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, $...


6

One possible reason could be jumps. Over the longer maturity, there could be more jumps so the jumps average out in a way; whereas over the short term, a jump can make a bigger difference and hence the risk of jump increases demand. This reasoning is used to justify Stochastic volatility with jumps models in some books.


5

I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more ...


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 ...


5

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 ...


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

You can convert the implied volatility to local volatility using this formula: $\sigma^2 \left(T,y\right)=\frac{\frac{\partial w}{\partial T}}{1 -\frac{ y}{w} \frac{\partial w}{\partial y}+\frac{1}{2}\frac{\partial^2 w}{\partial y^2}+\frac{1}{4}\left(\frac{ y^2}{w^2}-\frac{1}{w}-\frac{1}{4}\right)\left( \frac{\partial w}{\partial y}\right)^2}$ Where y is ...


3

Yes, that's what we wish to see from the correctly-specified model. Now, let me try to answer your 2nd and 3rd questions together as they are based on the same confusion. There are two different concepts: model-implied volatility and model-implied BSIV (Black-Scholes Implied Volatility). I think you are confused because of mixing them up. So yes, people ...


3

This is a big industry, but here are some alternatives(as usual, the best choice depends on purpose and desired accuracy): Fit a quadratic in delta space: $\sigma_{\Delta}=a + b \left( \Delta - \Delta_{ATM} \right) + c \left( \Delta - \Delta_{ATM} \right)^2$. When you have fitted this equation, you can input delta, and the function will return ...


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 ...


2

The skew/smile of long term options is flatter than short term options, the reason for this can be explained in several ways. The Vega of a shorter-dated option is smaller than a longer-dated option. Vega is the dollar value of a 1% change in implied volatility. i.e., 30d ATM option, $65 strike, .31 ivol = VEGA .07 30d 25 delta option,31% ivol = ...


2

You should use the out of the money options on either side of the curve as they carry the most information about the optionality part of the price.


1

As explained in the chapter 4.4 of I. Clark, you can estimate the weights by using the typical trading volumes. You can give more weight for dates with bigger trading volume which is logical.


1

It's obviously no calibration problem. It's just a numerical issue. The error resulting from solving the integral numerically is just to big for your really small option price. I would suggest to cut the wings of your volatility surface at an appropriate moneyness.


1

Apparently your heston model parameters should define the surface. You're fitting to options quoted in the market, thus a minimization exercise. Not like local vol, where it needs a abitrage free implied vol surface to garantee uniqueness


1

The concept behind showing volatility vs expected return is that a risk averse investor will wish to minimise risk, and maximise return. However, how good a proxy is volatility for risk? Given a normal distribution, risk and return (sigma and mu) alone will suffice. But especially for non-normally distributed returns (as alluded to by the mention of higher ...


1

The volatilities of short dated options are more sensitive to market changes as compared to those of long dated options. This is implied by square root of time rule. As such, volatility skew are larger for short dated options.


1

A call and a put option with the same strike and same expiry should have the same implied volatility by put-call parity indeed. A call and a put are essentially the same when you hedge the initial delta in terms of greek exposures


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 ...


1

In short no you don't need to estimate the values for these quantities to calibrate the model. Instead what you want to do is to perform a least squares optimization on the model parameters against market volatilities, this gives you the calibrated surface to the market prices and if you want to output the model quantities you mentioned then you can. So ...


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