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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?

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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.

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  • $\begingroup$ Thanks! Would you further explain why $E^P[S_\tau m_\tau] = S_0 E^{\bar{P}}[e^{-q\tau} m_\tau ]$? $\endgroup$ – Leslie Wu Jan 25 '19 at 22:37
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    $\begingroup$ 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 $A_t/S_t$. $\endgroup$ – Antoine Conze Jan 26 '19 at 6:53
  • $\begingroup$ 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 $\{ V_\tau | 0<\tau\leq T\}$. $\endgroup$ – Leslie Wu Jan 27 '19 at 18:43
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    $\begingroup$ 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. $\endgroup$ – Antoine Conze Jan 28 '19 at 8:02

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