English

Abstract

As it is well known an option is defined as the right to buy sell a certain asset, thus, one can look at the purchase of an option as a bet on the financial instrument under consideration. Now while the evaluation of options is a completely different mathematical topic than the prediction of future stock prices, there is some relationship between the two. It is worthy to note that henceforth we will only consider options that have a given fixed expiration time T, i.e. we restrict the discussion to the so called European options. Now, for a simple illustration of the relationship between true stock prices and options let us consider the following situation: if at the begging of January the SP in fact the VIX itself is both traded and sold in many ways. This is clearly not usable for the σ input as the foundations of the Black Scholes model, consistent with most mathematical models, are based on modelling a phenomenon based on logical behaviour as opposed to human erratic behaviour. A good example of this is how the VIX or S&P may change rapidly due to one word a single news reporter says, regardless if it was a truthful statement or not. In this study we seek to remedy this by using a simple comparison technique to identify in real time when the Black Scholes model truly is violating the assumptions of constant volatility with volatility defined more along the lines as market deviation from reality. To do this we create a linear regression model to predict the value of the S&P 500 using input values of macroeconomic measures such as GDP (Gross Domestic Product),M (total money aggregate), PPI (Producer Price). A correlation is proven here empirically and it is suggested that one can use the regression model as a red flag to caution the user in real time that at those points Black Scholes performance is suspect, with high or rapidly changing volatility implied. At these times one should use the Black Scholes with extreme caution, perhaps a regime switching approach, or look to alternate prediction methods often just common sense possibly just seeking secure assets. Keywords— Partial differential equations, parabolic, stochastic, financial mathematics. AMS classification: 35K10