# Tag Info

6

I basically agree with @John, let me expand: We want to model $y$ using a simple linear model, the most basic setup is $$y = c + \mathbf{X}\beta$$ with $y$ the $N$ observations, $c$ a constant, $\mathbf{X}$ the $N \times M$ matrix of regressors and $\beta$ a $M$-dimensional vector of coefficients. This model has $M$ parameters, the elements of $\beta$. ...

6

Let $R$ be a random vector of risky returns and let $r_f$ denote the risk free rate. Let vector of expected returns $\boldsymbol{\mu} = \operatorname{E}[R]$ and covariance matrix $\Sigma = \operatorname{Cov}(R)$. The maximum Sharpe ratio portfolio among risky assets is called the tangency portfolio. Quick method to tangency portfolio Let's find the ...

6

To complement @skoestimeier's answer on the shortselling-allowed case, I provide a vectorised version. Using the original notation in my post (you may change $r$ to something like $r-r_f$, but this doesn't affect the algebraic structure). Our goal is to find the maximiser for the problem $$\max_{w}f(w):=\frac{w^T r}{(w^T\Sigma w)^{1/2}}.$$ Let $$\phi: w\... 5 There are two cases, where short sales are allowed: With riskless lending and borrowing and without. As mentioned in the comments, you just have to solve a linear system. With riskless lending and borrowing The existence of a riskless lending and borrowing rate r_f implies that there is a single portfolio of risky assets, that is preferred to all other ... 4 We know that -1\le\rho_{imp}\le 1 so perhaps the simplest approach is to try the possible values \rho_{imp}=\{-1,-0.9,-0.8,\cdots,0.8,0.9,+1\}, to calculate resulting \sigma values, d± values, and M_{quote} values, then see which of these is closest to the observed market price. If desired you can then search a finer grid between two adjacent assumed ... 4 What you are talking about is called regression using fractional polynomials and it has its merits. The canonical reference is this one: Regression Using Fractional Polynomials of Continuous Covariates: Parsimonious Parametric Modelling by Royston and Altman (1994) From the abstract: The relationship between a response variable and one or more ... 3 The linear/non-linear classification is concerned about the dependent variables, and its derivatives. To verify whether the equation is linear, you should be checking that the equation is linear in each of these variables, and the coefficients of these are functions of the independent variables (t and x in your example). In your example, the dependent ... 3 Let \rho\triangleq\rho_{imp}. Note that:$$\frac{\partial \sigma}{\partial \rho}(\rho)=-\frac{\sigma_0\sigma_1}{\sigma(\rho)}<0$$Therefore \sigma is monotonic in implied correlation. In addition, the Margrabe pricing function M(\cdot) is also monotonic in volatility \sigma thus you can find an unique solution to the equation:$$\tag{1}M_{\text{...

3

Interesting idea. I'm guessing this isn't used for two reasons: First, the only algorithm I could find is $O(n^3)$, which is horrible if you're using a moderately-sized high-frequency dataset. Least squares is $O(nk^2)$ (n is the number of rows, and k is the number of predictors; typically $k<<n$). More relevantly, L1 regression is almost as ...

3

This book might be what you are looking for: Theory of Financial Risk and Derivative Pricing. From Statistical Physics to Risk Management by J.-P. Bouchaud and M. Potters As one reviewer from amazon wrote: Econophysics (the application of techniques developed in the physical sciences to economic, business and financial problems) has emerged as a ...

3

In full generality this is a very difficult question. The closest you will get to a general framework is Vapnik-Chervonenkis theory. You can read about this in Chapter 7.9 of "The elements of statistical learning" by Hastie, Tibshirani and Friedman which can be downloaded from their website . But be warned that this is a theoretical approach. Often more ...

3

The following is a good way to judge the quality of fits for a model. http://en.wikipedia.org/wiki/Akaike_information_criterion

2

Just brainstorming here, could you possibly approach risk of an option from a probabilistic perspective? Because the price of the option ($S - X$, where $S$ is lognormally distributed) is lognormally distributed with the same standard deviation as $S$ (aside from being truncated at 0 and having the probability go to infinity as $S$ decreases or $X$ ...

2

An approach to consider is: Computing the total return streams of all the instruments in the portfolio Calculate the risk parameters using 1 Weight appropriately (Equal risk contribution, min variance etc)

1

Try and think about linear and non-linear correlations in terms of joint probability density functions. What does it mean for two assets to be linearly or non-linearly correlated? Suppose we hypothesize a non-linear relationship between two (asset returns) variables as the following:  Y = \pm \sqrt{4 - X^2} + \epsilon , \quad \epsilon \sim \mathcal{N}(0, \...

1

I would argue as follows: In order to observe any type of resonant behaviour, the dynamics of the system you are looking at needs to be described by a second order differential equation. The equations of motions that come to mind in economics are clearly not: GBM: $dS=\mu S dt+\sigma S dZ$ OU: $dX=\theta(\mu - X)dt+\sigma dW$

1

As you suggest, in the case of non-stationary time series, the Hurst exponent is not suitable to measure the time seires persistence for the reasons you cited in the question. Particularly, when $H(q)$ is a non-linear function of q, as in the non-stationary time-series case, the time-series has to be analysed as it is a multi-fractal system (to deal this ...

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