Essentially a copy of the nominal version but taking a
different index and a set of pricers (not just one).
The payoff $ P $ of a capped inflation-rate coupon
with paysWithin = true is:
$$ P = N \times T \times \min(a L + b, C). $$
where $ N $ is the notional, $ T $ is the accrual
time, $ L $ is the inflation rate, $ a $ is its
gearing, $ b $ is the spread, and $ C $ and $ F $
the strikes.
The payoff of a floored inflation-rate coupon is:
$$ P = N \times T \times \max(a L + b, F). $$
The payoff of a collared inflation-rate coupon is:
$$ P = N \times T \times \min(\max(a L + b, F), C). $$
If paysWithin = false then the inverse is returned
(this provides for instrument cap and caplet prices).
They can be decomposed in the following manner. Decomposition
of a capped floating rate coupon when paysWithin = true:
$$
R = \min(a L + b, C) = (a L + b) + \min(C - b - \xi |a| L, 0)
$$
where $ \xi = sgn(a) $. Then:
$$
R = (a L + b) + |a| \min(\frac{C - b}{|a|} - \xi L, 0)
$$
Capped or floored inflation coupon.
Essentially a copy of the nominal version but taking a different index and a set of pricers (not just one).
The payoff $ P $ of a capped inflation-rate coupon with paysWithin = true is:
$$ P = N \times T \times \min(a L + b, C). $$
where $ N $ is the notional, $ T $ is the accrual time, $ L $ is the inflation rate, $ a $ is its gearing, $ b $ is the spread, and $ C $ and $ F $ the strikes.
The payoff of a floored inflation-rate coupon is:
$$ P = N \times T \times \max(a L + b, F). $$
The payoff of a collared inflation-rate coupon is:
$$ P = N \times T \times \min(\max(a L + b, F), C). $$
If paysWithin = false then the inverse is returned (this provides for instrument cap and caplet prices).
They can be decomposed in the following manner. Decomposition of a capped floating rate coupon when paysWithin = true: $$ R = \min(a L + b, C) = (a L + b) + \min(C - b - \xi |a| L, 0) $$ where $ \xi = sgn(a) $. Then: $$ R = (a L + b) + |a| \min(\frac{C - b}{|a|} - \xi L, 0) $$