Problem 4. Calibration of single-factor HJM models of interest rates Coordinators Miguel Carrión Álvarez - Banco Santander Gerardo Oleaga Apadula - Universidad.

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

Coordinators

Miguel Carrión Álvarez - Banco SantanderGerardo Oleaga Apadula - Universidad Complutense de Madrid

Participants

Antonio Bueno Universidad Complutense de MadridJavier García - Universidad Complutense de MadridSenshan Ji - Universidad Autónoma de BarcelonaSantiago López Vizcayno- Universidad Complutense de MadridAlejandra Sánchez - Universidad Complutense de MadridDaniel Neira Verdes-Montenegro - Universidad Complutense de Madrid Marco Caroccia - Università degli Studi di Firenze

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01 Introduction

- The time value of money

02 Concepts

- 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

Index

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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.

Money behaves just like a stock with a noisy price driven by a Brownian motion.

01

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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.

02

:tZ

T0t 1t it

3Time to Maturity M

1Time to Maturity Y

8Time to Maturity M

193% 94% 98% 99%

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ConceptsMarket Zero Coupon Bonds

Due to the fact ZT(T) = 1, Zt(T) can’t be a stationary process.

02

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ConceptsSynthetic Zero Coupon Bonds

However, we can define a synthetic constant-maturity bond whose price is

02

:tP

0t 1t jt 0t 1t

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ConceptsThe synthetic Zero coupon bond

The evolution of .This is a stationary process but highly autocorrelated because of price continuity.

02 ( )tP

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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

( )

( )t dt

t

P

P

02

( )1: log

( )t dt

tt

PX

P

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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

03

1( ) : log ( )t tR P

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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.

03

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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 the no-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)

03

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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

03

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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.

1( , )t k kF

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Forward correlation matrix

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PCA analysis for forward rates

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The arbitrage free model for the synthetic forwards

( ) ( ') 1( , ') ( ) ( ')

' 2t t

t t t t tdF dt dW

• 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!