Questions tagged [variance]
The variance tag has no usage guidance.
142
questions
45
votes
11answers
5k views
Lévy alpha-stable distribution and modelling of stock prices.
Since Mandelbrot, Fama and others have performed seminal work on the topic, it has been suspected that stock price fluctuations can be more appropriately modeled using Lévy alpha-stable distrbutions ...
31
votes
9answers
52k views
What is the difference between volatility and variance?
How do volatility and variance differ in finance and what do both imply about the movement of an underlying?
19
votes
5answers
19k views
Why would an investor trade a variance swap over a volatility swap?
Why would an investor trade a variance swap over a volatility swap? Is it simply related to the leverage involved in a Var (i.e. sigma-squared) or is there something else to it?
17
votes
4answers
943 views
Approximately what proportion of a stock’s volatility is explained by market movement?
Assume we decompose the daily (log) returns of a stock as beta times market movement plus an idiosyncratic part. If this is done ex-ante, what proportion of the variance is explained by the market ...
12
votes
2answers
443 views
Realized variance in SVJJ (Heston with jumps) model
I am working with the stochastic volatility model with jumps in both the price and volatility dynamics, ie. the risk neutral dynamics are of the form:
$$\mathrm{d}V_t = \kappa(\theta - V_t)\mathrm{d}t ...
11
votes
1answer
14k views
How to calculate the conditional variance of a time series?
I am reading a paper where the term conditional variance is mentioned, but I am not really sure what is meant by this and how this can be calculated:
Fig. 2 shows the conditional variances of the ...
10
votes
1answer
2k views
Variance replication using options
I would like to understand the intuition behind the following question:
Why a certain weighted sum of prices of put and calls is equivalent to the implied variance of an underlying?
A variance swap ...
10
votes
1answer
391 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 ...
10
votes
0answers
1k views
Formula for the efficient portfolios (mean-variance optimisation)?
Consider the setting of mean-variance portfolio optimisation: $n$ assets with expected returns $\overline{r}_1,...,\overline{r}_n$ and standard deviations $\sigma_1,...\sigma_n$.
For a certain fixed $...
9
votes
2answers
900 views
Why do low standard deviation stocks tend to have superior future returns?
I've recently stumbled on something that really surprised me. These papers (1, 2) find that past standard deviation of returns is inversely related to future returns. That is, portfolio of low ...
8
votes
2answers
3k views
Is the VIX more similar to a volatility swap or a variance swap?
I am reading the following paragraph on the VIX wikipedia article and I find it confusing:
The VIX is calculated as the square root of the par variance swap rate for a 30-day term[clarify] ...
8
votes
3answers
2k views
Why is a variance swap long skew?
I can appreciate the mathematical derivation, but can anyone explain this in a more intuitive sense?
I often come across the mistaken belief that due to the replicating portfolio being long more ...
8
votes
3answers
4k views
Variance of time integral of squared Brownian motion
I want to calculate the variance of
$$I = \int_0^t W_s^2 ds$$
I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get
$$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
8
votes
1answer
355 views
Do weights from portfolio theory contain bias?
I want to experiment with some portfolio modelling and I was wondering if you guys could help me with something. If I try to estimate and implement the traditional two-fund portfolio consisting of one ...
8
votes
1answer
338 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 ...
8
votes
1answer
5k views
Conditional or unconditional volatility?
I am reading a paper (reference below) that states "The conditional volatility for each underlying security (or for a market index) can be estimated using the standard deviation of the stock’s ...
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 ...
7
votes
3answers
2k views
Variance swap replication and variance vega
Noob here.
I've been trying to gain a better understanding of variance swaps and what better way than to replicate it with a portfolio of better understood instruments.
I have read the GS 1999 ...
6
votes
2answers
790 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 ...
6
votes
2answers
552 views
Choice of prior as a shrinkage target in portfolio construction?
There's various research showing how priors such as the minimum variance portfolio turn out to be a surprisingly effective shrinkage target in portfolio construction.
The sell point of these priors ...
6
votes
1answer
2k views
Static and Dynamic Hedging of Vol/Var Swaps
Why can a variance swap be perfectly statically hedged whereas a volatility swap requires dynamic hedging?
Possible reference request to the corresponding literature.
6
votes
1answer
212 views
How to scale option pricing components in regard to time
I am looking at closed-form options approximations, in particular the Bjerksund-Stensland model.
I have run into a very basic question. How should I scale the input variables in regard to time? My ...
5
votes
2answers
298 views
How to derive this approximation of the risk-neutral expectation of the variance?
On the paper Bollerslev, Tauchen and Zhou (2009 RFS) the authors say about equation (15):
The corresponding model implied risk-neutral conditional expectation
$$E^Q_t(\sigma^2_{r,t+1})=E_t(\sigma^...
5
votes
1answer
446 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 ...
5
votes
3answers
804 views
Negative high frequency intraday volatility - Zhou estimator
To estimate high frequency tick data stock intraday volatility, I have read Robert Almgren's notes7.pdf
http://www.cims.nyu.edu/~almgren/timeseries/notes7.pdf
where he talks about the bias free ...
5
votes
1answer
1k views
Calibrating Hull-White using volatility data
I would like to calibrate Hull-White model using volatility data.I am using [Park (2004)] paper as a reference.
