Questions tagged [variance]
Used for questions related to statistical measure "variance", i.e. a second central moment of a random variable. The variance is a risk measure.
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Multiplicative Metric Variance
I come from a math/statistics background and as learned some stuff as a data analyst
I learned a certain technique to calculate period to period variances between some metrics.
I was wondering if ...
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Why is quadratic variation path dependent? [closed]
In chapter 2 of The Econometrics of High Frequency Data the quadratic variation relative to a grid $\mathscr{G}$ of any process $X$ is defined as
$$
[X,X]_t^\mathscr{G} = \sum_{t+1\le t} (X_{t+1}-X_t)^...
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CBOE dispersion index formula
I came across the CBOE white paper Cboe S&P 500 Dispersion Index Methodology. The formula in Subsection Index Construction/Outline of the Dispersion Index Methodology on page 4 that defines the ...
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Forecasting Realized Volatility with Machine Learning [closed]
How is the daily realized variance calculated for an intraday one minute data.
How can realized volatility be forecasted using machine learning techniques such as neural network and LSTM. Any detailed ...
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negative portfolio variance? Creating a positive semi definite matrix in excel
I am attempting a portfolio optimization model and ended up generating negative portfolio variance using 2WaWbσaσbcorrel(a,b) or 2WaWb*Cov(a,b)
From reading the linked article where other users had an ...
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Is there any relationship between the Covariance(A, B) and the variance of the synthetic asset A/B?
Let's say we have 2 pairs of currencies: EUR/USD and GBP/USD. The cross-asset (or synthetic asset) would be (EUR/USD) / (GBP/USD) = EUR/GBP.
Is there any relationship between the covariance(EUR/USD, ...
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Why do we need the covariance when calculating portfolio VaR?
I was recently learning about value at risk and how to calculate it, and one of the steps was to calculate the covariance of the returns of the securities making up the portofolio.
This makes sense ...
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Covariance Matrix of Correlated Random Variable
Suppose I know or have estimated the covariance matrix for one random variable (for example an asset) and have:
$$
\begin{bmatrix}
<\text{spot, spot}> & <\text{atmv, spot}> \\
<\...
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Single outsized daily return value creates substantive discrepancy between annualized variance calculated from daily vs monthly returns
I am new here, and to the field. I hope my clunkiness in expressing myself can be forgiven.
My situation is as follows: I have around three years of daily return data for some financial asset. Out of ...
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Standard Deviation and Monotonicity property
I just read that standard deviation is a coherent risk measure, and therefore it should satisfy the monotonicity property:
$X_1 \geq X_2 \implies \rho(X_1) \leq \rho(X_2)$ where $X_1,X_2$ are asset ...
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Bessel Correction and Geometric Brownian Motion
Does it make sense to use bessel's correction for standard deviation and variance when fitting the drift and volatility parameters of geometric brownian motion to historical return data for a security....
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What is the intuition behind the VIX formula offset term [duplicate]
In the formula for the VIX we have that the spot value for the VIX is:
The first part of this equation is exactly the formula for the price of a variance swap which is the present value of realized ...
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constant dollar Gamma for variance swap
Here is the result of Gamma for Variance swap. I think $S_t$ is the only variable in $\sigma^2_{Exected,t}.$ Then how could we get the presentation of Gamma:
$$\dfrac{1}{S_t^2},$$
there should be some ...
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Does the expected return increase with variance for stocks? [duplicate]
I took a historical dataset of ~2600 stocks and computed the 30-day returns for non-overlapping windows, for a 9 year period. For the returns, I computed the mean and variance. Then I plotted the mean ...
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What is the Fair Strike in a Var/Vol Swap and how does it relate to its price? [closed]
I am a student trying to price volatility and variance swaps.
People who price those two products usually try to get the "fair strike", and don't seem to care about the price.
However, I ...
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How to derive numeric option VaR with delta-vega normal approach?
For an option with price C, the ΔC, with respect to changes of the underlying asset price S and volatility σ (first-order approximation), is given by
$\Delta C=\delta \Delta S+\nu\Delta\sigma$,
where ...
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Bias-Variance tradeoff for Covariance Estimation w/ Different Frequencies
In general, what does the bias-variance tradeoff look like when estimating covariance matrices with varying return frequencies (i.e. daily, weekly, monthly returns)?
