Derivation of Black-Scholes Equation Using Itô's Lemma

Year 2021, Volume: 3 Issue: 1, 38 - 49, 15.06.2021

Brandon Washburn , Mehmet Dik

https://doi.org/10.47086/pims.956201

Cited By: 2

Abstract

The Black-Scholes Equation is arguably the most influential financial equation, as it is an effective example of how to eliminate risk from a financial portfolio by using a hedged position.
Hedged positions are used by many firms, mutual funds and finance companies to increase the value of financial assets over time.
The derivation of the Black-Scholes equation is often considered difficult to understand and overly complicated, when in reality most confusion arises from misunderstandings in notation or lack of intuition around the mathematical processes involved.
This paper aims to take a simple look at the derivation of the Black-Scholes equation as well as the reasoning behind it.

Keywords

Stochastic, Black-Scholes Equation, Itô Calculus

Thanks

Thank you to Ben Stucky for advising me on this paper and Darah Chavey for leading my mathematics colloquium

References

• [1] Kiyosi Itˆo, RIMS (1994).
• [2] F. Black, and M. Scholes, The pricing of options and corporate liabilities, World Scientific (2019) 3–21.
• [3] T. Habb, What in the world will I ever use this for? Integration, Environmental Economics (2014).
• [4] J.C. Hull, Options futures and other derivatives, Pearson Education India (2003).
• [5] D. Khoshnevisan, and Y. Xiao, Stochastic analysis and related topics, Springer (2017) 179- 206.
• [6] R.C. Merton, Theory of rational option pricing, The Bell Journal of Economics and Man- agement Science (1973) 141–183.
• [7] A. Petters, and X. Dong, Stochastic calculus and geometric brownian motion mode, An Intro. Math. Finance. App. Springer (2016) 253-327.
• [8] K. Rubash, Myron Scholes and Fischer Black, Bradley University.
• [9] H. Stark, and J.W. Woods, Probability, random processes, and estimation theory for engi- neers, Prentice-Hall, Inc. (1986).