Hi,
I am currently trying to implement the Bergomi model to price forward skew sensitive options. In order to calibrate the stock price process, I need to compute the model implied spot skew ( difference between 95 and 105 moneyness implied vols). Since the stock SDE corresponds to a standard CEV model, I implemented the following two approaches (I use the CEV setup as specified by Bergomi here on page 5: http://papers.ssrn.com/sol3/papers.cfm? ... id=1493302 ):
1. Use the CEV closed form solution for std. European options and apply bisection algorithm ( see Schroeder(1989))
2. Use a perturbation expansion to get an approximation for BS implied vol directly ( see Hagen and Woodward (1998))
Unfortunately, for fixed large beta, both methods start to yield very different results. The perturbation based approach shows a skew that is convex in the level parameter sigma, the closed form shows concave behavior.
My question: Is there any literature about the asymptotic behavior of the implied vol surface for very large beta? Which behavior is intuitively more plausible? Can the perturbation error grow this large (I implemented it up to 4th order)?
I personally think the closed form behavior is more plausible, but the evaluation of the required non-central chi squared distribution is sometimes tricky, whereas the perturbation approach seems to be quite reliable for "reasonable" parameters.
Thanks in advance for any help!
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- JustusQuant
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Asymptotic behaviour of spot skew in CEV model
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Re: Asymptotic behaviour of spot skew in CEV model
Can a perturbation expansion fail so badly? Sure.
Here's what I get for the closed form, assuming (T=30 days), and taking your beta = 10.
So it looks like your closed form is probably correct. ([$]\sigma[$] is the effective lognormal volatility).
Here's what I get for the closed form, assuming (T=30 days), and taking your beta = 10.
So it looks like your closed form is probably correct. ([$]\sigma[$] is the effective lognormal volatility).
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Re: Asymptotic behaviour of spot skew in CEV model
Thanks a lot for the quick reply, Alan! That's what I feared, so I guess I'll have to use the closed form then.
Do you also get problems with the CEV closed form solution for small sigma/beta? For sigma less than 0.1, it blows up for a wide range of values for beta. If not: Which algorithm do you use to evaluate the non-central chi squared distribution?
Do you also get problems with the CEV closed form solution for small sigma/beta? For sigma less than 0.1, it blows up for a wide range of values for beta. If not: Which algorithm do you use to evaluate the non-central chi squared distribution?
Re: Asymptotic behaviour of spot skew in CEV model
Yes, my posted plot routine starts failing for [$]\sigma < 0.030[$]. It doesn't blow up, but the chart loses smoothness.
I am working in Mathematica, just integrating the put payoff over the transition density (and adding a term for the absorption).
If I wanted to cure the fails, I would rewrite the code, likely adding a high precision version to get the option value and also probably have to beef up my implied vol. routine. In your case, you already have a nice perturbation expansion for small [$]\sigma[$]. So, if I were you, I would switch to using that for small [$]\sigma[$].
I am working in Mathematica, just integrating the put payoff over the transition density (and adding a term for the absorption).
If I wanted to cure the fails, I would rewrite the code, likely adding a high precision version to get the option value and also probably have to beef up my implied vol. routine. In your case, you already have a nice perturbation expansion for small [$]\sigma[$]. So, if I were you, I would switch to using that for small [$]\sigma[$].
Last edited by Alan on July 18th, 2016, 3:45 pm, edited 1 time in total.
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Re: Asymptotic behaviour of spot skew in CEV model
Sorry, 'blows up' wasn't the right term. The bisection doesn't converge because the prices start violating no-arbitrage. Can you point me to any literature about the specifics of your or any other high-precision approach? I tried several methods, but all suffered from this caveat.
Thanks again for your help, Alan. I really appreciate it!
Thanks again for your help, Alan. I really appreciate it!
Re: Asymptotic behaviour of spot skew in CEV model
Any calculation in Mathematica can be performed at arbitrary precision. All their documentation is online.
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Re: Asymptotic behaviour of spot skew in CEV model
I initially did use a combined approach. However, there is a small range where the closed form doesn't work and the perturbation expansion already starts to look convex (i.e. wrong). I have to implement it in C++ and don't have access to Mathematica. Do they really precisely document online their approach mathematically? I couldn't find any specifics, only the general description of the corresponding function.
Re: Asymptotic behaviour of spot skew in CEV model
No, they don't. You won't be able to read the online docs (or any other docs about Mathematica) and reverse engineer their functions. Doing a high precision version in C++ sounds like a mess to me. Tell me the small range of [$]\sigma[$] where you need the closed form to work at machine precision and, if I get a chance (probably much later in the week), I'll see if I can get my code to work there (at machine precision). If so, I'll post it and you can then imitate it in C++. (Since this will not have any high precision stuff, the C++ port should be reasonably straight-forward).
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Re: Asymptotic behaviour of spot skew in CEV model
The perturbation starts looking odd for sigma > 0.03, but the closed form starts working for virtually all beta for sigma > 0.1. Although for beta close to the BS case, it is also unreliable for much higher values of sigma. It would be really great if I could take a look at your code!
EDIT: I just saw that Schroeder(1989) actually mentions an approximation (for very large inputs) to the non-central chi squared distribution, introduced by Sankaran(1963). I'll try to implement that and will let you know if this alleviates my problem. But thanks again for your help!
EDIT: I just saw that Schroeder(1989) actually mentions an approximation (for very large inputs) to the non-central chi squared distribution, introduced by Sankaran(1963). I'll try to implement that and will let you know if this alleviates my problem. But thanks again for your help!
Re: Asymptotic behaviour of spot skew in CEV model
I got the tricky case working. The option value is computed at machine precision, but I had to add some higher precision stuff to my implied vol routine. Attached also is my Mathematica notebook as a PDF, so readable without Mathematica. (Note my phi = - your beta).
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Re: Asymptotic behaviour of spot skew in CEV model
Looks great! Thanks, Alan, this will be really helpful!
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Re: Asymptotic behaviour of spot skew in CEV model
I implemented the approximation and the calibration seems to work fine now. However, if I use constant values for sigma_0 and beta, I get a non-stationary forward skew. In other words, the n months forward skew with, say, maturity 4 months seems to be a (linearly) decreasing function of n (see chart). Is this inherent to the standard CEV model or is my (Monte Carlo) algorithm flawed? I know that levy models produce a stationary skew, whereas stochastic volatility models generally do not, but I am not quite sure about the CEV model. For the illustrations I used sigma_0 = 0.3027 and beta = 3.57.
Any help would be appreciated!
Any help would be appreciated!
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Re: Asymptotic behaviour of spot skew in CEV model
How do you deal with the fact that, after n months, the implied vols may not exist because the stock has been absorbed at 0?
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Re: Asymptotic behaviour of spot skew in CEV model
Good point, Alan. I didn't treat this case in any special way. If you calculate the expected payoff, then these instances will just contribute zero to the sum. Can this have such a huge impact? If yes: Is there a way to alleviate this issue?
Re: Asymptotic behaviour of spot skew in CEV model
Given the possibility of absorption, I think the best you can do mathematically is define some kind of conditional forward skew -- conditional on no prior absorption. But there's no particular reason for that to be stationary or financially useful for anything. I don't know what you are trying to do with the model, but whatever it is needs to accommodate any absorption event. I'm not sure that notions like a "forward starting option" make much sense when the underlying has a high probability of effectively disappearing.