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 Dec30 comment FTAP a-la Harrison, Kreps and Pliska @user8: I meant this one Nov9 comment Stochastic Differential Like any other differential, this differential is defined in terms of its integral - that's a bit of an overstatement. It certainly is the case for most of stochastic differentials, but in real analysis the basic differential on a real line is often defined formally way before integrals. Nov9 comment What is a canonical book or article to learn pair trading? The book "Pairs trading" by Vidyamurthy is a standard reference Apr14 comment How can the Wiener process be nowhere differentiable but still continuous? I would be careful with such explanation, though. For example, a straight line is extremely self-similar on various scales, however it is perfectly smooth. Also, one can say that smoothness is exactly local similarity to straight lines, isn't it? Apr14 comment How can the Wiener process be nowhere differentiable but still continuous? @Probilitator The gif actually looks like some old-school flight simulator in the mountain area :) Apr11 comment Convolution copula? There is no need or requirement for the two copulas above to be the same. Do you mean here that $$\mathbb{P}(X\leq x,y_{1}\leq Y\leq y_{2})=C_1(F_{X}(x),F_{Y}(y_{2}))-C_2(F_{X}(x),F_{Y}(y_{1}))$$ with $C_1\neq C_2$ in general? Apr10 comment How can the Wiener process be nowhere differentiable but still continuous? @Probilitator: thanks. 3d - is some quant joke I fail to understand? Apr9 comment unique equivalent martingale measure in incomplete markets Sure :) also, what does these square mean? That you take an expectation/integral of squared R-N derivative? Apr8 comment unique equivalent martingale measure in incomplete markets Are you missing come expectations in the right-hand side? Apr8 comment How to choose a risk-neutral measure when the market is incomplete? What is the reason to pick up $\Bbb Q$ to be closest to $\Bbb P$ w.r.t. some metric? Apr7 comment Simple pricing example confusion I think I see your point: when we are making an equivalent change of measure, we have to restrict ourselves to finite intervals of time, otherwise changing the drift changes the null events. Thanks Apr4 comment PDE pricing of barrier options in BS Regarding PDE approach, I'd say Wilmott follows it everywhere in his books. Actually that's the same as martingale approach + Markovian structure, but without mentioning the latter two things too often (as Shreve does, in contrast) and using instead $\Delta$-hedging-like arguments, which of course leads to the same PDE as the martingale approach does. So I'd be interested in a book with a similar approach, but slightly more formal on the PDE side (not necessarily on a stochastic side). Maybe there are some known textbooks of that kind, if not - nevermind. Apr4 comment PDE pricing of barrier options in BS Thanks for your answer. I actually didn't mean solution of PDEs, especially an analytic one, just a PDE formulation. At least one advantage it gives is useful formulas for Greeks. My point is that the PDE for barriers in BS is "derived" using arguments like "value of option satisfies BS before hitting the barrier", so obviously we need to solve BS equation with an additional boundary condition on the barrier. As usual, it is this obvious step that can make the whole result being incorrect - so I just wondered whether there is a detailed explanation of this. Apr4 comment Why use implied volatility For your second argument: I've only traded on FX a couple of years ago, and there the frequency of data seemed quite enough to make good estimates of volatility just based on a 5-minute-wide window. Of course, the market is quite dynamical, but even for such a fast market 5 minutes did not seem to be such a big window. Although that's a historical data, it seems to be more relevant to "current volatility" given the latter is continuous, than the IV. Apr4 comment Why use implied volatility I see a point in your first argument - but as in my comment to @Richard, isn't that argument only true given that a lot of people on the market are using BS model for vanilla, or at least using IV? It seems, that IV is of the following feature: if everybody uses it, then it is also of value to you as you are playing against others. If nobody uses it, it does not give you a lot of information, though. Am I correct? Apr4 comment Why use implied volatility In that case, don't we completely exclude from our glance the situation when "market prices derivatives incorrectly" which we may think of taking advantage of? Apr4 comment Why use implied volatility Thanks for the answer. Can you clarify a couple of points? 1. It is useful: yes. Do you mean here, that people in fact use BS to price simple contracts often enough? Cause in that case, I'd agree that it is interesting to take a look at implied volatility. 2. BS-implied vol of the prices calculated by these models fits the BS-implied vol that can be observed on the market. Isn't that equivalent to saying that model prices coincide with market ones? Apr1 comment Fundamental Theorem of Asset Pricing (FTAP) Hi, you may be interested in this question of mine Mar27 comment Does risk-neutral measure have anything to deal with risk-neutrality in utility theory? Thanks, I'm assuming in your case $E = E_P$: the expectation over a market measure. In such case, the map $X_T \mapsto \mathsf E_P[X_T]$ is linear as well, so nothing distincts it from the case of a martingale measure. Mar26 comment Does risk-neutral measure have anything to deal with risk-neutrality in utility theory? @Probilitator: indeed, I've modified this part - better now?