Pricing a Libor Swap with a Conditional Rate Payment

Summary

The document considers a swap that receives three-month Libor plus a spread and pays a rate that depends on whether Libor is above a strike. The proposed simplification applies when Libor is fixed at the beginning of each three-month accrual period. Under that convention, the conditional payment can be represented as a combination of a cap and a digital option, so a full Libor Market Model or Monte Carlo simulation is unnecessary for the basic payoff.

The answer cautions that different fixing conventions can change the problem. A convexity adjustment may be needed when the rate is set differently, and path dependence can make a more elaborate model such as an LMM appropriate. The exchange gives no full valuation steps, market inputs, or numerical price, so implementation still requires discounting and consistent curves and volatility assumptions. Its main lesson is to inspect the reset timing and payoff dependence before choosing a pricing model.

Key ideas

• A conditional Libor payment may decompose into cap and digital-option components.
• The simplification assumes Libor is set at the start of each accrual period.
• Different fixing conventions can require a convexity adjustment.
• Path-dependent payoffs may justify an LMM or other more involved modeling approach.
• The document gives a model-selection principle rather than a numerical valuation.

Tags

# Multi-legged Swap pricing


# Multi-legged Swap pricing












can anyone guide me how to price a multi-legged swap and whether I need Monte Carlo / LMM based approach or if there is a closed form solution.

Receive leg "Libor 3m +1%"

Payment leg If Libor is greater than strike of 3%, then "Libor - 0.5%", else 3%

## Answer by ExIR (score 0, accepted)

https://quant.stackexchange.com/a/45646

typically the 3M libor is set at the beginning of a 3M calculation period. If so, your payout is a simple combination of cap + a digital option on 0.5%. You do not need LMM on this. It would be an overkill. However, if your 3M libor is set differently, you might have to use convexity adjustment, or even worse, a path-dependence is introduced, then LMM.

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