Heath Jarrow Morton a interest rate model for CVA calculations

For

Management staff Development

Alexis Maenhout 14/06/2013

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Contents

1 Context:......................................................................................................................3

2 Introduction:...............................................................................................................3

2.1 A real world model...............................................................................................3

2.2 A Risk neutral model for (derivative) pricing........................................................4

2.3 Multidimensional approach..................................................................................6

2.4 Observing Parameters..........................................................................................7

2.5 Volatility functions................................................................................................8

2.6 Practical aspects of Monte Carlo Simulation.........................................................8

2.6.1 Interpolation algorithm..................................................................................8

2.6.2 Numerical integration....................................................................................8

2.6.3 Principal component analysis........................................................................8

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1 Context:

Contrarily to what one might expect Heath Jarrow Morton is more a frame work than an “n-th” model next to other short rate models. For a start it is not a short rate model but an instantaneous forward rate model. Precisely to opt for the forward rate as fundamental quantity to model they provided an arbitrage free frame work for the stochastic evolution of the forward curve whereby the forward rate dynamics are fully specified by instantaneous volatility structures.

2 Introduction:

2.1 A real world model

First consider pricing a zero-bond paying 1 unit of currency at maturity based on the

instantaneous forward rate F ( t ,T ). This is the rate observed at time t paid at time T in

the future.

Z (t ,T )=exp ¿ (2.1)

Using Leibniz’ rule for differentiating integrals we obtain a basic expression for the forward rate.

F ( t ,T )=−∂∂TlnZ ( t , T )

(2.2)

We can obtain the differentiation for this forward rate:

dF (t , T )=d (−∂∂T lnZ ( t , T )) (2.3)

that we can manipulate this expression by displacing the minus sign and by interchanging the partial differentiation towards the T maturity with the total

differentiator d ( . ). The latter is possible regarding the fact that T must be seen as a

parameter and not as a variable in this context.

dF (t , T )=−∂∂Td ( lnZ ( t ,T ) )

(2.4)

Let us now concentrate on our zero-bond Z, we can also assign it a stochastic model to

describe its evolution. Herein T is parametric so d ( . ) is a variation of t keeping the

maturity T as fixed!

dZ ( t , T )=μ ( t , T )Z ( t ,T )dt+σ (t , T )Z ( t , T )dX (2.5)

This is nothing else than a single factor model, very much looking like a log normal

random variable.... Notice that by definition Z (t ,t )=1 this automatically implies σ ( t ,t )=0!

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Applying Itô’s differentiating lemma for a stochastic variable W=W (Z )=lnZ, keeping in

mind limt →0

dX2=dt and that the product of stochastic differentials can be neglected by

order of magnitude dtdX <dt ; dt2→0 we obtain

dW (Z )=∂W∂ Z

dZ+ 12∂2W∂ Z2

dZ2(2.6)

With Itô’s lemma (2.6) worked out based on (2.5) we get :

d ln (Z )= 1Z (t ,T )

(μ (t ,T )Z (t , T )dt+σ (t ,T )Z (t , T )dX )+ 12 ( −1Z2 (t ,T ) ) σ (t ,T )2Z2 (t , T )dt

(2.7)

After rearranging:

d ln (Z )=(μ ( t ,T )−12σ (t ,T )2)dt+σ ( t ,T )dX

(2.8)

Substitute this result in our forward rate expression (2.4) to obtain a “real world” SDE to describe the forward curve evolution

dF (t , T )=−∂∂T (μ (t , T )−1

2σ ( t , T )2)dt− ∂

∂Tσ ( t ,T )dX

(2.9)

2.2 A Risk neutral model for (derivative) pricing

In order to derive the risk neutral world we can start by looking how a hedged portfolio behaves. While we are still in a one factor model where we assume all maturities are perfectly correlated. We observe a portfolio of a bond hedged by another maturity bond.

Π=Z (t , T1 )−Δ∗Z (t ,T 2 ) (2.10)

Observing the portfolio evolution:

dΠ=dZ (t , T 1)−Δ∗dZ (t ,T 2 ) (2.11)

And substituting Error: Reference source not found while choosing delta Δ as an amount of hedging bonds:

Δ=σ (t ,T 1 )Z (t , T1 )σ (t ,T 2 )Z (t , T2 )

(2.12)

We set this portfolio evolution equal to its riskless return d Π=r Π dt and rearrange to obtain

μ ( t , T1 )−rσ (t ,T 1 )

=μ (t ,T 2 )−rσ (t , T 2)

(2.13)

This is only possible if both sides are independent of the bond maturity and thus the

market price of risk λ (t )is independent of the maturity

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μ ( t , T )=r ( t )+ λ ( t )σ (t , T ) (2.14)

Here from we see that the market price of risk must be zero for a risk neutral world where;

μ ( t , T )=r ( t ) (2.15)

In other words the expected return on any traded investment should be the riskless rate

r ( t ).

