The selection of a best element from some set of available alternatives. Typically consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function.

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1answer
80 views

Portfolio with a certain pay-off curve

I would like to find a relevant optimization option's portfolio models which can describe a certain pay-off curve (objective function) under same assumptions. For example, assumptions on how to limit ...
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2answers
159 views

Asset allocation problem using Hidden Markov Model

I am recently getting more interested in Hidden Markov Models (HMM) and its application on financial assets to understand their behavior. But what captured my attention the most is the use of asset ...
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0answers
30 views

Pricing options using particle swarm optimization (PSO)

I am currently trying to recreate some of the work done to price various types of options using particle swarm optimization. In particular, I am trying to price European options using a similar method ...
0
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1answer
96 views

EM for conditional Gaussian model

Let $$X_1\sim N(\mu_{X_1},\sigma_{X_2}^2)$$ $$X_2\sim N(\mu_{X_2}, \sigma_{X_2}^2)$$ where $\mu_{X_2}=c+aX_1$. Also, I have data $D$ (with missing values on $X_1,X_2$). How can I update/estimate the ...
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0answers
15 views

I have a test coming up and I could really use an explanation to this example problem

This test is on CAPM and portfolio optimization Suppose that investors A and B can invest in two risky assets and a riskless asset. The first risky asset has an expected annual return of 10%. The ...
2
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1answer
110 views

Starting values for constrOptim() in R

I want to perform a constraint optimization for Maximum Likelihood Estimation in R to forecast volatility of returns. The probleme is that my initial values aren't in the permitted region. Is there ...
2
votes
1answer
81 views

Quantum Computing for Quantitative Finance

It's been a while that quantum computing is looked as the next step in computational science. I somewhat always tought we were decade aways from it's happening but it appears I was wrong: ibm-quantum-...
6
votes
1answer
129 views

Stochastic control (HJB) for wealth process involving stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ where $\theta_t$ is the worth of holding at time $t$ and $c_t$ is the consumption stream. ...
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3answers
122 views

Why do we assume quadratic utility in portfolio theory?

In my text (Investments by BKM), the investor's mean-variance utility (given as $U = E[R] - \frac12A\sigma^2$) is stated to be the objective function we wish to maximize. Upon further digging, it ...
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1answer
40 views

How to understand this Risk Parity Algorithm?

I am trying to understand an optimization algorithm to achieve risk parity in a portfolio. I need some help figuring out the notation in the following formula: I found this on THIS paper. I ...
3
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0answers
149 views

A doubt about Evans and Jovanovic (1989) economic model for entrepreneurs with credit constraints

[I already posted this question on the math forum of stackexchange and I was advised that I should post this question here] In Evans and Jovanovic (1989) you will find a model for entrepreneurs with ...
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0answers
79 views

Optimize a trading strategy created in excel with R

I have created quite a complex back test in excel spanning 15 years with 17 parameters. I would like to optimize the parameters which would give me maximum return given a maximum draw-down percentage. ...
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11answers
13k views

What is the reference python library for portfolio optimization?

Does anyone know of a python library/source that is able to calculate the traditional mean-variance portfolio? To press my luck, any resources where the library/source also contains functions such as ...
15
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0answers
3k views

Algorithm to fit AR(1)/GARCH(1,1) model of log-returns

I am fitting numerically an AR(1)/GARCH(1,1) process to index and stock log-returns, $r_t=\log(P_t/P_{t-1})$, where $P_t$ is the price at time $t$, and thus far am not clear on where the observed log ...
2
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0answers
41 views

Using Market Prices of Bonds to Model the Discount Curve with a Polynomial (Math + R)

I have a small program I'm building to interpolate the discount curve from a portfolio of benchmark bonds. If anyone has any guesses as to whether it's my process, or my code that's messed up I would ...
1
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1answer
50 views

Portfolio optimization - maximize variance with exposure to risk factors equal to zero

Optimize a portfolio such that the exposure to risk factors is zero and the variance is maximized (instead of traditional minimization problem). so the optimization problem look like: $$maximize\;w^...
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0answers
19 views

