13 votes
Accepted

Two correlated brownian motions

First you need to correct the formula to: $$ W_t^2 = \rho W_t^1 + \sqrt{1-\rho^2} Z_t, $$ where $Z_t$ is a BM independent of $W_t^1$ If you calculate the variance and the covariance, then you see ...
Richi Wa's user avatar
  • 13.7k
12 votes

Two correlated brownian motions

Here is the general approach you can follow to generate two correlated random variables. Let's suppose, X and Y are two random variable, such that: $$X \sim N(\mu_1, \sigma_1^2)$$ $$Y \sim N(\mu_2, \...
Neeraj's user avatar
  • 2,238
11 votes
Accepted

What is the total correlation between assets in a portfolio?

This is indeed an interesting question. According to this website, a paper by Goldman Sachs [Tierens and Anadu (2004)] proposes three alternative methods for estimating average stock correlations: ...
vonjd's user avatar
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9 votes
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Correlation between stock prices given correlation between returns

We can obtain a closed-form expression for price correlation given (log) return correlation when the two stocks follow geometric Brownian motion: $$S_1(t) = S_1(0)e^{(\mu_1- \frac{1}{2} \sigma_1^2)t}...
RRL's user avatar
  • 3,650
8 votes

What is the total correlation between assets in a portfolio?

I just want to add to vonjd's answer some info on the comparison of the 3 methods. This is too big for a comment so I'm posting as a separate answer but please upvote his answer, not mine. Do the ...
msitt's user avatar
  • 741
8 votes
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How to annualize the correlation matrix?

No, because correlation is a unitless quantity. As you use volatilities to do the scaling, the $\sqrt{252}$ factor should already be taken into account in them. If you take a correlation of 1 between ...
siou0107's user avatar
  • 2,680
8 votes
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Which portfolio is more "diversified": the $\frac{1}{N}$, the MDP or the max decorrelation?

First of all, I am not sure what you mean by the ratio in your second point. However, I will try to give you a partial answer at least. There is a very comprehensive overview of these by EDHEC, page 4....
vanguard2k's user avatar
  • 2,915
7 votes
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What is the most stable, non-trivial dependence structure in finance?

It is hard to find a stable non-trivial dependence structure in financial data. Usually when such is found it is hard to rationalize. One of my favorite (although I am sure there are others) is the ...
phdstudent's user avatar
  • 8,306
7 votes
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Correlation basket equities

Let us consider a basket $B$ with components $S_1,\dots,S_n$ : $$B(t) = \sum_{i=1}^nw_iS_i(t)$$ At time $t$, each component has standard deviation $\sigma_i$, $i \in \{1,\dots,n\}$, and pairwise ...
Daneel Olivaw's user avatar
7 votes
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Creating a Covariance Matrix

here is how to get covariance matrix from correlations:
Valometrics.com's user avatar
7 votes

Estimate covariance matrix using prices

If you assume that a financial asset price has a change that is a wiener process then you can view the future value of that asset as the initial value plus the sum of the independent daily changes (...
Attack68's user avatar
  • 10.2k
6 votes
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Correlation of a lognormal asset and a normal asset

Let $(X_t)_{t\geq 0}$ denote a Geometric Brownian Motion $$ \frac{dX_t}{X_t} = \mu_X dt + \sigma_X dW^X_t,\ \ \ X(0) = X_0$$ such that $X_t$ is lognormally distributed $\forall t > 0$ $$ X_t = X_0 ...
Quantuple's user avatar
  • 14.6k
6 votes
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Simulating covariance matrices with nonzero correlation

What does 'simulate a covariance matrix' mean? If the question means, generate an arbitrary correlation matrix for 1000 stocks, then we can choose any symmetric matrix with all 1s down the diagonal, ...
StackG's user avatar
  • 3,016
5 votes

Why isn't it appropriate to use correlation between prices in a pairs trade strategy?

