Hull White shore interest rate model slide. This note is presented in the course of Interest Rate Theory and Practice at NCCU.

1. Hull-Whi(e Short Ra(e Model
A mid-'erm presen'ation in In'erest Ra'e
Derivative Pricing Theory Course @NCCU
Participants:
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2. To begin with, we need an critical
lemma for 'oday’s presen'ation.

3. Lemma 1 -
Consider this following equation:
y =
loge( x
m sa)
r2
yr2
= loge(
x
m
sa)
Me = X-mas
rryWe try 'o solve it:
eyr2
=
x
m
sa=>
meyr2
= x msa=>

4. Intro.
01.

5. Intro.
01.
Vasicek, O., 1977, “An Equilibrium Charac'erization of
the Term Structure,” Journal of Financial Economics
Basic Assumption-
(A.3) Ins'an'aneous Short Ra'e is the only s'a'e variable for
in'erest ra'e structure under efficient market hypothesis.
That is, r(t) shows all the information inclusive of present 'erm
structure, the knowledge of whole past development of ra'es
of all maturities.
(A.1) Ins'an'aneous spot ra'e follows a diffusion process.
(Continuous Markov Process)
(A.2) Bond price depend only on spot ra'e over it 'erms.

6. Mean Reversion Property is not
a theorem, It is not a hypothesis, or
a conjecture as well. It’s a his'orical fact.
Intro.
01.
Vasicek, O., 1977, “An Equilibrium Charac'erization of
the Term Structure,” Journal of Financial Economics

7. Orns'ein Uhlenbeck process
# Elastic Random Walk
dXt = Xtdt + dWt
Intro.
01.
Vasicek, O., 1977, “An Equilibrium Charac'erization of
the Term Structure,” Journal of Financial Economics

8. Intro.
01.
Vasicek, O., 1977, “An Equilibrium Charac'erization of
the Term Structure,” Journal of Financial Economics
Under risk-neutral probability measure (Q-measure),
drt = ( rt)dt + dWQ
tthe SDE of in'erest ra'e is :
Xt = rtekt
Then, we try 'o solve this differential equation. Let
By I'o’s Lemma, we could ob'ain: dXt = ekt
drt + krtekt
dt
Thus, the solution of this SDE shows that:
rT = rte k(T t)
+ (1 e k(T t)
) +
T
t
e k(T u)
dWQ
u
Consider expec'ation under Q-measure:
EQ
(rT) = rte k(T t)
+ (1 e k(T t)
)
T ∞
EQ
(rT) =

9. Intro.
01.
Hull and Whi'e. (1990). Pricing In'erest-Ra'e-Derivative
Securities. Review of Financial Studies
Under risk-neutral probability measure (Q-measure),
the SDE of in'erest ra'e is : drt = ( t rt)dt + dWQ
t
Xt = rtekt
Then, we try 'o solve this differential equation. Let
By I'o’s Lemma, we could ob'ain: dXt = ekt
drt + krtekt
dt
Thus, the solution of this SDE shows that:
rT = rte k(T t)
+ k
T
t
ue k(T u)
du +
T
t
e k(T u)
dWQ
u
Consider expec'ation under Q-measure:
EQ
(rT) = rte k(T t)
+ k
T
t
ue k(T u)
du
T ∞
t - Rela'ed

10. Let’s 'alk about the relation among
zero coupon bond price, ins'an'aneous
forward ra'e, and the'a function.
Intro.
01.

11. Intro.
01.
Show that zero coupon bond price:
= e rtB(t,T)+ T
t
(e k(T )
1)d +
2
4k3 [2k(T t) 1 e 2k(T t)
]
= e rtB(t,T)+ T
t
(e k(T )
1)d +
2
4k2 [2(T t) 2B(t,T)+kB(t,T)2
]
A(t, T)e rtB(t,T)
B(t, T) =
1 ek
(T t)
k
A(t, T) = e
T
t
(e k(T )
1)d +
2
4k3 [2k(T t) 1 e 2k(T t)
]
Under Q measure, it is trivial for us 'o underhand that:
dP(t, T)
P(t, T)
= rtdt B(t, T)dWQ
t # By I'o’s Lemma
P(t, T) = EQ
t (e
T
t
rudu
) # By definition of P(t,T) under Ft

12. Intro.
01.
Try 'o es'ablish the relationship between ins'an'aneous
forward ra'e and A(t,T). It is easier 'o implement our work.
= f(0, t)B(t, T) + ln
P(0, T)
P(0, t)
+
2
4k
(1 e 2kt
)B2
(t, T)
t = f(0, t) +
1
k
f(0, t)
t
+
2
2k2
(1 e 2kt
)=>
f(0, t) =
lnP(0, t)
t
= r0e kt
+ k
t
0
e k(t )
d
2
2k2
(1 + e 2kt
)
# By definition of f(0,t)1
f(0, t)
t
= k t kf(0, t)
2
2k
(1 e 2kt
)
Then, consider 1st fundamen'al theorem of calculus:2
lnA(t, T) =
T
t
(e k(T )
1)d +
2
4k3
[2k(T t) 1 e 2k(T t)
]
We could ob'ain the A(t,T) 'erm:3

13. Furthermore, how 'o implement
the calibration work in our research ?
Intro.
01.

14. Empirical
02.

15. Empirical.
02. Da'a Description
1st da'a set is in'erest ra'e da'a in the period
of 1982/01/04 - 2013/12/31 from Fed Online. (7993 samples)
2nd da'a set is tradable T-bond price in US
at 2013/12/31 from Da'aStream. (221 samples)
3rd da'a set is yield ra'e curve at 2013/12/31 from
Fed Online.
We couldn’t use time series
method (o calibra(e parame(ers.

