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Theoretically, such an asset would not exists with standard deviation less than the risk free rate. You have rightly pointed out that if risk free rate is higher than any rational investor would not invest in the instrument, given lower return along with some amount of risk, when the investor can earn more amount without any risk. In the real world we do ...


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The fact that Implied Vol rises has absolutely nothing to do with riskaversion. If market expects volatility before an upcoming uncertain earnings report, put option prices rise naturally. This is due to the asymmetric payoff profile of options, which always gain from volatility because the downside losses are capped but upside potentially unlimited.


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If you believe the process $Y_t$ to be stationary, you can try to profit from it via a mean-reversion strategy or any other way that exploits the stationarity. It doesn't matter whether $Y_t$ is obtained as a cointegrational combination of a few non-stationary processes, or as a linear combination of some processes that are stationary themselves. In the ...


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Jaeckel has a paper "Let's be rational" in which he "show how Black’s volatility can be implied from option prices with as little as two iterations to maximum attainable precision on standard (64 bit floating point) hardware for all possible inputs.". I guess it doesn't qualify as closed-form for you, though one might argue that having to apply a ...


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The method described in Hallerbach (2004) always worked well for me. We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [JBaF, 1996] estimator, has substantially higher approximation accuracy and extends over a wider region of moneyness.



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