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JustusQuant
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Re: Asymptotic behaviour of spot skew in CEV model

July 26th, 2016, 9:47 am

Thanks for all your input, Alan! As I mentioned briefly in the beginning, I am trying to implement the Bergomi forward variance model (http://papers.ssrn.com/sol3/papers.cfm? ... id=1493302). Here a quick overview of what I am trying to do:

Given a discrete tenor structure, I simulate discrete forward variances (replicable by buying/selling VS of corresponding maturities) modeled by two correlated Ornstein Uhlenbeck processes until their forward period ends (i.e. strike gets fixed, VS gets issued). For the lifetime (usually 1 month) of this VS , I then want to choose the CEV parameters (beta, sigma_0) such that the expected quadratic variation matches the simulated forward variance (i.e. fair VS strike) and such that my condition on the forward skew is met.
Basically, I have a different forward variance input in each simulation and each tenor ("vol of vol feature"), but I choose the CEV parameters always so that the desired forward skew in the respective tenor is reached.

Now, if the forward skew would be stationary in the CEV model, I could easily compute, given a certain forward skew target, the dependence of beta and sigma_0 on my input forward variance. As you can see at the top of page 5, the CEV setup Bergomi uses is slightly different (he resets the diffusion component at each tenor maturity) from the standard one. This also leads to the nice property (which stoch vol models generally lack) that the forward skew does not depend on the correlation between the spot and forward variance process.

Below you see my calibration for 8% target skew (which is only valid for all tenor dates if the forward skew is in fact stationary). Do you think that his CEV setup can lead to a stationary forward skew? His calibration (page 7) looks quite similar, but it would be pointless IMO if I would need a different calibration for each tenor to always match the desired forward skew?
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Alan
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Re: Asymptotic behaviour of spot skew in CEV model

July 26th, 2016, 3:19 pm

I don't know. The combination 'variance swap' and 'CEV model' may be fatally flawed. That's because [$]E \sum_{n=1}^N [\log S(t_n)/S(t_{n-1})]^2  = +\infty[$] for all [$]N[$] under all the parameter cases you are considering ([$]\beta > 1[$]).

Generally, the conventional VS does not exist in any model with a positive probability of a hit on S=0. To make them exist, you would have to regulate the terms of the swap payoff with a cutoff.
 
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JustusQuant
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Re: Asymptotic behaviour of spot skew in CEV model

July 29th, 2016, 8:18 am

You are absolutely right, Alan! I talked to Lorenzo Bergomi myself and he clarified that he is actually using a CEV model where the local volatility is capped (see picture attached). Unfortunately, he didn't mention that at all in his published paper (only in his book "Stochastic Volatility Modeling").

However, in his CEV specification, the forward skew is in fact independent of what the stock does during the forward period and the forward skew hence in fact stationary (which means I don't need a different calibration for each window if the tenor structure is equally spaced). The reason for my non-stationary skew results above is that I used the usual (uncapped) CEV process for calibration which, as you thankfully pointed out, cannot be easily simulated without creating a significant bias.
Remedy: I will now use the capped CEV specification for both calibration and simulation. Unfortunately, this entails that I have to calculate the forward skew and VS vol (which now exists) by PDE methods.

Thanks for all your help, Alan! Your hint to the absorption issue was really key in figuring this out!
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spot_skew.PNG
CEV_capped.PNG