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14

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 ...


10

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 ...


8

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 ...


5

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 $\...


5

If you take Quantuple's stuff a little further, you can really see whether you're long skew. You can pretty easily see the dependence on convexity too (though it should be obvious that you're long convexity). So first off, we need some smile parametrisation that lets us easily control convexity and skew. I just went with a made up one; $$\mathrm{convexity} ...


5

As I've mentioned in a comment, it would be wrong to think that a variance swap specifically amounts to being "long skew". What you can say however is that, in the absence of jumps (i.e. in a pure diffusion framework, see here and here for further info), the fair variance strike $K_{var}$ at which a variance swap with notional $N$ and payoff $$ N \times ( \...


5

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 ...


4

You do not state whether your evaluations will result in potentially implementing multiple strategies or just one of them. This matters because if you are going to be combining multiple ones then you need some reasonable capital allocation assumptions, which increases complexity immensely. Let's take the simpler case where you just want to choose one. ...


4

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


4

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 $f_G(x;\...


4

It is not the fact that volatility is time varying that creates the skew per se, but the fact that volatility is negatively correlated with the spot. That is to say, as the stock/index price declines volatility will tend on average to increase, and vice versa. Time varying volatility itself would create a more symmetric 'smile'. Edit: Suppose that you ...


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 cut-...


4

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 ...


4

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

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 ...


2

When I saw these curves they seemed very strange to me. I believe it is a data-quality issue.I went to Bloomberg and I retrieved the implied vols for 70 near ATM strikes of the weekly SPX options expiring November 27 2015 (I believe that is the yellow curve in your diagrams i.e. November 4th week). This was today 2015-oct-27 at about 15:00 New York time. As ...


2

The market does not follow Black-Scholes assumptions, as you clearly know : there is a skew and vol levels change. Neither does it follow any other particular known model. So when you say "dynamically hedge" you have to understand this as an approximate hedge that still leaves some significant risk. Vols will move, and not always together and in the way ...


2

Basically there are three steps to accomplish this. 1 - collect time series of options for several expirations and strikes. 2 - calculate implied volatility surface for every time period, and use model-based or model-free interpolation to create continuum of strikes / expirations. 3 - from the continuous surfaces you can calculate series of any specific ...


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

There's no best method. The question is : what is the behavior of the volatility structure (atm and skew) when the underlying moves? Each method assumes something different. In the real market, one method might work well for a period of time (in the sense that it minimizes residual p/l), but then another method might take over as best. Practitioners ...


1

The formula in your skew function is one of skew normal distribution. That distribution has a limit on skew parameter, while in the real world there is no such limit. From personal experience, few years ago I tried doing exactly what you described in your question. After comparing skew normal distribution on SPX with the real world, I concluded that there ...


1

The answer is that by definition, if the underlying stock obeys a lognormal distribution with std deviation parameter sigma, then the implied vol of options priced using this model will be sigma. Of course in the market we observe that options of different strikes have different vols - this just means that the underlying distribution is not perfectly ...


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_+} -\sigma_{...


1

In optimiazation system, you have to weight the price for the different maturities in a way that reflect your confidence in each data point (influenced by liquidity). One way to do so is to weight, each price by its Black-Scholoes Vega (see Tankov (2003)). So when minimazing the squared differences of the sum your weighted option prices, you can use the ...



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