This leads us to a new SDE for a riskless bond evolution very much like (2.5)

dZ ( t , T )=r (t )Z ( t ,T )dt+σ (t , T )Z ( t , T )dX (2.16)

Now let us rewrite our forward rate evolution SDE (2.9) as follows

dF (t , T )=m ( t ,T )dt+ν ( t , T )dX (2.17)

Based on the second term of (2.9) we easily recognize herein the forward rate volatility ν :

ν (t ,T )=−∂∂Tσ (t , T )

(2.18)

After integration and respecting σ ( t ,t )=0 we get, leaving the constant of integration

zero:

σ ( t ,T )=−∫t

T

ν (t , v )dv(2.19)

Based on the first term of (2.9) we obtain the forward rate drift:

m ( t , T )= ∂∂T ( 12 σ ( t , T )2−μ (t ,T )) (2.20)

Applying the partial derivation towards the maturity and forward rate volatility definition:

m ( t , T )=σ ( t , T ) ∂∂Tσ ( t , T )− ∂

∂Tμ ( t , T )

(2.21)

m ( t , T )=−∫t

T

ν (t , v )dv (−ν (t ,T ) )− ∂∂T

μ (t , T )(2.22)

For a risk neutral world our conclusion (2.15) must hold whereby r ( t ) is independent of T

and thus the last term of the forward rate’s drift vanishes. This has as a consequence that the drift and diffusion of the forward rate are dependent of the forward rate volatility

ν (t , T ). The model becomes:

dF (t , T )=¿(ν ( t , T )∫t

T

ν ( t , v )dv )dt+ν ( t , T )dX(2.23)

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For now we have described maturities as precise points in time, while in practice we describe market values in terms of a fixed term. There for we introduce the Musiela parameterisation where we switch from T to T−t=τ . We think about the forward rate volatility for a 20 yrs maturity instead of the June 15th 2033 maturity.

In other “words” the forward rate for fixed term (notice the superscript) becomes:

F ( t , τ )=F ( t , t+τ ) (2.24)

Our forward rate volatility becomes:

ν (t ,T )=ν ( t ,T−t ) (2.25)

The result we had with (2.23) now becomes with Musiela parameterisation:

d F (t , T )=[(ν (t , τ )∫0

τ

ν (t , s )ds)+ ∂∂ τ F (t , τ )]dt+ν (t , τ )dX(2.26)

The extra term in the drift, can be explained while evolving the curve towards the next present time (or observation time) we must respect the constant maturity and thus roll further on the curve.

2.3 Multidimensional approach

In order to reflect the particularity of contracts with dependence on multiple points on the curve we can better define the risk neutral forward rate curve with an N-dimensional SDE.

dF (t , T−t )=m (t ,T−t )dt+∑i=1

N

ν i ( t , T−t )d X i(2.27)

If we relate our HJM model to forward rate history we will have to determine the co-variances of the observed maturity CHANGES in forward rate’s (dF) time-series. Notice this requires clean and sufficiently populated data series. Once we have found this N*N

covariance matrix M we will obtain the N eigenvectors v i and eigenvalues λ i who obey

the matrix equation:

M v i=λ i v i (2.28)

Wherein index i refers to the i-th maturity forward rate. For example i=3 could refer to the instantaneous 1yr forward rate.

While the correlation matrix is symmetric and can be factorized as follows

M=VΛV T (2.29)

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Here we have Λ a diagonal matrix with the eigenvalues on its main diagonal and V (and

its transposed V T) are matrices of (column)eigenvectors

semi positive definite we can write this matrix as:

M=VΛ1 /2 (V Λ1 /2 )T (2.30)

Thus in a multi-dimensional setting we can rewrite our forward volatility factors as:

ν ( τ j )=(v i ) j√λ i (2.31)

Where τ j is the j-th maturity and (v i ) jis the j-the element of the i-th eigen(column)vector.

So the j-th element of the i-th eigenvector represents the movement of the j-th maturity. The i-th eigen value represents the variance of the movement of th ith component.

It is interesting to analyse how the volatility functions influence the evolution of our forward curve. Each eigenvalue’s corresponding eigenvector can be plotted by maturity, also described as principal components. The plotted eigenvector belonging to the largest eigenvalue will usually be a horizontal function. This reflects one’s intuitive suspicion that most curve movements are rather independent of maturity and represent a vertical shift of the existing curve. A second eigenvector could represent some positive entries and for longer maturities negative entries which result in tilting while positive movements on the short end would result in negative movements in the long run and vice versa.

Maybe insert a graf representing eigenvector fct of maturity here

Our N-dimensional model becomes:

d F (t , T )=[(ν (t , τ )∫t

T

ν (t , s )ds)+ ∂∂ τ F (t , τ )]dt+ν (t , τ )dX(2.32)

d F (t , T )=(∑i=1N

ν i (t , τ )∫0

τ

νi ( t , s )ds+ ∂∂ τF ( t , τ ))dt+∑i=1

N

ν i ( t , τ )d X i(2.33)

Using our correlation matrix decomposition, we obtain a practical form for each j-indexed maturity:

d F j ( t ,T )=(∑i=1N

√ λ i (v i ) j∫0

τ

√ λi (v i ) jds+∂∂ τF (t , τ ))dt+∑i=1

N

√ λ i (v i ) jd X i(2.34)

2.4 Observing Parameters

For the multidimensional model we have assumed we obtained a correlation matrix from statistical observation. This requires us to have sufficient time-series of the relevant T-

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maturity instantaneous forward rates. The production of these can be a cumbersome work, though central banks do publish them in more or less convenient formats.

As when we do not necessarily want to mime reality based on past observations we can

also impose our own volatility structure ν (t , τ ). This can be handy to create stressed

scenarios. The impact of the principal components described above guide us to simulate the desired dynamics. So as if one would not like the longer end of the curve to move one will choose accordingly smaller values for the principal components on the longer end.

2.5 Volatility functions

2.6 Practical aspects of Monte Carlo Simulation

2.6.1 Interpolation algorithm

In order to price instruments with precise maturities one will necessitate an interpolation algorithm to apply on the simulated discrete forward rate curves.

2.6.2 Numerical integration

An algorithm must be provided to integrate the volatility structure in order to calculate the drift factors.

2.6.3 Principal component analysis.

The determination of eigenvector and eigenvalues can be done efficiently by the iterative “power-loop” method. This orders eigenvalues automatically and also produces the eigenvectors.

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