Computations in the Hull-White Model (Rom-Poulsen)

8/11/2019 Computations in the Hull-White Model (Rom-Poulsen)

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Computations in the Hull-White Model

Niels Rom-Poulsen 1

October 28, 2005

1 Danske Bank Quantitative Research and Copenhagen Business School, E-mail:[email protected]


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1 Specications

In the Hull-White model, the Q dynamics of the spot rate is given by the following sto-

chastic differential equation (SDE) also know as the Ohrnstein-Uhlenbeck processdr (t) = (( t) r (t)) dt + dW (t) (1)

where ( t ) is the long term level to which the spot rate, r (t), is moving, is the rate atwhich the spot rate rate is pushed towards the long term level and W (t) is a Brownianmotion under Q . is the constant volatility of changes in the spot rate. is assumedconstant in this note.

Using the spot rate dened by (1), we construct a money market account by

A(t) = e t

0 r u du (2)

dA(t) = r t A(t)dt (3)

The specication of the spot rate, means that the Hull-White model belongs to theaffine class of interest rate models and thus prices of zero coupon bonds at time t for thetime T maturity have the following form

P (t, T ) = e ( t,T )+ ( t,T )r t (4)

The T -yield at t time t , y(t, T ) is dened as

y(t, T ) = ln P (t, T )T t

We generally want the model to be calibrated to the market today meaning that modelprices of zero coupon bonds today, P (0, T ) T , is equal to the prices observed in themarket. This can be achieved by choosing ( t) in equation (1) so that the initial yieldcurve is matched.

2 Closed Form Solution for Prices of Zero Coupon Bonds

We will now nd explicit formulas for the functions (t, T ) and (t, T ) in (4) and thus

closed form solutions for zero coupon bonds in the Hull-White model. First, however, wederive the fundamental partial differential equation for zero coupon prices in the Hull-White model. Start by nding the dynamics of zero coupon prices by employing Itoslemma.

dP (t, T ) = P t dt + P

r dr (t) + 12

2 P r 2 (dr (t))

2

Inserting the spot rate dynamics (1) yields

dP (t, T )P (t, T )

= (t, T )

t +

(t, T )t

r (t) dt + (t, T )dr (t) + 12

2(t, T ) (dr (t))2

= (t, T )

t + (t, T )

t r (t) + (t, T )( t) (t)r t + 12 2(t, T )2 dt

+ (t, T )dW (t)

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We now use the dynamics of the money market account given by (3) and the dynamics of the zero coupon bonds in (5) to nd the dynamics of deated zero coupon bond prices

dA(t)dP (t, T ) = A(t)dP (t, T ) + P (t, T )dA(t) +=0

dA(t)dP (t, T )Again inserting what is know we get

dA(t)dP (t, T )A(t)P (t, T )

= (t, T )

t +

(t, T )t

r (t) + (t, T )( t)

(t)r t + 12

2(t, T )2 r t dt + (t, T )dW (t)

Under the equivalent martingale measure, Q , deated prices are martingales. Accordingto the martingale representation theorem we must thus have that the dt-term must beequal to zero, and this holds for all t and r t . Thus

(t, T )t

+ (t, T )( t) + 12

2(t, T )2 = 0

(T, T ) = 0 (t, T )

t (t, T ) 1 = 0

(T, T ) = 0

(5)

(6)

(7)

(8)

We solve the two ordinary differential equations by rst postulating a solution for (t, T )

(t, T ) = 1

e (T t ) 1 (9)

It is easy to check that the solution in (9) in fact solves the ODE in (7) subject to theboundary condition in (8). Since the derivative of (t, T ) only depends on (t, T ) simpleintegration of (u,T )u between t and T is a solution to (5). Recall that

T t (t, T )t du = [(u, T )]T t = (T, T ) (t, T ) (10)and thus we have

(t, T ) = T

t (u, T )( u)du +

12

T

t 2(u, T )2du (11)

As mentioned above we want the model to match current zero coupon prices. This is doneby choosing ( u) in (11) so that the initial yield curve is matched. Instead of calibratingthe model to zero coupon yields directly, we calibrate the model to the term structure of forward rates. Forward rates are dened as

f M (0, T ) ln P (0, T )

T =

T

(0, T ) T

(0, T )r 0

From (9) we get

T

(t, T ) = e (T t ) (12)

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Using Leibnizs rule for differentiating integrals we have from (11), (6) and (8)

T (0, T ) = (T, T )( T ) +

T

0

T (u, T )( u)du

12

2(T, T )2 + 2 T 0 (u, T ) T (u, T )duInserting (9) and (12) yields

T

(0, T ) = T 0 e (T u )(u)du 2 T 0 e (T u ) 1 e (T u )du= T 0 e (T u )(u)du 22 12 1 e 2T 1 e T

Putting things together we get

f M (0, T ) = T 0 e (T u )(u)du (13)+

2

212

1 e 2T 1 e T + e T r 0 (14)

To isolate ( T ) we differentiate with respect to T

f M (0, T )T

= ( T )

