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It is the former. Martingales are defined by filtration and probability space. The probability space for the filtration need not be the same as the probability space for the martingale. I think? That's what my prof said (iirc).

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Could you please be more specific with your question and post the text here? This will be more helpful for other people visiting the site. Now as far as to where the 1/2 went, usually people put 1/2 in front of the second order term because this will simplify to 1 after the derivation: $$\frac{\partial x^2}{\partial x} = 2x$$ vs  \frac{1}{2} \cdot ...

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I tried to use the BPV/delta relashionship $\Delta = \frac{ \frac{\partial Z}{\partial r}-\frac{\partial P(t,T)}{\partial r}\frac{Z}{P(t,T)} } {\frac{\partial P(t,S)}{\partial r}}$ but it doesn't work as well.

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Here is a list of vendors connected to OPRA directly who can supply such data. http://www.opradata.com/related_info/vendor_list.jsp Like yourself I looked at Xignite. They have a nice REST API, though not streaming. But pricing is such that it is clear they only want to cater to institutions. Btw, with a \$30K budget you can probably become a Tier1 quote ...

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