# Taylor series expansion and hedging

I'm reading Options, Futures and other derivatives of Hull, ed.8. In the appendix to chapter 18, author uses Taylor series expansion to find the relationship between portfolio's price change and change of time and price of underlying. Then, he ingores terms which are of higher order then $\Delta t$, which is understandable for me. However, it is stated in the book without an explanation, that $\Delta S^2$ is of order $\Delta t$. Could someone explain to me why this is true?

• A warm welcome to Quant.SE and thank you for this important question. Please see my answer below. – vonjd May 3 '18 at 17:49

I would like to drive your attention to my paper which answers this question on page 12ff. (although I would encourage you to read the paper from the beginning):

von Jouanne-Diedrich, Holger, Ito, Stratonovich and Friends (May 18, 2017). Available at SSRN: https://ssrn.com/abstract=2956257 or http://dx.doi.org/10.2139/ssrn.2956257

Abstract
This exposition should provide you with the bigger picture of stochastic calculus, especially stochastic integrals. It heuristically and pedagogically develops key concepts and intuitions of one of the most important fields of applied mathematics today, namely quantitative finance. It demystifies ideas that a normally either too starkly dumbed down or hidden under highly technical details, so this text tries to fill a missing link in the literature where there seems to be no middle ground as of today. Additionally, the paper gives two results which cannot (to the best of my knowledge) readily be found in the classical literature: an illustration of the Ito correction term within binomial trees and a Taylor expansion for the Stratonovich integral.

I hope that this paper helps you to understand the involved concepts better. I would encourage you to give me honest feedback if things are still unclear or can be made clearer (also for future versions of this paper). Thank you.

• That's a very good reference! Thank you. Just for completeness a slightly more advanced reference (that I used to recommend before coming across your paper): epubs.siam.org/doi/pdf/10.1137/S0036144500378302 – phdstudent May 3 '18 at 18:15
• @vonjd Thank you very much for your paper, it was really helpful and gave the answer to my question – siwy9 May 3 '18 at 19:02
• @phdstudent: Thank you, I really appreciate that! Actually one of the very first books I had when I started out with Quant Finance was "An introduction to Financial Option Valuation" by Desmond Higham. – vonjd May 3 '18 at 19:03
• @siwy9: Thank you for your feedback and for accepting my answer. – vonjd May 3 '18 at 19:29