Discussion Papers In Economics
And Business

Pricing of a Chooser Flexible Cap
and its Calibration

Daisuke ITO

Masamitsu OHNISHI

Yasuhiro TAMBA

Discussion Paper 04-18

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPricing of a Chooser Flexible Cap
and its Calibration

Daisuke ITO

Masamitsu OHNISHI

Yasuhiro TAMBA

Discussion Paper 04-18
October 2004
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPricing of a Chooser Flexible Cap
and its Calibration
Daisuke ITO ∗†Masamitsu OHNISHI ‡Yasuhiro TAMBA §Abstract
In this paper, we deal with no-arbitrage pricing problems of a chooser flexible
cap (floor) written on an underlying LIBOR. The chooser flexible cap (floor) allows
a right for a buyer to exercise a limited and pre-determined number of the interim
period caplets (floorlets) in a multiple-period cap (floor) agreement. Assuming acommon diffusion short rate dynamics, e.g., Hull–White model, we propose a dynamic programming approach for their risk neutral evaluation. This framework issuited to a calibration from an observed initial yield curve and market price data of
discount bonds, caplets, and floorlets.
Keywords: chooser flexible cap, LIBOR, dynamic programming, Hull–White model,
calibration.
JEL classification: G13, G15, G21∗
This paper is partially based on the master thesis of the first author in the Graduate School ofEngineering Science, Osaka University. The authors would like to thank Dr. Hideo Nagai, Dr. Shigeki
Aida and Dr. Jun Sekine for helpful suggestions and comments.
†
Sumitomo Mitsui Bank‡
Graduate School of Economics, Osaka University, 1–7 Machikaneyama–machi, Toyonaka, Osaka 560–
0043, Japan, and Daiwa Securities Chair, Graduate School of Economics, Kyoto University, Yoshida–honmachi, Sakyo–ku, Kyoto 606–8501, Japan; E-mail: ohnishi@econ.osaka-u.ac.jp§
Graduate School of Economics, Osaka University, 1–7 Machikaneyama–machi, Toyonaka, Osaka 5600043, Japan; E-mail: tambayasuhiro@srv.econ.osaka-u.ac.jp

1

Introduction

Various types of exotic interest rate derivatives have appeared since the middle of 1990s.
One of the most traded interest rate derivatives is a Bermudan swaption. The Bermudanswaption is an option, which at each date in a schedule of exercise dates gives the holder
the right to enter an interest swap, provided that this right has not been exercised at any
previous time in the schedule. Because of its usefulness as hedges for callable bonds, theBermudan swaption is probably the most liquid interest rate instrument with a built-in
early exercise feature. There is another type of exotic interest rate derivatives, a chooser
flexible cap. The chooser flexible cap is a financial contract which allows to a buyer
the right to exercise dynamically at most l (1 ≤ l ≤ N ) out of N caplets whose i–th one(i = 0, · · · , N −1) is written on the LIBOR whose setting time and payment time are T i andTi+1 , respectively. The chooser flexible cap is less traded than the Bermudan swaption andmore traded in Europe than other countries. Although there are some different features
between the Bermudan swaption and the chooser flexible cap, these are the same type of
instruments in terms of having the early exercise feature. There are mainly two merits inthe chooser flexible cap compared to a more popular cap. The first merit is that its priceis cheaper than the cap’s because the chooser flexible cap has less exercise opportunities
than the cap. So an option holder can hedge the interest rate risk with the lower costthan the cap. The second merit is its flexibility. The holder of the chooser flexible cap canreflect her/his expectation of the future interest rate change in her/his hedging strategy.
For example, if s/he has the expectation that the floating interest rate will increase in thenext one year, s/he should exercise the option mainly in one year as s/he observes the realinterest rate values. So using the chooser flexible cap, the option holder can flexibly hedge
the interest rate risk with the low cost.
There are a lot of papers for pricing the Bermudan swaption because it is popular in themarket. The proposed pricing method of most papers is a Monte Carlo simulation becauseof its simplicity and applicability for a multifactor model with long maturity instruments.
In spite of its usefulness, there are some weaknesses of the Monte Carlo simulation to be
applied for pricing exotic derivatives. The most important drawback is that the Monte
Carlo simulation has difficulty in dealing with derivatives that contain early exercise features, like an American option and the Bermudan option. So we need some improvementof the Monte Carlo method for pricing the above early exercise derivatives. Longstaff
and Schwartz (1998) developed the least square method to overcome the weakness of theMonte Carlo simulation for pricing American option. Andersen (1999) used the Monte
Carlo simulation for pricing the Bermudan swaption, and derived a lower bound on the
Bermudan swaption prices considering less advantage strategies. Broadie and Glasserman(1997a, 1997b) also coped with the difficulties of the early exercise problem of the Monte
Carlo simulation.
Though the Monte Carlo simulation may be used for pricing the chooser flexible capwith some improvement of the method, there are different features between the Bermudanswaption and the chooser flexible cap. While the number of the exercise opportunity of
the Bermudan swaption is only one, the chooser flexible cap’s is generally greater than
two. Another difference is that while payoffs follow sequentially after the exercise in
the case of the Bermudan swaption, in the case of the chooser flexible cap the option
holder needs to choose exercise time dates, that is, the payoff dates. These different
features of the instruments affect a choice of pricing methods applied. The dynamic
programming approach may be more appropriate than the Monte Carlo simulation to
price the chooser flexible cap. The research of the chooser flexible cap pricing has been1

