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$$ Variance \, strike = E_t \left[ \int_t^T \sigma_u^2 du \right ] $$ $$ Volswap\, strike = E_t \left[ \sqrt{\int_t^T \sigma_u^2 du} \right ] $$ $$ VIX = \sqrt{E_t \left[ \int_t^T \sigma_u^2 du \right ]} $$ $$ VIX \, future = E_t \left [\sqrt{E_T \left[ \int_T^{T'} \sigma_u^2 du \right ]} \right ] $$ $$ Forward\, variance\, strike = E_t \left[ \int_T^{T'}...


6

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 ...


5

One of the most famous definition of Regimes and Regime Switching in Financial Markets comes from Wyckoff Cycle Wyckoff believed that prices judged by supply and demand, go through periods of advance, accumulation, decline an distribution based on the movement of smart money. In the quantitative world one can use state space models to model such regime ...


5

The ARMA(m,p) representation of GARCH(p,q) is : \begin{align*} \left[1-\alpha(L)-\beta(L)\right]r_{t}^{2} = w + [1- \beta(L)] v_{i} \end{align*} where \begin{align} &\alpha (L) =\sum_{i=1}^{q} \alpha_{i} L^{i} \qquad , \alpha (0)=0 \\ &\beta (L) =\sum_{i=1}^{p} \beta_{i} L^{i} \qquad , \beta (0)=0 \\ &m = \text{max}(p,q) \end{align} ...


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

@Quantuple's answer is correct. There is no exact closed-form formula for call prices or implied vols as a function of local vols, which is unique and ironic, given there are formulas the other way around. The closest thing to a practical exact formula, to my knowledge, is the harmonic short maturity expansion given in the answer. However, (and this is a ...


4

There is no simple way and you have to make correlation assumptions. For instance say you have a volatility surface for $\text{EURUSD}$ and another volatility surface for $\text{USDJPY}$ and you want to build a volatility surface for $\text{EURJPY}$. You start from the observation that a call with maturity $T$ and strike $K$ on $\text{EURJPY}$ with ...


4

Trading the skew is a common practice for traders specializing in options. Let's say you have a 3M skew curve like the blue one below (where I have highlighted a few key strikes) but you think the correct skew curve is more like the red one Let's further assume you unwilling to make a bet on the overall height (level) of the curve. You just want to bet on ...


4

It seems like he is assuming that the shorter term volatilities change more than the longer term ones and the relatively sensitivity is proportional to $1 / \sqrt{T}$. Thus, this hedge is not against a parallel shift of the surface. This is not an uncommon assumption and the corresponding vegas are often referred to as "time weighted vegas".


4

On average the implied volatility is higher than realized volatility because you can easily imagine that dealers will ask customers to pay a premium to write them options and risk manage them you can have a look at this paper for instance PIMCO-The Volatility Risk Premium Now you can investigate how realized volatility can be a signal for trading implied ...


4

There is no possibility to convert any two of your mentioned variables into the remaining one. For the compound and arithmetic return you can derive an inequality, but that's the best you can do. The definitions for your statements are: $$r_{\mathrm{compound}}= \prod_{t=0}^{n}{\left( 1+r_t \right)}$$ $$r_{\mathrm{arithmetic}}=\frac{1}{n} \sum_{t=0}^n{r_t}$...


4

You'll find here that in terms of European option prices, the absence of calendar arbitrage writes $$ \frac{\tilde{C}(k\, F(0,t_2),t_2)}{F(0,t_2)} \geq \frac{\tilde{C}(k \, F(0,t_1),t_1)}{F(0,t_1)}, \forall k \in \Bbb{R}, \forall \, 0 < t_1 < t_2 \tag{1} $$ where $\tilde{C}(K,t)$ denotes the undiscounted European call price for strike $K$ and time to ...


4

To answer your second question, per "Pricing, Hedging and Trading Financial Instruments" by Carol Alexander, the following approaches have been proposed in literature: cubic polynomials (Dumas et al., 1998) piecewise quadratic functions (Beaglehole and Chebanier, 2002) cubic splines (Coleman et al., 1999) hyperbolic trigonometric functions (Brown and ...


