10
A few thoughts.
Yes, your return series are autocorrelated (i.e., stocks don't exactly follow a random walk), so you should use Newey-West standard errors.
If you do this as a univariate regression $$R_{i,t} = \alpha_i + \beta_i R_{j,t-1} + \epsilon_{i,t}$$ then there's almost certainly an omitted variable inside $\epsilon$ that is moving both $R_i$ and $...
9
The best theoretical model for pricing vix futures and options is a variance gamma model.
However in practice that class of models is difficult to get robust results...
In practice, most floor traders in vix products base their hedging off of the SPX option chain. Vix is calculated from those options, in the first place, so this approach makes intuitive ...
9
VIX is a measure of implied volatility, specifically, model-free implied volatility, a concept originally developed by Demeterfi et. al. at Goldman Sachs in the 1990s. One of my recent questions, How to extrapolate implied volatility for out of the money options?, addressed some aspects of MFIV, and the papers mentioned in the question and answers will give ...
8
VIX is calculated from a basket of SPX options, and VIX futures expire into following expiration, e.g. September VIX futures that will expire next Wednesday will use SPX October options chain to calculate settlement value. If $B$ is the value of the basket then VIX value at expiration is $\sqrt{ B }$. Then VIX futures price is the expectation of the basket $...
8
There were two changes to the VIX; the first change in 2003 that switched from S&P 100 options to S&P 500, and from implied volatility to variance swap method. The second change was in 2014 when calculation included weekly options.
Before 2014 the first series used had to have at least one week to expiration. Then the next series was used without ...
8
$$ Variance \, strike = E_t \left[ \int_t^T \sigma_u^2 du \right ] $$
$$ Volswap\, strike = E_t \left[ \sqrt{\int_t^T \sigma_u^2 du} \right ] $$
$$ VIX = \sqrt{E_t \left[ \int_t^T \sigma_u^2 du \right ]} $$
$$ VIX \, future = E_t \left [\sqrt{E_T \left[ \int_T^{T'} \sigma_u^2 du \right ]} \right ] $$
$$ Forward\, variance\, strike = E_t \left[ \int_T^{T'}...
8
Put simply, VIX is a spot index (fair value to a variance swap on SPX of constant maturity) that you cannot own as a security. Market participants create futures for you to trade. Futures trade higher than the VIX -- if you long VIX futures, you lose when the futures contract converges to VIX. You therefore have a negative roll-down. VIX ETF doesn't avoid ...
7
In that white paper itself they quote where it came from: “More than you ever wanted to know about volatility swaps” by Kresimir Demeterfi, Emanuel Derman, Michael Kamal and Joseph Zou, Goldman Sachs Quantitative Strategies Research Notes, March 1999. This is a classic article which you should definitely read if you are trading volatility. While there might ...
7
If you look at tick data, you will probably get an even better analysis. However, vix correlation tends to be negative with spx but remember that this is generally more true for when spx tanks. When spx goes up, the correlation isn't as strong. Why? People panic after a drop, therefore leading to people buying options. They don't care about black scholes ...
7
Your question is an important one, but I am not aware of any particularly satisfying answer. There are several papers on this issue -- see Luo and Zhang 2009 and Zhang et al 2010, just for example.
One thing to note is that VIX futures are not always in contango -- after large jumps in the VIX, they can even be in rather steep backwardation. I have heard ...
7
Your questions about contango in VIX futures have close analogies in
options too. The Black & Scholes model suggests that all time
frames and all strikes should have the same implied volatility, but
they don't. I think one of the reasons is that the B&S model
assumes that stock returns are distributed in a normal (gaussian)
distribution, but ...
7
VIX is a measure of volatility -- something that changes explicitly with uncertainty. The chances of uncertainty arising tomorrow, is lower than the chances of uncertainty increasing in the longer term. A long-dated option should therefore have more "potential uncertainty" baked into the price.
When pricing normal futures, the price is a martingale, the ...
7
For How VIX works you can read this wonderful blog : http://onlyvix.blogspot.com/2011/09/intuitive-understanding-of-vix-formula.html
It provide wonderful non mathematical explanation of the how vix is actually computed.
Now comes to your last answer why vix is inversely related to market movement ?
In simple words, if market is more volatile then ...
7
These are 2 completely different ways of estimating volatility.
GARCH models are calibrated on historical time series i.e. information provided under the real-world measure $\mathbb{P}$. Although you can obviously use them for forecasting, the core information which is used to build the model is backward-looking in nature (historical behaviour of the stock)....
7
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 delta one contract on the VIX index), the exposure you gain is more comparable to the one of a vol swap in the following sense:
Consider a notional of 1 and a ...
