# Option pricing: Stock price, stock velocity and the acceleration Lagrangian

- Authors
- Journal
- Physica A Statistical Mechanics and its Applications 0378-4371
- Publisher
- Elsevier
- Publication Date
- Identifiers
- DOI: 10.1016/j.physa.2014.09.019
- Keywords
- Disciplines

## Abstract

Abstract The industry standard Black–Scholes option pricing formula is based on the current value of the underlying security and other fixed parameters of the model. The Black–Scholes formula, with a fixed volatility, cannot match the market’s option price; instead, it has come to be used as a formula for generating the option price, once the so called implied volatility of the option is provided as additional input. The implied volatility not only is an entire surface, depending on the strike price and maturity of the option, but also depends on calendar time, changing from day to day. The point of view adopted in this paper is that the instantaneous rate of return of the security carries part of the information that is provided by implied volatility, and with a few (time-independent) parameters required for a complete pricing formula. An option pricing formula is developed that is based on knowing the value of both the current price and rate of return of the underlying security which in physics is called velocity. Using an acceleration Lagrangian model based on the formalism of quantum mathematics, we derive the pricing formula for European call options. The implied volatility of the market can be generated by our pricing formula. Our option price is applied to foreign exchange rates and equities and the accuracy is compared with Black–Scholes pricing formula and with the market price.

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