Forums > Basics > Mid-curve swaptions

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 pj Total Posts: 3607 Joined: Jun 2004
 Posted: 2021-01-15 10:02 Hi,Mid-curve swaption is a swaption on a forward starting swap.How does one deal with them? The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken
 tbretagn Total Posts: 300 Joined: Oct 2004
 Posted: 2021-01-15 10:12 With a correlation factor for the vol triangle? Et meme si ce n'est pas vrai, il faut croire en l'histoire ancienne
 pj Total Posts: 3607 Joined: Jun 2004
 Posted: 2021-01-15 10:24 let's assume the correlation is given as a number The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken
 tbretagn Total Posts: 300 Joined: Oct 2004
 Posted: 2021-01-15 10:37 Then that gives you your implied volatility and you need to add a factor for the difference in roll-down of your underlying (ie let's say you have a 1y>1y1y mid-curve, the 1y2y, 2y1y and 1y1y dont have a similar roll-down). Et meme si ce n'est pas vrai, il faut croire en l'histoire ancienne
 tbretagn Total Posts: 300 Joined: Oct 2004
 Posted: 2021-01-15 10:38 I believe you should ping @mtsm who will be your man for this kind of thing. Et meme si ce n'est pas vrai, il faut croire en l'histoire ancienne
 nikol Total Posts: 1366 Joined: Jun 2005
 Posted: 2021-01-15 10:44 > let's say you have a 1y>1y1y mid-curve, the 1y2y, 2y1y and 1y1y don't have a similar roll-downtriangle works for underlying-spot (scalar). SWOPT ~ f(curve=vector). so the factor accounting for the discrepancy is vector too. At least. ... What is a man If his chief good and market of his time Be but to sleep and feed? (c)
 ronin Total Posts: 683 Joined: May 2006
 Posted: 2021-01-15 10:52 So, to be naive, option to receive forward <= option to receive long + option to pay short?More seriously, I'd go with @tbretagn - MC in a forward swap model. I'm sure you can write down some formulas by making assumptions about the shape of the curve, but I wouldn't trust those for trading. "There is a SIX am?" -- Arthur
 pj Total Posts: 3607 Joined: Jun 2004
 Posted: 2021-01-15 12:01 @ronin> So, to be naive, option to receive forward <= option to receive long + option to pay short?That's not that naive. Actually it is an option on a basket, not the basket of options. The older I grow, the more I distrust the familiar doctrine that age brings wisdom Henry L. Mencken
 ronin Total Posts: 683 Joined: May 2006
 Posted: 2021-01-15 13:27 I meant naive as in "stating the obvious"....But I did know salespeople who would look at that and go "The client needs one option, and I'm selling two? Challenge accepted."The reality is that you can't replicate it with vanilla swaptions, for the same reasons you can't replicate vanilla swaptions with caps. It's not just the correlations, it's that you need an extra dimension for the vol cube - a vol hypercube. And there are infinities of hypercubes which are all consistent with any given cube.Practically, it boils down to quantifying and tracking the vegas. Like with all exotics - one is a problem, a whole book is less of a problem. "There is a SIX am?" -- Arthur
 tbretagn Total Posts: 300 Joined: Oct 2004
 Posted: 2021-01-15 13:55 Other one which is fun is the changes in basis. Less a problem now but 2y against 6m. Hence fun fun to price 1y>1y1y kinda stuff.Obviously now everything is flat but pre 2015 that would get you in a world of trouble.But as the small time PMs I would work with at the time used to say: "Tanguy, you don't understand, it's just Black-Scholes." Et meme si ce n'est pas vrai, il faut croire en l'histoire ancienne
 nikol Total Posts: 1366 Joined: Jun 2005
 Posted: 2021-01-16 11:07 ronin >> there are infinities of hypercubesAt the end, once constant volatility assumption is relaxed, these hypercubes are present even in the case of spot-underlying, like stock or fx. We start taking about curve, surface, cube, etc correlations. It forces to impose assumptions about shapes of those objects and their dynamics and factorise them with one or two parameters. ... What is a man If his chief good and market of his time Be but to sleep and feed? (c)
 ronin Total Posts: 683 Joined: May 2006
 Posted: 2021-01-18 09:47 > these hypercubes are present even in the case of spot-underlyingThat is not quite what I meant.For spot underlyings, it is strike-maturity, i.e. a surface. E.g. option matures in 10 years.For rate underlyings, it is strike-maturity-tenor, i.e. a cube. E.g option matures in 5 years, it's an option on a 10 year swap.For forward rate underlyings, it is strike-maturity-start-tenor. E.g. option matures in 5 years, swap starts 2 years after that, it's a 7 year swap.The standard vol cube is basically the strike-maturity-0-tenor. There are many different strike-maturity-start-tenor hypercubes that, with 0 forward start, correspond to the same given cube. Off the top of my head, there seem to be a few obvious constraints on calendar arbitrage - e.g., 5y-2y-8y might imply negative forward variance relative to 5y-10y - but other than that, the cube does very little to fix the hypercube. "There is a SIX am?" -- Arthur
 nikol Total Posts: 1366 Joined: Jun 2005
 Posted: 2021-01-18 10:49 Understood ! No worries. You are correct.I am after the easiness of explanation. To my taste you immediately jump into very complex. Instead, I am starting from low dimensional problem like ivols of 3 stocks (spot), where we have the entire correlation function (3D?). It collapses into single correlation under assumption of constant vols (no smile). For IR-curves, we get those hyper-hyper-cubes because of smile and forward cross-correlations and interplay between them. So, dimensionality of corr-function explodes even further. ... What is a man If his chief good and market of his time Be but to sleep and feed? (c)
 silverside Total Posts: 1444 Joined: Jun 2004
 Posted: 2021-01-18 11:45 Its a while since i looked at these but I seem to rememberthere were two different ways of looking at the midcurve vol term structure (isn't tesseract the word for a 4-d cube?!)each midcurve swaption could be treated either as a spread option on two vanilla forward swaps (hence the important/extra parameter being the correlation)or as a vanilla swaption, but with early expiry (in which case the important thing is the term structure of forward volatility)maybe that's obvious - I never got close enough to pricing/trading to know if the choice of approach makes much difference?
