This is something that keeps bothering me about the Benchmark approach of Platen, which (very) shortly is as follows: Compare the development of an economic value with a growth optimal portfolio. Taking the expectation in the real world measure $\mathbb{P}$, conditioned on today's information, this should yield a fair and in some sense arbitrage free price, because you compare to the best possible (in expectation). Now, I want to know the price of a future on $X_T$ (hence $F(T,T)=X_T$). In discrete time, one just has to take the discounted future payments. For a time interval partition $\sigma$, this yields:
$\Sigma_t^{(\sigma)} = \sum_{i=1}^N\frac{1}{S^{\pi^*}_{t_i}}( F(t_i,T) - F(t_{i-1},T) )\\ = {\sum_{i=1}^N\frac{1}{S^{\pi^*}_{t_{i-1}}}( F(t_i,T) - F(t_{i-1},T) )} +{\sum_{i=1}^N\left(\frac{1}{S^{\pi^*}_{t_i}}-\frac{1}{S^{\pi^*}_{t_{i-1}}}\right)( F(t_i,T) - F(t_{i-1},T) )}$
Going to continuous time, this converges to $\int_t^{T} \left(\frac{1}{S^{\pi^*}_{s-}}\right) dF(s,T)+\int_t^{T}d\left[\left(\frac{1}{S^{\pi^*}}\right),F\right]_s$. Since the net position is zero and using the product rule for semimartingales, we get for the futures price at time t: $ \mathbb{E}\left[ \frac{F(T,T)}{S^{\pi^*}_{T}}-\int_t^{T}F(s-,T)d\left( \frac{1}{S^{\pi^*}_s} \right) \big|\mathcal{F}_t\right] = \mathbb{E}\left[ \frac{F(t,T)}{S^{\pi^*}_{t}} \big|\mathcal{F}_t\right] = \frac{F(t,T)}{S^{\pi^*}_{t}} $ To me it seems intractable. Is there any way to come the futures price process even close? I read some extensions to the usual equivalent martingale measure, cont., FV interest process one takes usually and which yields $F(t,T)=\mathbb{E}^{\mathbb{Q}}[F(T,T)|\mathcal{F}_t]$. But none of these can be applied here. Any ideas?