LPI swaps and inflation swaptions

UK limited price indexation swaps with inflation floored and capped each year, by a Black product of collars or Jarrow-Yildirim simulation, and European options on forward-starting zero-coupon inflation swaps by Black, Bachelier or Jarrow-Yildirim.

Inputs

Currency

Notional

Maturityy

Yearly floor%

Yearly cap%

Lognormal vol, yearly ratio%

Discount curve

Discount curve: flat rate% cc

Inflation curve

UK RPI today

1Y zero-coupon inflation%

2Y zero-coupon inflation%

5Y zero-coupon inflation%

10Y zero-coupon inflation%

30Y zero-coupon inflation%

Results

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Methodology

Conventions

LPI grows each year by inflation floored and capped (0% and 5% for UK pensions), swapped at maturity for (1 + K)^T − 1.
Black: independent lognormal yearly ratios, so the expected LPI is a product of collars. Jarrow-Yildirim: correlated ratios, simulated exactly.

References

Kenyon (2008), Inflation is Normal
Mercurio (2005), Pricing Inflation-Indexed Derivatives
Jarrow and Yildirim (2003), Pricing TIPS and related derivatives, JFQA

Formulas

LPI: LPI(T)/LPI(0) = Π min(max(I(tᵢ)/I(tᵢ₋₁), 1 + f), 1 + c)
Value: V = N D(T) (E[LPI(T)/LPI(0)] − (1 + K)^T)