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I have a question about the cost process of a trading strategy. Suppose we work in the finite discrete case and holding just one risky asset for simplicity. Let $\phi=(\theta,\eta)$, where $\theta$ should be predictable and models the amount of shares and $\eta$ models the bank account. Then the cost occuring over $k$ to $k+1$ are (working with discounted objects.)

$$\Delta C_{k+1}=C_{k+1}-C_k=(\theta_{k+1}-\theta_k)S_k + (\eta_{k+1}-\eta_{k})$$

Where $S_k$ is the price of the risky asset at time $k$. Now I have a very simple question. If $\Delta C_{k+1}\le 0$, i.e. the costs over this time are negativ, does this mean, I gain something? What would this mean in reality? The bank/broker would give some money for buying/selling my risky asset or borrow/pay back money to the bank?

thanks for your help

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Where have you seen a negative transaction cost? For that matter, where did you see $\phi$, $\theta$, $\eta$? What are these? How does someone model a bank account? – chrisaycock♦ Oct 3 '12 at 12:50
@chrisaycock I'm studying math. Usually you assume that the strategy is self-financing, i.e. $\Delta C_{k+1}=0$. But I want to know what the economical interpretation of a negative cost is. It's a general question and does not depend on $\phi,\theta,\eta$. It's a purely mathematical formulation and I want to know the economical interpretation of it. – hulik Oct 3 '12 at 12:54
I've never heard of a self-financing strategy. As per the FAQ, this site is intended for professional quants to ask practical questions. – chrisaycock♦ Oct 3 '12 at 12:59
@chrisaycock Although this question is not very deep, it makes perfect sense (at least for me). Self-financing strategy is an essential ingridient in risk-neutral option pricing and hedging, sometimes it's called self-financing portfolio see en.wikipedia.org/wiki/Self-financing_portfolio – Alexey Kalmykov Oct 3 '12 at 17:35
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@FKaria hulik doesn't say about negative hedging cost. $\Delta C_{k+1}\le 0$ doesn't mean negative hedging costs. It means that cost process is decreasing. – Alexey Kalmykov Oct 5 '12 at 11:46
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closed as not a real question by chrisaycock♦ Oct 3 '12 at 12:59

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, see the FAQ.