# 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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### 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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### 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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### 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 ...
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### Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)

For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could ...
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### Variance risk premium: When is realized vol higher than implied vol in practice?

Iām doing some work around the variance risk premium currently, and Iām interested in understanding the situations when realized volatility is > implied volatility in practice. I know in generally ...
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### varswap replication doubt

I have a doubt regarding the varswap replication- I know the portfolio of options with proper weights is a static one, and that there is a dynamic position required in underlying. My confusion is ...
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### 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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### 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; ...
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### 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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### 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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### 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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### Negative high frequency intraday volatility - Zhou estimator

To estimate high frequency tick data stock intraday volatility, I have read Robert Almgren's notes7.pdf http://www.cims.nyu.edu/~almgren/timeseries/notes7.pdf where he talks about the bias free ...
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### 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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### What is the difference between volatility and variance?

How do volatility and variance differ in finance and what do both imply about the movement of an underlying?
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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 $d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\... 1answer 70 views ### Variance of a spread for options on spreads I was reading the paper: https://people.umass.edu/nkapadia/docs/Negative_Vega.pdf In the equation$(5)$, he is defining the variance of the spread as: $$\sigma_1^2S_1^2 + \sigma_2^2S_2^2 - 2\... 0answers 30 views ### Show the expected return of the portfolio and how to derive return variance of the long-short portfolio? Show the expected return of the portfolio and how to derive the return variance of the long-short portfolio? (see picture) 1answer 138 views ### Fair Strike for Variance Swap with no Skew in IV Surface I am reading through Derman's 1999 research notes, "More than you ever wanted to know about Volatility Swaps." In equation B4 of Appendix B, the author takes the Taylor Series of the variance swap ... 1answer 50 views ### ARMA moments proof Consider a standard ARMA(1,1) process such as$$x_t - \beta x_{t-1} = \theta u_{t-1} + u_t$$where u_t is i.i.d. u_t \sim N(0,\sigma^2). I know how to derive mean and variance with stationary ... 2answers 159 views ### Prove Ļ(X,Z) = Ļ The covariance of two random variables X and Y is defined by:$$\mathrm{Cov}(X,Yļ¼= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\... 1answer 175 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 ... 3answers 2k views ### Value at Risk - Long/Short position I have a simple question on the VaR for a portfolio that consists of a long and short position. Say I have a portfolio consisting of the following positions: long 1000 shares of stock X short 1000 ... 0answers 57 views ### Variance swap correlation trade I found two highly correlated assets that have spread in 3M realized and implied vol at historical minimum. To go long on this spread I thought of using two variance swaps. Would it be cheaper to ... 0answers 43 views ### Is this the right way to compute "realized daily market return variance, annualized, over the preceding 126 trading daysā? Realized.Variance<-rollapply((log(Fama.French.daily$Mkt+1)^2) ,126,sum,by=1) So Fama.French.dail$Mkt is my daily Market return. To calculate the realized ... 1answer 183 views ### Antithetic sampling Monte Carlo In Peter Jaeckel, Monte Carlo in Finance book, I read the following sentence: Whenever the first realised moment of the underlying variate draws$\{z_i\}$has a strong impact on the result of the ... 1answer 235 views ### Vasicek model and spot interest rate parametrised by reversion rate By solving an SDE I want to derive the analytical results for mean and variance of the process of extended Vasicek model. $$dr(t) = \left(\eta - \gamma r(t) \right)dt + c dX(t)$$ where$\gamma$... 1answer 4k views ### What is the unconditional variance for a GARCH model? I want to use a Matlab script to calculate Heston Nandi GARCH prices. I found an appropriate script online and it asks for the "unconditional variance" as an input. How do I calculate the appropriate ... 1answer 55 views ### Condition expectation calculation examples and theory [closed] I want to ask you an advice about reading theory and examples of conditional expectation and conditional variance. I want to have my understanding deeper, because sometimes I can't understand ... 1answer 83 views ### Expectation and variance of standard brownian motion Assuming that the price of the stock follows the model$ S(t) = S(0) exp ( mt ā (Ļ^2/ 2) t + ĻW(t) ) , $where W(t) is a standard Brownian motion; Ļ > 0, S(0) > 0, m are some ... 0answers 92 views ### Residual Risk and Variance I've solved part a, but am struggling with b and c.$x_m$is the market portfolio vector, and I think$T\$ should be a diagonal matrix. Any hints greatly appreciated!
The expected logarithmic return of a portfolio is calculated as : $$šø_p = \log\left(\sum_i w_i e^{R_i}\right)$$ Therefore, I was wondering that how can I apply weight to use with the variance based ...