Continous-time portfolio allocation optimization for a given consumption rate

I have the following PDE $0 = V_t - c(t)V_x - \lambda^2 V_x^2/V_{xx} + rxV_x + 1/2\lambda^2x^2V_{xx}$ where $t\mapsto c(t)$ is some given function and $r,\lambda$ are given constants. If necessary, ...
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0answers
83 views

First step of Black-Litterman portfolio

I tried to implement Black-Litterman model. I have a covariance matrix, market capitalization for each asset. I assume a risk aversion factor to be 10. First I use the following code to get ...
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0answers
41 views

Bond portfolio optimization

Problem is that I want to match single country term structure return as closely as possible. What would be best way to construct proxy to do this, with less bonds than in original term structure? I ...
4
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1answer
406 views

How to optimize return in a moving average crossover algorithm

Moving average crossover strategy is a widely used strategy in algo trading. Is there a way to optimize return in a moving average crossover stratergy. I have used this site to backtest MA crossover ...
3
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2answers
45 views

Dealing with a constraint which is the square root of a quadratic form

I'm trying to maximize my portfolio, but don't know how to deal with the constraint which is on the form max $2u^Tx-x^T \Sigma x$ Subject to $e^Tx = 1$ $u^Tx - m (x^T \Sigma x)^{1/2} >= c $ ...
3
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1answer
63 views

Numerical Optimizer Matlab Calibration LMM

I am trying to mimimize the following function in order to calibrate the Libor Market Model $$\sum_{i=1}^{n} \left(\sigma_i^{market}-\sigma_i^{Reb}\left(a,b,c,d,\beta\right)/\sqrt{T_i}\right)^2,$$ ...
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1answer
81 views

Segmented investment to yield same monthly return in each segment

Not an investment specialist, so please excuse the very basic math. Given a lump sum, I need to distribute this lump sum over (x) segments, each lasting (y) years (years can be different for each ...
2
votes
1answer
129 views

How to scale $\alpha$, trading costs in a standard portfolio optimization problem

In the usual "portfolio optimization problem under linear constraints". Let me define the terms here. $$ \text{Find } w^*=\underset{w}{\text{argmax}} \ \ r^Tw - \lambda w^{T} \Sigma w - tradingCost(|...
3
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1answer
166 views

Portfolio with lots of subportfolios

An account manager has $N$ distinct, equally-sized pots of money, which will be used to make $N$ distinct subportfolios, each of which is drawn from a slightly different (but potentially overlapping) ...
3
votes
1answer
90 views

dynamic Markowitz portfolio

Let's take 4 assets, whose values are known during a period of time of 2 years. Then I calculate the expected returns for each of these 4 assets thanks to these 2 years - historical data. I deduce the ...
9
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1answer
188 views

The danger of using Principal Component Analysis (PCA) in Robust Optimization problems

I have received a reviewer's comment on a paper which applies PCA to reduce the size of a problem and the application is in the robust optimization field. The reviewer implies that "In robust ...
15
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4answers
5k views

Techniques to optimize the placement of orders in market making strategy?

Market making often requires placing and canceling a lot of orders. You have to buy and sell nearly simultaneously, so you need to move orders pretty often to beat other traders. But I would like to ...
0
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1answer
30 views

Equitable Allocation

This questions borders on the actuarial side of things but the general solution should have relevance in several situations. Suppose we have a set of $k$ people who will retire in $\{n_1,...,n_k\}$ ...
0
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1answer
40 views

Imposing MLE restrictions by logistic mapping

I am doing some Maximum Likelihood Estimation with a density that has time-varying parameters. I am using the fmincon function in Matlab, but I do not know how to ...
2
votes
1answer
114 views

Determining maximum strategy capacity and optimal order size for low frequency equity strategy

I have developed a low frequency equity trading strategy that seems to work well with stocks in the S&P 500. Someone asked me about the maximum capacity of the strategy (how much AUM I could ...
6
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5answers
2k views

Fastest solver possible for portfolio optimization

I am using quadprog in MATLAB for very simple mean-variance optimization, with less than 100 assets. It is quite fast but if I run a strategy with daily ...
4
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2answers
92 views

When would dedicated portfolios do better than 'immunized' portfolios?

