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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11
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0answers
1k views

Formula for the efficient portfolios in mean-variance optimisation?

Consider the setting of mean-variance portfolio optimisation: $n$ assets with expected returns $\overline{r}_1,...,\overline{r}_n$ and standard deviations $\sigma_1,...\sigma_n$. For a certain fixed $\...
0
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1answer
184 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 ...
3
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1answer
202 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 ...
0
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2answers
118 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 ...
0
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1answer
37 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 ...
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0answers
17 views

GARCH(1,1) variance forecast in one-step or multi-step?

Suppose I want to forecast 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 ...
1
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2answers
50 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
83 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\...
3
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1answer
183 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 ...
2
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1answer
64 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 ...
2
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1answer
130 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 ...
8
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3answers
5k views

Variance of time integral of squared Brownian motion

I want to calculate the variance of $$I = \int_0^t W_s^2 ds$$ I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get $$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
2
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2answers
142 views

Spot variance drift consequently to style drift

I am looking for some information on how to spot variance drift for a portfolio in accordance to its benchmarks, Let's say that we have returns of the portfolio $\textbf{P}=(P_1,...,P_t,...,P_n)$ and ...
1
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1answer
534 views

Terminal Variance in the Heston Model

I am trying to understand the basics of financial models. Random Walk as a model for asset prices. We use gaussian random numbers to generate a Gaussian Random walk. The variance of the terminal value ...
2
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1answer
88 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?
0
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1answer
76 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 ...
4
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1answer
151 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 ...
1
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0answers
60 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(...
2
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0answers
25 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_{...
10
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2answers
5k views

Is the VIX more similar to a volatility swap or a variance swap?

I am reading the following paragraph on the VIX wikipedia article and I find it confusing: The VIX is calculated as the square root of the par variance swap rate for a 30-day term[clarify] ...
13
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2answers
556 views

Realized variance in SVJJ (Heston with jumps) model

I am working with the stochastic volatility model with jumps in both the price and volatility dynamics, ie. the risk neutral dynamics are of the form: $$\mathrm{d}V_t = \kappa(\theta - V_t)\mathrm{d}t ...
0
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0answers
22 views

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 ...
2
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1answer
106 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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2answers
95 views

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 ...
2
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0answers
63 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. ...
2
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1answer
127 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
69 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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0answers
39 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 ...
1
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1answer
168 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 ...
9
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4answers
1k views

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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0answers
92 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 ...
32
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9answers
56k views

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?
3
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1answer
348 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
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\...
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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)
2
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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 ...
2
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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 ...
0
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2answers
159 views

Prove ρ(X,Z) = ρ

The covariance of two random variables $X$ and $Y$ is defined by: $$\mathrm{Cov}(X,Y)= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\...
2
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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 ...
-4
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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 ...
1
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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 ...
0
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0answers
43 views

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 ...
3
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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 ...
0
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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 $\gamma$ ...
2
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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 ...
0
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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 ...
1
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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 − (σ^2/ 2) t + σW(t) ) , $ where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
2
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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!
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0answers
142 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 ...
3
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0answers
66 views

How to Compute the payoff of Var Swaps, which I have replicated

I used Derman(1999) method, to calculate the fixed Kvar for Variance Swaps using actual option price data. The first Pic Shows the outcome. (ignore the 0s). Now the profit and loss of short var swaps ...