38
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 ...
33
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-...
29
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 *
...
21
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:
$$\text{Call} = (F-K)N(d_1) + \frac{\...
15
In addition to what vonjd already posted I would recommend you to look at the E.G. Haug's article - The Options Genius. Wilmott.com. You can find some aproximations of BS not only for vanilla european call and put but even for some exotics. For example:
chooser option: call = put = $0.4F_{0} e^{-\mu T}\sigma(\sqrt{T}-\sqrt{t})$
asian option: call = put = $...
15
The following papers may help.
A New Look at Minimum Variance Investing by Bernd Scherer
Minimum Variance Portfolio Composition by Clarke, De Silva & Thorley
Under a multifactor risk-based model, if the global minimum variance portfolio dominates the market portfolio, the implication is that the market portfolio is not multifactor efficient and that ...
15
I can think of three reasons.
First, and simplest, is that people care about variance.
Second, if you really do care about draw-downs, if returns are close to normally distributed, the distribution of draw-downs is just a function of the variance, so there's no need to include draw-downs explicitly in your portfolio construction objective. Minimizing ...
14
Markowitz's concepts attracted a great deal of interest from theorists (and still do), but never had much application in practice. The results from practical application were always disappointing (starting in the 1970's, well before DeMiguel, Garlappi, and Uppal (2007) study of $\frac{1}{N}$ portfolios), mainly because it is so difficult to provide accurate ...
12
Unlike the tangency portfolio on the efficient frontier (which represents the most efficient portfolio in terms of max expected sharp ratio), min var portfolios have no ex ante theory that suggests it should outperform a cap weighted market portfolio. The same can be said about other risk-weighted portfolio construction schemes, including equal risk ...
12
The unconstrained mean-variance problem $$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}\...
11
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 ...
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
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 ...
9
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/...
9
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....
9
There has been a split in the community ever since Mandelbrot published his paper "On the Variation of Certain Speculative Prices."
See:
Mandelbrot, B. (1963). The variation of certain speculative prices.
The Journal of Business, 36(4):394–419.
To understand why this is so important, you must first realize what economists are trying to do. When you ...
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 ...
8
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 ...
8
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 ...
8
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 (...
8
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 ...
8
The underlying problem: your ACTR constraints aren't convex
The $i$th constraint on your risk contribution can be written:
$$ w_i \sum_j \sigma_{ij} w_j \leq c_i s$$
And this isn't a convex constraint because of the $w_j w_i$ terms (a function $g(x,y)=xy$ isn't convex in $x$ and $y$). They're not convex constraints, so you won't be able to write them as ...
8
You seem to have two distinct problems:
How to generate random portfolios
How optimal portfolios are structured
Ad 1)
A straightforward way to simulate the weights of random portfolios is to use the Dirichlet distribution $Dir(\alpha_1,\ldots,\alpha_n)$. This is a distribution on the Simplex (i.e. on $S=\{x\in\mathbb{R}^n | \sum x_i =1, x_i\geq 0\}$, ...
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
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
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 ...
7
The term in sample and out of sample are commonly used in any kind of optimization or fitting methods (MVO is just a particular case).
When you make the optimization, you compute optimal parameters (usually the weights of the optimal portfolio in asset allocation) over a given data sample, for example, the returns of the securities of the portfolio for the ...
7
Check out following link. In page 23 you'll find the derivation.
http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf
7
Of course, optimal control is at the core of math finance. Take few applications:
Option Pricing: you have an exposure to a time dependent combination of market factors; you have some knowledge of their dynamics. They are partly deterministic, partly stochastic (i.e. random). At each "time step" you can adjust your portfolio at a given cost. Your goal is to ...
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