Risk-Neutral Pricing of Derivatives with Running Costs

Summary

The document explains why a derivative with a continuously paid running cost cannot be priced by treating its discounted ex-cost value as a martingale. The expected value includes both the discounted terminal payoff and the discounted stream of costs, so the derivative alone does not satisfy the usual self-financing assumption behind the martingale argument.

The proposed adjustment is to account for the accumulated costs when constructing the martingale quantity, then apply Itô’s lemma to that quantity to derive the pricing equation. The accepted response emphasizes that discounted prices are martingales for self-financing portfolios; fees or other cash flows must be included in the accounting. The discussion is conceptual and does not derive the resulting PDE or specify sign conventions for who pays the running cost, so those details depend on the contract setup.

Key ideas

• Discounted derivative value alone need not be a martingale when the contract has running cash flows.
• The martingale argument applies to self-financing portfolios under the risk-neutral measure.
• Running costs must be included in the portfolio value or cash-flow accounting before applying Itô’s lemma.
• The exact pricing equation depends on how the contract defines and assigns its running costs.

Tags

# pricing a derivative with a running cost


# pricing a derivative with a running cost












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.

## Answer by Mark Joshi (score 2, accepted)

https://quant.stackexchange.com/a/29945

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.