Pricing an American Asian Option with a Running Average Strike

Summary

The document considers an American Asian call whose strike is the underlying asset’s average price from inception to the exercise time. It presents a change of measure that rewrites the discounted payoff in terms of the ratio between the running average and the current asset price. Under the stock risk-neutral measure, the original problem becomes an American option problem on that ratio, with a payoff based on its difference from one.

The transformed ratio has a one-dimensional stochastic process, which makes numerical valuation with finite differences or an American Monte Carlo method practical. The answer does not provide a closed-form approximation; it says a useful one is not known. Its setup assumes the stated risk-neutral asset dynamics and does not give numerical results, implementation details, or a comparison of numerical methods.

Key ideas

• A running-average strike can be expressed through the ratio of the average to the current asset price.
• A change to the stock risk-neutral measure turns the discounted payoff into an expectation involving that ratio.
• The ratio follows a process with time-dependent drift and volatility proportional to the ratio.
• Finite differences or American Monte Carlo can value the transformed stopping problem.
• The document does not offer a closed-form approximation or numerical comparison.

Tags

# Pricing American style Asian option


# Pricing American style Asian option












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?

## Answer by Antoine Conze (score 1, accepted)

https://quant.stackexchange.com/a/43645

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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