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22

Volatility is mean reverting if the underlying security doesn't drop to zero. If the security has some underlying "value" then its price is co-integrated with that "value". The volatility is the uncertainty of that price as it tracks the security's "value". Edit 12/03/2011 ================================================= @pteetor, I may have missed ...


17

Many of them are on my website at emanuelderman.com. Others I probably have anyway. Feel free to email me


16

Volatility is typically unobservable, and as such estimated --- for example via the (sample) variance of returns, or more frequently, its square root yielding the standard deviation of returns as a volatility estimate. There are also countless models for volatility, from old applied models like Garman/Klass to exponential decaying and formal models such as ...


15

Volatility is mean reverting because you can prove by contradiction that it cannot be otherwise. You have an intuitive understanding of why, but you need something closer to a proof. Assume volatility is not mean reverting. At time t, the effect of the random component of the volatility on its level will be $\sigma \cdot \sqrt{t}$ For an arbitrarily ...


12

One of the reasons the ARCH family of models is used is that you only need price data to generate the model. These data exist back to the 1800s, so ARCH is great for looking at volatility over very long periods. I don't know that I'd say that the ARCH model has a lot of problems -- it solved the problem of not allowing volatility in time or in the level of ...


12

You may want to first broadly categorize volatility models before comparing between them within each class, it does not make sense to compare standard deviation models with an implied vol model. I would broadly classify as follows: Historical realized volatility: Those include standard deviation (sum of squared deviations), realized range volatility ...


11

Increased volatility (high VIX) signifies more risk. To keep their portfolio in line with their risk preferences, market participants deleverage. Since long positions outweigh short positions in the market as a whole, deleveraging entails a lot of selling and less buying. The relative increase in selling causes downward pressure on stocks.


11

The main underlying difference is in their definition. Variance has a fixed mathematical definition, however volatility does not as such. Volatility is said to be the measure of fluctuations of a process. Volatility is a subjective term, whereas variance is an objective term i.e. given the data you can definitely find the variance, while you can't find ...


11

I've read N. Taleb. Dynamic hedging for exactly the same reason and found it quite helpful. You can find a preview at Google Books to examine the content - the greatest thing about this book that N. Taleb tries to show how things work in pracice not just how to derive another formula (what is a subjsect for other great books on quantitative finance).


11

The price of a binary option, ignoring interest rates, is basically the same as the CDF $\phi(S)$ (or $1-\phi(S)$ ) of the terminal probability distribution. Generally that terminal distribution will be lognormal from the Black-Scholes model, or close to it. Option price is $$C = e^{-rT} \int_K^\infty \psi(S_T) dS_T$$ for calls and $$ P = e^{-rT} ...


11

The usual technique of computing the mean and standard deviation of returns happens to coincide with the maximum likelihood estimate when the data are regularly spaced. However, when the data are not regularly spaced, you can still do a maximum likelihood estimate. It's just more computationally intensive than before. That is to say, assume you have ...


11

Setting aside, that it's not pure riskless arbitrage, but rather statistical arbitrage: You can extract the profit by performing continuous delta hedging. If you constantly adjust your hedge position you gain/lose money by delta hedging. Being long option (gamma long), you sell at higher prices and buy at lower ones. Over the course of time you realize ...


11

Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance is defined by \begin{align*} \frac{252}{n}\sum_{i=1}^n \ln^2 \frac{S_{t_i}}{S_{t_{i-1}}}. \end{align*} Note that, for sufficiently small $x$, \begin{align*} ...


10

Great question! I think the most useful starting point is Stock Return Characteristics, Skew Laws, and the Differential Pricing of Individual Equity Options by Bakshi, Kapadia and Madan (2003). Their paper proposes a definition of model-free implied skewness (they originally called it risk-neutral skewness, but MFIS is more accurate), which they prove will ...


10

Some cynical but functional definitions: It's what you can't model if you're not using tick by tick data It's what proper quant pricing theory doesn't know how to model yet It's information (order book behavior) that reflects momentary fluctuations in the supply/demand of a given contract, rather than its underlying value (eg an arbitrage free price) ...