T

te (T u )(u)du

+2

e 2T e T e T r 0

Using (13) this can be written as

f M (0, T )T

= ( T ) f M (0, T ) + 2

12

1 e 2T ) 1 e T )

+ e T r 0 + 2

e 2T e T e T r 0

= ( T ) f M (0, T ) 2

21 e 2T

And thus

(T ) = f M (0, T )

T + f M (0, T ) +

2

21 e 2T (15)

Now that we know ( T ) we can plug it into (11) to nd an expression for (t, T ). Wecompute rst the integral

2

2 T t 2(u, T )du=

2

22

T

te (T u ) 1

2du

= 2

22 12

1 e 2 (T t ) + ( T t) 2

1 e (T t ) (16)

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Next we compute the integral

T

t (u, T )( u)du =

1

T

te (T u ) 1 (u)du

= 1

T t e (T u )(u)du 1 T t (u)duInserting (15) yields

1 T t e (T u ) f M (0, u)u + f M (0, u) du

1 T t f M (0, u)u f M (0, u)du

+2

2

T

te (T u ) 1 1 e 2u du (17)

Computing the last integral yields2

2 T t e (T u ) 1 1 e 2u du =2

231 e (T t ) +

12

e 2T e (T + t ) + 12

e 2t 2

22(T t)

We now have

T t (u, T )( u)du =1

T

t

e (T u )f M (0, u)

u du +

T

t

e (T u ) f M (0, u)du

1

f M (0, u)T t T t f M (0, u)du

+ 2

231 e (T t ) +

12

e 2T e (T + t ) + 12

e 2t

2

22(T t) (18)

Now use the integration by parts formula on the rst term on the right hand side inequation (18)

1

T

t e (T u ) f M (0, u)

u du =

1

e (T u )

f M

(0, u)

T

t

T t e (T u ) f M (0, u)duto get

T t (u, T )( u)du =1

e (T u ) f M (0, u)T

t T t e (T u )f M (0, u)du

+

T

t

e (T u ) f M (0, u)du 1

f M (0, T ) f M (0, t )

T

t

f M (0, u)du

+ 2

231 e (T t ) +

12

e 2T e (T + t ) + 12

e 2t 2

22(T t) (19)

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Now simplifying gives

T

t

(u, T )( u)du = f M (0, t ) (t, T )

T

t

f M (0, u)du

+ 2

231 e (T t ) +

12

e 2T e (T + t ) + 12

e 2t 2

22(T t) (20)

Combining the two integrals (16) and (20) we get

(t, T ) = T t (u, T )( u)du + 22 T t 2(t, T )du= f M (0, t ) (t, T ) T t f M (0, u)du

+

2

23 1 e (T t )

+

1

2e 2T

e (T + t )

+

1

2e 2t

2

22 (T t)

+ 2

22 12

1 e 2 (T t ) + ( T t) 2

1 e (T t )

Simplifying yields

(t, T ) = f M (0, t) (t, T ) T t f M (0, u)du+

2

4 2(t, T ) e 2t 1 (21)

We also have that

P (0, T ) = e T

t f (0 ,u )du

which leads to

ln P (0, T ) = T 0 f (0, u)duand thus

lnP (0, T )P (0, t )

=

T

tf (0, u)du

and we have

(t, T ) = f M (0, t ) (t, T ) + lnP (0, T )P (0, t )

+ 2

4 2(t, T ) e 2t 1 (22)

3 Solving the Stochastic Differential Equation

The solution to the SDE in equation (1) can be found by employing Itos lemma to ndthe dynamics of et r (t) and then integrating up. This yields the solution to (1)

r T = e (T t ) r t + T t e (T u )(u)du + T t e (T u )dW u

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Since EQ T t e (T u )dW u |F t = 0, we haveE Q

t [r

T ] = e (T t ) r

t +

T

te (T u )(u)du

Var Qt [r T ] = EQt r t E

Qt [r T ]

2

= 2 T t e 2 (T u )duNotice that

T t e (T u )(u)du = T t T (u, T )( u)duLiebnizs rule for differentiating integral gives

T T t (u, T )( u)du = (T, T )( T ) + T t T (u, T )( u)du

where the rst term on the right hand side is equal to zero according to (8). Thus we have

T t e (T u )(u)du = T T t (u, T )( u)duDifferentiating (20) with respect to T we get

T

te (T u )(u)du = f M (0, t )

T (t, T ) +

T T

tf M (0, u)du

2

22e (T t ) e 2T + e (T + t ) 1

= f M (0, t )e (T t ) + f M (0, T ) T t T f M (0, u) =0 2

22e (T t ) e 2T + e (T + t ) 1

Now add and subtract 2

2 2 (1 e kT )2 and simplify to get

T

te (T u )(u)du = (T ) (t)e (T t )

where

(t) = f M (0, t ) + 2

221 e kt

2

Thus the conditional expectation of the future spot rate can now be written as

E Qt [r T ] = e (T t )r t + (T ) (t)e

(T t ) (23)

And the conditional varians is given by

Var Qt = 22

1 e 2 (T t ) (24)

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4 Floaters and Caplets

In this section the Hull-White model is used to value a future stochastic rate today. We

want to compute the price of a payment received at time TA2 . This payment covers interestover the period from time T A1 to T A2 . The payment is not known until time T F1 where it is

xed as the simple rate over the period T F1 to T F2 .

