Olkhov, Victor (2020): Classical Option Pricing and Some Steps Further.

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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 2 – 5 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.
Item Type:  MPRA Paper 

Original Title:  Classical Option Pricing and Some Steps Further 
English Title:  Classical Option Pricing and Some Steps Further 
Language:  English 
Keywords:  Option Pricing; BlackScholesMerton Equations; Stochastic Volatility; Market Transactions; Expectations; Nonlinear equations 
Subjects:  G  Financial Economics > G1  General Financial Markets G  Financial Economics > G1  General Financial Markets > G12  Asset Pricing ; Trading Volume ; Bond Interest Rates G  Financial Economics > G1  General Financial Markets > G17  Financial Forecasting and Simulation 
Item ID:  99918 
Depositing User:  Victor Olkhov 
Date Deposited:  29 Apr 2020 07:26 
Last Modified:  29 Apr 2020 07:26 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/99918 