7
Heston - Change of measure
Consider the following Heston dynamics written under the real world measure $\Bbb{P}$
\begin{gather}
\frac{dS_t}{S_t} = \mu_t dt + \sqrt{v_t} dW_S^{\Bbb{P}}(t),\ S(0) = S_0 \\
dv_t = \kappa(\theta-v_t)dt + \xi \sqrt{v_t} dW_v^{\Bbb{P}}(t),\ v(0) = v_0 \\
d\langle W_S^\Bbb{P}, W_v^\Bbb{P} \rangle_t = \rho dt
\end{gather}
In order ...
6
Have you considered fitting ARIMA with exogenous regressors model? Linear regression with autocorrelated errors might be appropriate.
R can do this with the arima() function via specifying the xreg argument.
6
Short Answer : Futures don't have Greeks
Long Answer : Assuming a non strictly mathematical (i.e. false) point of view.
Well, having Greeks on VIX Futures is not relevant, VIX value is itself a "Greek" (and Futures don't have Greeks).
Sensitivity to
Price of the Underlying : Insensitive (ν = 0)
Volatility of the Underlying : Delta Δ = 1 (to Volatility of ...
6
You have to ask yourself what the ultimate purpose of this parameterization is. In their case, they imply the "end-goal is martingale pricing or maximum-likelihood estimation", both of which are ultimately about capturing long-period dynamics rather than intraday or interday behavior.
For this reason, the fact that intraday variance may, ahem, vary around ...
6
Due to the lack of a carry arbitrage, VIX futures are actually the direct hedge for VIX Index options
6
Strictly speaking, indices such as the VIX are built to approximate the expected variance (of log-returns) that would effectively realise under a pure diffusion setting (i.e. no jumps)
$$ \frac{dX_t}{X_t} = \mu(t) dt + \sigma(t,.) dW_t^{\mathbb{Q}} $$
Writing out the equations (*) yields the famous static replication formula in terms of strike-weighted OTMF ...
5
Simple explanation of VIX formula is that it is a square root of weighted sum of out of the money SPX options. If options are more expensive (implying that market is paying more for insurance from a major move) VIX will be higher, if options are cheap (implying that the market does not think that there will be a lot of movement in the SPX) VIX will be lower.
...
5
I'm going to go ahead an assume the spread you were looking at involved exchange traded options. As you presumably know, the actual implied volatility on your screen is a number derived from option prices by running the Black-Scholes model "backwards" from quoted option price to volatility. Higher prices imply higher volatility.
That last statement is the ...
5
Because VXX is not designed to track the spot VIX. http://blogs.cfainstitute.org/insideinvesting/2014/02/12/doing-what-it-says-on-the-tin-the-value-of-volatility-etps/
5
VIX futures tend to rise when the S&P 500 falls -- the correlation of returns is about -0.7. If there were no contango in VIX futures, everyone would buy them to get free insurance against stock market declines.
Here is a recent working paper making this argument -- note the last sentence of the abstract.
http://skinance.com/documents/...
5
The old VIX index is based on the Black-Scholes implied volatility of
S&P 100 options. To construct the old VIX, two puts and two calls for
strikes immediately above and below the current index are chosen. Near
maturities (greater than eight days) and second nearby maturities are
chosen to achieve a complete set of eight options. By inverting the
...
5
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 = \frac{1}{R_f}Var^Q(R_{t+1}) + \text{extra terms}$
As you correctly pointed out the expected return should be related with the risk neutral variance. The issue ...
5
The piece you are missing is an approximation via the Taylor formula of the logarithm:
$$\ln(1+x) \approx x-\frac{x^2}{2} \; .$$
Apply this to the first term in the final formula of the technical paper:
$$\frac{2}{T}\ln\frac{F_{0}}{S^{*}} = \frac{2}{T}\ln\left(1+\left(\frac{F_{0}}{S^{*}}-1\right)\right) \approx \frac{2}{T}\left(\left(\frac{F_{0}}{S^{*}}-1\...
4
Van Belle describes a basic correction for autocorrelation in a t-test, although it may be hard to wedge it into the regression t-test. For the 1-sample t-test of the mean, the correction is to multiply the t-statistic by $\sqrt{\frac{1 - \rho}{1 + \rho}}$, where $\rho$ is the 1-period autocorrelation (or estimate thereof).
4
to match the constant 30-day VIX horizon, I think you would want to trade two straddles in the first and second expiration cycles and delta hedge, gradually rolling the weight towards the second month straddle and then finally to a new straddle at/near expiration each month. Here are some problems I can imagine for this approximation.
Hedging error - the ...
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