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mekornilol
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Density implied by the SABR vol approximation

April 10th, 2018, 9:39 pm

I am trying to compute the probability density function of the forward rate implied by the SABR formula approximation in order to see how the density implied by the approximation has negative probabilities or explosive behaviour depending on the input parameters (especially for low strikes).


All I have is the stochastic SABR process and the closed form solutions with the first few terms of the asymptotic expansion of the term lognormal volatility.
But how would I go about computing the PDF for a range of points? I'm sure this is not a difficult problem to solve but I can't figure it out!
 
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MaxwellSheffield
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Re: Density implied by the SABR vol approximation

April 11th, 2018, 2:26 pm

Price butterflies on a granular strike grid.
 
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Alan
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Re: Density implied by the SABR vol approximation

April 12th, 2018, 3:10 pm

Agree, although a somewhat more exact way to say it is that, given the option formula

[$]c(K) = c_{BS}(K,IV(K))[$],

then from Breeden-Litzenberger

[$]p(K) = c''(K)[$]

and then it is just a (somewhat tedious) exercise in the chain rule. Hopefully, all the notation is self-explanatory.

Another interesting exercise might be to compare the [$]p(K)[$] you get with the exact [$]p(K)[$], which is known for a few cases: for example, for the lognormal SABR model with correlation. BTW, for that model and for another thread in this forum, I produced this animation which you might find interesting.

Image
Last edited by Alan on April 12th, 2018, 3:39 pm, edited 2 times in total.
 
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outrun
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Re: Density implied by the SABR vol approximation

April 12th, 2018, 3:21 pm

Yes, it works really good, and its fun to check yourself starting with

<Snipped wrong start>

(There might need to be an exp(rt) constant in front of this relation of it to make the probability estimate integrate to 1?)
 
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Alan
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Re: Density implied by the SABR vol approximation

April 12th, 2018, 3:43 pm

In general, yes, but for the model at hand, r=0. 

For clarity, I didn't mean use the chain rule to verify Breeden-Litzenberger, but to use it with the Black-Scholes formula on the r.h.s. of the first line I wrote.
 
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Collector
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Re: Density implied by the SABR vol approximation

April 24th, 2018, 7:21 pm

In general, yes, but for the model at hand, r=0. 

For clarity, I didn't mean use the chain rule to verify Breeden-Litzenberger, but to use it with the Black-Scholes formula on the r.h.s. of the first line I wrote.
A small side step how sensitive is Breeden-Litzenberger to approximations? Even for Black-Scholes where we get out Log-Normal from Breeden-Litzenberger (second partial derivative with respect to strike), how sensitive when we are very very far out in tails, because many (all?) Cumulative normal distributions functions are approximations, typically far outside practical relevant levels, but I just wonder. I plotted it far out without issues, but at some point it should run into problems, likely only of theoretical interest...
 
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Alan
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Re: Density implied by the SABR vol approximation

April 26th, 2018, 2:34 am

Not an issue I think.

For Black-Scholes, the result is essentially [$]p(K) \sim e^{-x^2/(2 \sigma^2 T)}[$], where [$]x = \log K/S_0[$] and this is not going to give any problems no matter how large or small [$]K[$] is. (Numerically, it will just eventually underflow, likely being interpreted as a zero, and all plots will look fine).

For real-world SPX and VIX [$]p(K)'s[$], see the third column plots in my Vol II book on pgs 160-162; all plotted with no problems down to very small [$]K's[$].   

The key is to make sure you have a smooth (twice-differentiable), arbitrage-free [$]IV(K)[$] expression. For example, in the animation I plot, while Paulot's series is smooth, it's not arbitrage-free. For some strikes, as you can see there are ranges where [$]IV(K) < 0[$], clearly a no-no. For the book plots I mention, [$]IV(K)[$] is the Gatheral SVI fit, which is smooth and, at least for the book examples, arbitrage-free.
 
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outrun
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Re: Density implied by the SABR vol approximation

April 26th, 2018, 7:35 am

The method I'm really fond of is to start out with a flexible density function like Gaussian Mixtures, compute theoretical prices (which is very easy for GMM), and then minimize some loss between market prices and theoretical prices f(bid,ask,theoretical price) by changing the GMM density parameters. It offer both good guarantees and flexibility in shapes and objective.
https://www.deep.fund/2018/04/02/modell ... re-models/
a nice loss function is to use deviation from the mid but with an additional penalty when the price moves outside the bid/ask.
 
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Alan
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Re: Density implied by the SABR vol approximation

April 27th, 2018, 3:03 pm

Very nice!  

 You have a nice method that happens to use a 15-parameter fit, if I read it right. Given that there are probably a couple hundred quotes for a given expiration, I think that's fine. I will guess you would need a few more Gaussians to fit the puts down to, say, K=500 or wherever the last put bid drops to 0.05. 

The important thing about the risk neutral density is that it is a true thing, a property of the market that nicely summarizes the option chain. If anybody else has some different method, as long as they produce a smooth fit inside all the bid-asks, their density [$]Q(K)[$] will essentially agree with yours. So, the particulars of the parameterization to arrive at [$]Q(K)[$] don't really matter, except that it should be easy to compute and I'm sure yours is. 
 
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outrun
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Re: Density implied by the SABR vol approximation

April 27th, 2018, 8:08 pm

Thanks!
Yes indeed 15 (with 14 dof):  in the example it uses 5 mixture components and each has a weights, location and scale parameter. The choice of 5 will be different for different curves, it's still a bit of an heuristic (look at error & bid-ask violations as a function of number of components). 

What's interesting is that if you assume underlying *returns* to have a Gaussian Mixture distribution, then the underlying price will have a mixture of LogNormals distribution, and so the theoretical option price will be a mixture of B&S prices! Brigo, Mercurio & Rapisarda have this same result but starting with a SDE mixture of the underlying. I think implied vol doesn't need a underlying process, I prefer to limit the context of modelling implied to just implied.

What you said about the tails in indeed an important aspect. The Gaussians distributions decay too fast, and there are various ways to try and address that: more mixture components, different base distribution in the mixture,.. or a utility of wealth view. Deep out the money puts are probably still worth something because it's a small cost in the scheme of things to protect against extreme events. Also I doubt people have a clear view on the low extreme probability as a basis for bidding. So modelling the uncertainly of the tail probability as an additional risk factor might be a nice model elements, and also the utility of getting some extra trading capabilities if you have good downward protection. When I was a market maker we had a policy to alway have a net positive number of options at the low strikes. 

..and then there is the resolution of the market: you can't get any lower than [zero bid, 0.05 offer]. This is one of the reason I like optimisation based methods that allow you to use creative cost functions like "anything between bid-ask is fine".

A new thing I'm current working on is a Bayesian/probabilistic extension: fit curves to past data and get a distribution of curve shape parameters. Using that distribution we can then 1) better interpolate when you only have few quotes and 2) get a full distribution of curves that fit, model the uncertainty as well. As you said there is a whole family of curves that fit inside the bid-ask. I've tried to model that distribution before with different curve parameterisations but keeping it arbitrage free has been a challenge.  Using a density as a starting point eliminates that model risk. It should end up looking like "Gaussian Process Regression" below -where the dot are quote constraints- but with an empirical curve shape distribution (GPR typically uses a-priori chosen curve families), and curves always being arbitrage free.
Image