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Since it does not have a closed form solution for the price, it's unlikely to be a martingale. However, on the other hand, if we represent the price as a function of the current stock price and the average price so far, we can write down the formula: $$v_n(s,y)=(1+r)^{-1}[\tilde{p}v_{n+1}(us,(y*(n+1)+us)/(n+2))+\tilde{q}v_{n+1}(ds,(y*(n+1)+ds)/(n+2))]$$

where $S_n=s,\frac{1}{n}\sum_{k=0}^nS_k=y$ and $u,d$ are up/down factors. This seems to be a martingale.

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[Short answer]

Yes, one can always represent option prices as martingales, hence expectations, assuming there is no free lunch.

When there is no closed-form formula for an option price, it just means that the latter expectation is not analytically tractable.

[Long answer]

In the absence of arbitrage opportunities, one can always appeal to the existence of equivalent probability measures under which option prices (I should rather say the discounted price of all self-financing strategies attainable using marketed securities) emerge as martingales. This is sometimes known as the fundamental theorem of asset pricing.

More specifically, assuming the market model is complete (which is the case for the binomial tree pricing model), each of these measures can be shown to correspond to a unique choice of so-called numeraire, a numeraire being an asset $N_t > 0$, such that the price $V_t$ of any tradable option, when expressed in numeraire units i.e. $V_t/N_t$, is a martingale.

The equivalent martingale measure associated to the money market account numeraire $$ N_t = B_t = e^{\int_0^t r (u) du}$$ Is best known as the risk-neutral measure.

In the absence of arbitrage opportunities, one can thus always write: $$ V_0 = \mathbb {E}^{\mathbb {Q}^B}[ e^{\int_0^t r (u) du} V_T ] \tag{I}$$ where the terminal value of the option $ V_T$ is simply its payoff by absence of arbitrage, payoff which can be written as $$ V_T = f (S_{t_1},...,S_{t_N}) $$ where $f (.) $ is a completely generic function of $N $ equity fixing values observed over the horizon $[0,T]$ ($N $ need not even be finite).

Not having a closed form formula for an option price does not mean we cannot write the price in martingale form $(I)$, hence as an expectation, but simply means that the expectation in question is not analytically tractable.

This can happen because the underlying diffusion model is intractable per se. For instance, even European call/put options have no closed-form formula under the local volatility model or the binomial tree pricing model: one has to restor to numerical methods.

This can also happen for a tractable diffusion model because of the exotism of the target option: e.g. there is no closed-form formulas for discrete Arithmetic Asian options in the simple Black-Scholes model, while there exist formulas for both the Geometric and continuous arithmetic version.

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