14
votes
Accepted
Derivation of VIX Formula
The piece you are missing is an approximation via the Taylor formula of the logarithm:
$$\ln(1+x) \approx x-\frac{x^2}{2} \; .$$
Apply this to the first term in the final formula of the technical ...
- 1,537
9
votes
What are the significant implications of the long-run average variance rate and why Engle won the Nobel Prize for ARCH model development?
The best answer to your question is probably given by the Nobel prize committee itself in "The Prize in Economic Sciences 2003 - Advanced Information" document. You should read it in full. Below is an ...
- 1,714
9
votes
What is the difference between squared returns and variance?
Usually the formula for the sample variance of a stock is given by:
\begin{equation}
Var(R_{i}) = E (R_t - E(R_t))^2
\end{equation}
If you are using daily data to compute the variance then the ...
- 8,621
8
votes
Accepted
CAPM model as a regression
If you really believed the CAPM's prediction that $\alpha=0$, then imposing $\alpha=0$ in your estimation would indeed lead to your 2nd formula.
The problems?
The CAPM doesn't work so imposing a ...
- 7,024
7
votes
How do you find variance of a sde?
Here are two approaches that you could take to compute the variance of $X_t$. I am not making the conditioning explicit as it just complicates the notation but doesn't really add any additional ...
- 6,094
7
votes
Accepted
Risk, required return and expected volatility - what is the relationship?
I think you may be interested in this QJE forthcoming article by Ian Martin. The key idea of the article (page 5) is that the expected return on the market can be decomposed as
$E_t[R_{t+1}]-R_f = \...
- 1,896
7
votes
Variance of time integral of squared Brownian motion
Here's another take on the question:
\begin{align}
\int_0^t W_s^2 ds &= \int_0^t \int_0^s d(W_u^2) ds \\
&= 2 \int_0^t \int_0^s W_u dW_u ds + \int^t_0 \int^s_0 du ds \tag{Itô's lemma}\\
&...
- 14.8k
6
votes
Accepted
Jim Gatheral's assertion on ATM implied volatility vs. square root variance
Below are my 2 cents only, but this was too long for a comment.
As he shows in the next lines (see also Variance Swaps chapter of Bergomi's book)
$$ \sigma_{VS}^2(T) = \int_{-\infty}^{+\infty} \tilde{\...
- 14.8k
6
votes
Accepted
Corwin-Schultz estimator of bid-ask spread
If you have access to intraday data, they are better ways to estimate the bid-ask spread.
If you have Open, High, Low and Close price on each 5min bin $b$ (or any other interval): the Close of the ...
- 12.6k
5
votes
Accepted
How to compute the variance of a Long-Short Equity Portfolio?
We have weights $w_A$, $w_B$ and $w_C = 1 - w_A - w_B$ that sum to $1$.
With de-meaned returns $r_A$, $r_B$, and $r_C$, the portfolio variance is
$$E\{[w_A r_A + w_B r_B + (1 - w_A - w_B)r_C]^2 \} = ...
- 3,730
5
votes
Lévy alpha-stable distribution and modelling of stock prices.
I asked this question 6 years ago, and in the meantime I came across this little volume:
Lévy Processes in Finance: Pricing Financial Derivatives by Wim Schoutens (2003).
- 1,537
5
votes
Accepted
What is the difference between overlapping and non overlapping returns
An example of non-overlapping one month returns: the return in January, the return in February, the return in March, etc.
An example of overlapping 30 day returns: the return from January 1 to ...
- 9,440
5
votes
Terminal Variance in the Heston Model
From the equations of the model it is clear that $v_t$ is the instantaneous variance of the log-returns, not the terminal annualised variance of the log-asset price.
Put differently, you are you ...
- 14.8k
5
votes
Accepted
For portfolio variance, why doesn't $Var(X w) = w^\top \Sigma w$?
I'm not a Python programmer, however, reading the reference manual of np.var, you're using the "biased" version of the variance estimator. Instead use the unbiased variance estimator:
...
