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Is there any approximation of American style Asian option (with strike equal to the running averaging from 0 to $t$) pricing based on analytical closed form formula?

I see the price difference between an European Asian option and an American Asian can be given by some integral form. Is there any way to simply it so it is easier to use?

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I don't think there is any good approximation to the american option $\max_{\tau}E^P\left[e^{-r \tau}(S_{\tau} - M_{\tau})^+\right]$ where $M_t = \frac{1}{t}\int_0^t S_u du$ is the running average, but you can compute it quite efficiently by noting that $$ \max_{\tau}E^P\left[e^{-r \tau}(S_{\tau} - M_{\tau})^+\right] =S_0\max_{\tau}E^P\left[\frac{e^{-r \tau} S_{\tau}}{S_0}(1 - m_{\tau})^+\right] =S_0\max_{\tau}E^{\tilde{P}}\left[(1 - m_{\tau})^+\right] $$ where $\tilde{P}$ is the stock risk neutral measure defined as $d\tilde{P}/dP=\frac{e^{-r t} S_{t}}{S_0}$, and $m_t=M_t/S_t$. From the original stock price dynamics under $P$ $$ dS_t=r S_t dt + \sigma S_t dW_t $$ a bit of Ito calculus and Girsanov theorem application yields the stochastic dynamics for $m_t$ under $\tilde{P}$ $$ dm_t=\left(\frac{1-m_t}{t} - r m_t \right)dt + \sigma m_t d\tilde{W}_t $$ You're left with computing an american option on $m_t$ with the above dynamics, using a finite differences scheme or an american monte carlo method.

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