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Questions tagged [variance]

The tag has no usage guidance.

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1answer
47 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
34 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
46 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
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1answer
52 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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2answers
118 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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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 ...
8
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1answer
213 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
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0answers
20 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
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2answers
222 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
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1answer
24 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
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1answer
51 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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0answers
26 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
148 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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0answers
75 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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0answers
116 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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0answers
111 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
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1answer
53 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
40 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 ...
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1answer
98 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
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0answers
27 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
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3answers
363 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
116 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 ...
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1answer
113 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
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2answers
111 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 ...
0
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1answer
60 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
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1answer
257 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 ...
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3answers
651 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 ...
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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 ...
1
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1answer
62 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
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1answer
349 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
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1answer
72 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
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1answer
280 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 ...
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0answers
135 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
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1answer
238 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
299 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
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0answers
60 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{...
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2answers
1k 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
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1answer
72 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.
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1answer
94 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?
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0answers
210 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
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1answer
111 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
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1answer
374 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
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1answer
372 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
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1answer
167 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 ...
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1answer
95 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 ...
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0answers
95 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 ...
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2answers
701 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 ...
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1answer
74 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 ...
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0answers
150 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 ...
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4answers
665 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).