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.
195
questions
0
votes
0
answers
27
views
Minimum variance portfolio [closed]
Suppose we have n stocks, covariance, expected returns. I got efficient Frontier weights from scipy SLSQP. Does minimum variance weights change if we change expected returns?
0
votes
0
answers
42
views
The use of the trading time variance/volatility curve
In trading, how is the trading time variance/volatility curve and spread curve such as depicted and parameterized on p. 277 of Jim Gatheral and Roel C.A. Oomen, Zero-intelligence realized variance ...
- 2,471
2
votes
1
answer
89
views
Deriving the variance of G2++ Model
I'm studying G2++ Model in Brigo(2007)'s book.
The model constructed as follows,
$$
r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\
$$
with the dynamics of $dx(t)$ and $dy(t)$ described by:
\begin{align}...
- 303
0
votes
0
answers
47
views
Hedging an option with a stock and forward variance
Following Deep Hedging under Rough Volatility (https://arxiv.org/abs/2102.01962) they construct a hedging portfolio consisting of a stock and a so-called forward variance to hedge a European call ...
1
vote
0
answers
55
views
How much compensation need to take on risk?
Quant Firm Interview Question
We roll three, 8 sided dice. If same face appears 3 times we win 80 dollars. We have a bank of 10,000 dollars. How much are we willing to pay to play? What if we increase ...
- 61
1
vote
1
answer
137
views
Variance of the price from returns variance
Let's say that we have the variance of the daily return at $t_0$:
$$\sigma_{r_{t_0}}^2=\text{Var}[r_{t_0}]=\text{Var}[\frac{S_{t_0}-S_{t_0-1}}{S_{t_0-1}}]$$
for price process $S_t$. Is there a way to ...
0
votes
1
answer
77
views
Verify numerically relation between mean deviation and standard deviation
I was reading "We Don’t Quite Know What We Are Talking About When We Talk About Volatility" by Goldstein and Taleb, and I was trying to quickly verify numerically the relation between mean ...
2
votes
1
answer
117
views
Kelly Criterion — maximize expected value and minimize the variance in card game with $x$ red and $y$ black cards
You have $x$ red cards and $y$ black cards. I flip them over one at a time. The probability of flipping a particular colour is proportional to the amount of those coloured cards left. You start with $...
- 155
0
votes
1
answer
59
views
Variation of the trading range
Example: The trading range (in points) for each of the last 5 trading days for asset A is: 5,21,2,15,32 and for asset B is: 5,6,5,5,5. Is there an indicator that ranks assets based on variation of ...
user59094
0
votes
1
answer
112
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:
$$ Z_{CV} = \frac{...
- 25
1
vote
1
answer
151
views
Why is $Z_t$ uncorrelated with $X_{t-1}$ in $X_t=\theta X_{t-1}+Z_t$?
In a solution to the problem below, the teaching assistant solves it by calculating $\mathbb{E}[X_t^2]$ and ends up with also having to calculate $\mathbb{E}[X_{t-1}Z_t]$ after expanding the square. ...
- 221
1
vote
0
answers
84
views
Computing the Variance Risk Premium
The Variance Risk Premium (VRP) is defined as:
$$VRP(t,t+\Delta t) \equiv RV(t,t+\Delta t)^2 - IV_t(t,t+\Delta t)^2$$
where $RV^2$ is the realized variance between $t$ and $t + \Delta t$ and $IV_t^2$ ...
- 181
7
votes
2
answers
383
views
Heston: Variance of Integrated Variance
Consider the standard Heston model\begin{align*}
dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\
dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\
dBdW&=\rho dt.
\end{align*}
Computing $\mathbb{E}\...
- 686
0
votes
0
answers
78
views
Monte Carlo Simulation of GBM Process has a Very High Variance - Explanation Needed as to why?
I use Geometric Brownian Motion (GMB) to simulate a share price from March 24, 2020 to March 24 as follow:
\begin{equation}
S_t=S_{t-1}exp((rf-0.6\sigma^2)*(2)+\sigma*sqrt(2)*\mathcal{N}(0,1))
\end{...
- 21
3
votes
1
answer
169
views
Market-maker's gain variance
I am reading the book "Trades, Quotes and Prices" by JEAN-PHILIPPE BOUCHAUD and have stuck in the very beginning with understanding the formula of variance of MM's gain per trade (see ...
- 33
0
votes
1
answer
58
views
Show that the following result holds true for the variance of the return of a portfolio of shares
Start with a portfolio $p$ of $n$ shares, each with weight $x_i = \dfrac{1}{n}$ (for $i$ ranging from $1$ to $n$, discretely). Its return is given by:
$$R_p=x_1R_1+\ldots+x_nR_n=\sum_{i=1}^{n}=x_iR_i\...
0
votes
1
answer
385
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 ...
- 3
1
vote
2
answers
152
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 ...
- 121
0
votes
0
answers
35
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
1
answer
98
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 ...
- 1
2
votes
0
answers
173
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 ...
- 21
1
vote
1
answer
138
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 ...
- 11
0
votes
2
answers
83
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 ...
- 101
2
votes
1
answer
313
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 ...
- 6,860
-1
votes
1
answer
94
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 ...
- 11
0
votes
1
answer
541
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 ...
- 136
1
vote
1
answer
227
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 ...
- 267
2
votes
1
answer
90
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 ...
- 355
0
votes
0
answers
78
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 ...
- 17
-3
votes
1
answer
225
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 $X w$ ...
- 2,835
0
votes
1
answer
326
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 ...
- 23
0
votes
2
answers
108
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 ...
- 111
2
votes
1
answer
755
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 ...
- 1,708
-1
votes
2
answers
176
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 ...
- 520
3
votes
1
answer
536
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 ...
- 43
2
votes
0
answers
110
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 ...
- 516
0
votes
2
answers
143
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 ...
- 105
1
vote
0
answers
120
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 ...
- 2,471
0
votes
1
answer
133
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 ...
- 1
3
votes
1
answer
281
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 ...
- 127
1
vote
3
answers
99
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 ...
- 13
2
votes
1
answer
266
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\...
- 2,471
2
votes
1
answer
99
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 ...
- 136
3
votes
1
answer
385
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 ...
- 1,788
2
votes
1
answer
157
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 ...
- 191
2
votes
1
answer
133
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?
- 23
0
votes
1
answer
188
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 ...
- 47
1
vote
0
answers
140
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(...
- 23
2
votes
0
answers
29
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_{...
- 489
2
votes
1
answer
324
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 ...
- 181