Statistics: Posted by fomisha — July 14th, 2017, 4:53 pm

]]>

Hi guy, I am trying to fit an SSVI model in order to calculate local vols. I have found quite a lot discussion on how to implement the original SVI formulation, but very few on SSVI. The main literature I am referring to is Gatheral and Jacquier (2013). I also managed to find some sample code that is based on this paper, about which I have some doubts. Any comments are welcome.

The major steps of the approach are as follows:

1) compute the ATM implied variance with linear interpolation.

2) estimate the SSVI model to whole vol data set.

3) convert to SVJ Jump Wing parameters, and re-fit to vol data slice by slice to eliminate potential arbs across slices.

4) linearly interpolate SVJ-JW parameters to get the the parameter set of any intermediate maturity slice.

Regarding the above procedure, it seems that SSVI formulation is only used to generate an initial guess. Also, the rationale of linearly interpolating parameters of SVJ-JW looks less obvious to me. Anyone could share his comments? According to this paper, interpolating in time dimension in an arb-free way could be very subtle.

Another more straightforward approach here is due to Jacquier himself. Anyone has tried it or has any ideas on its performance? Thanks in advance!

Another point to add. Gatheral and Jacquier (2013) Lemma 5.1 provides a way to do time interpolation in an arb-free way. But it seems to be an interpolation on option price (eq. (5.1)), instead of on implied variance (or vol). This would invoke BS formula evaluation and implied vol calculation for each query of intermediate implied vol, which I suspect could be quite a computational burden for local vol calculation using Dupire's formula.

Statistics: Posted by miniwolfy — July 14th, 2017, 7:35 am

]]>

The major steps of the approach are as follows:

1) compute the ATM implied variance with linear interpolation.

2) estimate the SSVI model to whole vol data set.

3) convert to SVJ Jump Wing parameters, and re-fit to vol data slice by slice to eliminate potential arbs across slices.

4) linearly interpolate SVJ-JW parameters to get the the parameter set of any intermediate maturity slice.

Regarding the above procedure, it seems that SSVI formulation is only used to generate an initial guess. Also, the rationale of linearly interpolating parameters of SVJ-JW looks less obvious to me. Anyone could share his comments? According to this paper, interpolating in time dimension in an arb-free way could be very subtle.

Another more straightforward approach here is due to Jacquier himself. Anyone has tried it or has any ideas on its performance? Thanks in advance!

Statistics: Posted by miniwolfy — July 14th, 2017, 7:19 am

]]>

The longer answer is that I have a few pages of discussion in a recent book, and it is easier to just post an excerpt for you, which I have done below. The last 3 pages, and in particular Sec 4.4.1, get to the issue of the effect of jumps on the interpretation of the VIX. I know this wasn't directly your question, but it was one that interested me. You might want to track down the cites at the bottom of the last page, too.

Statistics: Posted by Alan — July 12th, 2017, 4:02 am

]]>

Why would jumps impact the variance? The authors state this result is model-free (page 19 of the report: "Equation 26 makes precise the intuitive notion that implied volatilities can be regarded as the market’s expectation of future realized volatilities. It provides a direct connection between the market cost of options and the strategy for capturing future realized volatility, even when there is an implied volatility skew and the simple Black-Scholes formula is invalid."). It seems to me variance would only depend on the $X_{i}$ or $R_{i}$ (prices or returns) and jumps would be handled adequately.

If the Carr-Madan replication result perfectly replicates the log contract, why is there any error at all?

Statistics: Posted by jayjo — July 11th, 2017, 5:01 pm

]]>

I think this MIT lecture give a very good intro about Recurrent Neural Networks (RNN), which is the equivalent to the latent vol state in stock vol models.

If you are interested in RNNs applied to stochastic volatility models, "A Neural Stochastic Volatility Model" might be worth reading.

I have tried to use a simpler version as a generative model, but never really managed to get it to learn/calibrate in a stable fashion.

Statistics: Posted by mkoerner — July 9th, 2017, 12:48 pm

]]>

]]>

]]>

]]>

]]>

Anyway, my suggestion is to first make sure the PDE and MC agree on the fully linearly interpolated [$]\sigma(S,t)[$]. Then, move on to see if they agree on the less smooth case from the calibration (linearly interpolated in S, but only piecewise constant in t)

You can see from the exact solution to BS in the case of piecewise constant (in time) vol that the solution only depends on the integrated variance (integrated in time). So, actually bounded discontinuities for the local vol vs t should be tolerated by a good MC and a good (backwards) PDE, and they should both reproduce this exact solution. Try it -- say with

[$]\sigma(t) = \sigma_0 1_{\{t \le T_0\}} + \sigma_1 1_{\{t >T_0\}} [$] with an option expiration at say [$]T = 2 \, T_0[$].

In reality we do see evidence of such vol discontinuities, say when a company releases earnings, so the theory (and numerics) needs to accommodate it.

Statistics: Posted by Alan — July 7th, 2017, 1:44 pm

]]>

Well, it's your Monte Carlo and your FDM. The most benign choice to me is [$]\sigma(S,t)[$] continuous and bounded -- implying (among the choices you presented) piecewise linear S-

I would also begin to use piecewise linear interpolation in time starting with 6/2/2017 and later. Then, [$]\sigma(S,t)[$] will be continuous and bounded for your whole local vol surface. With that, again I believe both procedures will lead to the same option prices for all strikes, through the furthest expiration of your data.

Finally, if you want to consider t > tmax, where tmax is the furthest expiration of your data, I would just use constant extrapolation in time. Again this will assure [$]\sigma(S,t)[$] will be continuous and bounded -- now for [$](S,t) \in (0,\infty) \times (0,\infty)[$].

It looks as if we are talking about conditions for the SDE to have a L2 bounded unique solution?

In that case, mathematical niceties demand that the coefficients be measurable as well as satisfying Lipschitz and linear growth conditions?

Linear interpolation is not very smooth in general. Is that a problem here?

Statistics: Posted by Cuchulainn — July 7th, 2017, 11:06 am

]]>

I would also begin to use piecewise linear interpolation in time starting with 6/2/2017 and later. Then, [$]\sigma(S,t)[$] will be continuous and bounded for your whole local vol surface. With that, again I believe both procedures will lead to the same option prices for all strikes, through the furthest expiration of your data.

Finally, if you want to consider t > tmax, where tmax is the furthest expiration of your data, I would just use constant extrapolation in time. Again this will assure [$]\sigma(S,t)[$] will be continuous and bounded -- now for [$](S,t) \in (0,\infty) \times (0,\infty)[$].

Statistics: Posted by Alan — July 7th, 2017, 1:19 am

]]>

Statistics: Posted by caperover — July 6th, 2017, 8:17 pm

]]>

Statistics: Posted by Alan — July 6th, 2017, 7:10 pm

]]>