only seen in Pedersen and Sidenius (1998) utilizing an optimality equation of the dynamic
programming approach. This method is rather appropriate for pricing exotic derivatives
because we can solve the problem by a backward induction setting up a recombining tree.
Although Pedersen and Sidenius (1998) made an important contribution in terms of using
the optimality equation for the pricing, they did not describe much about a calibration.
The other feature of their paper is that they used the spot LIBOR for the tree construction.
In this paper we focus on the pricing problem of exotic interest rate derivatives, inparticular the chooser flexible cap and the chooser flexible floor, based on the observed
market prices of rather simple interest rate derivatives, e.g., caplets and the floorlets,
etc. We extend the pricing method in some points compared with Pedersen and Sidenius(1998). One of them is that deriving the theoretical prices of bond, the caplet and thefloorlet we use these prices for a calibration. Another difference is that we use the short
rate for the tree construction instead of the LIBOR. This point has advantage in that the
short rate model like Hull–White has some convinced features matching the interest rate
movement in the real world such as a mean reversion property. We also show numericalexamples and comparative statics.
The paper is organized as follows. In Section 2, we introduce the various notations
about interest rate. In Section 3, we derive theoretical no-arbitrage prices of discountbonds in an affine term structure. In Section 4 and 5, we discuss the pricing methodof the chooser flexible cap and the chooser flexible floor with the optimality equations.
Section 6 shows how to calculate theoretical no-arbitrage prices of the caplet and thefloorlet. Section 7 is devoted to a calibration of model parameters. In Section 8, we
describe a construction of a trinomial tree for the chooser flexible cap pricing. In Section
9, we show numerical examples and discuss comparative statics. Section 10 concludes thepaper.

2

Various Notations about Interest Rates

In this section we explain notations of various types of interest rates.
Let D(t, T ) 0 ≤ t ≤ T ≤ T ∗ be the time t price of the discount bond (or zero–couponbond) with maturity T , in brief T –bond, which pays 1–unit of money at the maturity T(where D(T, T ) = 1 for any T ∈ T∗ ). For 0 ≤ t ≤ S < T ≤ T ∗ ,ln D(t, T ) − ln D(t, S)
D(t, S)
R(t; S, T ) := −⇐⇒ exp{R(t; S, T )(T − S)} =(1)
T −S
D(t, T )is the (continuous compounding based) forward rate prevailing at time t which covers timeinterval (S, T ]. For 0 ≤ t < T ≤ T ∗ ,11
Y (t, T ) := R(t; t, T ) = −ln D(t, T ) ⇐⇒ exp{Y (t, T )(T − t)} =(2)
T −t
D(t, T )is the (continuous compounding based) spot rate prevailing at current time t or yield which
covers time interval (t, T ]. The map T 7→ Y (t, T ) is called the yield curve at time t. For
0 ≤ t ≤ T ≤ T ∗,f (t, T ) := lim R(t; T, U ) = −
U ↓T