You could, and it doesn't hurt for you to test this yourself. Some of my best work has come from drawing the opposite conclusion to conventional wisdom or stylized "facts" in publications. That said, ...
madilyn's user avatar
  • 5,240
5 votes
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Correlation Matrix - NaN Values

Your biggest problem is with computing the pairwise correlations of returns. Suppose for simplicity that you have 2 assets A and B. For asset A, you have closing prices for all 3 days $t_0$, $t_1$, ...
Dimitri Vulis's user avatar
5 votes
Accepted

Correlation between brownian motions and Cholesky decomposition

I assume, the first equation is about creating 2 correlated standard normal random variables. Then $X_1 = Z_1$ and $X_2 = \rho Z_1 + \sqrt{1- \rho^2}Z_2 $ are correlated with correlation $\rho$. One ...
Gábor Pálovics's user avatar
5 votes
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Information Coefficient (IC) Formulae Differences

Paraphrasing some quote: "they are different but same but still different" In reality the number of correct bets $N_c$ is the number of times the analyst was correct predicting the ...
emot's user avatar
  • 876
5 votes

correlation of 1/X

A few identities will be helpful to remember. $$\log(1/X)=\log(X^{-1})=-\log(X)$$ $$ cor(X,Y)=\dfrac{cov(X,Y)}{\sqrt{var(X)var(Y)}} $$ $$ var(X)=cov(X,X)\\ $$ $$ cov(aX,bY)=ab\times cov(X,Y)\\ var(aX)=...
Dave's user avatar
  • 353
5 votes

How to design a strategy whose PnL is proportion to correlation?

You are looking for a correlation swap. There is a PhD thesis on static replication. Sébastien Bossu. Static replication of European options and dynamic replication of correlation swaps. Business ...
Yike Lu's user avatar
  • 241
4 votes

Correlation between prices or returns?

There is a good explanation here. Summarizing, what we are computing with Pearson correlations are relations between deviations respect to the means, which is no meaningful using prices. So, you ...
Federico Caccia's user avatar
4 votes
Accepted

Average Correlation

He is forced to use some tricks because Excel can only take average of a rectangular area, but he wants the avg of upper non-diagonal elements of the matrix only. So he subtracts $\frac{1}{n}$ (the ...
Alex C's user avatar
  • 9,372
4 votes
Accepted

What is the covariance of two correlated Ornstein-Uhlenbeck processes?

Using https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process#Solution $$X^i_t = (X^i_0 + \int_0^t\sigma_i e^{a_i u} dB^i_u)e^{-a_it} $$ and $$ X^i_t-\mathbb{E}[X^i_t] = e^{-a_it} \int_0^...
M. Jeunesse's user avatar
  • 2,422
4 votes

Pearson correlation coefficient based on OHLC data

Given only OHLC information, with no timing information as when H and L occured in relation to one another, the covariance between any two assets is only defined for O and C since you know when these ...
David Addison's user avatar
4 votes
Accepted

Control for non-synchronous trading in correlations

I contacted one the authors of the original paper. He confirmed that the overlapping three day log returns are to be used on both stock and market returns.
Mike Haye's user avatar
  • 193
4 votes

Why isn't it appropriate to use correlation between prices in a pairs trade strategy?

If you are correlating prices that would imply that you are sizing positions based on the number of shares in each position. This can result in a book that is very biased in terms of dollars invested....
amdopt's user avatar
  • 4,338
4 votes

How to calculate implied correlation via observed market price (Margrabe option)

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, ...
Alex C's user avatar
  • 9,372
4 votes
Accepted

Brownian Motions theorems

For the first part looks quite obvious, since independence implies that the covariance is zero and since the correlation is just the covariance divided by the product of the standard deviations, it ...
FunnyBuzer's user avatar
  • 1,012
4 votes

Calculating Correlation of Two portfolios?

You may be over-thinking it. It is a straightforward calculation using matrices, as easy as turning the crank of a sausage-making machine. The standard deviation matrix is ...
Alex C's user avatar
  • 9,372
4 votes

How to test ESG score as a factor against traditional factors

Preliminary/Warning: A correlation test is not an appropriate method for analyzing potential risk-factors! Let's (very precisely) recall, what a risk-factor is (see Bali/Engle/Murray (2016), p.173f.),...
skoestlmeier's user avatar
  • 2,916
4 votes
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Correlation Gold and SPX in BBG

I do not think that you were terribly wrong thinking that gold and SPX (or equity market in general) are negatively correlated. The reason behind this is that gold and stocks are in fact negatively ...
AK88's user avatar
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