Fixed Income

Explaining the evolution of interest rates: Basis, Libor and multiple curves

30/11/2012

Massimo Morini, Understanding and Managing Model Risk

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When the market model changes. Example from Rates

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Swaps are the main interest rate derivatives. They are priced ‘without a model’ via replication The simplest swap is the Forward Rate Agreement (FRA). We can price it using information in the spot interbank market, where loans are made from bank to bank. If the prevailing rate is , when a lender gives 1 at t he receives If instead he gives

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he receives 1 at T.

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Replicating a FRA

If

a FRA fixes in α and pays in 2α, the payoff in 2α is : replicate it, lend at 0 to your counterparty

To

until α, so that you get 1 at α. At α lend again to counterparty 1 until 2α, getting at 2α :

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At 0 you also borrow

until 2α, so that you

receive at 2α :

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Putting together, payoff at 2α is :

30/11/2012

Massimo Morini, Understanding and Managing Model Risk

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Replicating a FRA

30/11/2012

Massimo Morini, Understanding and Managing Model Risk

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The equilibrium FRA rate and the Basis Spreads

The price of a FRA is the cost of its replication. Recalling that in modern markets the discount “bond” comes actually from interbank rates:

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and the equilibrium K is:

.

30/11/2012

Massimo Morini, Understanding and Managing Model Risk

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The equilibrium FRA rate and the Basis Spreads

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With many FRA paying from

to

we have swap value:

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The frequency of payment, and consequently the tenor of the rates paid, does not enter the valutation of floating legs. This implies null spreads for Basis swaps, that exchange two floating legs of different frequency and different rate indexation (for example 1y vs 6m).

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30/11/2012

9 August 2007: the market changes

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