# Tag Info

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The minimum variance solution loads up on securities that have low variances and co-variances. Theoretically you are correct that this should have a low expected return profile. However, it turns out - in contradiction to modern portfolio theory - that securities that have low-volatility or low-beta experience higher returns than high-volatility or high-...

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This is just to expand a bit on vonjd's answer. The approximate formula mentioned by vonjd is due to Brenner and Subrahmanyam ("A simple solution to compute the Implied Standard Deviation", Financial Analysts Journal (1988), pp. 80-83). I do not have a free link to the paper so let me just give a quick and dirty derivation here. For the at-the-money ...

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This one is the best approximation I have ever seen: If you hate computers and computer languages don't give up it's still hope! What about taking Black-Scholes in your head instead? If the option is about at-the-money-forward and it is a short time to maturity then you can use the following approximation: call = put = StockPrice * 0.4 * ...

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Sorry for not being able to give more than one hyperlink, please do some web search for the project pages. Portfolio optimization could be done in python using the cvxopt package which covers convex optimization. This includes quadratic programming as a special case for the risk-return optimization. In this sense, the following example could be of some use:...

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In addition to what vonjd already posted I would recommend you to look at the E.G. Haug's article - The Options Genius. Wilmott.com. You can find some aproximations of BS not only for vanilla european call and put but even for some exotics. For example: chooser option: call = put = $0.4F_{0} e^{-\mu T}\sigma(\sqrt{T}-\sqrt{t})$ asian option: call = put = \$...

