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18

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 ...


16

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 ...


15

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 ...


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 ...


10

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 ...


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 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 ...


8

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 ...


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

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 ...


5

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 ...


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

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 ...


3

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, ...


3

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


2

Suppose X is a random variable representing the returns of an asset having finite mean $\mu$ and variance $\sigma^2>0$. Variance $\sigma^2$ represents the expected squared deviation of $X$ from $\mu$. Intuitively, this is a measure of how dispersed returns are about the mean. If returns are measured in $\%$, then the units of variance are $\%^2$. ...


2

It sounds like all you need is to run a logistic regression, with the sign of $Y$ as your dependent variable instead of $Y$ itself. This will only give weight to the sign of the variable, and not to the magnitude. Once you have reformulated your question in more general terms (sign and magnitude of $Y$, rather than direction and volatility), you may be ...


2

If the variances are known to be $\sigma_0$, $\sigma_1$ and $\sigma_2$ and the correlations are $\rho_{01}$, $\rho_{02}$ and $\rho_{12}$ then you can do exactly as you suggest - write down the variance of the total portfolio as a function of your holdings $x_0$, $x_1$ and $x_2$ and set the partial derivatives with respect to $x_1$ and $x_2$ to zero. You end ...


2

I would recommend to use simple standard deviation (among the 2 options you offered). You are performing time series analysis of historical data points, you are not forecasting. Thus, why exposing yourself to a much more computationally intensive method? May I also point you to a related (not duplicate) thread: Why are GARCH models used to forecast ...


2

I don't know what you mean by "any scaling" rule. For the square-root of time I can say that it only needs uncorrelated returns. Assume that the return from time point $1$ to $T$ is called $r_{1,T}$ and that it is given as $r_{1,T} = r_1 + r_2 + \cdots + r_T = \sum_{t=1}^T r_t$ where $r_t, t=1,\ldots,T$ are the one-period (e.g. one day) returns. The ...


1

Wikipedia gives: $\sigma(x,y) = E[xy] - E[x]E[y]$ and $\sigma(ax+by,cz) = ac\, \sigma(x,z) + bc\, \sigma(y,z)$ (paraphrasing the $\sigma(ax+by,cW+dV)$ rule). So $\sigma(I,A) = \sigma([aA+bB+cC+dD],A)$ $\sigma(I,A) = a\,\sigma(A,A) + b\,\sigma(B,A) + c\,\sigma(C,A) + d\,\sigma(D,A)$ $\sigma(I,A) = a\,\sigma^2(A) + b\,\sigma(B,A) + c\,\sigma(C,A) + ...


1

definition of a variance swap is $ \int^{T+\Delta}_T \mathbb{E}_t[v_s] ds $ where $v_s$ is the variance and $\mathbb{E}_t[v_s]$ is the expectation of the variance of time s at time t. therefore, pnl is: $ (\int^{T+\Delta}_T \mathbb{E}_t[v_s] ds - \int^{T+\Delta}_{T} \mathbb{E}_{t-\delta}[v_s] ds)*d\delta $


1

A variance swap has a set of fixing times, and the volatility between those times has no specified effect. Therefore you end up wanting to apply a model. For a model-free approximation, though, your formula works up to a constant.


1

First of all, I'm not sure I got it right. You're directly shrinking the final result (vector of asset weights), right? If that's the case, it may not be exactly what you're looking for, but you may still have a look at some papers by Ledoit and Wolf. Specifically, in Honey, I Shrunk the Sample Covariance Matrix they propose that shrinkage always be ...



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