Classical Option Pricing and Some Steps Further
Abstract
This paper modifies single assumption in the base of classical option pricing model and derives further extensions for the BlackScholesMerton equation. We regard the price as the ratio of the cost and the volume of market transaction and apply classical assumptions on stochastic Brownian motion not to the price but to the cost and the volume. This simple replacement leads to 2dimensional BSMlike equation with two constant volatilities. We argue that decisions on the cost and the volume of market transactions are made under agents expectations. Random perturbations of expectations impact the market transactions and through them induce stochastic behavior of the underlying price. We derive BSMlike equation driven by Brownian motion of agents expectations. Agents expectations can be based on option trading data. We show how such expectations can lead to nonlinear BSMlike equations. Further we show that the Heston stochastic volatility option pricing model can be applied to our approximations and as example derive 3dimensional BSMlike equation that describes option pricing with stochastic cost volatility and constant volume volatility. Diversity of BSMlike equations with 25 or more dimensions emphasizes complexity of option pricing problem. Such variety states the problem of reasonable balance between the accuracy of asset and option price description and the complexity of the equations under consideration. We hope that some of BSMlike equations derived in this paper may be useful for further development of assets and option market modeling.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.13708
 Bibcode:
 2020arXiv200413708O
 Keywords:

 Quantitative Finance  Pricing of Securities;
 Quantitative Finance  Statistical Finance
 EPrint:
 16 pages