Hamilton's principle and delta neutral portfolio
Date of Issue2009
School of Mechanical and Aerospace Engineering
Black Scholes formula is a Nobel Prize winning formula for determining the price of option based on no arbitrage principle. In its derivation, Black and Scholes came up with a model based on setting up a delta neutral portfolio. Applying no arbitrage principle on the delta neutral portfolio, the Black Scholes partial differential equation is obtained. The partial differential equation is solved by transformation to a form similar to heat diffusion equation and an appropriate solution for heat diffusion equation is applied. This project analyzes the delta neutral portfolio and no arbitrage principle and applies the Hamilton’s principle on to obtain the general expression for the return of delta neutral portfolio based on profit optimization method. Further analysis of the assumption of Black Scholes model and applying the Black Scholes formula, the profit expression is given an exact form which is different from the implied profit of delta neutral portfolio based on Black Scholes model. The delta neutral return equation obtained in this project is based on the payment schedule of a treasury note and includes arbitrage measure which utilizes Black Scholes formula for measuring the deviation of real option price from a “correct” price. The inclusion of arbitrage profit explains why investors will want to invest in delta neutral portfolio over a risk-free government bond investment and can be used as a more accurate yardstick for making investment decision. The fundamental inconsistency of Black Scholes model is not solved by applying Hamilton’s principle in Black Scholes model. A new model based on Hamilton’s principle is proposed. Based on Hamilton’s Principle, the spread between the price of a derivative, representing a potential earning in the future, and the current asset on which the derivative is based should be a minimum. The potential application of the new model is as an indicator for stability of a derivative market.
Final Year Project (FYP)
Nanyang Technological University