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The Black-Scholes 'normal-vol' formula leads quickly to a similar approximation to the one described by olaker. Click here for a paper which contains a formal derivation of the call and put prices based on a normal model (ie a brownian motion rather than a geometric brownian motion). The formula for the call price is: $$\text{Call} = (F-K)N(d_1) + \frac{\... 12 The following papers may help. A New Look at Minimum Variance Investing by Bernd Scherer Minimum Variance Portfolio Composition by Clarke, De Silva & Thorley Under a multifactor risk-based model, if the global minimum variance portfolio dominates the market portfolio, the implication is that the market portfolio is not multifactor efficient and that ... 11 I can think of three reasons. First, and simplest, is that people care about variance. Second, if you really do care about draw-downs, if returns are close to normally distributed, the distribution of draw-downs is just a function of the variance, so there's no need to include draw-downs explicitly in your portfolio construction objective. Minimizing ... 10 I haven't had the time to try them personally, but if I were you I'd try The Solver Foundation or maybe you can find something useful within these libraries. What I did was to compile a MATLAB algorithm and used the produced DLL. UPDATE: I read yesterday in the Wilmott Magazine that the NAG Library is also available for .Net now. Again, I haven't used it ... 10 Unlike the tangency portfolio on the efficient frontier (which represents the most efficient portfolio in terms of max expected sharp ratio), min var portfolios have no ex ante theory that suggests it should outperform a cap weighted market portfolio. The same can be said about other risk-weighted portfolio construction schemes, including equal risk ... 9 Hey, it's early days yet. After all it is still called MODERN portfolio theory. I think there are two main issues and they are both really cultural: 1) specifying alphas 2) wild results Alphas I agree with Gappy that alphas are the key thing you need to have effectiveness (unless you are doing minimum variance). Having a vector of expected returns is ... 9 Minimum variance can be solved simply and efficiently via a quadratic optimizer as the only key input is a covariance matrix. Drawdown or Sortino cannot be optimized via a covariance matrix unless you assume some functional relationship between co-variances/variances and your risk metric of interest. Likely you'll wind up with a similar portfolio to the ... 8 You can use https://projects.coin-or.org/Clp Other options: http://sourceforge.net/projects/lpsolve/ and in R http://cran.r-project.org/web/packages/lpSolve/index.html They all solve pure linear, integer and mixed problems 8 The PortfolioAnalytics package will create weights without reference to current weights, if that's what you want. It should also have much of the reporting that you like from Rmetrics fPortfolio. There is a longer seminar presentation on Portfolioanalytics from 2010's R/Finance conference here: Complex Portfolio Optimization with Generalized Business ... 7 Tools from the field of stochastic optimization are best suited for these problems. In particular, attached is a paper on non-parametric density estimation for stochastic optimization that describes an algorithm if state variables can be associated with draws from the predictive distribution. Here's another approach by Kuhn. These are all one-period ... 7 This is the website to the R/Finance conference this year. Tons of great links. http://www.rinfinance.com/agenda/ Brian Peterson's slide (Building and Testing Quantitative Strategy Models in R) mentions Portfolio-Analytics (which I think is based on R/Metrics). And here is a paper based on Portfolio-Analytics. http://cran.r-project.org/web/packages/... 7 The minimum variance optimization framework does not guarantee positive return whatsoever. As a matter of fact what you are trying to do is something close to the following:$$\underset{w}{\arg \min} \quad w' Q w \quad \text{s.t} \quad Aw \leq b,\quad \sum_i w_i=1$$The fact that you get positive return is a nice result that you get from your backtest (i.... 7 In my experience, a VaR or CVaR portfolio optimization problem is usually best specified as minimizing the VaR or CVaR and then using a constraint for the expected return. As noted by Alexey, it is much better to use CVaR than VaR. The main benefit of a CVaR optimization is that it can be implemented as a linear programming problem. Another option I have ... 7 Bernd Scherer has done exactly this test in his text "Portfolio Construction and Risk Budgeting 4th Edition". There is an SSRN paper by Scherer called "Resampled Efficiency and Portfolio Choice (2004)" you can take a look at as well. I would suggest you skip re-sampling (especially if you have a long-only portfolio) and take a look at Meucci's Robot ... 6 I don't know if you can really improve, the point of Market Making is that you don't know when you'll be executed. It also depends a lot on the type of product you're trading, it's not the same business Market Making far from the money options (where you will never be executed but just offer a reference price and answer traders phone calls) and MM on Bonds/... 6 You can also have a look at ALGLIB or DotNumerics. It would help though if you clarified what kind of optimization problem you have or what kind of algorithm you look for. And if by free you mean GPL or something more like MIT? 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 Using solve.QP in R, a straightforward approach is to add a binary exposure vector as an inequality constraint to your Amat matrix for each group that you want to constrain. The only catch is that values in the exposure and b_0 vectors should be negative, since the function is really satisfying the constraints: A^T b >= b_0. For a simple mean-variance ... 6 First, we are few quants and academics to use the full toolkit of machine learning: stochastic algorithms, to optimal trading. Here are at least two papers: Optimal split of orders across liquidity pools: a stochastic algorithm approach, Sophie Laruelle (PMA), Charles-Albert Lehalle, Gilles Pagès (PMA) Optimal posting distance of limit orders: a stochastic ... 6 The VaR constraint is convex and quadratic and can be handled with any solver supports quadratic constraints, like Guribi, cplex (from IBM) or xpress (from FICO). The CVaR can be formulated as a linear program if you are able to perform monte-carlo simulations on the returns. Briefly, the LP model is \begin{eqnarray*} c &\ge& \alpha + {1 \over (... 6 I think the original reference of mean-variance portfolios being “error maximizing portfolios” is: Michaud, R. (1989). “The Markowitz Optimization Enigma: Is Optimization Optimal?” Financial Analysts Journal 45(1), 31–42. The reason is that even small changes in the estimated means can result in huge changes in the whole portfolio structure. Have a ... 6 I believe there are several ways you can tackle your problems. First, you mentioned that your perform several optimizations. One solution that comes to mind instead of speeding up the optimization itself is to perform the optimizations in parallel, so you could look at Mathwork's Parallel Computing Toolbox. Second, providing the optimizer with a good ... 6 The unconstrained mean-variance problem$$w_{mv,unc}\equiv argmax\left\{ w'\mu-\frac{1}{2}\lambda w'\Sigma w\right\} $$can easily be found by taking the derivative$$\frac{\partial}{\partial w}\left(w'\mu-\frac{1}{2}\lambda w'\Sigma w\right)=\mu-\lambda\Sigma w $$setting it to zero, and solving for w. This gives$$w_{mv,unc}\equiv\frac{1}{\lambda}\...

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There are many portfolio optimization paradigms that include a preference for skewness. These are generally alternatives meant to replace the modern portfolio management mean-variance framework developed by Markowitz. Skewness (or, more generally, higher moments) are only relevant in portfolio optimization if (a) assets are not normally distributed, and (b)...

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The blog post http://www.portfolioprobe.com/2011/10/03/predictability-of-kurtosis-and-skewness-in-sp-constituents/ suggests that there is some predictability in kurtosis, but it isn't clear (to me at least) that there is enough predictabiilty to be useful. If there is a place for higher moments, my guess is that it is in asset allocation problems where ...

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