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It's really quite simple. It's just a matter of the fact that we can change measure on the stochastic volatility while not changing the fact that the stock is a martingale. Once we can do this, we have payoffs that have different values under different measures, so the market can't be complete. For clarity, just consider a stock S, a money market account ...

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I think a sketch of the proof would look like this Let's say you start from $$S_t = S_t \odot (\mu_t dt + \sigma_t dW_t)$$ where $S$ is an vector valued process of your $n$ risky assets prices, $W$ a standard $k$-dimensionnal brownian motion under the historic probability, $\sigma_t$ an $n \times k$ matrix valued process and $\odot$ is the Hadamard ...

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The paper by Marc Romano and Nizar Touzi, Section 3, contains a general proof that a stochastic volatility model cannot be complete in the sense that the addition of the option completes the market (in the sense of Harrison and Pliska) generated by the underlying and risk-free borrowing/lending: ...

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TLDR: Massive expansion of credit fuelled by rehypothecation, a general shift to repo, then the scale tips and everyone pays as credit collapses. Quants were there, but I don't think they can be simply blamed for all the ills of the world. There is a general disagreement about what caused what, so some of this is guesswork. I'm marking this a community wiki ...

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