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Problem 4. Calibration of single-factor HJM models of interest rates

Problem 4. Calibration of single-factor HJM models of interest rates. Coordinators Miguel Carrión Álvarez - Banco Santander Gerardo Oleaga Apadula - Universidad Complutense de Madrid Participants Antonio Bueno Universidad Complutense de Madrid

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Problem 4. Calibration of single-factor HJM models of interest rates

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  1. Problem 4. Calibration of single-factor HJM models of interest rates Coordinators Miguel Carrión Álvarez - Banco Santander Gerardo Oleaga Apadula - Universidad Complutense de Madrid Participants Antonio Bueno Universidad Complutense de Madrid Javier García - Universidad Complutense de Madrid Senshan Ji - Universidad Autónoma de Barcelona Santiago López Vizcayno- Universidad Complutense de Madrid Alejandra Sánchez - Universidad Complutense de Madrid Daniel Neira Verdes-Montenegro - Universidad Complutense de Madrid Marco Caroccia - Università degli Studi di Firenze

  2. Index 01Introduction - The time value of money 02Concepts - Time value of money - Interest rate - Model features 03 The HJM framework - Analysis of the forward rates - Forward correlation matrix - PCA analysis of forward rates - Arbitrage free model for the synthetic forwards

  3. 01 IntroductionThe time value of money • Time is money. A dollar today is better than a dollar tomorrow. And a dollar tomorrow is better than a dollar next year. Is every day worth the same or will the price of money change from time to time? • The interest rate market is where the price of money is set. What does “price of money” mean? It is the cost of borrowing and lending money. It is usually quoted by means of “rates” per unit of time (1% per annum, 2% per annum). • The price of money depends not only on the length of the term, but also on the moment-to-moment random fluctuations of the market. • Moneybehaves just like a stock with a noisy price driven by a Brownian motion.

  4. 02 ConceptsMarket Zero coupon bonds We denote by Zt(T) the value on date t of one monetary unit deliverable on date T. By definition, ZT(T) = 1, Zt(T) < 1 for all t< T.

  5. 02 ConceptsMarket Zero Coupon Bonds Due to the fact ZT(T) = 1, Zt(T) can’t be a stationary process.

  6. 02 ConceptsSynthetic Zero Coupon Bonds However, we can define a syntheticconstant-maturity bond whose price is

  7. 02 ConceptsThe synthetic Zero coupon bond The evolution of .This is a stationary process but highly autocorrelated because of price continuity.

  8. 02 ConceptsThe synthetic Zero coupon bond. If we consider we obtain a stationary and not autocorrelated process. Any function of this variable is stationary and not autocorrelated too. So, we define

  9. 03 Theoretical Concepts Constant maturity yields • Yield. Given a discount bond price Z at time t, the yield R is given by: • The forward rates and can be written in terms of the bond prices as: • Instantaneous forward rates

  10. 03 Theorical ConceptsModel features We want to consider stochastic models of interest rates with the following features: • They have as few underlying stochastic factors as possible. • They are consistent with absence of arbitrage opportunities (“there is no free lunch”). • They can potentially accommodate any observed term structure of interest rates.

  11. 03 The HJM framework • The Heath–Jarrow–Morton theory ("HJM") is a general framework to model the evolution of interest rate curve - instantaneous forward rate curve in particular. • The key to these techniques is the recognition that the drifts of theno-arbitrage evolution of the instantaneous forwards can be expressed as functions of their volatilities, no drift estimation is needed. • The general parameterization of continuous stochastic evolution due to HJM is: • Where Wt is basic Wiener process, so that Wt~N(0;T)

  12. 03 The HJM framework • In the risk neutral probability measure the drift changes as . • Choosing the cash bond Bt to discount prices, the no-arbitrage condition implies: • Where is the log-volatility of the discounted bond price

  13. Analysis of the forward rates We cannot obtain the instantaneous forward rates from the data, but we are able to analise the forwards between two consecutive : We have a stochastic variable for each k, so we proceed to a principal component analysis of the forwards. This allows to construct a “discretised” HJM model with no arbitrage.

  14. Forward correlation matrix

  15. PCA analysis for forward rates

  16. The arbitrage free model for the synthetic forwards • This arbitrage-free model is obtained from the HJM condition imposed to our synthetic variables. • In the simplest setting, the volatilities are estimated from principal component analysis. • There is only one risk factor involved. • For future work, a more complex model for the volatilities is needed, and more factors may be included. • Thank you!

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