Revisiting the Black-Scholes model related to the volatility assumptions

Abstract

The Black-Scholes (BS) or Black-Scholes-Merton (BSM) formula is the most popular model that is used to price vanilla options. However, some of the assumptions underpinning this formula have been found to be false (Dumas, 1998). One such assumption is stated below:"There are no transaction costs on the underlying asset" (Wilmott, 2006).This assumption advances the idea that markets are frictionless and therefore there are no transaction costs incurred in those markets. This in fact is not true. It is costly to carry out financial transactions in financial markets. Taxes are normally applied to transactions that take place in various financial jurisdictions (Wilmott, 2006).One other BS assumption says that "there is a single constant volatility for the stochastic pro-cess followed by the spot" (Austing, 2014). If the world were truly BS in nature, the volatility quoted for each of the options with different strike prices in a liquid market would be the same (Austing, 2014), (Kwok, 2008). In other words, if the implied volatility quotes were plotted against the strike prices, a straight horizontal line would be obtained (Austing, 2014).However, after the financial crisis of October 1987, it was discovered that when the ImpliedVolatilities (IV) of a group of vanilla options of the same maturity were plotted against strike prices, the graph had the shape of a smile (Dupire, 1994). This is called the volatility smile. This was not expected. The shape of the graph that was expected was that of a straight hori-zontal line.The above issues and others have resulted in the values of options that are calculated by the BS not being fully accurate. This has created a gap in the body of published literature which needs to be filled.In addition to the above issues, another of the inaccuracies in the BS has been attributed to the IV parameter. All the parameters that are input into the BS formula are observable except the IV (Kwok, 2008), (Wilmott, 2006), (Dupire, 1994). Therefore, those parameters that areobservable can have their values determined accurately from observable evidence whereas the value of the IV does not enjoy this privilege.In fact, the IV is normally determined by the sentiment of the financial markets. This is not very accurate. The quest of this research is to develop a mathematical model that would determine the values of the IVs more exactly which can be used to obtain more accurate results from the BS formula.This approach will be unique and has not been evidenced in literature. It will help to fill the gap that currently exists in the body of knowledge. The idea is to use this mathematical model to calculate more precise values of the IV which will be input into the BS formula in order to improve the accuracy of the BS formula.The secondary benefit from the approach taken by this research is being able to determine the strike prices that match the given IVs. Sometimes the strike prices agreed upon by the market participants can be out of kilter with the other parameters in the BS formula. This re-search would be able to help resolve that.

Date of Award	2 Jun 2017
Original language	English
Awarding Institution	University of Bedfordshire
Supervisor	Sandar Win (Supervisor) & Rodolphe Ocler (Second supervisor)