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.
175
questions
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1answer
75 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 ...
0
votes
1answer
202 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 ...
0
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2answers
54 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 ...
0
votes
1answer
85 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 ...
0
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1answer
54 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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1answer
47 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 ...
0
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1answer
61 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 ...
2
votes
1answer
42 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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0answers
57 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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votes
1answer
173 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 $X w$ ...
2
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1answer
130 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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0answers
36 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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2answers
99 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 $Ļ_i^2$ of the expected return of an asset $i$.
In particular, how can I derive the following formula?
$$Ļ_i^2 = Ī²_i^2 ...
2
votes
0answers
112 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
164 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
votes
1answer
149 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 ...
2
votes
3answers
82 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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0answers
56 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 ...
12
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0answers
2k 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 $\...
3
votes
1answer
210 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
128 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
50 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
193 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
72 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
votes
1answer
135 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
votes
3answers
6k 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
votes
2answers
144 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
vote
1answer
584 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
votes
1answer
97 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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votes
1answer
94 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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votes
1answer
204 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
81 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
votes
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_{...
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2answers
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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
votes
2answers
594 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 ...
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0answers
24 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 ...
3
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1answer
133 views
Variance risk premium: When is realized vol higher than implied vol in practice?
Iām doing some work around the variance risk premium currently, and Iām interested in understanding the situations when realized volatility is > implied volatility in practice.
I know in generally ...
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2answers
100 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
86 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
votes
1answer
158 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
votes
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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0answers
42 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
237 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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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
132 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 ...
33
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9answers
57k 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
620 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}$$
We have
$d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\...
1
vote
1answer
72 views
Variance of a spread for options on spreads
I was reading the paper:
https://people.umass.edu/nkapadia/docs/Negative_Vega.pdf
In the equation $(5)$, he is defining the variance of the spread as:
$$\sigma_1^2S_1^2 + \sigma_2^2S_2^2 - 2\...
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0answers
36 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
165 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 ...
