4
$\begingroup$

Can someone point me to any notes on how to derive the closed form formula for Asian geometric average option with payoff $\text{max}\left(\text{log}\left(\frac{A_T}{K}\right), 0\right)$ where $A_T$ is given by

$$ A_T=\text{exp}\left(\frac{1}{T}\int_0^T \text{log}(S_u) du \right) $$

We can assume that the stock follows a GBM as in the black scholes model.

$\endgroup$
  • $\begingroup$ I changed the small $t$ to capital $T$ in the definition of $A_T$. $\endgroup$ – Gordon Jul 8 '15 at 13:11
5
$\begingroup$

Note that \begin{align*} \int_0^T\ln S_u du &= \int_0^T\Big[\big(r-\frac{1}{2}\sigma^2\big)u + \sigma W_u \Big] du\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + \sigma\int_0^T\int_0^u dW_s \,du\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + \sigma\int_0^T\int_s^T du \,dW_s\\ &=\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T^2 + \sigma\int_0^T(T -s) \,dW_s,\\ \end{align*} which is normally distributed. Then \begin{align*} \ln \frac{A_T}{K} &= \frac{1}{T}\int_0^T\ln S_u\, du -\ln K\\ &\sim N\left(\frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T-\ln K, \ \Big(\frac{\sigma T}{\sqrt{3}}\Big)^2 \right)\\ &=\mu + \Sigma\, \xi, \end{align*} where $\mu = \frac{1}{2}\big(r-\frac{1}{2}\sigma^2\big)T -\ln K$, $\Sigma = \frac{\sigma T}{\sqrt{3}}$, and $\xi$ is standard normal random variable. Consequently, the option payoff has value \begin{align*} e^{-rT} E\left(\max\Big(\ln \frac{A_T}{K}, \, 0 \Big) \right) &= e^{-rT} E\left(\max\big(\mu + \Sigma\, \xi, \, 0 \big) \right)\\ &=\frac{e^{-rT}}{\sqrt{2\pi}}\int_{-\frac{\mu}{\Sigma}}^{\infty} (\mu+\Sigma \, x) e^{-\frac{1}{2}x^2}dx\\ &=e^{-rT}\bigg[\mu \Phi\Big(\frac{\mu}{\Sigma}\Big)+\frac{\Sigma}{\sqrt{2\pi}} \, e^{-\frac{\mu^2}{2\Sigma^2}}\bigg], \end{align*} where $\Phi$ is the cumulative distribution function of a standard normal random variable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.