From my observations I've noticed ...
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Calculating variance of long/short portfolio
Say I have a portfolio of stocks, stock A, stock B and stock C, with the below positions:
stock A: long 100 USD
stock B: long 50 USD
stock C: short 200 USD
How do I calculate the portfolio variance ...
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Smile wings and varswap pricing
Is it true that far wings of the volatility smile have an outsized influence on the price of a variance swap? Is there a mathematical argument demonstrating this idea? What do we generally refer as ...
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Compute monthly realized variance for Fama-French factor
I need to compute monthly realized variance from daily data for Fama-French factors. Knowing that Fama-French factors is the difference of return between different type of stocks, for example SMB ...
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Minimizing variance of a long short equity portfolio in practice
I understand the finance 101 explanation of how to minimize variance of a long-short portfolio using a covariance matrix. I also know that it doesn't really work because the covariance matrix is ...
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Calculating monthly UST3M returns
I am trying compute a variance/covariance matrix for 5 stocks using monthly returns. I plan to use CAPM (using monthly log returns) as the expected return and the US 3-month Treasury Bill (UST3M) ...
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Compute monthly realized variance from daily data
I am confused about the correct formula to compute monthly realized variance from daily data. What is the first sigma in the picture: sum or average? I mean, after subtracting each observation from ...
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How does autocorrelation bias annualizing variance?
I read somewhere that autocorrelation prevents someone from annualizing variance. But how does it bias it? Let's say you have daily returns. If autocorrelation is high, should that overstate or ...
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How is the variance schedule used?
Let $v(t)$ be the instantaneous variance of an underlying stock or index at time $t\in[0,1]$ between the open at $t=0$ and close at $t=1$ of an exchange. Usually $v(t)$ achieves local maxima at $t\in\{...
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Daily vs Monthly vs. other return for volatility calculation?
I thought I read/heard somewhere that annualized volatility, using monthly returns vs daily returns is usually lower. With that said, I can't seem to find any papers on this.
Does anyone have any ...
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Best way to extrapolate on implied volatility
I am doing some standard svd calibration to mark market implied vols in difference to a previous volatility surface.
For longer term maturities where there is no market data, I am extrapolating ATM ...
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Deriving the variance of G2++ Model
I'm studying G2++ Model in Brigo(2007)'s book.
The model constructed as follows,
$$
r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\
$$
with the dynamics of $dx(t)$ and $dy(t)$ described by:
\begin{align}...
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How much compensation need to take on risk?
Quant Firm Interview Question
We roll three, 8 sided dice. If same face appears 3 times we win 80 dollars. We have a bank of 10,000 dollars. How much are we willing to pay to play? What if we increase ...
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Variance of the price from returns variance
Let's say that we have the variance of the daily return at $t_0$:
$$\sigma_{r_{t_0}}^2=\text{Var}[r_{t_0}]=\text{Var}[\frac{S_{t_0}-S_{t_0-1}}{S_{t_0-1}}]$$
for price process $S_t$. Is there a way to ...
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Verify numerically relation between mean deviation and standard deviation
I was reading "We Don’t Quite Know What We Are Talking About When We Talk About Volatility" by Goldstein and Taleb, and I was trying to quickly verify numerically the relation between mean ...
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Kelly Criterion — maximize expected value and minimize the variance in card game with $x$ red and $y$ black cards
You have $x$ red cards and $y$ black cards. I flip them over one at a time. The probability of flipping a particular colour is proportional to the amount of those coloured cards left. You start with $...
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Variation of the trading range
Example: The trading range (in points) for each of the last 5 trading days for asset A is: 5,21,2,15,32 and for asset B is: 5,6,5,5,5. Is there an indicator that ranks assets based on variation of ...
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Control Variates - Option pricing
I am trying to reduce the Monte Carlo variance with Control Variates technique. In practice, I am able to reduce it with a generic European Call option, with the following formulas:
$$ Z_{CV} = \frac{...
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Why is $Z_t$ uncorrelated with $X_{t-1}$ in $X_t=\theta X_{t-1}+Z_t$?
In a solution to the problem below, the teaching assistant solves it by calculating $\mathbb{E}[X_t^2]$ and ends up with also having to calculate $\mathbb{E}[X_{t-1}Z_t]$ after expanding the square. ...
