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9

Just to be painfully clear, it only seems to make sense to consider the logarithm of returns, i.e. $X=\log (1+\frac r{100})$ for a simple return of $r\%$ in an arbitrary period because this is what sums when returns are temporally aggregated. A basic property of cumulants is that cumulants of all orders are additive under convolution, for which a proof can ...


9

There have been many studies of the shape of volatility skew. There are two simple empirical approaches that have proven most popular: Spline (or piecewise polynomial) fits of observable implied vols Parabolic parametric fits. and there is one other approach that is fairly simple yet reliable: Edgeworth expansions Other empirical approaches also ...


6

Skew is indeed a widely used word and can represent one of the following: Skew(ness) - 3rd standardized moment that represents assymetry of the distribution (olaker metioned it his answer). (Volatility) skew - is observable property of implied volatility surface that can be seen on the market after the 1987 crash. It shows that OTM puts (high demand) are ...


4

Generally, you should ignore deep in-the-money option prices because they are far less liquid (due to their low leverage). That lets you use just the calls on the high strikes and just the puts on the low strikes. For your purposes, you can cap the volatility at, say, the highest mid-market volatility of all puts having both a bid and an offer. This ...


4

First we must define what we mean by implied volatility. Let $c_{BS}(t,S(t),K,T;\sigma)$ denote the price of the call option with strike price $K$ and maturity $T$ in the Black-Scholes model with the volatility $\sigma$ (emphasized in the argument). Furthermore, let $c_{MA}(t,S(t),K,T;\sigma)$ denote the corresponding price on the market. The volatility ...


4

Scott Mixon argues in What Does Implied Volatility Skew Measure that among all measures of implied volatility skew, the (25 delta put volatility - 25 delta call volatility)/50 delta volatility is the most descriptive and least redundant (volatility is Black-Scholes implied volatility). His paper, recently published in the Journal of Derivatives, gives a ...


3

VG belongs in the family of variance-mean mixture models. Given a horizon $T$ the distribution of log-returns $f$ is a mixture of Gaussians $f_G$ with randomised mean and variance. The randomisation density is $g$ and its mean and variance increase with $T$. For the VG process this randomised factor is Gamma-distributed. More concretely, denote with ...


3

Implied volatility skew is simply collection of implied volatilities on the same underlying instrument for a given expiration. Term "implied volatility skew" is only loosely connected to statistical definition of skewness. Implied volatility surface is the collection of implied volatilities on the same underlying for several expirations. If BS formula were ...


3

The skew of a distribution is a measure of its assymetry. Let $X_n$ be a discrete process (say, of daily returns) with mean 0 and noncentered volatility $\sigma$. Then the noncentered skew is defined as $$\frac{1}{n}\sum_{k=1}^{n}\frac{X_k^3}{\sigma}.$$ It will be positive is $X_n$ and $X_n^2$ are positively correlated and negative if they are negatively ...


3

Judging by the math in a paper by Vahamaa (1999), you should measure the slope using the options closest to the strike you are examining. In other words, suppose that you are trying to come up with the skew-adjusted delta of an SPY option with strike of 130. Then the skew slope should be based on the 129 and 131 strike options. If these points happen to ...


2

You have to remember that implied volatility comes from a "wrong" model to give the right answer. Option prices are determined by supply and demand (subject to a few arbitrage bounds). A higher implied volatility for OTM/ITM options relative to ATM options simply means that the prices of these options are higher than the Black-Scholes model would imply ...


2

put call parity guarantees that the implied volatility of a call and put with the same strike is the same. So the smile graph is the same as well and so are all quantities derived for it. In more detail, $$ C(K) = P(K) + F(K) $$ The value of $F(K)$ is model independent and does not depend on volatility. So knowing the implied of $C(K)$ gives you the price ...


1

Well the terminal FX rate is lognormally distributed and lognormals are skewed. So this is not surprising.


1

To expand on pbr142, If the implied volatility (vis. Black & Scholes) is persistently higher for short-expiry contracts away from the money, the problem is the model, not the thing that's modeled. The price of a contract at a given point in time is the "correct" price at that point in time (or we should move this to philosophy.stackexchange.com). So ...


1

This is a common convention. If your spot is $S$ and you're looking at options maturity in $T$, it is natural to look at the the strikes $S_\pm=S.exp^{-\frac12\sigma^2T\pm\alpha\sigma\sqrt T}$ for a fixed $\alpha$. So your skew measure will be something like $$ \frac{\sqrt T(\sigma_{S_+} -\sigma_{S_-} )}{\log (S_+/S_-)} = \frac{\sqrt T(\sigma_{S_+} ...



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