- 4,921
5
votes
Accepted
Market-maker's gain variance
The probablility of a jump of $J = \phi$. ( in either direction so I'll assume $\frac{\phi}{2} = $ probability of J and $\frac{\phi}{2} = $ probability of -J ). The probability of a jump of $0 = (1-\...
- 1,178
5
votes
Accepted
Heston: Variance of Integrated Variance
Studying zero-coupon bond prices in the CIR (1985) short rate model, $\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\xi\sqrt{r_t}\text{d}W_t$, Hirsa (2013, Section 1.2.6.2) states that the characteristic ...
- 16.3k
5
votes
Accepted
Is there such a thing as monthly/yearly skewness and kurtosis?
Assuming $n$ independent and identically distributed returns $X_i$, all moments of the distribution of the cumulative return $Y=\sum_{i=1}^n x_i$ scale with $n$:
$$
\mathrm{E}(Y^k)\propto n\mathrm{E}(...
- 7,140
4
votes
Variance of time integral of squared Brownian motion
A few hints I would like to suggest:
How is $Var(W_t^2)$ computed? Note that
\begin{align*}
W_t^2 = 2\int_0^t W_s dW_s + t.
\end{align*}
Then
\begin{align*}
Var(W_t^2) &=E\left(W_t^2-t)^2\right)...
- 21.3k
4
votes
Accepted
Using CAPM to find correlation of two assets with each other
The solution provided can be derived using the CAPM. For asset $A$ you have:
$$R_A-R_f = \alpha_A +\beta_A(R_M-R_f)+\epsilon_A$$
Similarly for asset B:
$$R_B-R_f = \alpha_B +\beta_B(R_M-R_f)+\...
- 440
4
votes
Question regarding the purchase of a Variance Swap
The options will form a static replication - and yes - they should expire on the same day as the variance swap. You should be sure to do all of your analytics in business time. Also, typically a ...
4
votes
What are the significant implications of the long-run average variance rate and why Engle won the Nobel Prize for ARCH model development?
$V_L$ is the long-run variance (or the unconditional variance) if and only if $\gamma=1-\sum_{i=1}^n \alpha_i$, because the long-run variance compatible with the model
$$
\sigma_n^2 = \gamma V_L + \...
- 3,281
4
votes
Accepted
Intuition Behind Scaling Factor in Variance Swaps
I believe you want to know why the VIX is a weighted portfolio of calls and puts with weights proportional to $\frac{1}{K^2}$ (NB: obviously the T is there to adjust for time to maturity, hence is not ...
- 1,896
4
votes
Accepted
What is the difference between standard deviation, volatility and quadratic variation?
Using only words and no equations:
Knowing the Variance (or standard deviation) of a Brownian Motion we can calculate the uncertainty in the future position of a particle. Knowing $\sigma^2$ and ...
- 11.8k
4
votes
Variance risk premium: When is realized vol higher than implied vol in practice?
Since you talk about a variance risk premium, a word about some of the details involved might be important. Technically, your variance premium is the difference between the expected volatility under ...
- 2,536
4
votes
Accepted
Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$
Just to expand on Alex answer.
Empirically it is simply not true. Focusing on the diagonal of the variance-covariance matrix, we know that there is a large variance risk premium. Take a look at table ...
- 8,621
4
votes
Accepted
Show that the following result holds true for the variance of the return of a portfolio of shares
The variance part is correct.
For the covariance part we can observe the following: There are $n$ variance terms in the $n \times n$ covariance matrix. This implies that there must be $n^2-n$ ...
- 4,921
4
votes
Accepted
Why is $Z_t$ uncorrelated with $X_{t-1}$ in $X_t=\theta X_{t-1}+Z_t$?
$E(X_{t-1}Z_t) = 0$ in the causal case $|\phi | < 1$, but not in the non-causal case $|\phi | >1$.
Causal case $(|\phi| < 1)$
In this case, the unique stationary solution to the AR(1) ...
- 225
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