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\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} &= \sqrt{\mathrm{E}_t \left[ \int_t^T \sigma_u^2 du \right ]} \\ \text{VIX future} &= \mathrm{E}_t \left [\sqrt{\mathrm{E}_T \left[ \int_T^{T'} \...

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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 to ...

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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\... 9 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 ... 9 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 ... 9 If your strategy truly has no directional bias, then the benchmark should be cash (ie whatever you would earn using the capital in your trading account and taking no risk). 8 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 ... 8 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 ... 8 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 ... 8 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 ... 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 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 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 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).... 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 ... 6 You could compare it, over the historical period of interest, to 1000 randomly generated VIX strategies which are: Flat on 60 Percent of days (randomly chosen days) Long VIX futures on 20% of days Short VIX futures on 20% of days (You would adjust these percentages to the characteristics of your strategy. I guessed these values from your comment). The ... 6 Gonzalez-Perez (2015) Model-free volatility indexes in the financial literature: A review makes some remarks on this topic in section 2.2. Andersen, Bondarenko & Gonzalez-Perez (2013) identify a new error source in VIX that generates a significant number of jumps in the volatility index unconnected with the underlying volatility process and that weakens ... 6 Yes, the VIX took a sharp downfall on 2020/03/02, from 40.11 to 33.42 (-6.69). But that is not what the 2020/04/15 Put options are based on, they are based on the 2020/04/15 VIX Futures (VIJ20), these went from 23.025 on 2020/02/28 to 23.325 on 2020/03/02 an increase of 0.3. The Vix options are based on the futures, not the spot Vix value. 6 Long story short: yes both might introduce static arbitrage opportunities if performed blindly. There are 3 types of static arbitrage to consider: Calendar arbitrage: total (implied) variance should be an increasing function of time for fixed (forward) moneyness. Vertical arbitrage (or call spread arbitrage): call spreads should have a positive price ... 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 In your implementation, you are approximating continuous integrals over the strike domain by Riemann sums. This introduces an error. More specifically, for a fixed time to expiry \tau, you're integrating (scaled) OTM price curves (see equations (3)-(4)-(5)). As volatility increases, these curves will stretch over wider and wider spatial domains. When ... 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 Interactive Brokers offers historical data for VIX options. You need to have an account , other than that - historical data downloads are available for no extra charge. You have to write some code to download the data and deal with pacing violations and API request restrictions (no more than 2000 bars returned per request), so you'd have to split time ... 5 Unfortunately, there is no publicly available reliable source for historical VIX options data. Also, probably, in addition to VIX options you may need corresponding underlying futures data to do analysis (not only the corresponding VIX spot price). You can always go with an established commercial provider, of course. 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 If you are developing this strategy to use personally, I would benchmark it against your next best option. If the strategy has been developed to attempt to manage other peoples money I would benchmark it against the HFRX RV: Volatility Index. This is an index of alternatives that a Vol investor would consider versus investing in your strategy. From HFRX ... 5 Variance Premia; disentangled. Let me address this question a bit differently and bring the question forward: What part (i.e. 'side') of the volatility smile attracts a significant premium in relationship to the underlying uncertainty that it is trading? To this end, let us define the physical, i.e. empirical (average realized) return semivariance as:$$ SV_\...

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Just for future visitors of this post: You can also buy VIX option data directly from the exchange where they are traded, the Cboe: [https://datashop.cboe.com]

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