3

You can only infer forward vol by pairing a mid-curve option with a spot option. It's easier to go through an example (I'll use 5y x 5y vol since I have the sketch below handy...) One decomposition of the 5y5y spot vol is as follows: 1y forward 4y x 5y vol: this is the implied vol of an option starting in 1 year, expiring 4 years thereafter, and eventually ...


3

You can think of both ( difference and ratio ) indicators as some aggregated measure of difference between flat vol (ATM vol) and "total vol" than includes skew and kurtosis effects.


3

The following source contains detailed answers to your questions in a research paper from ETH Zürich. van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model" http://resolver.tudelft.nl/uuid:029cbbc3-d4d4-4582-8be2-e0979e9f6bc3


3

The price volatility of a bond option is the implied volatility using a Black type model, so it is exactly analogous to an equity option, using the bond price instead of the equity price. Because long dated bonds have naturally more price volatility than short dated bonds due to the extra duration , using price volatility is not very helpful when ...


3

@q.t.f. 's answer is 100% correct. As an OMM, I wanted to add some reasoning behind this. The practice of trading ATM options has been established for over a century now, and before formal mathematical methods were developed, traders have developed many heuristics for pricing ( proportional to vol ) and hedging ( delta = 1/2 ) . Typically in a newly ...


3

In short , this claim does not hold under all circumstances. There are a few ways to break down such approximation. The options under consideration have very long expiry, i.e. $T$ is very large As expiration date approaches, the volatility smile becomes more pronounced, i.e. $v$ becomes relatively large. Under extreme market condition, the magnitude of $\...


3

The VIX Index is computed from option prices on S&P Index. A VIX-like Index for other industries would first require to have a liquid option market.


3

"Why would someone want to do calculate the value of the option with different spot prices?" Because Delta is the derivative of the option price function wrt spot. Since the option price function is not linear in spot according to the BS model you cannot get the option prices for different spot just by Delta. What you are asking about is simply using a ...


3

It is hard to know what "inherent volatility" refers to, as this term is somewhat non-standard. I will interpret it as the long term equilibrium level of volatility $\bar{\sigma}$ to which all volatilities are expected to revert. Clearly a short term vol of 60 and a long term vol of 34 is a highly unusual situation. The market expect volatility to be very ...


3

VeV is simply the scale parameter $\sigma$ such that the returns follow the $N(-\dfrac{1}{2} \sigma T, \sigma^2T)$ distribution and is obtained by inverting the VaR formula under this assumption. Have a look at this question where the full derivation of VeV is covered.


3

What are the underlying assumptions for doing this Assumption: Historical returns are lognormally distributed with no autocorrelation. can those assumptions be tested statistically Testing: $\sqrt{xy} = \sqrt{x} \sqrt{y}$ Substitute time $t$ and variance $\sigma^2$ for $x$ and $y$ respectively $\sqrt{t\sigma^2} = \sqrt{t} \sqrt{\sigma^2} = \sigma\...


2

One of the most famous definition of Regimes and Regime Switching in Financial Markets comes from Wyckoff Cycle. Wyckoff believed that prices judged by supply and demand, go through periods of advance, accumulation, decline an distribution based on the movement of smart money. In the quantitative world one can use state space models (ARIMA+Markov) to model ...


2

You can find a greast summary on volatility estimation here. I also suggest to get familiar with Garman-Klass volatility. Discussed here and in this article.


2

Looks correct to me, the open of the last bar divided by the close of the previous bar. def clLog = log(open / close[1]); For intraday it would be... def clLog = log(open / close); Also, to answer your comment, when combining plots all lines are scaled to use the full vertical space of the graph (i.e. 0-100%) which is why a lower number value from one ...


2

There is quite a recent paper by Rolloos and Arslan (Wilmott, January 2017) that shows how to obtain a good approximation of the volswap strike from the implied volatility smile without having to specify any model, i.e. a model-free approximation that is immune to correlation to first order. Their method is much easier to follow and implement than Carr and ...


2

You are correct. The midcurve swaption expresses the volatility of the forward swap rate , not the "forward volatility". The latter refers to the price of an option whose strike price will be determined at a future date.


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In general, if you have a model of relation between $y$ and $x$ whereby the relation is not perfect but measured with errors: $$y_t = f(x_t) + \varepsilon_t,$$ where errors $\varepsilon$ are assumed to be additive but need not be, you are free to choose the distribution of these errors to better fit the reality. That is where GARCH enters as a great ...


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