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The SABR model of Hagan is described by the following Stochastic differential equations: $$\begin{align} & d{{f}_{t}}={{\alpha }_{t}}f_{t}^{\beta }d{{W}_{t}}^{1} \\ & d{{\alpha }_{t}}=v\,{{\alpha }_{t}}d{{W}_{t}}^{2} \\ & {{E}^{Q}}[d{{W}_{t}}^{1},d{{W}_{t}}^{2}]=\rho dt \\ \end{align}$$ In these equations, $f_t$ is the forward rate, ...


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I would say that $\log K/F$ points towards a log-normal type model. If I were you I would experiment with the moneyness defined as $K-F$ instead. This would make it consistent with normal dynamics. An alternative would be to define an 'interest rate floor', say $L=-200bp$ and take relative changes relative to that rather than zero, ie define moneyness as ...


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The first thing you have to understand that volatility is an abstraction, and there are different possible implementations of this abstraction in terms of trading. When someone writes "short spot index volatility, long on implied volatility" they mean something like take a position in options (implied vol) and delta hedge in the underlying instrument, ...


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Yes, your broker could have used one or combination of many factors: estimated volatility surface from historical returns of your target index, historical returns of similar indexes, implied volatility of similar indexes, existing inventory,etc. Check out these two approaches to deriving surfaces from returns starting slide 14


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Please read "Volatility's Impact On Market Returns" at http://www.investopedia.com/articles/financial-theory/08/volatility.asp. It is important to remember that VIX is a volatility index comprised of options and not stocks. It predicts volatility of future prices. It is not a measure of the present stock market. For a more thorough understanding see the ...


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Not sure what they use in Thinkorswim but I would assume some variation of a Cubic Spline, taking points along the skew and interpolating between them.


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Implied Volatility means the option is overpriced versus the model price. It means I'm paying more to buy an option than normally. For example, when a gap happens on the stock, the option price is high. It has no bearing on what will happen, but more on what did. The Bid and Ask price are set by people with expectations. And those prices are out of line with ...


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You can show that "the implied variance of an ATM short maturity option is equal to the expectation under the risk neutral measure of the integrated variance over the life of the option." As you move away from the assumptions: ie not ATM, longer maturity, risk neutral measure far from true, then the forecasting power diminishes. (Google 'stochastic ...


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1 is wrong. The implied vol is a convenient way to look at the option price, nothing more. 2 is an observed fact for equities in general but not the case for some other assets eg commodity futures. 3 is also an observed fact for equities generally (but not for single stocks with short time to expiry). If 1 and 2 were true, then 3 would naturally ...


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To my point of view, the answer is hidden in your question. You correctly stated some of the BS assumptions and empirically it is proven that they are not true (volatility is not constant and the assumption regarding the distribution of returns is unrealistic due to fat tails). The model is as good as its assumptions are. Given that volatility is the ...


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Well, hopefully your calculations are right. There are a few things to remember: The carry can be higher than what you are thinking. Very often you will get charged if you are long or short. That can cost a lot depending on the name. Implied is theoretically always higher than realized. You are selling insurance. You should collect a premium more ...


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Well... this is simply a picture to illustrate what is written in the text. It is not an absolute truth. The author just chose 2 implied volatility smiles that share the same ATM volatility level for clarity. One exhibits negative skew (typical of equity markets) and the other one is flat (you'll never observe that in practice, although it is exactly what ...


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The answer by @HenriK is certainly correct. However, for justification, technique such as the Jensen inequality is needed. For example, since $x^+$ is a convex function, assuming zero interest and zero divdiend, \begin{align*} E\big((S_{T_{2}}-K)^+ \mid \mathcal{F}_{T_1} \big) &\ge \big(E(S_{T_{2}} \mid \mathcal{F}_{T_1})-K\big)^+\\ &=(S_{T_1}-K)^+. ...


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I think (1) is the issue. You need to compare market normal vols to normal vols implied by the sabr model. (2) is not the issue - these vols look reasonable. By the way we express normal vols in bp per annum, not percent!


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See also this dissertation by Durrleman.


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This is a result of Ledoit et al Lemma proof in Appendix of http://www.ledoit.net/9-98.pdf



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