I'm faced with the task of pricing CMS-Spread options. Therefore I'm trying to use the copula-approach introduced by Mourad Berrahoui in his wilmott article "pricing cms spread options and digital cms spread options with smile". This approach makes use of the sabr-model and requires the calculation of probabilities Prob(S1 > K1) and Prob(S2 > K2), where S1 and S2 are for example the cms10y-rate and the cms2y-rate. I'm calculating these probabilities as prices of a call spread, i.e. Prob(S > K) = {Call(K-e, sabr(K-e)) - Call(K+e, sabr(K+e))}/(2e). Now the problem is that these probabilities become weird and implausible for large maturities and low strikes K. My question is: could anyone provide me with a workaround or give me a hint how to tackle this problem?Thanks in advance!

You shouldn't use the asymptotic formula for the implied volatility in the SABR model to calculate the probabilities you are interested in. Instead, you can use the asymptotic formula for the probability density derived in the old paper by Pat Hagan, Diana Woodward, and myself "Probability distribution in the SABR model of stochastic volatility".

Thank you very much, Andrew. That's a good hint.I've found two papers entitled "Probability distribution in the SABR model of stochastic volatility". One is from 2004 and the other one is from 2005. So with "old paper" you mean the 2004-paper? As far as the implied volatility formula is concerned, should I also use the results of the paper "Probability distribution in the SABR model of stochastic volatility" or is it okay to work with your original wilmott-article from 2002?Thanks again and kind regards.

These are different drafts of the same thing, use the more recent one. For the implied vol, use the original Wilmott article.

Thank you very much again, Andrew.Just one last question please: are the probabilities given by "Probability distribution in the SABR model of stochastic volatility" consistent with the probabilities that I can derive with a Monte Carlo simulation of the sabr model? More precisely, I would use an Euler-scheme for the sabr dynamics and count 1 for every path where S(maturity) > K. Kind regards.

Yes, they are, to within the validity of the asymptotic expansion (the parameter of the SABR expansion is volvol^2 x time to expiration). A good way of monitoring it is to keep track of the total volume of the asymptotic probability distribution (it should not deviate from 1 too much). Btw, you can do a bit better than the Euler scheme: the SDE for the beta vol can be integrated in closed form.

Thank you very much, Andrew!

Hi, BustopherJones,I'm also working on the CMS spread option by Mourad Berrahoui's paper. Can you tell me where I can find the 2005 paper "Probability distribution in the SABR model of stochastic volatility " and what's the 2002 wilmott-article you are referring to? Thanks.Another question, when you appliy the Gaussian copula for two CMS, what's the correlation you input for the coupla? Is it the instantaneous correlation of the two CMS?Thanks again.

Do any of you find the "Tests" in section V of the Berrahoui paper at all useful? The sample data gives the illusion that you can replicate his results, but you can't.For example, there is a sample swap curve and vol surface, but the surface starts at a strike that is already above the ATM rates? How can you claim to price the option with "ATM vol" then? And for which underlying did he define the vol surface, for the 6M, or the 3M forward? Or is it one surface for all underlying forwards?And for the 20Y option on the 6M-3M spread (Table 1), you would need to know the 20Y6M and the 20Y3M forwards, but his sample curve stops at the 20Y par rate? Same goes for the table where he shows a 20Y option on the 20Y-2Y spread. Am I missing a crucial page here that is only available to Wilmott subscribers?The paper contains very neat ideas, but has it been reviewed thoroughly? I mean, in any normal scientific journal, even if your 1st draft is 100%, the anonymous reviewers always find sth to complain about. In this case, he paper looks like an early draft, but it has still been published just like that? With all due respect, it is not a finished/polished product.pls correct me if i am wrong, and shed some light on how the dude arrived at his sample results.

Dear Prof. Lesniewski, thank you very much for your papers on the subject. You were probably the first to use the language of geometry in finance. Using perturbation theory, you have computed in your article, the first coefficients of the so called Minakshisundaram-Pleijel expansion of the Heat-Kernel associated with the SABR model. However, there are three natural issues:1) The approximation is not necessary positive.2) The approximation does not integrate to 1.3) I don't know any bound for the error.What to do to remedy to this problems? It depends on what you want.1) If you really want your probability density function to be positive, then just consider the order 0. You are 100% sure that your function is positive since at this order your diffusion is gaussian.2) If you want a better approximation and the integral to be as close as 1, it's better to go to order 1 but you will lose the positivity of the functionRegarding any majoration of the error, I don't know any good reference on this subject.

I am afraid I don't have really good answers to your questions... These are asymptotic expansions, which means that they provide approximations to the (exact) solutions only within a certain range of parameters. Asymptotic expansions are often very useful, when set up so that that range contains the useful regime (think QED, fluid dynamics,...). Specifically, 1) I agree, it is not positive for all conceivable values of market parameters. However, for typical markets (including the recent turbulent markets), the kernel is positive.2) I agree again. However, if you calculate the total mass for typical markets, not-too-long expires, and not too close to 1 (away from the lognormal model), it integrates to a value close to 1.3) It is not easy to get a useful rigorous bound of the error beyond stating what is the order of magnitude in terms of the small parameter.QuoteOriginally posted by: GrunspanDear Prof. Lesniewski, thank you very much for your papers on the subject. You were probably the first to use the language of geometry in finance. Using perturbation theory, you have computed in your article, the first coefficients of the so called Minakshisundaram-Pleijel expansion of the Heat-Kernel associated with the SABR model. However, there are three natural issues:1) The approximation is not necessary positive.2) The approximation does not integrate to 1.3) I don't know any bound for the error.What to do to remedy to this problems? It depends on what you want.1) If you really want your probability density function to be positive, then just consider the order 0. You are 100% sure that your function is positive since at this order your diffusion is gaussian.2) If you want a better approximation and the integral to be as close as 1, it's better to go to order 1 but you will lose the positivity of the functionRegarding any majoration of the error, I don't know any good reference on this subject.

Hi,Why not move away from SABR and use as input to your copula a more decent model which will have a well-defined cumulative?I agree that asymptotic expansions are good to get a better intuition of the effects of all the parameters in a model, but one should not rely on them for the actual pricing. Especially when it comes to using a copula which requires a proper cumulative function.If you use an approximation to get you cumulative function, what will happen if you are in a configuration where your approximation breaks down?Are you going to tell your trader that you cannot price his spread option because Alpha^2*T is too big and that the client should choose a shorter maturity?