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Assume I pricing some commodity derivative that has a running cost with $\$c$ being paid per unit of time. So I define the price under the risk neutral measure to be $$P(t,S_t)=\tilde{\mathbb{E}}\left[e^{-r(T-t)}F(S_T)+\int_t^T c\,e^{-r(s-t)}\text{ds}\,|\,\mathcal{F}_t\right]$$ Now if I multiply both sides by $e^{-rt}$, I have "discounted price of a derivative" on the left, and from the fact that discounted price functions are martingales under risk neutral measure I could calculate $d(e^{-rt}P(t,S_t))$ and set the $dt$ term to zero.

But clearly, that would be a wrong pde in this case. I can see that if I break $\int_t^T=\int_0^T-\int_0^t$ and move the second part to the left, I will get a martingale and I should do Ito on that one. So does the argument "all discounted traded derivatives are martingales under risk neutral measure" doesn't apply in this case? And the argument valid only for the derivatives that have a payoff at maturity $T$? I have not see that mentioned anywhere in EEM pricing.

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  • $\begingroup$ I am not sure what you mean by running cost. Does one have to pay $c$ per unit time for holding the derivative? $\endgroup$
    – Mark Joshi
    Sep 2, 2016 at 2:27
  • $\begingroup$ @Mark Joshi: yes, for example storage fees. $\endgroup$
    – Medan
    Sep 2, 2016 at 2:34
  • $\begingroup$ normally the fees are holding the underlying rather than the derivative so I am a little confuse.d $\endgroup$
    – Mark Joshi
    Sep 2, 2016 at 2:40
  • $\begingroup$ @Mark Joshi: could also be accumulated credit, or medium term notes, I think those pay something along the life... $\endgroup$
    – Medan
    Sep 2, 2016 at 2:43

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self financing portfolios have discounted prices that are martingales. So if the products involves paying fees, these have to taken account of to get a martingale. The product is not a self financing portfolio if these are ignored.

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  • $\begingroup$ can you be more specific how to identify the self-financing portfolio? I thought this notion would apply to the hedging portfolio, for example in a case of plain European option, the replicating portfolio of stock and cash would be self-financing. $\endgroup$
    – Medan
    Sep 2, 2016 at 13:19
  • $\begingroup$ Consider $N$ tradable assets $S_i$, $\Pi = \sum_{i=1}^{N} w_i S_i$ is self-financing iff the infinitesimal P&L incurred when holding $\Pi$ over a time interval $dt$ is: $$ d\Pi = \sum_{i=1}^{N} w_i dS_i $$ $\endgroup$
    – Quantuple
    Sep 5, 2016 at 17:49
  • $\begingroup$ Knowing this, consider a non-dividend paying stock $S$. The portfolio which consists in going long the former stock is clearly self-financing since $d\Pi=dS$. Should the stock $S$ now be assumed to pay dividends - e.g. a continuous dividend yield $q$ w.l.o.g. - then the portfolio consisting in going long the stock will yield an infinitesimal PnL: $$ d\Pi = dS + Sqdt $$ meaning that it will not be self-financing anymore. Do you see the similarities with your original question? $\endgroup$
    – Quantuple
    Sep 5, 2016 at 17:49

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