/2308-944X-2021-9-3-7-26
Abstract
By employing a randomisation procedure on the variance parameter of the standard geometric Brownian motion (GBM) model, we construct new families of analytically tractable asset pricing models. In particular, we develop two explicit families of processes that are respectively referred to as the randomised gamma (G) and randomised inverse gamma (IG) models, both characterised by a shape and scale parameter. Both models admit relatively simple closed-form analytical expressions for the transition density and the no-arbitrage prices of standard European-style options whose Black-Scholes implied volatilities exhibit symmetric smiles in the log-forward moneyness. Surprisingly, for integer-valued shape parameter and arbitrary positive real scale parameter, the analytical option pricing formulas involve only elementary functions and are even more straightforward than the standard (constant volatility) Black-Scholes (GBM) pricing formulas. Moreover, we show some interesting characteristics of the risk-neutral transition densities of the randomised G and IG models, both exhibiting fat tails. In fact, the randomised IG density only has finite moments of the order less than or equal to one. In contrast, the randomised G density has a finite first moment with finite higher moments depending on the time-to-maturity and its scale parameter. We show how the randomised G and IG models are efficiently and accurately calibrated to market equity option data, having pronounced implied volatility smiles across several strikes and maturities. We also calibrate the same option data to the wellknown SABR (Stochastic Alpha Beta Rho) model.
About the Authors
G. CampolietiCanada
Giuseppe (Joe) Campolieti — Professor at Graduate Faculty. He received PhD from McGill University in 1989. He joined Wilfrid Laurier University in 2002 as associate professor of Mathematics and as SHARCNET chair in Financial Mathematics. He also founded a financial software and consulting company in 1998. 75 University Ave W, Waterloo, ON, Canada
H. Kato
Canada
Hiromichi Kato — PhD student at Graduate Faculty (Mathematical and Statistical Modelling). 75 University Ave W, Waterloo, ON, Canada
R. Makarov
Canada
Roman Makarov — Associate Professor at Graduate Faculty; Chair, Mathematics. He received PhD in Computational Mathematics from the Russian Academy of Sciences in 2000. He joined Laurier in 2003. 75 University Ave W, Waterloo, ON, Canada
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Review
For citations:
Campolieti G., Kato H., Makarov R. Option Pricing under Randomised GBM Models. Review of Business and Economics Studies. 2021;9(3):7-26. https://doi.org/10.26794/2308-944X-2021-9-3-7-26