We just learned about cash-matching through dedicated portfolios (using risk free bonds) in my class that concerned mathematical programming. However, in an aside one of the notes said: It should be ...
2
votes
3answers
175 views

What are the canonical books on optimization methods?

I am looking for some literature devoted to optimization methods in finance (portfolio optimization, asset pricing etc). Could you please recommend some books (perhaps, essentially non elementary: I ...
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vote
2answers
457 views

Optimizing Principal Component factor weightings over time

I was given the returns of a cross-asset class portfolio of ETFs and I conducted PCA to obtain factors on dates from T-n, T-3, T-2,..., T. What I would like to do is decompose the market moves from T+...
4
votes
3answers
2k views

Markowitz mean-variance optimization as “error maximization”

I hear it said a lot that standard MV optimization "maximizes errors". But I can't find a good explanation for what exactly they mean by this "maximization" of estimation error. I understand that if ...
0
votes
1answer
157 views

Parameters for numerically fitting t-distribution to log-returns

I am fitting the t-distribution to log-returns numerically (not using R, MATLAB, Stata, etc.), but rather using general programming. Assuming the log-return values are $r_t$, and the $t$-variates are ...
0
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0answers
19 views

Question in the proof of “Optimization of conditional value-at-risk”

I'm reading the paper "Optimization of conditional value-at-risk" by Rockafellar and Uryasev. The state two theorems within the paper which are proven in the appendix. Let me introduce some notation ...
9
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1answer
151 views

Are there references about liquidation, transaction, market impact costs in portfolio optimization

I am looking for some references treating of what I would call liquidation cost market impact cost transaction cost(*) in the usual "portfolio optimization problem under linear constraints". Let ...
9
votes
4answers
2k views

What .NET library can I use to solve optimization problems?

I'm working with C# and I start being bored writing optimization algorithm. Do you know any free library containing this sort of algorithms? In particular I'm currently working with Semidefit ...
2
votes
1answer
65 views

How to calculate a hypothetical minimum-variance point?

If we have $N$ assets which are uncorrelated, but have the same mean return of $\mu$ but the variances are different where $\sigma_i^2$ is the variance of each asset $i = 1, 2,...,N$ how can you write ...
33
votes
12answers
16k views

Why does the minimum variance portfolio provide good returns?

I've been a researching minimum variance portfolios (from this link) and find that by building MVPs adding constraints on portfolio weights and a few other tweaks to the methods outlined I get ...
8
votes
0answers
207 views

Max option leverage strike

Since options represent leveraged stock investments, at which strike $K$ does a European option provide maximum leverage? Hereby define leverage $L$ as ratio of Delta/Optionprice: $$L(K)=\frac{\...
1
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0answers
84 views

How to hedge an ETF position with a basket of its underlying components

In practice, when one takes on a large equity ETF position, I would imagine it's not necessarily "optimal" to hedge using a basket of all the constituents even though that should be a perfect hedge. ...
2
votes
0answers
46 views

Given (past) stock values for N assets, how to find the maximum - theoretical - profit?

In the past few days I have been thinking about a question which seems trivial, yet I can't think of any efficient way to find the optimal solution... Here is the problem: imagine you have a ...
0
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0answers
75 views

Discrete Hedging of Options

Assume that a stock $S_t$ follows simple geometric Brownian motion. Let's say we sold option whose payoff is $f(S_T)$. Now, we are only allowed to trade 2 times in the interval [0,T]. What kind of ...
0
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2answers
129 views

Ledoit-Wolf Shrinkage estimator not giving positive definite covariance matrix

I used ten year daily data for 407 stocks and computed the daily and monthly covariance matrices. Since I have more variables than observations for the monthly matrix, I wasn't surprised to find the ...
4
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1answer
89 views

Full Kelly portfolios having same weights as tangency portfolios

I'm currently comparing empirically the differences between Markowitz and Kelly portfolios. I calculated the Kelly weights for monthly return observations over 10 years for a sample of 50 stocks from ...
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0answers
53 views

Smoothening yield curve by minimizing forward curve slope

I am using government bullet bond data and have bootstrapped a yield curve by solving the following optimization which minimizes unweighted price error: $$\text{min}\sum_i\left(P_i-\sum_t\frac{F_{it}}...