An empirical study of Black Scholes option pricing model and its relevance in real option valuation

Abstract

Purpose: The objective of the study is to test the validity of the Black Scholes option pricing model by comparing the theoretical option prices derived from the model with the actual market price of the options. We also try to investigate if the pricing error caused by the difference between the theoretical actual option prices vary with respect to the moneyness (in-the-money or out-of-the-money) and time left to expiry of the option. We will also try to model the strength of market efficiency in the various option trading exchanges through statistics on the relative pricing error. We also do a detailed study of risk-neutral pricing and explain why it fails in incomplete markets. The second part of the project is focused on real options and their analysis. In real options analysis, we define real options, look at practical illustrations of real options, look at methods to value them and finally, explain why evaluating them using Black-Scholes is not always the accurate thing to so. Method: We will gather all the necessary data required for pricing the option using the Black Scholes option from the market. The data on historical option parameters are available from Bloomberg. We then derive the theoretical option price using the Black Scholes model and compare it with the actual market price to obtain the relative pricing error. The relative pricing error is used for further analysis and hypothesis testing. Generally the Index options have European style of exercise and the Stock Options have American Style of exercise. Since the Black Scholes model is used to value European options, we have looked at the index option prices for the 5 indices across US and Europe. These are AEX Index, DJX Index, OMX Index, SPX Index & UKX Index. We look at practical examples of real options for the second part and evaluate them using Black-Scholes. We also give a back of the envelope method for evaluation and explain the disparity in results. Conclusion: Almost 69% of our observations had less than 90 days left to maturity. 53% of the samples were in-the-money. It turned out that in most of the cases the options were in-the-money and overpriced by the Black Scholes model. We also found that the relative pricing error is generally higher for out-of-the-money options rather than in-the-money options. The scatter plot of the relative pricing error and the time left to expiry indicates that options with shorter time left to maturity are generally contribute to a higher pricing error compared to options with longer time left to maturity. We understand the importance of real options in contemporary finance and conclude that Black-Scholes formulation for real options is inaccurate because of the underlying assumptions in Black-Scholes formulation. We also try and analyze why risk-neutral pricing is inaccurate for incomplete markets.