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.
152
questions
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1answer
67 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
33 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
19 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_{...
0
votes
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
votes
1answer
92 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 ...
4
votes
1answer
91 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
50 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
108 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
67 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
38 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
129 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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0answers
69 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 ...
0
votes
1answer
157 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
259 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
66 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:
$$\sigma_1^2S_1^2 + \sigma_2^2S_2^2 - 2\...
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0answers
25 views
Show the expected return of the portfolio and how to derive return variance of the long-short portfolio?
Show the expected return of the portfolio and how to derive the return variance of the long-short portfolio? (see picture)
2
votes
1answer
50 views
ARMA moments proof
Consider a standard ARMA(1,1) process such as
$$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 ...
2
votes
1answer
120 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
147 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 ...
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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 ...
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0answers
54 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 ...
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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
votes
1answer
168 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
1answer
206 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)
$$
where $\gamma$ ...
0
votes
1answer
53 views
Condition expectation calculation examples and theory [closed]
I want to ask you an advice about reading theory and examples of conditional expectation and conditional variance. I want to have my understanding deeper, because sometimes I can't understand ...
2
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0answers
86 views
Residual Risk and Variance
I've solved part a, but am struggling with b and c. $x_m$ is the market portfolio vector, and I think $T$ should be a diagonal matrix. Any hints greatly appreciated!
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0answers
123 views
Applying portfolio variance weight based on logarithmic returns?
The expected logarithmic return of a portfolio is calculated as :
$$𝐸_p = \log\left(\sum_i w_i e^{R_i}\right)$$
Therefore, I was wondering that how can I apply weight to use with the variance based ...
1
vote
1answer
77 views
Expectation and variance of standard brownian motion
Assuming that the price of the stock follows the model
$ S(t) = S(0) exp (
mt −
(σ^2/
2)
t + σW(t)
)
,
$
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
0
votes
2answers
159 views
Prove ρ(X,Z) = ρ
The covariance of two random variables $X$ and $Y$ is defined by:
$$\mathrm{Cov}(X,Y）= \operatorname{E}(X-\operatorname{E}(X))(Y-\operatorname{E}(Y))=\operatorname{E}(XY)-\operatorname{E}(X)\...
3
votes
0answers
62 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 ...
11
votes
1answer
616 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
24 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
653 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
46 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
99 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 ...
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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
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1answer
298 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
95 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
137 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
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0answers
320 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
55 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 ...
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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
119 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
36 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
1k 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 ...
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0answers
263 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
193 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
129 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
2answers
93 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
418 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 ...
