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The skew/smile of long term options is flatter than short term options, the reason for this can be explained in several ways. The Vega of a shorter-dated option is smaller than a longer-dated option. Vega is the dollar value of a 1% change in implied volatility. i.e., 30d ATM option, $65 strike, .31 ivol = VEGA .07 30d 25 delta option,31% ivol = ... 0 The volatilities of short dated options are more sensitive to market changes as compared to those of long dated options. This is implied by square root of time rule. As such, volatility skew are larger for short dated options. 5 One possible reason could be jumps. Over the longer maturity, there could be more jumps so the jumps average out in a way; whereas over the short term, a jump can make a bigger difference and hence the risk of jump increases demand. This reasoning is used to justify Stochastic volatility with jumps models in some books. 0 Given the variation, ATM vol = alpha * F ^(beta-1), if your stochastic process for forward price dF= alphaF^beta dW, that means your effective beta, CEV, is 1. This gives horizontal backbone of the vol surface. I think it all depends on whether this is what you expect to see - the vol surface is stickey under shocked price scenarios. 5 Your question is twofold How a market maker should adjust its quotes on a vol surface with respect to his inventory? How to adjust the vol surface when a new trade is observed on the markets? Let me focus on the market making question, and that for you need to be familiar with optimal trading and optimal market making literature: A breakthrough has been ... 2 Yes, that's what we wish to see from the correctly-specified model. Now, let me try to answer your 2nd and 3rd questions together as they are based on the same confusion. There are two different concepts: model-implied volatility and model-implied BSIV (Black-Scholes Implied Volatility). I think you are confused because of mixing them up. So yes, people ... 1 One way to think about this is to forget about equities (for a moment), and think about credit. 90% of the time, credit just gets paid. 5% of the time, credit still gets paid; but a booming economy means that rates rise, so the increased certainty of getting paid is worth less than the decline to NPVs from the coupon becoming worth less. 3% of the time, ... 2 The Black-Scholes model was based on assuming lognormal stock price fluctuations with a constant volatility. However, the modern practice is to use the Black-Scholes formula not as a prediction but merely as a parametrization of option prices, where the observed price of a given option at a given time translates to a "local" implied volatility (IV). Thus, ... 2 No, and this is wrong. The implied vols (from market prices) are actually not necessarily convex but yet may be still arbitrage-free, there are many examples of this for various equities. Furthermore, preserving convexity is not necessarily enough either. In terms of implied variance$w(y)=\sigma^2 T$as a function of log-moneyness$y=\ln\frac{K}{F}$, the no ... 1 If you recall the derivation from Breeden and Litzenberger (1978), all you need (other than no-arbitrage and infinitely many call options) is the following$\max\{S_0e^{−qT} − Ke^{−rT} , 0\} \leq C(S_0,K,T) \leq S_0e^{−qT}$for all strikes$K \geq0$,$\frac{\partial C(S_0,K,T)}{\partial K}\geq -e^{-rT}$for all strikes$K \geq0$,$\lim\limits_{K\to\infty}\...

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As @XiaotianDeng mentioned, the simple at-the-money approximation you mention does not always hold: it works only if you assume that $\alpha^2 T, \nu^2 T$ are small, typically $o(1)$. I wanted to add that there is really no need for such an approximation, except, possibly, to do calculations in your head, or for understanding the scale of $\alpha$ against \$\...

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Using a cubic spline or worse, SVI is overkill to find the at-the-money (ATM) volatility when it is not quoted by the market: both approaches are global in the sense that a small change of one of the quotes far from the money will have a not so small impact on the at-the-money implied vol. Yes, one solution is to truncate the range of option strikes ...

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