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Computing the Variance Risk Premium
The Variance Risk Premium (VRP) is defined as:
$$VRP(t,t+\Delta t) \equiv RV(t,t+\Delta t)^2 - IV_t(t,t+\Delta t)^2$$
where $RV^2$ is the realized variance between $t$ and $t + \Delta t$ and $IV_t^2$ ...
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Heston: Variance of Integrated Variance
Consider the standard Heston model\begin{align*}
dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\
dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\
dBdW&=\rho dt.
\end{align*}
Computing $\mathbb{E}\...
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Monte Carlo Simulation of GBM Process has a Very High Variance - Explanation Needed as to why?
I use Geometric Brownian Motion (GMB) to simulate a share price from March 24, 2020 to March 24 as follow:
\begin{equation}
S_t=S_{t-1}exp((rf-0.6\sigma^2)*(2)+\sigma*sqrt(2)*\mathcal{N}(0,1))
\end{...
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Market-maker's gain variance
I am reading the book "Trades, Quotes and Prices" by JEAN-PHILIPPE BOUCHAUD and have stuck in the very beginning with understanding the formula of variance of MM's gain per trade (see ...
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Show that the following result holds true for the variance of the return of a portfolio of shares
Start with a portfolio $p$ of $n$ shares, each with weight $x_i = \dfrac{1}{n}$ (for $i$ ranging from $1$ to $n$, discretely). Its return is given by:
$$R_p=x_1R_1+\ldots+x_nR_n=\sum_{i=1}^{n}=x_iR_i\...
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Covariance Shrinkage - Am I getting the right variances?
I am looking into a quite simple task: shrinking the sample covariance matrix of a minor sample of monthly returns data on 5 different assets.
I am using Python to process my data and have been using ...
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looking for recommendation for a var/vol swap trading book
I am aware this book - volatility trading by Euan Sinclair, and it's nice book. But I am looking for book focus on var/vol swap trading, i.e., introduce about trading strategy/ideas by using var/vol ...
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Statistical Inference of Variance Risk Premia
Good afternoon,
I am currently following Carr and Wu (2009) to compute variance risk premia from options written as (RV-EV)*100 for the payoff of a long var swap position.
Now I want to see whether my ...
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Is scaling the standard deviations in the VaR formula (parametric) equivalent to scaling the VaR figure at the end?
I have come across people calculating parametric VaR who scaled the standard deviations by say square root of 10 to scale up to a 10 day horizon.
Elsewhere I have seen textbooks suggesting that it is ...
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Variance of Log Returns
Consider an asset held for $n$ time periods with weakly stationary log-returns $r_t$, $1≤t≤n$.
Show that
$var(r_1 +r_2 +r_3 +r_4)=var(r_1 +r_2 +r_3)+var(r_1)(1+2ρ_3 +2ρ_2 +2ρ_1)$,
where $ρ_k$ is the ...
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Equivalence of Standard Deviation and Variance as a risk measure - WRONG?
In Modern Portfolio Theory, I often see that people seem to view Standard Deviation and Variance as equivalent. Example from Markowitz himself:
"Thus far I have used the standard deviation ...
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Estimating the variance of returns with aggregated data
Say I have an asset return time series:
Jan2020: -5%
Feb2020: +5%
Mar2020: -5%
Apr2020: +5%
May2020: -5%
Jun2020: +5%
Q3 2020: +20%
Oct2020: +5
Nov2020: -5
Dec2020: +5
Note that 3 months of data is an ...
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T-statistics on monthly returns vs annualized monthly returns
eqI am very confused about a very basic question. This is probably more statistics than quantitative finance, but still, should be useful for this stackexchange board as well.
Let's assume I have ...
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Portfolio variance $<=$ weighted average of individual variances [closed]
In portfolio theory, I often (with some justifications but the message is the same) come across the following statement:
"The most important quality of portfolio variance is that its value is a ...
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Corwin-Schultz estimator of bid-ask spread
I am reading a paper "A Simple Way to Estimate Bid-Ask Spreads from Daily High and Low Prices" cf.A Simple Way to Estimate Bid-Ask Spreads from Daily High and Low Prices
The authors proposed ...
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