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16

You can use changepoint analysis to identify regime change. You can also look at large angle differences in the eigenvectors between your most up-to-date/recent covariance matrix and the covariance matrix from the prior window. Another way to identify regime change is using a factor model. If the returns on a particular set of factors is X standard ...


7

What you refer to as the 99.5th percentile is known as the "Value-at-Risk." You are correct that you will need to make a distributional assumption, and there is a popular and well-researched approach to this problem, though I'm not certain it could be called "standard." I would recommend you use the "truncated Levy flight" distribution. James Xiong at ...


7

I would suggest a multivariate garch model as a possibility. We aren't exactly overrun with wonderful software for that, but with just bivariate data I would think that the in-sample correlation estimates would be reasonably robust over models and estimation. It would be good to try two or three ways of doing it to make sure I'm right about that. You may ...


6

Very interesting question. I am not an expert on the subject, however, I was able to find a collection of papers on the subject that should get you started. Here is a good and very informative paper that walks you through several tick by tick volatility estimators that seek to reduce the volatility imposed by market micro-structure: Efficient estimation of ...


6

Estimating $MA(q)$ models is significantly harder than $AR(p)$ models. Eviews, MATLAB and R can use multiple algorithms which are all based on some form of maximum likelihood estimation. You can look at the source of MATLAB and R or the excellent Eviews documentation. However, I strongly advise against rolling your own since efficient and well tested ...


6

Nonlinear optimization algorithms are very susceptible to starting points, so some problems with same structure can become difficult to solve compared to others. A few suggestions: For a few instances where you are having difficulty in getting answers, try using another solver. You can try Excel, Matlab or R, all of which can be used for fitting. Try adding ...


6

I personally use the simple Garch(1,1) for volatility filtering in the risk management area. In fact in most cases I don't even estimate the parameters, I stick 0.94 for mean reversion, 0.04 for the squared error and I get the constant by matching the series variance. My experience is that there is no point pretending to finetune parameters when vol is ...


6

There is no standard approach to this problem to the best of my knowledge. Different approaches exist and each has its own pros and cons as usual. To mention a few: Information-based methods: these aim at "risk-neutralising" the distribution observed under the physical measure relying on some information criterion e.g. minimising the KL divergence, or ...


5

The blog post http://www.portfolioprobe.com/2011/11/21/asynchrony-in-market-data/ explains a bit more about the problem and it also points to a paper that shows that a moving average model is the way to make the adjustment that Tal is seeking. The paper is presented in the context of a multivariate garch model. That is gratuitous, really -- the MA estimate ...


5

I think an extremely interesting strand of research on this topic is represented by extensions of vine copulas with time-varying parameters. For vine copulas in general have a look at this site from the Technische Universität München: Vine Copula Models One of their research projects, which is the most relevant in this context, is:Time varying vine copula ...


5

The use of kernels to estimate volatility using intraday data is "nothing more" than combining: intraday volatility estimation kernel smoothing Thus you have to take care about the "usual pits" of these two approaches. Intraday volatility estimation. I hope you know the "signature plot" effect. Of course if you use the proper estimation method, it should ...


4

The answer to the original question is simple: the Chopra-Ziemba paper is highly flawed and unreliable. Note that the framework is in-sample and based on a utility function. It has nothing to do with out-of-sample behavior of the mean vs. the covariance in an optimization. Estimation error grows linearly in the mean but quadratically in the covariance. At ...


4

Did you try solving for $w_k$? $$\bar{r}_t = \sum_{k=0}^p w_k r_{t-k}$$ $$\bar R = W R$$ Since you probably have $t>>k$, you can solve for $W$ using OLS $$\bar R = W R +\varepsilon$$ -- UPDATE You can try applying Kalman filter. Here, your state evolution is $$r_t=\mu+\varepsilon_t$$. You introduce new vector $x_t=(r_t, r_{t-1}, \dots, r_{t-p+1})$...


4

Thanks @Aksakal for suggesting Kalman Filter. Here I provide more details. We will view it as a state-space model: $$ \begin{split} z_t &= A_t z_{t-1} + B_t u_t + \epsilon_t, \\ y_t &= C_t z_t + D_t u_t + \delta_t, \\ \epsilon_t &\sim \mathcal{N}(0, Q_t),\ \delta_t \sim \mathcal{N}(0, R_t), \end{split} $$ where $z_t$ is the latent variable, $y_t$...


4

If $\log{(|R_t|)}$ is your first term, I'm not sure why this is a matrix. Modulus (determinant herein) applied to a matrix $R_t$ gives a scalar. If your implementation in python produces a matrix, that's likely because modulus is treated as an element-wise abs() function for each element of a matrix. It may be easier and faster to use rugarch (univariate ...