2

∂
ln D(t, T )∂T(3)

is the (instantaneous) forward rate prevailing at current time t with the maturity time T .
The map T 7→ f (t, T ) is called the forward rate curve at time t. For t ∈ T ∗ ,r(t) := f (t, t) = lim Y (t, T ) = −T ↓t∂ln D(t, T )∂T
T =tis the short rate at time t. For 0 ≤ t ≤ T ≤ T ∗ ,Z T
B(t, T ) := exp
r(s)ds

(4)

(5)

t

is the risk–free bank account at time T with unit investment capital at time t (whereB(t, t) = 1).
For N ∈ Z+ , let
0 ≤ T0 < T1 < · · · < Ti < Ti+1 < · · · < TN −1 < TN ≤ T ∗(6)be the sequence of setting times and payment times of floating interest rates, that is, fori = 0, · · · , N − 1, the floating interest rate which covers time interval (T i , Ti+1 ], is set attime Ti and paid at time Ti+1 . For convenience, we let
Ti+1 − Ti = δ (= constant ∈ R++ ),i = 0, · · · , N − 1.

(7)

For i = 0, · · · , N − 1, we define the simple (or simple compounding based) interest rate
which covers time interval (Ti , Ti+1 ] by


1
1
−1 .
(8)
LTi (Ti ) :=
δ D(Ti , Ti+1 )This amount is set at time Ti , paid at time Ti+1 , and is conventionally called as a spot
LIBOR (London Inter–Bank Offer Rate). For i = 0, · · · , N − 1,1
D(t, Ti )LTi (t) :=−1(9)
δ D(t, Ti+1 )is the simple (or simple compounding based) interest rate prevailing at time t (∈ [0, T i ])which covers time interval (Ti , Ti+1 ], and is called as a forward LIBOR.
The i–caplet is a financial contract in which, at time Ti+1 , the seller pays to the buyer
the amount of money corresponding the difference between the spot LIBOR L Ti (Ti ) anda predetermined upper–limit exercise rate K (∈ R) if the former exceeds the latter:δ[LTi (Ti ) − K]+ (= δ max{LTi (Ti ) − K, 0}) .

(10)

It could be considered as a call option written on the underlying LIBOR for hedging its
upside risk. The i–floorlet is a financial contract in which, at time Ti+1 , the seller pays
to the buyer the amount of money corresponding the difference between the spot LIBOR
LTi (Ti ) and a predetermined lower–limit exercise rate K (∈ R) if the former falls below
the latter:δ[K − LTi (Ti )]+ (= δ max{K − LTi (Ti ), 0}) .

(11)

It could be considered as a put option written on the underlying LIBOR for hedging its
downside risk.
3

3

No–Arbitrage Prices of Discount Bonds

3.1

Affine Term Structure Model

We consider a continuous trading economy with a finite time horizon given by T ∗ := [0, T ∗ ](T ∗ ∈ R++ ). The uncertainty is modelled by a filtered probability space (Ω, F , P; F). In
this notation, Ω denotes a sample space with elements ω ∈ Ω; F denotes a σ-algebra onΩ ; and P denotes a probability measure on (Ω, F ). The uncertainty is resolved over T ∗according to a 1–dimensional Brownian (motion) filtration F := (F (t) : t ∈ T∗ ) satisfying
the usual conditions. W := (W (t) : t ∈ T∗ ) denotes a 1–dimensional standard (P; F)–
Brownian motion.
Consistent with the no-arbitrage and complete market paradigm, we assume the existence of the risk neutral equivalent martingale measure P∗ with a bank account as anuméraire in this economy. Assume that, under the risk neutral probability measure P ∗ ,the short rate process r = (r(t) : t ∈ T∗ ) follows a diffusion type (Markovian) SDE:
dr(t) = µ(r(t), t)dt + σ(r(t), t)dW ∗ (t),t ∈ T∗ ,(12)

where the drift coefficient µ(r(t), t) and the diffusion coefficient σ(r(t), t) are deterministic
functions of r and t with suitable regularity conditions. Since the short rate processr = (r(t) : t ∈ T∗ ) is Markovian, we have, by the risk neutral evaluation method,
1∗
D(t, T ) = E
F (t)B(t, T )

 Z T


∗
= E exp −r(s)ds r(t)t=: D(t, T ; r(t)),

0 ≤ t ≤ T ≤ T ∗.

