Questions tagged [variance]
The variance tag has no usage guidance.
142
questions
0
votes
1answer
121 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
votes
1answer
132 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
61 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:
$$\sigma_1^2S_1^2 + \sigma_2^2S_2^2 - 2\...
1
vote
0answers
21 views
Show the expected return of the portfolio and how to derive return variance of the long-short portfolio?
Show the expected return of the portfolio and how to derive the return variance of the long-short portfolio? (see picture)
2
votes
1answer
45 views
ARMA moments proof
Consider a standard ARMA(1,1) process such as
$$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 ...
2
votes
1answer
90 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
votes
1answer
103 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 ...
0
votes
0answers
40 views
Varswap replication product
I would like to ask about a product that some Flow desks sell : Varswap replication strategies. I know that it consists of weighted basket of calls and puts , however I would like to know how does it ...
1
vote
0answers
49 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
votes
0answers
39 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
votes
1answer
133 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
votes
0answers
31 views
How can i fit the following regression in R? Why is the coefficient [second Columns] for R so low?
'Rwml' is the monthly log return
So the first column is clear, I got nearly the same values, at least the same magnitude. But: If I regress on the variance, my input values are way too low to get a ...
0
votes
1answer
133 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)
$$
where $\gamma$ ...
0
votes
0answers
61 views
How is Kalman Filter used to estimate Term structure Models
I am implementing "The Term Structure of Variance Swap Rates and Optimal Variance Swap Investments" . This paper is using kalman filter to estimate the state and the mean variance and a parameters on ...
0
votes
1answer
52 views
Condition expectation calculation examples and theory [closed]
I want to ask you an advice about reading theory and examples of conditional expectation and conditional variance. I want to have my understanding deeper, because sometimes I can't understand ...
2
votes
0answers
74 views
Residual Risk and Variance
I've solved part a, but am struggling with b and c. $x_m$ is the market portfolio vector, and I think $T$ should be a diagonal matrix. Any hints greatly appreciated!
1
vote
0answers
93 views
Applying portfolio variance weight based on logarithmic returns?
The expected logarithmic return of a portfolio is calculated as :
$$𝐸_p = \log\left(\sum_i w_i e^{R_i}\right)$$
Therefore, I was wondering that how can I apply weight to use with the variance based ...
1
vote
1answer
70 views
Expectation and variance of standard brownian motion
Assuming that the price of the stock follows the model
$ S(t) = S(0) exp (
mt −
(σ^2/
2)
t + σW(t)
)
,
$
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
0
votes
2answers
157 views
Prove ρ(X,Z) = ρ
The covariance of two random variables $X$ and $Y$ is defined by:
$$\mathrm{Cov}(X,Y）= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\...
3
votes
0answers
57 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 ...
10
votes
1answer
393 views
Derivation of VIX Formula
I've read a lot of derivations about VIX formula. I can say it is -adjusted- fair strike of variance swap. But I can't see how it goes from variance swap rate to VIX formula. In particular I can't see ...
2
votes
0answers
23 views
Rolling sum of conditional variance
I have a model to compute the conditional expectation and variance for a return series, given various factor returns. Initially attempted to trade the deviations of actual return for the day from the ...
3
votes
2answers
430 views
How is volatility different from variance?
I always thought volatility was just variance ^ (1/2). Now I'm reading this book and it's saying that the two are different concepts. Excerpts include:
Partly due to its use in Black-Scholes, ...
0
votes
1answer
32 views
variance unsystematic component
I was wondering how to calculate the variance of the unsystematic component in an asset.
For example, if an asset's expected return is 10% with standard deviation of 6% and a beta of zero. What ...
2
votes
1answer
85 views
Variance Swap : dividends and rates
In a simplified world you can assume that the var swap is replicated by a continuous set of calls and puts and interest rates are equal to zero. So your PNL is only sensitive to the volatility.
But in ...
1
vote
0answers
27 views
Variance of integrated dynamical system
Define time increment $\mu:=t_{k+1}-t_{k}$. Consider the signal $x(\mu)-\mathbb{E}[x(\mu)]$ defined as
$x(\mu)-\mathbb{E}[x(\mu)]=\frac{1}{\mu}\int_{t_{k}}^{t_{k+1}}\int_{0}^{\tau}e^{A(\tau-\delta)}...
1
vote
1answer
254 views
Options on realized volatility / variance
If I'd like to price options on variance/volatility in the Heston model. Is MC simulation and/or finite difference the only way to do it? Or is there an analytical expression for the probability ...
2
votes
0answers
90 views
Mean-Variance portfolio: How do I compute the variance when the portfolio is normalized
Let's consider the very basic of a Mean-Variance Portfolio:
$$
\text{max}_{x}
(1-\lambda)\sum_i^n\mu_ix_i-\lambda\sum_i^n\sum_j^n x_i Q_{ij}x_j
$$
$$\text{ s.t. }\sum_i^nx_i=1 \text{ , } x_i \geq ...
3
votes
0answers
130 views
Usages of variance swap
I’m interested in variance swap.
