A financial derivative is an instrument whose payoff is derived from the behavior of another underlying asset. One of the most commonly used derivatives is the option which gives the right to buy or to sell an underlying asset at a pre-specified price at (European) or at and before (American) an expiration date. Finding a fair price of the option is called the option pricing problem and it depends on the underlying asset prices during the period from the initial time to expiration date. Thus, a โgoodโ model for the underlying asset price trajectory is needed. In this work, we are interested in European call options. We propose a new Constant Elasticity of Variance (CEV) model that covers the post-crash situations. First, we set up the modified CEV model for markets with high volatility. Then we find a numerical solution for the stochastic differential equation of the underlying price. The risk-neutral valuation method shows that the option price can be written as an expected value of the discounted underlying asset price at maturity. Then we use Monte Carlo methods for finance this to find a numerical solution for the price of a European option under a CEV model with high volatility. Keywords
