Chris IJ Hwang

I am a Quantitative Analyst/Developer and Data Scientist with backgroud of Finance, Education, and IT industry. This site contains some exercises, projects, and studies that I have worked on. If you have any questions, feel free to contact me at ih138 at columbia dot edu.

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#Intro

Taylor Expasion is very useful especially in measuring risk and returns of the portfolio with non-linear functions.

Definition: A Taylor expansion about a particular point $x_0$ provides an approximation from a polynomial function in the region of $x_0$.

$n^{th}$ order Taylor Approximation of $f(x)$ about $x_0$ :

$f(x) \approx f(x_0) + (x-x_0){f}'(x_0) + \frac{(x-x_0)^2}{2!}{f}''(x_0) + ... + \frac{(x-x_0)^n}{n!}{f}^{(n)}(x_0)$

Now, change a little bit as the below:

$f(x+\Delta x) - f(x) \approx f'(x_0)\Delta x + \frac{1}{2}{f}''(x_0)(\Delta x)^2 + ... + \frac{f^{(n)}(x_0)}{n!}(\Delta x)^n$

Applications

delta-gamma approximation in Option
duration-convexity aproximation in Bond
datla-gamma-vega approximation in Options : Multivariate Taylor Expansion case

References:

[1] Alexander, Carol. Market Risk Analysis. Vol. I. Chichester, England: Wiley, 2008. Print.