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.

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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 ...
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1answer
83 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 ...
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2answers
89 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 ...
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46 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 ...
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46 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. ...
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1answer
94 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; ...
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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 ...
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1answer
65 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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36 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
115 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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921 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 ...
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1answer
145 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 ...
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59 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 ...
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9answers
54k 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?
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239 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}+\...
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1answer
65 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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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)
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1answer
108 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 ...
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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 ...
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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)\...
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1answer
142 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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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 ...
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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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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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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 ...
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1answer
163 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 ...
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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 ...
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1answer
190 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 $\gamma$ ...
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1answer
4k views

What is the unconditional variance for a GARCH model?

I want to use a Matlab script to calculate Heston Nandi GARCH prices. I found an appropriate script online and it asks for the "unconditional variance" as an input. How do I calculate the appropriate ...
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64 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 ...
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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 ...
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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 ...
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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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121 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 ...
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4k 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] ...
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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 ...
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1answer
584 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 ...
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1answer
158 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 ...
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2answers
606 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, ...
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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 ...
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1answer
45 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 ...
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1answer
16k 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 ...
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1answer
98 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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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)}...
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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{\...
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1answer
290 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 ...
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135 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 ...
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93 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 ...
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301 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 ...
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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 ...