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Questions tagged [variance]

The tag has no usage guidance.

45
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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 ...
30
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8answers
50k 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?
18
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5answers
18k 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
934 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
367 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
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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 $...
10
votes
1answer
12k 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 ...
9
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1answer
1k 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 ...
9
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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 ...
9
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2answers
886 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
1answer
336 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
3answers
591 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 ...
8
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1answer
194 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 ...
8
votes
1answer
251 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
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3answers
3k 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{\...
6
votes
2answers
542 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
368 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 ...
6
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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
1answer
209 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
682 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 ...
5
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2answers
284 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
3answers
760 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
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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
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2answers
1k 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
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.
4
votes
2answers
109 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
505 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
370 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
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 ...
3
votes
1answer
360 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 + \...
3
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1answer
915 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
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2answers
2k views

GARCH variance vs standard deviation for volatility

in my series of questions related to GARCH and volatility I finally think I've got a decent grasp on it. You guys have been great help clearing up my questions for me. My next question is just a ...
3
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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
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1answer
163 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
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2answers
387 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
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1answer
218 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
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2answers
212 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
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2answers
288 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
4answers
650 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).
3
votes
2answers
313 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
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1answer
284 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
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0answers
47 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
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0answers
114 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
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0answers
212 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
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2answers
125 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
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1answer
144 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
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3answers
335 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
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1answer
3k 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. ...