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Usually it's aggressiveness of your orders + short term alpha. While it's equities not FX, I think you may find something interesting in the recent Barclays LX darkpool lawsuit documents.

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The first thing you have to understand that volatility is an abstraction, and there are different possible implementations of this abstraction in terms of trading. When someone writes "short spot index volatility, long on implied volatility" they mean something like take a position in options (implied vol) and delta hedge in the underlying instrument, ...

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They are equivalent. From Definition 1, note that $S_1^0 = S_0^0(1+R)$. Then \begin{align*} x_0 S_1^0 + \cdots x_n S_1^n &= x_0 S_0^0 (1+R)+ x_1 S_1^1 + \cdots x_n S_1^n\\ &= (-x_1 S_0^1 - \cdots -x_n S_0^n) (1+R)+ x_1 S_1^1 + \cdots x_n S_1^n\\ &= x_1 \left(S_1^1 - S_0^1(1+R) \right) + \cdots + x_n \left(S_1^n - S_0^n(1+R) \right)\\ ...

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Short answer: They seem not equivalent. On the first definition, the risk-free rate is part of the portfolio, and enters with quantity $x_0$. In the second definition, they discount the $S_1^n$ quantities by the risk free rate, implying that you have an arbitrage only if the gain you get on your portfolio (at time 1), is greater than the gain you could have ...

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To my point of view, the answer is hidden in your question. You correctly stated some of the BS assumptions and empirically it is proven that they are not true (volatility is not constant and the assumption regarding the distribution of returns is unrealistic due to fat tails). The model is as good as its assumptions are. Given that volatility is the ...

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Well, hopefully your calculations are right. There are a few things to remember: The carry can be higher than what you are thinking. Very often you will get charged if you are long or short. That can cost a lot depending on the name. Implied is theoretically always higher than realized. You are selling insurance. You should collect a premium more ...

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The answer by @HenriK is certainly correct. However, for justification, technique such as the Jensen inequality is needed. For example, since $x^+$ is a convex function, assuming zero interest and zero divdiend, \begin{align*} E\big((S_{T_{2}}-K)^+ \mid \mathcal{F}_{T_1} \big) &\ge \big(E(S_{T_{2}} \mid \mathcal{F}_{T_1})-K\big)^+\\ &=(S_{T_1}-K)^+. ...

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This question is extremely interesting and not that straightforward. See answer here. From a financial perspective this is very much like pricing an American call (stopping rule = intrinsic value from exercice (i.e. current cash earned) > continuation value (i.e. what you can expect to gain). Note that you can never win more than 13 nor lose (at worst you ...

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