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Fair prices under a unified lattice approach for interest rate derivatives

  • S.I. : Recent Developments in Financial Modeling and Risk Management
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Abstract

An open question in interest rates derivative pricing is whether the price of the contracts should be computed by means of a multi-curve approach (different yield curves for discounting and forwarding) or by using a single curve (just one yield curve both for discounting and forwarding). The answer is of primary importance for financial markets as it allows to define a class of fair contracts. This paper calculates and compares the price of a simple swap within both multi-curve and single curve approaches and proposes a generalization of the lattice approach, which is usually used to approximate short interest rate models in the multi-curve framework. As an example, I show how to use the Black et al. (Financ Anal J 46(1):33–39, 1990) interest rate model on binomial lattice in multi-curve framework and calculate the price of the 2–8 period swaption with a single (LIBOR) curve and two-curve (OIS+LIBOR) approaches. Such technique can be used for pricing any interest rate based contract.

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Notes

  1. The statistic derives from the Semiannual OTC derivative statistics of the Bank for International Settlements.

  2. The most popular vanilla interest rate derivatives are interest rate swaps (fixed-for-floating), cross currency swaps, interest rate caps, swaptions (options on swaps), bond options, forward rate agreements, interest rate futures.

  3. This is called the swap coupon frequency.

  4. Indeed, the best current proxy of the risk free rate is the Overnight Indexed Swap (OIS) Rate—EONIA in the Euro-zone—which is a unique discounting curve for all tenor forward curves commonly used to discount collateralized derivative deals.

  5. IRS is a contract in which two counterparties agree to exchange interest payments of different character based on an underlying notional principle amount that is not exchanged.

  6. Coupon swaps exchange fixed rate for floating rate instruments in the same currency, Basis swaps exchange of floating rate for floating rate instruments in the same currency, Cross Currency interest rate swaps exchange of fixed rate instruments in one currency for floating rate in another.

  7. We follow the Mercurio (2008).

  8. A different notation has been used to stress the fact that \(K_{a,n,c,m}\) is the rate computed in (7) whereas K is the rate agreed in the European swaption contract.

  9. For example, see Tse (1995) and Ball and Torous (1999).

  10. We use LIBOR for both discounting and forwarding.

  11. We use OIS rate for discounting and LIBOR for forwarding.

  12. I consider LIBOR rate.

  13. I used LIBOR tree for both discounting and forwarding.

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Authors and Affiliations

  1. Department of Economics and Finance, LUISS University, Rome, Italy

    Giacomo Morelli

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Correspondence to Giacomo Morelli.

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I am grateful to the Guest Editors Rita D’Ecclesia and Roy Cerqueti. I would like to thank Carol Alexander, Damiano Brigo, Maria Chiarolla, Marco Dall’Aglio, Xavier Freixas, Enrico Saltari, Alessio Sancetta, Marco Scarsini and Paolo Santucci de Magistris for insightful comments and suggestions that improved the quality of this work. Also, I thank two anonymous referees for a detailed review of the paper. I acknowledge the research support of the European Commission project: CREA-DLV-766463.

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Morelli, G. Fair prices under a unified lattice approach for interest rate derivatives. Ann Oper Res 299, 429–441 (2021). https://doi.org/10.1007/s10479-019-03209-y

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