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Whereas when you marketmake on a last-look basis: - You, the marketmaker, are sending indicative prices to the ECN - The ECN sends orders to you and is at risk (since you have the option to reject, hopefully rarely) When you marketmake on a no-last-look (NLL) basis: - the ECN is sending indicative prices - You, the marketmaker, send orders to the ECN and ...

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To answer your question: I mean in a theoretical sense: If we have a particular market model (which I guess we may assume is complete or frictionless if need be) where shorting and fractional purchases are allowed, does presence of arbitrage necessarily make all kinds of derivatives have zero value? The answer is no. See example below. Went over your ...

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I am also interested in the answer to this question, and would like to expand a little bit on it as well. First of all, let me add some value in terms of a partial answer: There are restrictions on when short selling is allowed. According to the SEC, and the "Alternative Uptick Rule" short selling is not allowed on "a stock that has dropped more than 10 ...

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What if you write $$P[R_{n+1} = d|F_n] = 1 - P[R_{n+1} = u|F_n] ?$$ Let us write $P(u) = P[R_{n+1} = u|F_n]$ Then the part to show is $$u \bar{S}_n P(u) + d \bar{S}_n (1-P(u))$$ and this $$\bar{S}_n \left(d +(u-d)P(u) \right),$$ where we just expanded terms and then extracted the coefficients.

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You should make your borrow cost sufficient to dissuade unlimited short selling. In practice, each short would require you to borrow shares from your broker. This is usually handled when computing transaction cost. You should account for this in your trading algorithm or in the factor model itself. A simple method would make shorts some N% more expensive ...

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