16. Empirical.
02.
S'ep1. We set ins'an'aneous forward ra'e f(t,T) as cubic polynomial:
Method 1
f(t, T) = a(T t)3
+ b(T t)2
+ c(T t) + d
S'ep2. f(t,T) is fit'ed by the 3rd da'a set, yield ra'e curve.
a b c d
polyfit 1.4061E-09 -1.2764E-06 0.0004 -0.0021
Af'er that, the polynomial of f(t,T)
is de'erministic as follow:

17. Empirical.
02.
S'ep3. Cross Sectional Calibration
Method 1
B(0, T) = c
T
t=1
P(0, t) + FP(0, T)
A coupon bond is combination of several zero coupon bonds:
Given the market prices of Treasury bonds, the parame'ers
are calibra'ed by minimizing SSE.
ˆ = arg min
n
i=1
(Bmarket
i Bmodel
i )2
n
k σ r0
Cross
Sectional
3.616E-10 0.0306 0.033

18. Empirical.
02.
S'ep4. Bonds price simulation is that :
Method 1

19. Yield Ra'e Curve can not capture the
behavior of ins'an'aneous forward.
Empirical.
02.

20. Empirical.
02.
S'ep1. We set ins'an'aneous forward ra'e f(t,T) as cubic polynomial:
Method 2
f(t, T) = a(T t)3
+ b(T t)2
+ c(T t) + d
S'ep2. Consider the definition of zero coupon bond as follow:
P(t, T) = e
T
t
f( )d = (T, t) = T twhere:
Thus, P(t,T) indica'es P(t,T,a,b,c,d).
Let’s consider coupon bond as follow:
B(0, T) = c
T
t=1
P(0, t) + FP(0, T) Hence, B(0,T) implies B(0,T,a,b,c,d).

21. Empirical.
02.
S'ep3. We could decide the coefficient of a, b, c, d by SSE condition:
Method 2
ˆ = arg min
n
i=1
(Bmarket
i Bmodel
i )2
n
Af'er that, the polynomial of f(t,T) is de'erministic as follow:
# minimize the s'andard error of mean
a b c d
polyfit 1.6129E-05 -8.2566E-04 0.0125 -0.0057
t = f(0, t) +
1
k
f(0, t)
t
+
2
2k2
(1 e 2kt
)
P(t, T) = A(t, T)e rtB(t,T)
f(t, T) = a(T t)3
+ b(T t)2
+ c(T t) + d
lnA(t, T) = f(0, t)B(t, T) + ln
P(0, T)
P(0, t)
+
2
4k
(1 e 2kt
)B2
(t, T) B(t, T) =
1 ek
(T t)
k
P(t,T) = P(t,T,k,σ)

22. Empirical.
02. Method 2
S'ep4. Once again, coupon bond price shows that
B(0, T) = c
T
t=1
P(0, t) + FP(0, T) Hence, B(0,T) = B(t,T,k,σ)
Take the cross sectional calibration condition in'o our account:
ˆ = arg min
n
i=1
(Bmarket
i Bmodel
i )2
n
# minimize the s'andard error of mean
Af'er that, we ob'ain the parame'ers:
k σ r0
Cross
Sectional
2.8414E-12 0.0112 0.2655

23. Empirical.
02. Method 2
S'ep5. Bonds price simulation is that :

24. Next, we 'end 'o calibra'e all the
parame'ers 'ogether one time.
Empirical.
02.

25. Empirical.
02. Method 3
S'ep1. We set ins'an'aneous forward ra'e f(t,T) as cubic polynomial:
f(t, T) = a(T t)3
+ b(T t)2
+ c(T t) + d
S'ep2. Consider the definition of zero coupon bond as follow:
B(0, T) = c
T
t=1
P(0, t) + FP(0, T) B(0,T,a,b,c,d,k,σ)
t = f(0, t) +
1
k
f(0, t)
t
+
2
2k2
(1 e 2kt
)
P(t, T) = A(t, T)e rtB(t,T)
lnA(t, T) = f(0, t)B(t, T) + ln
P(0, T)
P(0, t)
+
2
4k
(1 e 2kt
)B2
(t, T) B(t, T) =
1 ek
(T t)
k
where: P(0, T) = e
T
0
f( )d
& P(0, t) = e
t
0
f( )d
Therefore, under Hull Whi'e model
1
2
3

26. Empirical.
02. Method 3
S'ep3. Once again, consider cross sectional calibration condition:
ˆ = arg min
n
i=1
(Bmarket
i Bmodel
i )2
n
# minimize the s'andard error of mean
Af'er that, we ob'ain the parame'ers:
k σ r0 a b c d
Calibration 0.1905 0.0315 0.0012 1.3084E-06 6.5163E-05 -0.0033 -0.0035

27. Empirical.
02. Method 3
S'ep3. Bonds price simulation is that :

28. Eventually, method 3 is more sui'able for
calibrating these parame'ers in our work.
Empirical.
02.

29. Empirical.
02.
Calcula'e the Yield Ra'e
ˆP(t, T) = e
ˆR(t,T)(T t)
k,
We could get the model
zero-coupon bond price.
ˆP(t, T) = ˆA(t, T)e rT
ˆB(t,T)
ˆR(t, T) =
lnˆP(t, T)
T t