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### New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **August 3rd, 2021, 1:42 pm**

by **Alan**

The SABR model has been around for a long-time and it often studied via a small-maturity power series. In fact, all stochastic volatility models have a double power series expansion, for the Black-Scholes implied volatility, in the time-to-maturity T and the log-moneyness. With my coauthor Dan Pirjol, we prove that this series (taking the lognormal SABR model at the at-the-money point) is strictly asymptotic: non-convergent for any [$]T>0[$]. Paper is

here
For some background, the non-analyticity of the diffusion kernel factor, [$]e^{-d^2/T}[$] at [$]T=0[$] leads many to believe strict asymptotics in [$]T[$] is the only possibility for option values and related solutions for diffusion models. As we explain in the paper, that's not correct. For example, a simple case is the at-the-money Black-Scholes option value [$]V_{BS}(T) = \mbox{Erf}(2^{-3/2} \sqrt{T})[$]. If you divide the right-hand-side by a [$]\sqrt{T}[$], you get an entire function of [$]T[$], and so an associated power series in [$]T[$] that converges for any [$]|T| < \infty[$]. In principle, that or analyticity for [$]|T| < R[$], where [$]R>0[$] is a convergence radius, could hold in the SABR model. We prove it doesn't.

As always, any comments on the paper are appreciated.

### Re: New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **August 3rd, 2021, 7:55 pm**

by **Paul**

Asymptotic series and convergent series are different things. Asymptotic series don't have to converge. Is that relevant here? (I haven't read the paper!)

### Re: New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **August 3rd, 2021, 8:01 pm**

by **Alan**

Right, so I am calling asymptotic series that don't converge: strictly asymptotic.

### Re: New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **May 20th, 2022, 3:25 pm**

by **Alan**

The SABR model has been around for a long-time and it often studied via a small-maturity power series. In fact, all stochastic volatility models have a double power series expansion, for the Black-Scholes implied volatility, in the time-to-maturity T and the log-moneyness. With my coauthor Dan Pirjol, we prove that this series (taking the lognormal SABR model at the at-the-money point) is strictly asymptotic: non-convergent for any [$]T>0[$]. Paper is

here
For some background, the non-analyticity of the diffusion kernel factor, [$]e^{-d^2/T}[$] at [$]T=0[$] leads many to believe strict asymptotics in [$]T[$] is the only possibility for option values and related solutions for diffusion models. As we explain in the paper, that's not correct. For example, a simple case is the at-the-money Black-Scholes option value [$]V_{BS}(T) = \mbox{Erf}(2^{-3/2} \sqrt{T})[$]. If you divide the right-hand-side by a [$]\sqrt{T}[$], you get an entire function of [$]T[$], and so an associated power series in [$]T[$] that converges for any [$]|T| < \infty[$]. In principle, that or analyticity for [$]|T| < R[$], where [$]R>0[$] is a convergence radius, could hold in the SABR model. We prove it doesn't.

As always, any comments on the paper are appreciated.

This paper has now been published online at Quantitative Finance and will appear in their print edition in due course.

For limited (first 50 clicks) free access to the published version, follow this

link and look for the sentence "The final published version ..."

### Re: New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **May 23rd, 2022, 7:43 am**

by **gatarek**

Dan is a serial model killer. He's already killed LMM, quadratic Cheyette and now SABR. What's next?

### Re: New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **May 23rd, 2022, 2:55 pm**

by **Alan**

The SABR model is fine -- mathematically. Our work just means the small time series (for the lognormal special case) for the implied volatility definitely does not converge. This result was probably the expected behavior. Finite and arbitrarily accurate results are still available from numerics. Finally, the truncated asymptotic series results can be quite accurate, too, for small enough times to maturity and not too many terms in the series. That's typical of asymptotic series generally and, again, probably the expected prior for this model.

### Re: New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **May 23rd, 2022, 4:31 pm**

by **dpacc**

Dan is a serial model killer. He's already killed LMM, quadratic Cheyette and now SABR. What's next?

Will all due modesty, my reputation as a model-slayer is undeserved. I prefer to think of it as mapping the limits of applicability of these models.

Both the LMM and log-normal type Cheyette are perfectly fine, for sufficiently small maturity and volatility.

Outside of these limits,

Eurodollar futures convexity adjustments become unrealistically large,

MC paths may diverge - in other words, use at own risk.

### Re: New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **May 24th, 2022, 7:20 am**

by **gatarek**

Let's not forget about eurodollar futures in Black and Karasinski.

### Re: New paper -- Proof of non-convergence of the short-maturity expansion for the SABR model

Posted: **May 27th, 2022, 3:33 pm**

by **dpacc**

The divergence of the Eurodollar futures prices in the Dothan and Black-Karasinski models (treated in continuous time) was shown a long time ago by

Hogan and Weintraub (1998). However it was not clear how to reconcile this with the finiteness of the Eurodollar futures in discrete time. This question was discussed in this

paper about the Black-Derman-Toy model (which is a discrete time version of Dothan, and a limiting case of BK). It turns out that the ED convexity adjustments are well-behaved for sufficiently small vol not too fine time grids, but become quickly very large as the time step decreases or as the vol increases above some threshold. These explosions seem to be a typical phenomenon for stochastic interest rates models with log-normal distributions.

This is a bit off-topic, and more appropriate for another thread.