Callable rates exotics

Callable range accruals on a rate or a CMS spread, snowballs, TARNs, ratchet floaters, callable FRNs, sinking fund bonds, knock-out swaps, flexi swaps and balance-guaranteed swaps, by Monte Carlo on Hull-White, G2++, Cheyette local vol or a LIBOR market model, with issuer calls and holder options by Longstaff-Schwartz. Price, call value, call probabilities and expected life.

Inputs

InstrumentCurrency

Notional

Maturityy

Coupon%

Step added each period%

Leverage on the index

Callable by the issuer

First cally

Mean reversion

Short-rate vol (normal)%

Paths

Discount curve

Discount curve: flat rate% cc

Results

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Methodology

Conventions

Monte Carlo on a model fitted to today's curve: each path carries its discount factors and its zero-coupon bond prices, from which every fixing, swap rate and CMS spread is read.
Issuer calls and holder options by Longstaff-Schwartz: continuation values regressed on the swap rate to maturity and the coupon state.
LMM event dates snap to the index tenor grid. Model parameters are inputs, not calibrated.

References

Longstaff and Schwartz (2001)
Brigo and Mercurio, Interest Rate Models, chs. 4 and 6
Andersen and Piterbarg, Interest Rate Modeling, vol. 2

Formulas

Hull-White: dx = −a x dt + σ dW, r = x + φ(t)
G2++: r = x + y + φ(t), dx = −a x dt + σ dW₁, dy = −b y dt + η dW₂
Cheyette: dx = (y − a x)dt + σᵣ(x) dW, dy = (σᵣ² − 2a y)dt, σᵣ = σ + βx
LMM: d(Fⱼ + s)/(Fⱼ + s) = μⱼ dt + σ dWⱼ, ρᵢⱼ = e^(−β|Tᵢ − Tⱼ|)