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.
187
questions
0
votes
1answer
58 views
Control Variates - Option pricing
I am trying to reduce the Monte Carlo variance with Control Variates technique. In practice, I am able to reduce it with a generic European Call option, with the following formulas:
$$ Z_{CV} = \frac{...
1
vote
1answer
138 views
Why is $Z_t$ uncorrelated with $X_{t-1}$ in $X_t=\theta X_{t-1}+Z_t$?
In a solution to the problem below, the teaching assistant solves it by calculating $\mathbb{E}[X_t^2]$ and ends up with also having to calculate $\mathbb{E}[X_{t-1}Z_t]$ after expanding the square. ...
1
vote
0answers
61 views
Computing the Variance Risk Premium
The Variance Risk Premium (VRP) is defined as:
$$VRP(t,t+\Delta t) \equiv RV(t,t+\Delta t)^2 - IV_t(t,t+\Delta t)^2$$
where $RV^2$ is the realized variance between $t$ and $t + \Delta t$ and $IV_t^2$ ...
7
votes
2answers
331 views
Heston: Variance of Integrated Variance
Consider the standard Heston model\begin{align*}
dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\
dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\
dBdW&=\rho dt.
\end{align*}
Computing $\mathbb{E}\...
0
votes
0answers
51 views
Monte Carlo Simulation of GBM Process has a Very High Variance - Explanation Needed as to why?
I use Geometric Brownian Motion (GMB) to simulate a share price from March 24, 2020 to March 24 as follow:
\begin{equation}
S_t=S_{t-1}exp((rf-0.6\sigma^2)*(2)+\sigma*sqrt(2)*\mathcal{N}(0,1))
\end{...
3
votes
1answer
131 views
Market-maker's gain variance
I am reading the book "Trades, Quotes and Prices" by JEAN-PHILIPPE BOUCHAUD and have stuck in the very beginning with understanding the formula of variance of MM's gain per trade (see ...
0
votes
1answer
49 views
Show that the following result holds true for the variance of the return of a portfolio of shares
Start with a portfolio $p$ of $n$ shares, each with weight $x_i = \dfrac{1}{n}$ (for $i$ ranging from $1$ to $n$, discretely). Its return is given by:
$$R_p=x_1R_1+\ldots+x_nR_n=\sum_{i=1}^{n}=x_iR_i\...
0
votes
1answer
104 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 ...
1
vote
2answers
113 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 ...
0
votes
0answers
45 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 ...
0
votes
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 ...
0
votes
1answer
51 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
votes
0answers
111 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 ...
1
vote
1answer
79 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 ...
0
votes
2answers
66 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 ...
2
votes
1answer
158 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 ...
-1
votes
1answer
62 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
votes
1answer
232 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 ...
1
vote
1answer
136 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
55 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 ...
0
votes
0answers
63 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 ...
-3
votes
1answer
195 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 $X w$ ...
0
votes
1answer
186 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 ...
0
votes
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
votes
1answer
415 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 ...
-1
votes
2answers
168 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
329 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
0answers
61 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 ...
0
votes
2answers
133 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 ...
1
vote
0answers
62 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 ...
0
votes
1answer
89 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
votes
1answer
228 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 ...
1
vote
3answers
86 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
votes
1answer
203 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
votes
1answer
81 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 ...
3
votes
1answer
225 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
votes
1answer
144 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
votes
1answer
112 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
votes
1answer
144 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 ...
1
vote
0answers
115 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
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_{...
2
votes
1answer
234 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 ...
5
votes
1answer
324 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 ...
2
votes
0answers
117 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
231 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
74 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 ...
0
votes
0answers
44 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
vote
1answer
381 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 ...
1
vote
0answers
216 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 ...
3
votes
1answer
860 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}+\...
