23
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:
$$\text{Call} = (F-K)N(d_1) + \frac{\...
19
Markowitz's concepts attracted a great deal of interest from theorists (and still do), but never had much application in practice. The results from practical application were always disappointing (starting in the 1970's, well before DeMiguel, Garlappi, and Uppal (2007) study of $\frac{1}{N}$ portfolios), mainly because it is so difficult to provide accurate ...
15
I can think of three reasons.
First, and simplest, is that people care about variance.
Second, if you really do care about draw-downs, if returns are close to normally distributed, the distribution of draw-downs is just a function of the variance, so there's no need to include draw-downs explicitly in your portfolio construction objective. Minimizing ...
13
The unconstrained mean-variance problem $$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 ...
11
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 ...
10
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 ...
10
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 ...
10
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 $S=\{x\in\mathbb{R}^n | \sum x_i =1, x_i\geq 0\}$, ...
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 ...
9
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
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
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
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 ...
8
It is more complicated than that: It is not the optimization per se that leads to inferior results but the data you use.
Kritzman et al. makes a strong case in defense of optimization vs. 1/N in this popular paper:
In Defense of Optimization: The Fallacy of 1/N, Financial Analysts Journal, Vol. 66, No. 2, 2010 by Mark Kritzman, Sebastien Page and David ...
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
6
Meucci covers this example precisely in his paper "Fully Flexible Views: Theory & Practice". You can find his code here for three examples related to the paper. The Butterfly Trading example covers the CVAR scenario.
6
The logistic distribution approximates the normal distribution function used in the Black-Scholes. The drawbacks to the normal cumulative distribution function are that it cannot be computed exactly through elementary functions, it cannot be inverted algebraically (i.e., the inverse bijection cannot be solved algebraically), and it is computationally ...
6
Actually, Ralph Vince's Leverage Space Trading Model does utilise draw down. A short introductory pdf is available here, and the R-forge package is here.
Briefly, a genetic algorithm is used to model the maximum expected portfolio return based on a joint probability distribution of the portfolio component returns, subject to an overall maximum draw down ...
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
You can use empirical distribution and use Mean-CVaR as a target function. CVar ("Expected shortfall") is considered a better risk metrics than VaR if we depart from the light-tailed normal distribution.
The code below is in R and is taken from the book "Portfolio Optimization with R/Rmetrics" By Diethelm Wuertz, Yohan Chalabi, William Chen, Andrew Ellis. ...
6
Without the discrete constraints, the minimum tracking error/variance problem is a quadratic program. If you constrain the tracking error, you have a convex quadratically-constrained problem which is solved as an SOCP by modern commercial solvers. SOCP does not address discrete constraints like cardinality of assets or minimum investment levels. SOCP ...
6
Portfolio optimization techniques are used quite a bit by hedge funds. I think you misunderstand how portfolio optimization operates in the context of an active trading strategy. Your question suggests a view of portfolio optimization as a tool to adjust portfolio weights arrived at by a separate, active strategy. Under that approach, you are correct, the ...
6
That's a pretty heavy question for this forum, and its answer is worthy of a semester-long discussion in a university course. The short answer is that (for convex optimization) the dual problem can give you a lower bound on your objective function (for minimization).
In addition, the values of the dual variables are related to the sensitivity of your ...
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\...
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 ...
5
Any explanations? Yes. Within each asset category we find that stocks may be:
Unattractively underperforming the category norm
Attractive as they meet the expected norm
Unsustainable as their returns exceed the category norm and may suffer mean reversion
By focusing on low variance, we exclude type (3) stocks that damage portfolio performance through high ...
5
With minimum variance, the covariance matrix does not change when you change the holdings. So all the optimizer needs to do is change the weights. This makes it easy to calculate the gradients.
To construct the drawdown statistic, you would need the distribution of returns in each period to your horizon. You would then need to calculate the path of profits ...
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