 ronin Total Posts: 683 Joined: May 2006
 Posted: 2021-01-19 20:14 So how does the spread thing work?For swaptions, you would use dv01 as the numeraire. But here you have two legs with different dv01 - so what is the numeraire?Do you actually have to simulate every single cashflow then?And it's not more user friendly as a spread option either. A spread of two lognormals would usually be normal. But this one would be lognormal, or something. "There is a SIX am?" -- Arthur
 silverside Total Posts: 1444 Joined: Jun 2004
 Posted: 2021-01-19 23:04 Oh this wasn't so fancy as to think about numeraires or change-of-measure!Just if you have a 5x5x10 midcurve (expiry in 5 years, on the 10y swap which starts 10 y from today) - forget about skew, and assume a flat rate curve, Then the relevant forward swap is the difference between the 5y15y and the 5y5y swaps (with the same nominal) This can be reexpressed as A * 5y15y rate - B * 5y5y rateYou know the vol of both rates and assume a flat correlation (and assume A and B are constant) which, using basic methods for the volatility of a sum (easier if you work using BP vols) gives you a crude estimate for the midcurve vol - without having to explicitly model the forward volatility term structure. Or something like that... I've a feeling I haven't remembered some of the details correctly - for any mistakes I blame covid ðŸ˜‚
 ronin Total Posts: 683 Joined: May 2006
 Posted: 2021-01-20 11:44 Yes, I got that. But now say you see some skewed prices. So you want to fit to them, and you use Monte Carlo.So you propagate to 5y, you see the 10, you see the 5. In a normal swaption, you would just say that's it. Rate minus strike, floored at zero, times dv01. But in this, you don't know the dv01. So you have keep going to simulate all cash flows for the next 10 years and you can only subtract them in the end. So it's even nested Monte Carlo. Like I say, it's not not-doable. But it just gets more and more complicated the more you think about it, beyond the basic approximation. "There is a SIX am?" -- Arthur
 deeds Total Posts: 516 Joined: Dec 2008
 Posted: 2021-01-20 11:49 Does some kind of clever compound, reset or cliquet option get us an approximation to nested MC? quicker for calibration, at least to ballpark it
 silverside Total Posts: 1444 Joined: Jun 2004
 Posted: 2021-01-20 12:10 Can you not just use BGM to model the rate curve until expiry (assuming midcurve swaptions are European style)?You still have the problem of specifying the vol term structure and the correlation matrix of the forwards but I don't see that the simulation is necessarily that much more difficult than for vanillas?On a related note, I think Henrard has done some work on the differences between cash - settled and physically - settled swaption values (and shown it to be small, at least for vanilla swaptions). My feeling though is that we're discussing minor details when the key point from a mathematical perspective is how the forward variance is distributed.
 ronin Total Posts: 683 Joined: May 2006
 Posted: 2021-01-20 15:45 > Can you not just use BGM to model the rate curve until expiry (assuming midcurve swaptions are European style)?Well, no - that's where the optionality is. If you just read them off the curve, you'd get a swap, not a swaption. You need them to be volatile. "There is a SIX am?" -- Arthur
 silverside Total Posts: 1444 Joined: Jun 2004
 Posted: 2021-01-20 17:29 do you mind expanding on that?so with my example of the 5x5x10 midcurveyou evolve the rate curve for 5 years (N times where N is 10,000 or whatever)on each path at that point you have the curve which gives you the 5y10y forward (and all the DV01's) - exercise decision is simply Max(Fwd Rate - Strike,0) * DV01 on that curveam I missing something ?
 ronin Total Posts: 683 Joined: May 2006
 Posted: 2021-01-21 09:18 No, you're fine - I was talking nonsense. I was thinking about something else.The problem with the dv01 is different - it's about how you evolve the 5x5 and 5x10 consistently. The only way to do it is to do the 5x5 and 5x5x5 separately, and make 5x10=5x5+5x5x5, otherwise you have scenarios in which you can arb 5x5 against 5x10. And it doesn't matter if we are talking forward BGM-style models or short rate models, you always end up back where you started - with the 5y option on the (then) 5x5 forward. "There is a SIX am?" -- Arthur
 nikol Total Posts: 1366 Joined: Jun 2005
 Posted: 2021-01-21 14:37 @silversideronin talks about syntetics, such that underlying evolution cancels out, hence specific model becomes irrelevant. ... What is a man If his chief good and market of his time Be but to sleep and feed? (c)
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