# 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 $E^P[S_\tau m_\tau] = S_0 E^{\bar{P}}[e^{-q\tau} m_\tau ]$? – Leslie Wu Jan 25 '19 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 $A_t/S_t$. – Antoine Conze Jan 26 '19 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 $\{ V_\tau | 0<\tau\leq T\}$. – Leslie Wu Jan 27 '19 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. – Antoine Conze Jan 28 '19 at 8:02