Keep the RS implied-standard-deviation approximation finite

[Abhayindia]

blackFormulaImpliedStdDevApproximationRS returns NaN over a sizeable part of ordinary parameter space, and blackFormulaImpliedStdDevLiRS inherits it because it seeds its iteration from this function.

Found by round-tripping: price an option with blackFormula, then feed the price back to each implied-standard-deviation inverse and check the library accepts its own output. Over 1488 priced cases the exact solver, the plain approximation and Chambers are all clean; RS returned a non-finite value 304 times.

On a plain grid of forward 1.0, call and put, moneyness 1.01 to 10 and standard deviation 0.01 to 1.5, 11 of the 49 cells were NaN.

Two separate causes

beta underflows to exactly zero for an in-the-money option at low volatility, so gamma = -pi/2 * log(0) is infinite and sqrt(gamma+y) - sqrt(gamma-y) evaluates as inf - inf. Writing that difference in its conjugate form, 2y / (sqrt(g+y) + sqrt(g-y)), is algebraically the same and tends to zero as gamma grows.

C = (R2 - (ey-1)^2) * ((ey+1)^2 - R2) / ey2 is exactly zero when the price sits on intrinsic, since R equals ey - 1 there. Rounding can leave it slightly negative instead, for example -8.2e-17 at forward 1.0, strike 1/1.1, and then beta is negative and log(beta) is NaN. Clamped at zero.

Why NaN is the wrong answer rather than an inaccurate one

In the failing cells the price has collapsed onto intrinsic, and blackFormulaImpliedStdDev returns exactly 0 there. After this change RS returns 0 too:

forward/strike	true std dev	exact	RS before	RS after
5	0.1	0	NaN	0
10	0.1	0	NaN	0
5	0.4	0.4	0.393236	0.393236
10	0.4	0.4	0.40932	0.40932

Values that were already finite are unchanged: the conjugate branch is only taken when a square root is not finite, and the clamp only engages where C was negative.

Verification

testImpliedStdDevApproximationRSStaysFinite covers moneyness 1.01 to 10 against standard deviations 0.01 to 1.5, calls and puts. Checked that it fails on master without the library change: 27 of its 28 assertions fail.

After the change the 1488-case sweep is clean for RS, and the full test suite passes: 1548 test cases, 614293 assertions.

Related, not addressed here

blackFormulaImpliedStdDevLiRS still throws on 336 of the 1488 cases. Before this change the failure was boost::math::quantile ... Probability argument is -nan, propagated from the RS guess. With the guess now finite it fails differently, boost::math::erfc_inv Overflow Error, when the normalized price reaches the 0 or 1 boundary and MaddockInverseCumulativeNormal overflows. That is a change inside the iteration rather than to a closed-form expression, so I have left it alone; happy to look at it separately if you would like.

[coveralls]

coverage: 78.252% (+0.001%) from 78.251% — Abhayindia:fix/rs-implied-stddev-nan into lballabio:master

[lballabio]

Either hi or lo are not finite here. Shouldn't we replace this with the limit?

[Abhayindia]

Replaced it with the limit - gamma is only non-finite when beta underflows, and sqrt(g+y)-sqrt(g-y) tends to 0 there, so the difference branch now returns 0 directly (the sum branch already yields +inf on its own). Over 1920 gridded cases gamma goes infinite 1182 times, and in every one the price is either exactly the intrinsic value or denormal, so 0 is the limit rather than a fallback.
The finiteness test still reports 27 failures without the guard and none with it.