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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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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1answer
63 views
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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2answers
93 views
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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24 views
Mean variance portfolio optimization with long short positions
Sorry if this has been asked before. Can someone point me to some places explaining how to set up the mean variance optimization on long short portfolios? Classical formulation has long only.
Is it as ...
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0answers
30 views
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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1answer
46 views
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 ...
2
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0answers
106 views
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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1answer
76 views
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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2answers
63 views
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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1answer
123 views
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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1answer
58 views
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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1answer
170 views
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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1answer
91 views
PCA and K-means clustering on returns
I am running a PCA on a set of returns and I would like to cluster the results of the output to group stocks that have similar factor exposures.
However when I run the PCA on the covariance of the ...
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1answer
50 views
How to find the price variance of an infinitely expanding Binomial Tree?
How to find the price variance of an asset in a Binomial Tree Model? Suppose the price of the Stock is $S_t$ at time $t$ and it has a probability of $p$ that will go up $u$ times to $u \cdot S_t$ and ...
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61 views
Using stock prices as control variate
In this paper , the author suggested using terminal stock price as control variates. However, I do not understand as we only observe stock price distribution at the terminal, and we do not have any ...
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1answer
187 views
For portfolio variance, why doesn't $Var(X w) = w^\top \Sigma w$? [closed]
From multivariate asset returns $X$, we can calculate the sample covariance matrix $\Sigma$.
The definition of (any) portfolio variance is $w^\top \Sigma w$, where $w$ are portfolio weights.
If $X w$ ...
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1answer
124 views
Calculating Daily Realized Variance with Non-Constant Sampling
I was able to obtain some tick data on a particular asset and I wanted to calculate the daily realized variance of the asset. After browsing through a few threads here, it seems the formula to ...
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2answers
103 views
Nonsystematic risk in a random rate of return [closed]
Good evening,
I am studying the CAPM and I have a doubt regarding the variance $σ_i^2$ of the expected return of an asset $i$.
In particular, how can I derive the following formula?
$$σ_i^2 = β_i^2 ...
2
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1answer
283 views
Valuation of Corridor Variance Swaps
Given that the payout of the Corridor Variance Swap (CVS) is $V \left(\frac{\sum_{n=0}^{N}I}{T_2 - T_0} (\sigma^2 - K^2) \right)$, where $\sigma^2$ is the realized variance within the pre-specified ...
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2answers
165 views
The best “risk measure” for an investor who does not want to lose any of his seed money
Question
There is an investor who is afraid of losing any of his seed money (initial investment).
Variance of investment returns is not a problem to him. He is willing to take variance as long as he ...
3
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1answer
250 views
Can we model Implied volatility using GARCH?
Can I use Implied volatility as a dependent variable in a GARCH model? I believe my IV data shows ARCH effects and hence can I use it to model volatility of the volatility? I know literature has used ...
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0answers
59 views
Sensitivity to total variance for an option
In the famous article of Demertifi, Derman et al (1999), the authors, in the appendix, show that it it necessary to have options weighted inversely proportional to the Square of the Strike in order to ...
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2answers
132 views
Empirical equivalent for implied vol
Implied volatility is supposed to show volatility of the underlying over next k days where k - maturity of the option. Say our stock price is $S_t$ and percentage return is $r_t$. Then which empirical ...
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0answers
53 views
GARCH(1,1) variance forecast in one-step or multi-step?
I would like to forecast the daily variance of a stock using GARCH(1,1) model while I have high frequency data of 5 minute returns. What is the difference between applying GARCH(1,1) in one-step ...
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1answer
78 views
Someone help me understand why for portfolio variance or Parametric Value at Risk we have to compute the covariance matrix?
I understand that portfolio variance is computed through $w'Cw$, where w is the vector of weights, $C$ being the covariance matrix. However, what I don't get is this: why can't this portfolio variance ...
3
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1answer
220 views
How to reduce variance in Monte Carlo using Control Variates when spot prices are decreasing?
I'm trying to use the Control Variates technique to reduce the variance of the estimate obtained from a Monte Carlo simulation for option pricing. As suggested in the book by Glasserman I'm using this ...
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3answers
85 views
Change in variance time series
I am analysing a time series (stock returns) and I am trying to check whether variance in the second half of my sample is different from the first half. I assigned a period to the observations. Here ...
2
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1answer
175 views
GARCH model using high frequency price return
I would like to forecast variance at time length $k\delta$ based on a price (return) time series of time step length $\delta$. I will apply a GARCH(1,1) model to subsamples at time intervals length $k\...
2
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1answer
77 views
forecasting hourly variance with higher resolution data available
Assume one has price data $P_{1}, P_{2}, \dots, P_{n}$ with one hour resolution and aims to forecast the variance for one hour ahead return. The first approach to try is ARCH or GARCH models. There ...
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1answer
216 views
Minimizing variance vs. expected shortfall: distributions where the difference is salient
In portfolio theory in finance, given a set of $n$ assets to choose from, one often selects portfolio weights so as to maximize expected return and minimize some measure of risk, e.g. variance or ...
