# Develop an option pricing equation by Ornstein Uhlenbeck process

I know that Black-Scholes equation is based that the Equity price has a Geometrical Brownian movement. Can I develop from the same principles( now with transaction cost) that Black Scholes is developed an eqution that the equity price is based on the Ornstein Uhlenbeck process? Thanks

• This is too general. What is motivation for such modeling choice? What is the economic or financial sense of the model? What would you like to achieve, and what are your assumptions? – Gordon Sep 27 at 18:54
• The Ornstein Uhlenbeck process has the mean reversion property that geometric Brownian does not has – Hernan Sep 27 at 18:59
• I'm not sure if this is valid but you can try setting up a hedging portfolio consisting of a short position in the option, delta shares of the stock, and a position in the money market. The infinitesimal P&L of this portfolio should be $0$, which gives the Black Scholes PDE. So you can do something similar using the OU process but now there will be a source term from the drift. – Slade Sep 27 at 20:19
• Can you use the Giranov theorem to change the measure and make a Martingale? – Hernan Sep 27 at 20:41
• Are you looking for a Black&Scholes-style "closed form" option pricer or just any numerical approach will do? You can do OU dynamics with transaction costs very easily in FDM for example – James Spencer-Lavan Sep 28 at 5:57