10

The volatiltiy surface is just a representation of European option prices as a function of strike and maturity in a different "unit" - namely implied volatility (while the term implied volatility has to be made precise by the model used to convert prices (quotes) into implied volatilities - for example: we may consider log-normal vols and normal vols). ...


10

I think you are interpreting too much into the matter. The $-\frac12\sigma^2$ is just a correction term that comes from Jensen's inequality. You need this when switching from supposedly symmetric returns (normal distribution) to the skewed price process (log-normal distribution). I think there are no deeper truths to be found here.


9

By volatility people usually refer to to annualized standard deviation of an asset. For an asset it's usually quoted as a percentage of the asset price (i.e. the return volatility). For a portfolio, it is often quoted in currency units. Variance is the square of the standard deviation. It is usually not quoted directly because it doesn't have an intuitive ...


9

Implied volatility is the volatility implied by some model. You will have a skew if your model is implying different volatilities for different strikes. However, the realized volatility of the underlying will be the same for all strikes. So, when you are dealing with realized vol, you can drop the "moneyness" axis. Volatility cones can help you compare ...


9

It seems that your question refers to the microstructure noise defined in papers about intraday volatility estimates. Originally, it comes from the bid-ask bounce, i.e. the fact that even if the volatility is zero, you have buyers and sellers at this price and consequently you observe prices at Bid or Ask prices, and not at mid-price. Because of that, if ...


9

The main issue measuring intraday volatility is called "signature plot": when you zoom in, the volatility measure (i.e. empirical quadratic variations) explode. Similarly you have the "Epps effect" for correlations: when you zoom in, the correlations collapse (it is at least a mechanical effect). For the volatility a lot of models can correct this: - first ...


9

I do not have the time right now to write up a summary concise enough but at the same time trying to really touch on all the points that have to be made to delineate the above. Instead I point you to couple papers that are concise enough to skim over in a matter of minutes in order to understand the differences. Jim Gatheral on Local vs Stoch Vols: ...


9

There is no "plain Black Scholes implied surface" because implied volatilities come from options market prices (calls and put). If you had a whole continuum of call prices $C : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$, $(T,K) \mapsto C(T,K)$ you would get a implied volatility function $\sigma_I : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ ...


8

GARCH(1,1) is a "standard approach for modeling volatility" mainly in academic literature. Most of us in the real world don't use it. Volatility forecasting tends to come more from looking at more-liquid comparables for future market volatility than from fitting fancy retrospective models. As for ignoring the dependence of residuals, well, folks are ...


8

The optimal growth portfolio is obtained by applying the Kelly criterion which is one of the pillars of the sound risk management. Ed Thorp's weekend forays to Las Vegas to play blackjack were one of the first historically documented cases of successful practical implementation of the Kelly strategy. Since then this method and its modifications have been ...


8

Technically, yes, the VIX is a measure of implied volatility. But practically speaking, it is a measure of market uncertainty: when market participants are uncertain of the future, they buy options to protect their positions, driving up option premiums and increasing implied volatility. The broader market hates uncertainty, however, so that same uncertainty ...


8

Intraday seasonality is a major factor in comparing volatility at different times of day. Most time series display significantly higher volatility in the morning EST than mid-day. For US exchange-traded products, volatility picks up again just before 4:00 PM EST. This is known as the u-shaped volatility pattern for exchange-traded products. A proper ...


8

The expression you have is fine. But more generally, for the intraday volatility, I don't think there "the correct definition". More like, whatever works in the given context. I found the following notes by Almgren pretty useful: http://cims.nyu.edu/~almgren/timeseries/notes7.pdf


8

This is correct: "The general idea of cleansing a correlation matrix via random matrix theory is to compare its eigenvalues to that of a random one to see which parts of it are beyond normal randomness." This is not correct: "These are then filtered out and one is left with the non-random parts." The term "filtering", although used extensively in the ...


8

The way market makers mark their volatility curves is by using models which 'fill in the gaps', i.e. they will make a price for a given option even if they do not believe this option is going to get a lot of volume. They are still willing to go long/short because they have a strategy to hedge their overall position (i.e. by managing their greeks and ...



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