29 votes
Accepted

Variance replication using options

Let $t_0, t_1, \ldots, t_n$ be observation dates, where $0=t_0 < \cdots < t_n = T$, and $\{S_t \mid t \geq 0\}$ be the equity price process without dividend payments. Then the realized variance ...
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  • 20.5k
27 votes

Is the VIX more similar to a volatility swap or a variance swap?

\begin{align*} \text{Variance strike} &= \mathrm{E}_t \left[ \int_t^T \sigma_u^2 du \right ] \\ \text{Volswap strike} &= \mathrm{E}_t \left[ \sqrt{\int_t^T \sigma_u^2 du} \right ] \\ \text{VIX}...
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14 votes

What is the difference between volatility and variance?

Volatility = Variance^1/2 = Standard Deviation
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  • 141
13 votes

How to calculate the conditional variance of a time series?

Let’s take a simple example to answer a broad but interesting question: Imagine that we have a daily return serie denoted $r_{t}$ ( which is assumed to be stationary) and let's take a little time to ...
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  • 2,514
13 votes
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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 ...
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  • 1,487
9 votes
Accepted

Is the VIX more similar to a volatility swap or a variance swap?

The price/value of the VIX index is more akin to the strike/price of a variance swap expressed in vol units than to the strike/price of a vol swap. However, if you are to trade a VIX future (i.e. a ...
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  • 469
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 ...
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  • 1,654
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 ...
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  • 6,810
8 votes

Why is a variance swap long skew?

As I've mentioned in a comment, it would be wrong to think that entering a variance swap specifically amounts to being "long skew". What you can say however is that, in the absence of jumps (i.e. in ...
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  • 14k
8 votes

Why is a variance swap long skew?

If you take Quantuple's stuff a little further, you can really see whether you're long skew. You can pretty easily see the dependence on convexity too (though it should be obvious that you're long ...
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  • 2,426
8 votes
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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 ...
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  • 6,354
7 votes
Accepted

Negative high frequency intraday volatility - Zhou estimator

In any finite sample, it is always possible for the Zhou estimator to return a negative number, even though we know the unobservable parameter being estimated is non-negative. This is a well known ...
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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}\\ &...
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  • 14k
6 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 ...
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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 ...
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  • 10.6k
6 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 ...
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  • 13.9k
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).
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  • 1,487
5 votes

Why is this delta-hedging/P&L example on a variance swap call correct?

To answer your questions: Is the trading p&l meant to be the delta-hedging p&l? Yes, in his example it concerns delta hedged pnl. how come p&l is raising steadily even when stock price ...
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  • 1,411
5 votes
Accepted

How to derive this approximation of the risk-neutral expectation of the variance?

We first list the assumptions. \begin{align*} g_{t+1} &= \mu_g + \sigma_{g, t} z_{g, t+1}, \tag{1}\\ \sigma_{g, t+1}^2 &= a_{\sigma} + \rho_{\sigma} \sigma_{g, t}^2 + \sqrt{q_t} z_{\sigma, t+1}...
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  • 20.5k
5 votes

Why would one prefer variance swaps over other instruments?

The vega of an option is very dependent on the spot price. The vega of a variance or volatility swap is not.
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  • 6,763
5 votes
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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 = \...
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  • 1,856
5 votes
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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 ...
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  • 9,097
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 \} = ...
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  • 3,365
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 ...
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  • 14k
5 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} \...
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  • 14k
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: ...
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  • 3,733
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-\...
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  • 1,017
4 votes
Accepted

Static and Dynamic Hedging of Vol/Var Swaps

There has been a lot of work in recent years on the pricing and hedging of volatility derivatives, leading to some non-obvious, even startling results. It is summarized in Mark Joshi's book More ...
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  • 9,097
4 votes
Accepted

GARCH variance vs standard deviation for volatility

If your question is: "Given all the information available up to time $t$, if I compute the 1 period ahead forecast $r_{t+1}$, is the conditional volatility over $[t,t+1[$ given by $\sqrt{r_{t+1}}$?", ...
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