Mathematics of parallel stratagems - Jalan, Priyal (2017)
Parallel stratagems are used as hedging strategies by investors to minimise their exposure to risk in the financial markets. Options are one of the most popularly used instruments for effective hedging. A number of mathematical models have been developed for predicting the price of option contracts that enable investors to optimally hedging their portfolios. Three of the most widely used models are the Black-Scholes for pricing European options, Binomial Tree for pricing American options and Monte Carlo Simulations for pricing Asian options. All of them are built on fundamental financial theories like the no-arbitrage principle, efficient market hypothesis and risk-neutral measures. However, these theories are not completely realistic and showcase certain discrepancies from the real world behaviour of financial markets. This is one of the root causes for the inaccuracies in the model‟s predictions. The empirical tests of the Black-Scholes model indicate that though it is fairly accurate in pricing all European options, the model performs best in short-time horizons and at times of low market volatility. Moreover, the model is not well-suited for pricing American and Asian options as it gives unfavourable results when compared with the other two models of Binomial Tree and Monte Carlo. In addition to that, critical evaluation of the model highlights a major mathematical error in its derivation that renders it logically incoherent. This report aims to correct that mathematical error and in general, proposes plausible modifications or extensions to the existing Black-Scholes model so as to make it empirically stronger as well as suitable for pricing options of styles other than European. ii The report makes three proposals out of which two (Proposals 1 and 3) are deemed to be successful as they are supported by logical reasoning and generate favourable results. Proposal 1 corrects the model‟s mathematical error by assuming that the time-integrated value of an option‟s delta is a constant, and Proposal 3 aims to make the model a better fit for pricing Asian options by minimising the fluctuations in the value of a delta hedged portfolio throughout its lifetime.