# Pricing an Asian style forward contract with early exercise feature

Is there an analytic way to price or approximate a contract with payout $$A_t - K$$, where $$A_t$$ is the running average price of the underlying asset from $$[0, t]$$ and $$K$$ is (fixed) strike.

If this is an European style contract, then I think we can replicate it using put-call parity. What if it is American (early exercise)? How to price/approximate the value of the early exercise option in such contract?

Welcome to Quant SE. Unfortunately there is no closed form formula for computing the american contract value $$\max_{\tau}E^P\left[e^{-r\tau}(A_{\tau} - K)\right]$$, so you have to resort to an american monte carlo method or a 2 dimensional PDE finite differences scheme for the joint dynamics $$dS_t/S_t = (r - q) dt + \sigma dW_t \\ dA_t = d\left(\frac{1}{t} \int_0^t S_u du\right) = \frac{S_t-A_t}{t} dt$$ In the case where $$r=0$$ the problem reduces to computing $$\max_{\tau}E^P\left[A_{\tau} \right] - K = \max_{\tau}E^P\left[S_{\tau}m_{\tau} \right] - K = S_0\max_{\tau}E^{\tilde{P}}\left[e^{-q\tau}m_{\tau} \right] - K$$ where $$m_t=A_t/S_t$$, $$\tilde{P}$$ is the stock risk neutral measure, and the dynamics for $$m_t$$ under $$\tilde{P}$$ is $$d m_t=\left(\frac{1-m_t}{t}+qm_t\right)dt+\sigma m_t d\tilde{W}_t$$ and you can resort to a 1 dimensional PDE finite differences scheme.
• Thanks! Would you further explain why $E^P[S_\tau m_\tau] = S_0 E^{\bar{P}}[e^{-q\tau} m_\tau ]$? Jan 25, 2019 at 22:37
• The stock risk neutral measure is defined as $d\tilde{P}/dP|_t = e^{-(r-q)t}S_t/S_0$ so that for any random variable $X$ measureable wrt the filtration on $t$, $E^{\tilde{P}}[X] = E^P[(d\tilde{P}/dP) X]=E^P[e^{-(r-q)t}(S_t/S_0) X]$. Next to obtain the dynamics of $m_t$ under $\tilde{P}$, first apply Girsanov to get $dS_t/S_t=(r-q + \sigma^2) dt + \sigma d\tilde{W}$, then apply Ito to $A_t/S_t$. Jan 26, 2019 at 6:53
• Thanks for the explanation. I guess in practice I can first evaluate $V_\tau = E^P[A_\tau - K]$ for each day $\tau \in (0, T]$ using Asian call and Asian put options (put-call parity), assuming I have an Asian option pricer, and then take the maximum of $\{ V_\tau | 0<\tau\leq T\}$. Jan 27, 2019 at 18:43
• Doing that would give only a lower bound to the price, as it amounts to choosing at time 0 a fixed exercise date $\tau$. To get the exact solution you need, same as for any american option, a backward algorithm (finite differences scheme or american monte carlo) that starts from maturity and goes backward in time checking for optimal exercise at each point in time. Jan 28, 2019 at 8:02