Alternative methods to evaluate the arbitrage opportunities

In this paper, we present alternative methods to evaluate the presence of the arbitrage opportunities in the market. In particular, we investigate empirically the well-known put-call parity no-arbitrage relation and the state price density. First, we measure the violation of the put call parity as the difference in implied volatilities between call and put options that have the same strike price, the same maturity and the same underlying asset. Then, we examine the nonnegativity of the state price density since its negative values immediately correspond to the possibility of free-lunch in the market. We evaluate the effectiveness of the proposed approaches by an empirical analysis on S&P 500 index options data. Moreover, we propose different approaches to estimate the state price density under the classical hypothesis of the Black and Scholes model. In this context, we use two different methodologies to evaluate the conditional expectation and its relationship with the state price density. Firstly, we examine the real mean return function using local polynomial smoothing technique. Then, we evaluate the conditional expectation using the real probability density. According to the hypothesis of the Black and Scholes model, we are able to derive a closed formula for approximating the conditional expectation with the risk neutral probability. Finally, we propose a comparison among two estimators of the state price density.