Density Estimation For Options Arbitrage By Kernel Density Method

The option contract is a right to buy or sell another asset at a given price within a specified period of time, In the case of an European call option,a hedge portfolio is constructed by establishing a long position in the option and a short position in the underlying stock.The effect of diffusion in the stock price is thus eliminated and with continual adjustment of the hedge composition the value of the hedge at maturity becomes rikless.In practice it is clearly impossible to rebalance the portfolio continuously considering the transaction costs.If the hedge portfolio was adjusted discretely,the hedge portfolio return becomes risky.The mismatch between the payoffs of the stock and that of the option would make an arbitrage.The paper is as follows.First, I get an approximate expression for the return on the hedge when rebalancing occurs at discrete intervals. Second,using matlab generate a series of stock price and some dates.Third,I use Kernel density estimation and develop an approximation to the conditional distribution by two types of discrete hedging strategies:time-based and move-based strategy,and compare their advantage and disadvantage.I find that for low volatility levels no hedging is optimal,for high volatility levels very-frequent rebalancing is optimal,and for intermediate volatility levels rebalancing when the option hedge ratio changes by a specified amount(0.25).