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

[Aleksis A. R. <ale...@go...>]

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... <mailto: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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