(13)

In equation (12), we assume that the drift coefficient function µ(r(t), t) and the diffusion
coefficient function σ(r(t), t) have the forms:µ(r(t), t) = m0 (t) + m1 (t)r(t);σ 2 (r(t), t)= s0 (t) + s1 (t)r(t).
2

(14)
(15)

Then, at time t, no–arbitrage price D(t, T ; r(t)) of T –bond has the following the Affine
Term Structure (ATS), see Filipović (2002):
D(t, T ; r(t)) = exp{−a(t, T ) − b(t, T )r(t)},0 ≤ t ≤ T ≤ T ∗,(16)

where a(t, T ) and b(t, T ) are solution functions of the following system of simultaneous
differential equations:∂a(t, T ) = −m0 (t)b(t, T ) + s0 (t){b(t, T )}2 ;∂t∂b(t, T ) = −1 − m1 (t)b(t, T ) + s1 (t){b(t, T )}2
∂t

(17)
(18)

with the terminal conditions:a(T, T ) = 0,b(T, T ) = 0.
4(19)

In this case, the forward rate satisfies∂ln D(t, T ; r(t))∂T∂∂a(t, T ) + r(t) b(t, T ),∂T∂Tf (t, T ; r(t)) := −=3.2

0 ≤ t ≤ T ≤ T ∗.

(20)

Hull–White Model

The Hull–White model, Hull and White (1990), is one of the most popular short rate
model with the ATS feature in practice because it has desirable characters of the interest
rate such a mean reversion property. It can also be fitted with an observable initial term
structure. The Hull–White model assumes that, under the risk neutral probability measureP∗ , the short rate process r = (r(t) : t ∈ T∗ ) satisfies the following special form of SDEwith the ATS property:
dr(t) = {α(t) − βr(t)}dt + σdW ∗ (t),t ∈ T∗ ,(21)

that is,m0 (t) = α(t),σ2s0 (t) = ,2

m1 (t) = −β;(22)s1 (t) = 0.

(23)

Under the Hull–White model, the time t price of T –bond can be explicitly derived asfollows:
D(t, T ; r(t)) = exp{−a(t, T ) − b(t, T )r(t)},0 ≤ t ≤ T ≤ T ∗,

(24)

where
Z
Z T
σ2 T
2
a(t, T ) = −{b(s, T )} ds +α(s)b(s, T )ds;
2 tt
1 − e−β(T −t)b(t, T ) =
.
β

(25)
(26)

Then, the initial forward rate can be derived explicitly as:∂∂a(0, T ) + r(0) b(0, T )
∂T
∂T
Z T
σ 2 −βT∂∂2= − 2 (e− 1) +
α(s) b(s, T )ds + r(0) b(0, T ),
2β
∂T
∂T
0

f (0, T ; r(0)) =0 ≤ T ≤ T ∗.

(27)

4

Pricing the Chooser Flexible Cap

4.1

Pricing the Chooser Flexible Cap under the Risk Neutral
Probability P∗

The chooser flexible cap is a financial contract which allows to a buyer the right to exercise
dynamically at most l (1 ≤ l ≤ N ) out of N caplets whose i–th one (i = 0, · · · , N − 1) is
5

written on the LIBOR whose setting time and payment time are Ti and Ti+1 , respectively.
Let W (Ti , r(Ti ), l), i = 0, · · · , N − 1, l = 1, · · · , M be the fair (no–arbitrage) price of thechooser flexible cap when, at time Ti , at most l exercises are remained to the buyer. The
optimal equation can be derived by using the Bellman Principle. We can use the short
rate as the state variable instead of the LIBOR. This is possible because the LIBOR is theincreasing function in the short rate and these are one to one relation as we see later inequation (40). In this subsection, we derive the optimality equation under the risk neutral
probability measure P∗ with a bank account as a numéraire.
Optimality Equation:(i) For i = N − 1 (Terminal Condition):W (TN −1 , r(TN −1 ), l) = D(TN −1 , TN )δ[LTN −1 (TN −1 ) − K]+ ,l = 1, · · · , M ;(28)(ii) For i = N − 2, · · · , 0:nh W (T , r(T ), l − 1)
i
i+1
i+1
r(Ti ) ,W (Ti , r(Ti ), l) = max D(Ti , Ti+1 )δ(LTi (Ti ) − K)+ + E∗
B(Ti , Ti+1 )h W (T , r(T ), l)
io
i+1
i+1
E∗
r(Ti ) , l = 1, · · · , M,
(29)
B(Ti , Ti+1 )where
W (Ti , r(Ti ), 0) = 0,

