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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53 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
42 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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56 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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60 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
39 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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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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167 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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49 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
98 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 ...
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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
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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 ...
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140 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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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
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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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35 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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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 ...
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1answer
208 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
78 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 ...
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1answer
122 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\...
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1answer
68 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
192 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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133 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 ...
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94 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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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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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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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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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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1answer
123 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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186 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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78 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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150 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; ...
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1answer
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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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1answer
216 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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118 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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185 views

Uncovering patterns in price timeseries using linear regression

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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527 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
71 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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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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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
159 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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193 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 ...
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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 ...
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Is this the right way to compute "realized daily market return variance, annualized, over the preceding 126 trading days”?

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 ...
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207 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 ...
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281 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 $\gamma$ ...
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1answer
56 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 ...
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101 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!
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155 views

Applying portfolio variance weight based on logarithmic returns?

The expected logarithmic return of a portfolio is calculated as : $$𝐸_p = \log\left(\sum_i w_i e^{R_i}\right)$$ Therefore, I was wondering that how can I apply weight to use with the variance based ...
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1answer
91 views

Expectation and variance of standard brownian motion

Assuming that the price of the stock follows the model $ S(t) = S(0) exp ( mt − (σ^2/ 2) t + σW(t) ) , $ where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...