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 ...


5

@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 ...


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

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


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

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 ...


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 ...


4

The Exponentially Weighted Moving Average (EWMA for short) is characterized my the size of the lookback window $N$ and the decay parameter $\lambda$. The corresponding volatility forecast is then given by: $$ \sigma_t^2 = \sum_{k = 0}^N \lambda^k x_{t-k}^2 $$ Sometimes the above expression is normed such that the sum of the weights is equal to one. ...


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

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 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

"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

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\...


3

Forget for a moment that your option is delivering the immediate entrance in a swap (if the swaption is physically settled) or the cash amount of the swap (if the swaption is cash-settled), as your question doesn't depend on this fact, and take a "general" 1Y option. Your today's (date $t_0$) cube loses the "swap tenor dimension" and becomes a today's ...


3

Look at the infinitesimal version of the change in variance: $$ d\sigma^2 = 2\sigma d\sigma + (d \sigma)^2 $$ The Ito term $(d\sigma)^2$ is non-zero for stochastic processes, and is of order $dt$, but if we ignore that then we get the approximate relation $$ d\sigma^2 \approx 2 \sigma d\sigma $$ which is where the factor $2 \sigma$ comes from in the ...


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

Yes, it mostly makes sense. The process you are outlining would give you a VaR estimate using the assumption that the returns of the cryptos are Normally distributed, and have a zero drift value. I think those assumptions are a bit of a stretch for cryptos in practice. I would multiply the variance matrix of the daily changes by 365. 365 would be the best ...


2

Take a look at the following paper by Wystup: https://mathfinance.com/wp-content/uploads/2017/06/CPQF_Arbeits20_neu2.pdf For a more modern take, look at the book by Ian Clark on FX Options. There are several interpolation schemes you can use to consistently reprice both 10 and 25 delta quotes such that the surface is consistent with both BF and RR.


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

The most straightforward would be trading volatility swaps. If not liquid you can create a proxy for being long/short volatility using options such as calls/ puts for example.


2

The characteristic function of the Heston model is known in closed form. To obtain the option prices you have to perform numerical integration though. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.139.3204&rep=rep1&type=pdf you can also check the already asked (and answered) question Problem on Characteristic function in Heston model ...


2

Well it all depends how theta is calculated in the first place. Depending on your pricing scheme those could be very different things. Anyways assuming that you are dealing with european vanilla then the BS theta is an instantaneous quantity that assumes that volatility does not change so you definetely don’t get any carry effect from this quantity. Now ...


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