4.2

i = 0, · · · , N − 1.

(30)

Pricing the Chooser Flexible Cap under the Forward Neutral
Probability PTN

We can also derive the optimality equation under the forward neutral probability P TN witha TN –bond as a numéraire.
Optimality Equation:(i) For i = N − 1 (Terminal Condition):W (TN −1 , r(TN −1 ), l) = D(TN −1 , TN )δ[LTN −1 (TN −1 ) − K]+ ,l = 1, · · · , M ;(31)(ii) For i = N − 2, · · · , 0:nW (Ti , r(Ti ), l) = max D(Ti , Ti+1 )δ(LTi (Ti ) − K)+hi
TN W (Ti+1 , r(Ti+1 ), l − 1)r(Ti ) ,+D(Ti , TN )E
D(Ti+1 , TN )hio
TN W (Ti+1 , r(Ti+1 ), l)D(Ti , TN )Er(Ti ) ,D(Ti+1 , TN )l = 1, · · · , M,

(32)

where
W (Ti , r(Ti ), 0) = 0,

i = 0, · · · , N − 1.
6

(33)

4.3

Pricing the Chooser Flexible Cap under Varying Forward
Neutral Probabilities PTi (1 ≤ i ≤ N)

In this subsection we write the optimality equation under forward neutral probabilities
PTi varying at each period with a Ti –bond as a numéraire. This optimality equation isdifferent from the both optimality equations of Subsection 4.1 and 4.2 that have the fixedprobability measures at all periods.
Optimality Equation:(i) For i = N − 1 (Terminal Condition):W (TN −1 , r(TN −1 ), l) = D(TN −1 , TN )δ[LTN −1 (TN −1 ) − K]+ ,l = 1, · · · , M ;(34)(ii) For i = N − 2, · · · , 0:nW (Ti , r(Ti ), l) = max D(Ti , Ti+1 )δ(LTi (Ti ) − K)+l = 1, · · · , M,

+D(Ti , Ti+1 )ETi+1 [W (Ti+1 , r(Ti+1 ), l − 1)|r(Ti )],oD(Ti , Ti+1 )ETi+1 [W (Ti+1 , r(Ti+1 ), l)|r(Ti )] ,

(35)

where
W (Ti , r(Ti ), 0) = 0,

5i = 0, · · · , N − 1.

(36)

Pricing the Chooser Flexible Floor

The chooser flexible floor is a financial contract which allows to a buyer the right to exercise
dynamically at most l (1 ≤ l ≤ N ) out of N floorlets whose i–th one (i = 0, · · · , N − 1) iswritten on the LIBOR whose setting time and payment time are Ti and Ti+1 , respectively.
Let V (Ti , r(Ti ), l), i = 0, · · · , N − 1, l = 1, · · · , M be the fair (no–arbitrage) price of thechooser flexible floor when, at time Ti , at most l exercises are remained to the buyer.
We derive the optimality equation for the chooser flexible floor under the risk neutral
probability measure P∗ by the same way as Subsection 4.1. The optimality equations
under PTN and PTi can be obtained by the same ways as Subsection 4.2 and 4.3.
Optimality Equation:(i) For i = N − 1 (Terminal Condition):V (TN −1 , r(TN −1 ), l) = D(TN −1 , TN )δ[K − LTN −1 (TN −1 )]+ ,l = 1, · · · , M ;(37)(ii) For i = N − 2, · · · , 0:nh V (T , r(T ), l − 1)
i
i+1
i+1
V (Ti , r(Ti ), l) = max D(Ti , Ti+1 )δ(K − LTi (Ti ))+ + E∗r(Ti ) ,B(Ti , Ti+1 )h V (T , r(T ), l)
io
i+1
i+1
E∗
r(Ti ) , l = 1, · · · , M,
(38)
B(Ti , Ti+1 )where
V (Ti , r(Ti ), 0) = 0,

i = 0, · · · , N − 1.
7

(39)

