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
1
vote
1answer
97 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
163 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
72 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 ...
14
votes
1answer
848 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
34 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
885 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
59 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
123 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
30 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
351 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
98 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
151 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
375 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
59 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
44 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 ...
0
votes
1answer
128 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
48 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
2k 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
345 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
230 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
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
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 ...
9
votes
1answer
507 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
3k 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 ...
5
votes
2answers
3k 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
74 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
628 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
111 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
586 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 ...
1
vote
0answers
210 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
668 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
620 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
80 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{...
-4
votes
3answers
2k views
Value at Risk - Long/Short position
I have a simple question on the VaR for a portfolio that consists of a long and short position. Say I have a portfolio consisting of the following positions:
long 1000 shares of stock X
short 1000 ...
2
votes
1answer
95 views
Typical SPX variance GARCH(1,1) coefficients
Can someone provide a typical numerical values of GARCH(1,1) coefficients $(\omega,\alpha,\beta)$ for estimating SPX index variance? I will appreciate it if some references could be provided.
-3
votes
1answer
99 views
Derivation of arithmetic variation of a portfolio over multiple periods [closed]
I am very confused on how to derive the attached equation (15).
Would someone be kind enough to walk me through the proof?
1
vote
0answers
389 views
Does delta adjusted exposure make sense for an equity variance swap?
The software vendor that I am using for the calculation of the market risk exposure claim that they cannot compute the delta adjusted of the equity variance swap positions since there is no specific ...
2
votes
1answer
140 views
minimum variance hedge with stochastic processes
Problem set up:
asset S: $$\frac{dS}{S} = \mu dt+\sigma dz$$
Hedged using a forward contract: $F = F(S,t).$
Hedge portfolio: $$P = S+nF$$
I want to find the variance of $dP$, and then minimize that ...
4
votes
1answer
428 views
Portfolio diversification and Sharpe ratio
I have a given trading strategy T and say 3 assets in my universe. The hold time is one day. The trading strategy can general signals for the 3 assets in any given day (so signal can trigger for any ...
5
votes
1answer
566 views
Replicating Log Contract - Errors Introduced by Jumps
In the GS Research Note about Volatility Swaps, it is shown that you can replicate a pure variance exposure (hedge) with only vanilla calls and puts, primarily thanks to the Carr-Madan formula of ...
3
votes
1answer
262 views
Intuition Behind Scaling Factor in Variance Swaps
In More Than You Ever Wanted to Know About Volatility Swaps the fair value of a future variance swap can be replicated from market prices for calls and puts. The fair put and call strike is shown to ...
1
vote
1answer
112 views
Variance of returns on a portfolio
This must be very basic, but I don't seem to be able to express the variance of returns on a portfolio in terms of variances-covariance sum of returns of its constituents, which seems to be what is ...
0
votes
0answers
103 views
Units of Risk: Variance vs Standard Deviation
Suppose you are trading two mean-reverting assets, A and B, and that $Covar(A, B) > 0$. You are currently long one unit of A, and are considering buying one unit of B. Compared to the situation ...
6
votes
2answers
954 views
What are the significant implications of the long-run average variance rate and why Engle won the Nobel Prize for ARCH model development?
In a ARCH(m) model we have
$$
\sigma_n^2=\sum_{i=1}^{m} \alpha_i u_{n-i}^2
$$
where $u_i$ is defined as the continuously compounded return during day $i$ (between the end of day $i-1$ and the end of ...
0
votes
1answer
92 views
Question regarding the purchase of a Variance Swap
Imagine I price a Variance Swap for an investor and the observation date starts tomorrow and ends in 30 days. If I use dynamic replication with options to price my variance swap do I use options with ...
0
votes
0answers
224 views
Black Variance Surface
I came across black variance surface in quantlib code. For options, usually volatility surface is used for pricing. When you will use variance surface for pricing or any advantages over volatility ...
2
votes
4answers
1k views
Why is variance problematic as a risk measure?
I am looking for a simple example which explains why variance as a risk measure can be problematic (with a long-only portfolio with no options).
1
vote
0answers
276 views
Quantitative Strategy on Variance Swap (master thesis)
I am doing a master thesis on Variance Swap and my dear friend told me I could find some valuable help on the "Quantitative Finance Stack Exchange".
I would like to apologise beforehand, if my ...
1
vote
1answer
2k views
variance of log return
Suppose $C_i$ is i-day's closed price, when drift is small,
we have the close to close variance
$$\sigma^2 =\dfrac{1}{n}\sum\limits^n_{i = 1}\left(\log\left(\dfrac{C_i}{C_{i-1}}\right)\right)^2.$$
If ...
2
votes
1answer
629 views
How do you find variance of a sde?
I know how to find the mean of an SDE: write it on integral form, take derivative, solve a simple ODE.
But what to do when we want a variance?
In my case, $$X_{T + \delta t} = X_T + \int_T^{T + \...