TF1 TA1 TA2 TF2

The following notation is used

T F1 : Start date of xing period

T F2 : End date of xing periodT A1 : Start date of accrual periodT A2 : End date of accrual period 1 : Accrual period in years (T A2 T A1 ) 2 : Fixing period in years (T F2 T

F1 )

At time T F1 the spot LIBOR rate over the xing period is given by

L(T F1 , TF2 ) =

1 2

1P (T F1 , T F2 )

1 (25)

Since the accrual period is 1 the payment received at time T A2 is equal to 1L(T F1 , TF2 ).The value at time t of the unknown payment is given by

V (t) = E Qt e T

A2

t r u du 1L(T F1 , TF2 ) (26)

= E Qt e T

F1

t r u du 1L(T F1 , TF2 ) E

QT F1

e T

A2

T F1

r u du (27)

= E Qt e T

F1

t r u du P (T F1 , TA2 ) 1L(T

F1 , T

F2 ) (28)

Changing from the risk-neutral measure to the forward measure with the zero-couponbond maturing at time T F1 as numeraire yields

V (t) = P (t, T F1 )ET F1t P (T

F1 , T

A2 ) 1L(T

F1 , T

F2 ) (29)

which is the same as

V (t) = P (t, TF1 ) 1 2

E TF1

tP (T F1 , T A2 )P (T F1 , T F2 )

P (T F1 , TA2 )

= 1 2

P (t, T F1 )ET F1t

P (T F1 , TA2 )P (T F1 , TF2 )

P (t, T A2 ) (30)

In the Hull-White model, the ration P (T F1 , T A2 )/P (T F1 , TF2 ) is equal to

P (T F1 , T A2 )P (T F1 , T F2 )

= e (T F1 ,T

A2 ) (T

F1 ,T

F2 )+( (T

F1 ,T

A2 ) (T

F1 ,T

F2 )) r T F

1

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Which is used to get

V (t) = 1

2P (t, T F1 )e

(T F1 ,TA2 ) (T

F1 ,T

F2 ) E T

F1

t e( (T F1 ,T

A2 ) (T

F1 ,T

F2 )) r T F

1 P (t, T A2 ) (31)

The only unknown object in Equation (31) is ETF1

t e( (T F1 ,T

A2 ) (T

F1 ,T

F2 )) r T F

1 , but this ex-

pectation can easily be computed since r T F1 is normally distributed.

E TF1

t exp ( (TF1 , T

A2 ) (T

F1 , T

F2 )) r T F1 =

exp ( (T F1 , TA2 ) (T

F1 , T

F2 ))E

T F1t [r T F1 ] +

12

( (T F1 , TA2 ) (T

F1 , T

F2 ))

2Var TF1

t [r T F1 ] (32)

which can be inserted into (31) to get a closed form solution for the value of a single oatpayment.

Fixing at an Average Rate

V (t) = E Qt e T

A2

t r u du 1M

M

i=1

L(T F1i , TF2i )

= 1M

M

i=1

E Qt e T

A2

t r u du L(T F1i , TF2i )

The last expression has exactly the same form as the expression found in the previoussection and its value is thus know in closed form.

Caplet Pricing with Unnatural Time Lag We will now nd the value of a capletthat caps the simple compounded interest rate given in Equation (25). The rate is xedover the period from T F1 to T F2 but it is paid out at time T A2 .

cpl(t, T F1 , TF2 , T

A2 ) = E

Qt e

TA2

t r u du 1 L(T F1 , TF2 ) K

+

= 1P (t, T A2 )ET A2t L(T

F1 , T

F2 ) K

+

= 1 2

P (t, T A2 )ET A2t

1P (T F

1, TF

2)

(1 + 2K )+

= 1 2

P (t, T A2 )ET A2t e

(T F1 ,TF2 ) (T

F1 ,T

F2 )r T F

1 (1 + 2K )+

= 1 2

P (t, T A2 )e (T F1 ,T

F2 ) E T

A2

t e (T F1 ,T

F2 )r T F

1 e (TF1 ,T

F2 )(1 + 2K )

+

Since rT F1 (M r , V 2

r ), where M r and V 2r is the mean and variance of rT F1 respectivelyand is the standard cumulative normal distribution function, we have (T F1 , TF2 )r T F1( (T F1 , T F2 )M r , (T F1 , TF2 )2V 2r ). From Brigo and Mercurio (2001) we have the followingfor a lognormally distributed stochastic variable X with mean M and variance V

E (X K )+ = eM +12 V

2

M ln(K ) + V 2

V K

M ln(K )V

(33)

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References

Brigo, D. and F. Mercurio (2001): Interest Rate Models Theory and Practice,

Springer, rst edition.

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Computations in the Hull-White Model (Rom-Poulsen)

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