4

If you estimate your model via Maximum Likelihood method, you are forced to re-estimate the full model. This is due to the fact that estimates are values which maximize the full likelihood, the latter being based on a recursive algorithm which use all observations (including the new one) and implies that a new observation may also impact likelihood values of ...


4

Not really. For infinite maturity bonds we have $Price = coupon/yield$ so your approximation is actually correct. However for short dated bonds it is not a good approximation. For example , a 1 year annual pay bond gives $Price=(1+coupon)/(1+yield)$ which is very poorly approximated by $coupon/yield$.


3

One approach would be Engle (2002) dynamic conditional correlations. Taking your $Y_t$ and $X_t$, I will make the simplifying assumption that the mean equation of these is: $$\boxed{Y_t = \mu_y + \varepsilon_{y,t}}$$ $$\boxed{X_t = \mu_x + \varepsilon_{x,t}}$$ with $\varepsilon_{y,t} = z_{y,t} \sigma_{y,t} \sim N(0,\sigma_{y,t})$, $\varepsilon_{x,t} = ...


3

There are many techniques, but I would begin with Stambaugh Analyzing Investments Whose Histories Differ in Lengths. The full information maximum likelihood approach he describes basically involves regressing the short history series against the long history series to obtain the covariance with the longer history securities and adding back the covariance of ...


3

The pdfs of Student-t distributions have asymptotically Paretian tails, and the tail shape parameter (aka the maximal moment exponent) is equal to the distribution's degrees of freedom parameter. Assuming you have enough observations, you could estimate the Pareto parameter using the so-called Hill method (named after Bruce Hill, 1975). A word of caution: ...


3

Since you're asking on a quant finance forum, the mathematical approach would be Decide on a model that the stock price follows, and Compute the expected value of the price, conditional on the most recent price. A famous model, made ubiquitous by Black, Scholes and Merton, is a geometric Brownian motion. Under this model, the stock price $S_T$ at time $T$ ...


3

$\alpha=0$ does not imply constant volatility. Consider just a simple Garch(1,1): $$\sigma^2_t = \omega + \alpha \eta_t^2 + \beta \sigma^2_{t-1}$$ Note that: $$\sigma^2_t = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2- \sigma^2_{t-1})$$ Now add $\eta_{t+1}^2$ to both sides: $$\eta_{t+1}^2 = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2-...


3

First, you should use an exponential moving average, since the amount of state you need to keep is much smaller than for a simple moving average. Second the well known estimator of volatility, $$ \hat{\sigma} = \sqrt{\frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})^2} $$ is not very robust, since the squaring amplifies the contribution of outliers (which is why ...


3

Carr-Madan formula tells you that the European-style payoff $f(F_T)$ can be decomposed as: $$f(F_T)=f(\kappa) + f'(\kappa) [(F_T - \kappa)^+ - (\kappa - F_T)^+] + \int_0^{\kappa} f''(K) (K-F_T)^+ \ d K + \int_{\kappa}^{\infty} f''(K) (F_T-K)^+ \ d K$$ for any positive $\kappa$ of your choice. Assuming deterministic rates w.l.o.g. as well as a complete ...


3

Based on an my updated understanding of your problem you have a portfolio consisting of $N$ illiquid assets. Valuations are not real time and usually lagged, by say, upto 3 months (or slightly longer), but at least valuations correspond to a consistent timestamp (or otherwise you interpolate a consistent timestamp). You want to construct a predictive model ...


2

Here's an answer from a purely statistical point of view: http://www.duke.edu/~rnau/regnotes.htm#constant And another from Cross Validated: https://stats.stackexchange.com/questions/7948/when-is-it-ok-to-remove-the-intercept-in-lm The lean in both cases is to include the intercept unless there is a strong theoretical reason. A more satisfying answer ...


2

Time-series regression is not a great method for determining betas on individual securities. Rather, the most common method used by the commercial risk model providers is called "predicted beta" or "fundamental beta." The leader in this area is Barra. The way they define the predicted beta, it appears that they include the constant in the regression.


2

Unless it is due to random chance, there seems to be a bias in your estimation method for $\kappa$, and this bias appears to depend on the size of the sample. This may be revealing a deeper underlying problem with your technique that will ultimately make it clearer what the tradeoff is between accuracy and sample size. I do not believe it should be the ...


2

Once we start building time-varying copulas like Lopes suggests in that paper, I think we are better off venturing into the world of state space models. When viewed in a bayesian context, the similarities between the approaches are striking to me. The advantage of the copula, as I understand it, is that it is a quick and dirty way to understand the ...


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