7

The reason is that, as shown in Proposition 2.1 of that paper, in order to exclude static calendar arbitrage, the total variance has to be strictly increasing in forward moneyness. See also the below to links for details on this result. The intuition is that for European options, only the distribution of the terminal spot price is relevant. Furthermore, $...


5

This is not quite true, in either direction. If you have an arbitrage free implied vol surface, you might not have a well-defined local vol surface. An example comes from a discrete model. Consider a spot dynamics where the spot is a martingale that jumps up or down by integer amounts. The spot distribution is discrete, with zero density in between ...


4

Maybe it would help you to think of it the following way. The strike $\sigma^2(T)$ of a fresh-start variance swap of maturity $T$ in the Heston model only depends on parameters $(v_0,\theta,\kappa)$, see related question here. More specifically \begin{align} \sigma^2(T) &= \Bbb{E}_0^\Bbb{Q}\left[ \frac{1}{T} \langle \ln S\rangle_T \right] \\ &= \...


3

Let's take a step back to look at what implied volatility (IV) really is. If we know the price of a call option, the interest rate (we can use the spot rate corresponding the option maturity) then Implied volatility is that level of volatility that will result in the option price when putting into the Black-Scholes formula for a call option value. If we ...


2

I just checked this one. I saw the picture you are referring to and I think the frown is not real. Much of it looks like noise created by wide bid ask spreads and extremely high implied vols. The data here is very poor and the options are spread very far apart. The underlying is $4.60 and the options are struck a dollar apart. That would be like SPX ...


2

In black-scholes world, correlation between volatility and spot is zero. From the above details you can estimate how the implied volatility for a given option (note options have FIXED strikes) might change for a given move in spot. If when stock goes up, the option's implied vol goes down, this would be a violation of the black-scholes model (which scenario ...


1

See https://en.wikipedia.org/wiki/Foreign_exchange_date_conventions for details. In summary expiry = T+tenor for weekly tenors and expiry = ((T+2)+tenor)-2 for monthly and yearly tenors, with all the appropriate business day adjustments.


1

See the paper "FX Volatility Smile Construction, Dimitri Reiswich and Uwe Wystup" http://janroman.dhis.org/finance/FX/FX%20Volatility%20Smile.pdf for a comprehensive construction of the FX volatility surface, and in particular converting deltas into strikes. In particular beware that even the notion of ATM may have a different meaning depending on the ...


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