Numerical Techniques for Determining Implied Volatility in Option Pricing

Abstract

One of the assumptions of the Black–Scholes model is that volatility is constant in the market. However, in reality, volatility cannot be constant. In this paper we examined an inverse problem of determining the time-dependent non-constant volatility from the observed market price of a European put/call option with a strike price.These unknown non-constant volatilities are from the time the option contract was signed to the time it was exercised (strike price) where the fluctuations of option price is known but the non-constant volatilities that was generated by option price within that time period is unknown.Although many studies have used the portfolio stocks to reconstruct volatility, no study have retrieved volatility from the option market. The aim is to recover unknown non-constant volatilities in vector form from one option contract period using manufacturer observed market data, that is, data from agreed contract period using Black–Scholes Partial Differential Equation (BSPDE). This would be done by using the chain and product rule to take the derivative with respect to volatility in the theoretical(Black-Schoes PDE) model to obtain the non-constant volatilities. By keeping the fluctuations data from observed option prices constant,we differentiate the volatility in the BSPDE to obtain a gradient equation and discretize the gradient equation with the finite difference method to obtain model simulated data. It is expected that non-constant volatility that would be recovered from the market using the gradient descent method would be matched with the non-constant market volatility from the model simulated data and if they are the same or approximately same, the volatility would have deem recovered. All these would be done in the least square sense.