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20

I recently read "Modeling financial data with stable distributions" (Nolan 2005) which gives a survey of this area and might be of interest (I believe it was contained in "Handbook of Heavy Tailed Distributions in Finance"). Another more recent reference is "Alpha-Stable Paradigm in Financial Markets" (2008). I'm not aware of anything covering "risk of ...


17

There are several application of Lévy alpha-stable distributions to finance, especially in insurance and reinsurance. I believe that Embrechts-Kluppelberg-Mikosh's "Modelling Extremal Events for Insurance and Finance" is still an excellent reference. However, in the modeling of stock prices, this line of research is essentially inactive. The reason is that ...


17

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*} \...


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 ...


13

Statistically you would apply Bessel's correction to address the bias you point out. However, that misses the point that the variance-covariance matrix is non-stationary, suffers from the curse of dimensionality, and that the noisy mean return estimates have significantly more impact than a biased covariance matrix on portfolio weights. The best ways to ...


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 ...


10

I am still a beginner to this topic, and have been working through Cont and Tankov's textbook Financial Modelling With Jump Processes (2003), which is a fairly elementary treatment of the subject. I think a revised second edition is to come out later this year. One interesting area of applications that has become more prominent with a recent wave of papers ...


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

I just ran across an interesting presentation comparing the effectiveness of Normal, Cauchy, and Student's-t distributions in modeling the S&P. It concludes that the normal distribution underestimates extreme movements, the Cauchy overestimates them (although a comment on the presentation points out that Mandelbrot used different parameters than the ...


9

I am implementing a method in Java to calculate the variance, covariance, and value at risk for a portfolio, which should be flexible for use with any number of assets in a portfolio. I am struggling with how to calculate the covariance of the assets as I can only find formulae to do so for two or three sets of values. Are you sure you are ...


7

Var and vol swaps are very similar products, with the leverage (convexity) being the biggest theoretical difference, yes. In the actual market however they are very different. After the 2008 debacle var swaps in the single stock space are not too common, whereas single stock vol swaps are regularly quoted. One interesting perspective is trading one versus ...


6

I know you're really looking for some empirical work on this topic, but I think the following theoretical paper puts your question into proper perspective.* Risk-Based Asset Allocation: A New Answer to an Old Question by Wai Lee, JPM 2011. Overall, he finds that supposedly risk-based approaches to portfolio construction are really making implicit ...


6

Derman et al has a long note on this from 1999. Variance swaps are actually the more natural choice. It has nothing to do with leverage. From the linked article: Although options market participants talk of volatility, it is variance, or volatility squared, that has more fundamental theoretical significance. This is so because the correct way to ...


6

PCA gives you a decomposition of the covariance matrix of the form $$ \Sigma = V \Lambda V^T $$ where $\Lambda$ is diagonal with the eigenvalues in the diagonal. Your portfolio variance is $$ w^T \Sigma w = (V^T w )^T \Lambda (V^T w) $$ On the other hand if you take your return matrix $R$ and define $$ F = V^T R $$ then the covariance matrix of these so ...


6

We first list the assumptions. \begin{align*} g_{t+1} &= \mu_g + \sigma_{g, t} z_{g, t+1}, \tag{1}\\ \sigma_{g, t+1}^2 &= a_{\sigma} + \rho_{\sigma} \sigma_{g, t}^2 + \sqrt{q_t} z_{\sigma, t+1}, \tag{2} \\ q_{t+1} &= a_{q} + \rho_q q_t + \varphi_q \sqrt{q_t} z_{q, t+1}. \tag{3} \end{align*} Moreover, \begin{align*} r_{t+1} &= -\ln \delta +\...


5

As you know both var swap & vol swap are traded on vol. The difference comes in convexity. Although variance swap payoffs are linear with variance they are convex with volatility. Because of the convexity, a variance swap will always outperform a contract linear in volatility of the same strike. This convexity is the reason that variance swaps strikes ...


5

Volatility = Variance^1/2 = Standard Deviation


5

The term 'rule of thumb' is ambiguous here. Because I don't think there are any rule of thumb, you just need to do the number crunching. However there are some stable characteristic through time linked to correlation. For instance it is a common fact that the hierarchy of correlation within different market is relatively stable. US equities are less ...


5

So my first answer was off base. For some reason I was thinking first moment (idiosyncratic returns), but he's looking for second moment (idiosyncratic volatility). There is a line of research on the returns to portfolios sorted on idiosyncratic volatility and I was hoping that there were descriptive statistics that said "fraction $\rho$ of stock/portfolio ...


5

I just want to give a qualitative assessment to your question: Volatility of a market is different than the volatility of a stock. Similarly like Copeland and Antikarov (2001) say that "...the volatility of a gold mine is different than the volatility of a gold..." If you want to quantitatively compute the percentage of a stock's volatility affected by ...


5

The given matrix can not represent a covariance matrix since it would imply that asset 1 is negatively correlated to asset 2 and asset 3. But asset 2 is negatively correlated to asset 3 which contradicts the first statement. In general a covariance matrix has to be positive semi-definite and symmetric, and conversely every positive semi-definite symmetric ...


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

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

The vega of an option is very dependent on the spot price. The vega of a variance or volatility swap is not.


4

An Axioma research paper from August 2011, Using Multiple Risk Models for Superior Portfolio Management… A Practice Not Just For Quants, answers exactly your question, I believe. Note the graphs at the top of page 8. They compare their medium-horizon fundamental and statistical factor models from January 2008 to January 2009. At the start of the period, ...


4

Clearly, from a theoretical point of view, a varswap is a better way of capturing volatility change, since as mentioned by Mark Joshi a varswap has, by construction, a Vega that does not vary with the stock price. For a single option on the other hand the Vega is at maximum at a stock price $S^*$ roughly comparable to the strike price X and decays in a "bell ...


3

the only difference between volatility and variance is the square. everything else is bs, as concept that apply to one applies to the other (historical vs implied, blabla)


3

Here are the exact steps to calculate TSRV. I also like this paper.


3

Let’s take a simple example to answer a broad but interesting question: Imagine that we have a daily return serie denoted $r_{t}$ ( which is assumed to be stationary) and let's take a little time to define main concepts : Mean Process (First moment process) The unconditional mean of $r_{t}$ denoted $u$ is just its expectation $E(r_{t})$. It is not time ...



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