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1answer
139 views
Variance convex risk measure
I hope you can help me with this question that I really struggle with.
Is variance a convex risk measure? I guess not, but I find it really hard to find a counter example.
Here are my thoughts.
I ...
2
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1answer
103 views
Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$
I'd like to understand why $\Sigma$ is the same under both measures $\mathbb{P}$ and $\mathbb{Q}$.
Is it an assumption or a general fact based on theoretical concepts?
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1answer
106 views
How can we unwind a Index ( SPX ) Variance swap?
Client A comes to dealer to trade variance notional $1m at T=0. The trade is executed with dealer short volatility with strike of 20.
term Payoff of dealer = notional*( Stike^2 - realized vol^2 )
now ...
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104 views
Conditional and unconditional variance, autocovariance and autocorrelation of an ARMA process
Given an ARMA(1,1) process $x_t = a + bx_{t-1} + \varepsilon_t + \theta\varepsilon_{t-1}$, how can we
find the conditional variance, i.e. $Var_{t-1}(x_t)$,
find the unconditional variance, i.e. $Var(...
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0answers
26 views
Expression for the expectation of Integrated variance in case of GARCH(1,1) process
I have the following SDE (GARCH(1,1)) for the instantaneous variance:
$$ d\sigma_t^2 = \kappa (\theta - \sigma_t^2) dt + \psi \sigma_t^2 dW_t $$
I would like to find an expression for $IV_t = E[\int_{...
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1answer
185 views
Variance risk premium: When is realized vol higher than implied vol in practice?
I’m doing some work around the variance risk premium currently, and I’m interested in understanding the situations when realized volatility is > implied volatility in practice.
I know in generally ...
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1answer
279 views
Hedging vega risk with varswaps
I have encountered a statement that in summary reads like this:
Varswaps became popular after the LTCM meltdown due to high levels of implied volatility the market was seeing at the time. Hedge funds ...
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0answers
103 views
CBOE Skew Index Intuition
I was recently reading (and very much struggling to understand) the CBOE white paper on their Skew Index (CBOE Link), I thought it might be useful as I'm trying to better understand volatility skews. ...
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1answer
193 views
Deriving the VIX formula
I am having trouble filling in a few steps in the derivation.
From Martin (2017), we get the following assumptions:
Constant continuously compounded rate $r$;
The underlying doesn't pay dividens;
...
2
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1answer
73 views
Calculating covariance from three variances
I have been asked to look to refactor some code.
There is a line shown below:
$\text{implied covariance} = -\frac{(\text{var}_1 - \text{var}_2 - \text{var}_3)} {2}$,
where $\text{var}_1$ is the ...
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43 views
Variance/VaR calculation for a Portfolio
I'm considering a portfolio of multiple stocks (>2), and calculating their Standard Deviation/Variance and VaR for the portfolio. My question is about the below two ways to calculating them
Consider ...
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1answer
340 views
What is the difference between standard deviation, volatility and quadratic variation?
What is the difference between standard deviation, volatility and quadratic variation?
As I know, volatility is the standard deviation of the log returns, so they are basically the same. (One of ...
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197 views
Expected value and variance of the stock log-returns under Local Volatility framework
I want to calculate the expected value and the variance of the stock process log-returns in the Local Volatility setting (and the realized/terminal correlation but let us begin in the one-dimentional ...
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1answer
225 views
Uncovering patterns in price timeseries using linear regression [closed]
I have some minute-bar data which my professor suggested I resample to 5 minute bars and then separate it into timeseries per bar period. For example, I get one time series for 12:00, another one for ...
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1answer
804 views
How to calculate the mean and variance of this Ito integral?
I tried to calculate this integral use Ito's lemma, $W_{t}$ is the Wiener Process.
$$I_{T}=\int_{0}^{T}\sqrt{|W_{t}|}dW_{t}$$
We have
$d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\...
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1answer
78 views
Variance of a spread for options on spreads
I was reading the paper:
https://people.umass.edu/nkapadia/docs/Negative_Vega.pdf
In the equation $(5)$, he is defining the variance of the spread as:
$$\sigma_1^2S_1^2 + \sigma_2^2S_2^2 - 2\...
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0answers
44 views
Show the expected return of the portfolio and how to derive return variance of the long-short portfolio?
Show the expected return of the portfolio and how to derive the return variance of the long-short portfolio? (see picture)
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1answer
50 views
ARMA moments proof
Consider a standard ARMA(1,1) process such as
$$x_t - \beta x_{t-1} = \theta u_{t-1} + u_t$$
where $u_t$ is i.i.d. $u_t \sim N(0,\sigma^2)$. I know how to derive mean and variance with stationary ...
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1answer
197 views
Fair Strike for Variance Swap with no Skew in IV Surface
I am reading through Derman's 1999 research notes, "More than you ever wanted to know about Volatility Swaps."
In equation B4 of Appendix B, the author takes the Taylor Series of the variance swap ...
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1answer
240 views
Realized Volatility Methods
Can someone explain to me which of these two methods is more accurate or commonly used to calculate Realized Volatility? I'm seeing both used, but I get very different results from them.
1) Standard ...