He suggests to minimize the following objective function:
where the first term 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 ...
4
votes
2answers
116 views
Does heteroskedasticity of returns depend on the time frame?
Similarly to my last question, for which I obtained very interesting and useful answers, I would like to know if there has been any study regarding heteroskedasticity and time-frames of the returns.
...
4
votes
1answer
514 views
Linear regression and assets direction prediction
I have the following asset returns Y and the predictions for the same periods Y':
Y = { 10, 200, -1000, -1, -7 }
Y' = { 1, 2, -3, -4, -5 }
The OLR R-squared for ...
4
votes
1answer
386 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 ...
3
votes
1answer
1k views
How to compute the variance of a Long-Short Equity Portfolio?
I am calculating the historical portfolio variance of various long-short equity portfolios. For simplicity, assume the portfolio is long stock A with weight 1.0 and short stock B with weight -0.5. ...
3
votes
1answer
1k views
Why is this delta-hedging/P&L example on a variance swap call correct?
I'm looking into this article about var swaps: http://sbossu.com/docs/VarSwaps.pdf and not sure how to correctly interpret Exhibit 2.1.1.
"In this example an option trader sold a 1-year call struck ...
3
votes
1answer
207 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 ...
3
votes
2answers
424 views
Variance of a Stock price and relationship with volatility
A bit of background. I know that the forward price of a stock (or its expected price) is given by $\mathbb{E}[S_T]=S_te^{(r-q)(T-t)}$. Here, $r$ and $q$ are not constant, but follow a curve. I was ...
3
votes
1answer
130 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}$$
We have
$d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\...
3
votes
1answer
133 views
Antithetic sampling Monte Carlo
In Peter Jaeckel, Monte Carlo in Finance book, I read the following sentence:
Whenever the first realised moment of the underlying variate draws
$\{z_i\}$ has a strong impact on the result of the ...
3
votes
1answer
385 views
Criticise GARCH relative to Realized Volatility
I would like to have your opinion about a simple question.
While GARCH would be useful to calculate the conditional volatility, and the RV being in some sense the "historical" volatility, what would ...
3
votes
2answers
420 views
How is volatility different from variance?
I always thought volatility was just variance ^ (1/2). Now I'm reading this book and it's saying that the two are different concepts. Excerpts include:
Partly due to its use in Black-Scholes, ...
3
votes
2answers
407 views
Estimation Risk-Neutral Variance of Returns
I am trying to find a method which allows me to estimate $Var_{\mathbb{Q}}\left(\frac{S_{t_{i+1}}}{S_{t_i}}\right)$ where $S$ denotes the price process of an underlying stock (which has to be assumed ...
3
votes
2answers
342 views
How to estimate the variance of this stochastic process?
I have an unobservable stochastic quantity $\lambda(t)$, which I analytically know the variance of, that is
$$\text{Var}(\lambda(t))= \frac{\theta \sigma^2}{2\kappa}$$
My goal is to get an estimate ...
3
votes
1answer
293 views
Why are there different estimators for stock volatility? (realized variance, RAV, etc)
I am very confused about why different volatility estimators (RV, RAV, BPV, etc) exist. If the goal is to find the best estimator for stock volatility, and volatility is latent, how do I know which ...
3
votes
0answers
57 views
How to Compute the payoff of Var Swaps, which I have replicated
I used Derman(1999) method, to calculate the fixed Kvar for Variance Swaps using actual option price data. The first Pic Shows the outcome. (ignore the 0s).
Now the profit and loss of short var swaps ...
3
votes
0answers
130 views
Usages of variance swap
I’m interested in variance swap.
Considered from its feature, variance swap is used for betting the (historical) volatility of underlying asset.
If we use it for hedge tool of Vega or Volga, does it ...
3
votes
0answers
230 views
Variance swap “fast” models
As far as I understand, Variance Swap (VS for short) function as follows :
no payment when entering the contract
at maturity the VS buyer pays a strike $K^2$ and is paid (by the VS seller) the ...
2
votes
2answers
1k views
Covariance for arbitrarily large portfolios
I am implementing a method in Java to calculate the variance, covariance, and value at risk for a portfolio, which should be flexible for use with any number of assets in a portfolio. I am struggling ...
2
votes
2answers
134 views
Why would one prefer variance swaps over other instruments?
I understand that an investor who has a view on an underlying's variance would be tempted by a variance swap. But why would one prefer such a contract over another instrument whose value is based on ...
2
votes
1answer
431 views
How do you find variance of a sde?
I know how to find the mean of an SDE: write it on integral form, take derivative, solve a simple ODE.
But what to do when we want a variance?
In my case, $$X_{T + \delta t} = X_T + \int_T^{T + \...
2
votes
1answer
154 views
Risk, required return and expected volatility - what is the relationship?
Return required from risk averse agents from risky investments are proportional to expected return variance. That is from the textbook, you take the portfolio with the highest return to standard ...
2
votes
3answers
692 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 ...
2
votes
1answer
4k views
Using CAPM to find correlation of two assets with each other
I stumpled upon an exercise in an investments book:
The data below describe a three-stock financial market that satisfies
the single-index model.
...
