Review of Models used to Price Options: Stock-Price vs Option Price and Implied Volatility vs Actual Volatility Comparison for Different Option Pricing Methods

This paper will review LVM models, GP-LVM, CEV model, DD model, Crank-Nicolson, and other finite difference methods, Greeks, SABR, martingales, and LSMC option pricing. The effects of changing the volatility on paths generated by Bachelier, Black-Scholes proved no difference between these two models. Implied volatility for all the models was higher when compared to actual volatility for:BS,BSM, and Bachelier. Crank-Nicolson method for ATM, ITM,OTM showed higher intrinsic value for the price after the initial stock price, but with diminishing returns, intrinsic minus extrinsic value is zero at the last price. In Greeks analysis t was observed no put and call parity for different values of :Delta,Gamma,rho. GP-LVM forecast proved to be closest to the actual stock price.