The derivatives of Asian call option prices

TL;DR

This paper derives integral formulas for pricing Asian options and their sensitivities, focusing on the distribution of the time integral of geometric Brownian motion, which is crucial for understanding Asian option pricing.

Contribution

It introduces integral representations for Asian option prices and Greeks, advancing the understanding of the distribution of the time integral of geometric Brownian motion.

Findings

Provides explicit integral formulas for Asian option prices.

Derives formulas for Greeks: delta, gamma, theta, vega.

Enhances methods for analyzing Asian options' dependence on parameters.

Abstract

The distribution of a time integral of geometric Brownian motion is not well understood. To price an Asian option and to obtain measures of its dependence on the parameters of time, strike price, and underlying market price, it is essential to have the distribution of time integral of geometric Brownian motion and it is also required to have a way to manipulate its distribution. We present integral forms for key quantities in the price of Asian option and its derivatives ({\it{delta, gamma,theta, and vega}}). For example for any $a>0$ $\mathbb{E} [ (A_t -a)^+] = t -a + a^{2} \mathbb{E} [ (a+A_t)^{-1} \exp (\frac{2M_t}{a+ A_t} - \frac{2}{a}) ]$, where $A_t = \int^t_0 \exp (B_s -s/2) ds$ and $M_t =\exp (B_t -t/2).$