Considered from its feature, variance swap is used for betting the (historical) volatility of underlying asset.
If we use it for hedge tool of Vega or Volga, does it ...
1
vote
0answers
219 views
Overlapping Data
I have a daily time series data spanning over 22 years. I need to compute some meaningful yearly standard deviation statistics / generate probability distribution and estimate tail risk. 22 years ...
1
vote
1answer
54 views
What's the correct graphical comparison in a GARCH fit?
Suppose that the stationary series $r_t$ is well fitted by an $ARMA(p,q)+c$ and $GARCH(r,s)$ model, where $GARCH(r,s) = \sigma_t ^2$
If in the testing sample I have to graphically compare the ...
1
vote
0answers
42 views
Total Variance of an asset in case of stochastic rates
Let's suppose the underlying S follows a BS dynamic with the drift being the short rate that follows a short dynamic model.
the "local volatility" of the equity should be the implied volatility from ...
-1
votes
1answer
111 views
Minimizing variance when searching for Cointegration
This paper by Meucci explains that in order to find a combination leading to cointegration of several series $X$, you have to find the vector $w$ which minimise the quantity $\textrm{Var}(w'X)$. I do ...
2
votes
0answers
32 views
Use of second similar European Option as control variate to simulate a European option
I understand the idea and math behind the concept of control variate for the sake of variance reduction, but I struggle to apply it to option pricing.
I need to simulate an European option of a stock ...
2
votes
3answers
707 views
Negative variance?
Using the formula w*Cov*t(w) I can generate a negative portfolio variance. What are the implications of a negative variance? Should I just assume it's zero? A negative variance is troublesome ...
1
vote
0answers
193 views
VarSwap PnL formula
I came across this formula for the varswap PNL:
let $r_i$ be the log return over $[t_i,t_{i+1}]$ and suppose we risk manage the VS at a fixed implied volatility sigma, the PnL of (the payoff) over ...
-2
votes
1answer
163 views
Variance covariance matrix - number of periods required
Hi I am reviewing the example of Barra risk model in the following document page 23 there is the statement:
"Estimating a covariance matrix for, say, 3,000 stocks requires data
for at least 3,...
2
votes
2answers
128 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 ...
0
votes
1answer
65 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 ...
8
votes
1answer
342 views
Jim Gatheral's assertion on ATM implied volatility vs. square root variance
In Jim Gatheral's book The Volatility Surface Section Dependence on Skew and Curvature on page 138, he asserts that
We know that the implied volatility of an at-the-money forward option
in the ...
7
votes
3answers
1k views
CAPM model as a regression
The CAPM model states that the returns of a stock are-
$r_s=r_f+\beta (r_m-r_f)+\varepsilon_s$
The $\beta$ defined above is then calculated as $\frac{cov(r_s,r_m)}{var(r_m)}$. My question is ...
4
votes
2answers
2k views
What is the difference between squared returns and variance?
I am trying to calculate 1-day ahead volatility forecasts using the exponentially weighted moving average, however I am unsure on how to read the formula provided within Risk-Metrics Technical ...
1
vote
1answer
67 views
An ad hoc portfolio optimization scheme
Say at each time $t$ I have a covariance matrix for the next period. Call this $\Sigma_{t+1}$. If I choose portfolio weights $w$ to minimize the variance, subject to the constraint that $\sum_i w_i = ...
1
vote
1answer
476 views
Excel Add-In Volatility Interpolation I am trying to Understand
The Microsoft Excel at my investment bank has an .xll add-in with a function whose coded functionality I cannot observe. This function is called VolInterp and as the name suggests, calculates the ...
2
votes
1answer
77 views
How to Deal With Betas when variance is Zero?
To calculate a beta, I was using the following formula(Considering $ra$ as returns of $a$ and $rb$ as returns of $b$):
$$
\beta = { cov(ra, rb) \over var(rb)}
$$
As a software developer, I ...
1
vote
1answer
387 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 ...
1
vote
0answers
160 views
How can I compute a realized variance for raw instead of log returns?
Whenever I read about calculating realized variance, people are using log returns. However, I was asking myself whether it is possible to calculate realized variance also for simple, raw returns.
...
3
votes
1answer
391 views
Criticise GARCH relative to Realized Volatility
I would like to have your opinion about a simple question.
While GARCH would be useful to calculate the conditional volatility, and the RV being in some sense the "historical" volatility, what would ...
3
votes
2answers
408 views
Estimation Risk-Neutral Variance of Returns
I am trying to find a method which allows me to estimate $Var_{\mathbb{Q}}\left(\frac{S_{t_{i+1}}}{S_{t_i}}\right)$ where $S$ denotes the price process of an underlying stock (which has to be assumed ...
2
votes
0answers
64 views
Determine GARCH(1,1) from a mean reverting time series recursion
Let $(v_t)$ be a discrete time series of variance obeying a mean-reverting variance process $v_t$, which is actually the discrete version of the Heston model in finance.
\begin{align}
x_t &= \sqrt{...
