Re: [Quantlib-users] LinearTSR CMS pricer vs replicated swaptions

[Peter C. <pca...@gm...>]

Hi Aleksis,

some additional points to check are maybe

- the extrapolation of the volatilities, if you use linear
extrapolation volatilities might be become unreasonably large outside the
quoted region, especially is the skew is steep
- the integration boundaries used in the LinearTSR pricer - these should
correspond to the range of replication scenarios you use I assume

Kind regards
Peter


On Sat, 15 Aug 2020 at 11:57, Aleksis Ali Raza <ale...@go...>
wrote:

> That’s quite helpful Peter.
>
> My primary concern was the unsavoury exponential behaviour of the TSR
> model I was seeing with increasing vol skew (the third plot of my original
> email) leading to unrealistic values of the TSR convexity adjustment. There
> doesn’t appear to be much in the literature regarding the sensitivity of
> these models to pure vol skew (although I admit I haven’t braved through
> Piterbarg’s work).
>
> However, your suggestion of playing around with the mean reversion
> parameter yielded some much more agreeable results (by setting the mean
> reversion to much higher levels than I was originally using):
>
>
> Many thanks, Aleksis
>
>
> On 14 Aug 2020, at 21:32, Peter Caspers <pca...@gm...> wrote:
>
> Hi Aleksis,
>
> the LinearTSR model assumes
>
> (*) P(t,T) / A(t) = a S(t) + b
>
> where t = fixing time, T = payment time, P(t,T) = discount factor, A(t) =
> annuity, S(t) = swap rate and a and b are model parameters. b is determined
> by a no arbitrage condition. a is implied from the mean reversion
> parameter. See the Piterbarg Interest Rate Modeling books for more details
> on this.
>
> You compute your static replication portfolio using parallel shifts of the
> rate curve if I understand correctly. This not exactly the same as (*).
> This might be one possible source of the mismatch under extreme
> market conditions. I suppose different choices of the mean reversion will
> affect the result as well. Other possible reasons certainly include the
> numerical integration scheme of the LinearTSRPricer, you might pass in a
> custom Integrator to do a check on this.
>
> But in short, I just wouldn't expect a perfect match since the two
> approaches represent different models from the start.
>
> Kind regards
> Peter
>
> On Fri, 14 Aug 2020 at 14:06, Aleksis Ali Raza via QuantLib-users <
> qua...@li...> wrote:
>
>> Hi.
>>
>> I’ve been looking at the LinearTSR pricer for CMS in Quantlib and
>> comparing it with a replication portfolio of OTM swaptions.  To set things
>> up I use: a flat rate term structure of 5% (both for discounting and
>> forwards); a normal vol cube with simple symmetric smile (ATM vols around
>> 50bps and skew +/- 15bps for each +/- 1% OTM). I consider a CMS swaplet
>> (fix in adv, pay in arrears with an annual tenor) with a maturity of 5y on
>> a 20y swap reference rate (and compare it with a 5y20y annual/3mL vanilla
>> swap on a 100mm nominal).  After solving for the appropriate CMS nominal
>> and the corresponding nominal weights of the replication swaptions (using a
>> parallel shift range of +/- 10% in 50bps shifts i.e. using 20 payer+20
>> receiver swaptions), the convexity adjustments are a close match and the
>> overall fit looks pretty decent:
>>
>> <Figure_1.png>
>>
>> Now playing around with both the vols and the vol skew, the scenarios of
>> the TSR pricer and the replicated swaption portfolio have the behaviors
>> shown in the plots below (granted up to quite extreme vol levels!). At
>> these more extreme levels of vol and skew, increasing the range of the
>> replication portfolio gives a closer match for the vol shift case, and
>> somewhat improves the match for the skew scaling case - however the
>> discrepancies are still apparent even using a large replication set of
>> swaptions.  Note that skew scaling here just means making the vol smile
>> more acute by scaling the OTM spreads by a scale factor.
>>
>> Not knowing enough about what the LinearTSR pricer is actually doing it’d
>> be helpful if someone could comment on the deviations shown below at the
>> high vol/skew levels and what model parameters might potentially
>> address/explain them. Or, more significantly, if the below behaviors are
>> more of a reflection of a flaw in my replication methodology?
>>
>> Thanks, Aleksis
>>
>> <Figure_2.png><Figure_3.png>
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>
>