6

No–Arbitrage Prices of the Caplet and the Floorlet

To calculate the price of the chooser flexible cap using the Hull–White model, we needto decide the values of the model parameters. The parameters are decided as observable
simple derivatives (the caplet and the floorlet) in the market fit theoretical prices derived
by the model. So, for the calibration we need to derive theoretical prices of the caplet andthe floorlet based on the Hull–White model.
We assume that the short rate r(t) follows the Hull–White model (21). From theresults (24), (25) and (26) the forward LIBOR LTi (t) can be represented as
D(t, Ti ) − D(t, Ti+1 )δD(t, Ti+1 )
n
o
n
o
1 1
= − + exp − a(t, Ti ) + a(t, Ti+1 ) exp − b(t, Ti ) + b(t, Ti+1 ) r(t).(40)δ δLTi (t) :=Definingno1
exp − a(t, Ti ) + a(t, Ti+1 ) ,δh(t, Ti , Ti+1 ) := −b(t, Ti ) + b(t, Ti+1 ),1LTi (t) := LTi (t) + ,δ

g(t, Ti , Ti+1 ) :=

(41)
(42)
(43)

we can represent LTi (t) as
LTi (t) = g(t, Ti , Ti+1 )eh(t,Ti ,Ti+1 )r(t) .

(44)

From the Itô’s Lemma LTi (t) follows the SDE
n g(t, T , T )dLTi (t)ii+1 t=+ h(t, Ti , Ti+1 )t r(t) + h(t, Ti , Ti+1 )(α(t) − βr(t))
g(t, Ti , Ti+1 )LTi (t)o
12 2+ h(t, Ti , Ti+1 ) σ dt
2+h(t, Ti , Ti+1 )σdW ∗ (t),(45)
where g(t, Ti , Ti+1 )t and h(t, Ti , Ti+1 )t are partial differentials of g(t, Ti , Ti+1 ) and h(t, Ti , Ti+1 )with respect to t. g(t, Ti , Ti+1 )t and h(t, Ti , Ti+1 )t can be calculated asg(t, Ti , Ti+1 )t =h(t, Ti , Ti+1 )th σ2 noib(t, Ti ) + b(t, Ti+1 ) − α(t) h(t, Ti , Ti+1 )g(t, Ti , Ti+1 );2= h(t, Ti , Ti+1 )β.

(46)
(47)

Hence, we obtainh h(t, T , T )σ 2 no 1idLTi (t)ii+1=b(t, Ti ) + b(t, Ti+1 ) + h(t, Ti , Ti+1 )2 σ 2 dt + h(t, Ti , Ti+1 )σdW ∗ (t)22
LTi (t)hn σ  1oi= h(t, Ti , Ti+1 )σb(t, Ti ) + b(t, Ti+1 ) + h(t, Ti , Ti+1 )σ dt + dW ∗ (t)
n 2
o 2
∗
= h(t, Ti , Ti+1 )σ σb(t, Ti+1 )dt + dW (t) .
(48)
8

Changing the probability measure bydW Ti+1 (t) = σb(t, Ti+1 )dt + dW ∗ (t),(49)

we obtain that LTi (t) follows the SDE
dLTi (t)= h(t, Ti , Ti+1 )σdW Ti+1 (t),LTi (t)0 ≤ t ≤ Ti ,(50)

where W Ti+1 (t) is the Brownian Motion under the forward neutral probability PTi+1 .
By utilizing the risk neutral evaluation method, we know that the fair (no–arbitrage)
price of i–caplet at time t (∈ [0, Ti ]), CplTi (t), is given by the expectation under the risk
neutral probability measure P∗ :∗ δ[LTi (Ti ) − K]+
CplTi (t) = E
F (t) , 0 ≤ t ≤ Ti .
(51)
B(t, Ti+1 )So under the forward neutral probability measure PTi+1 we can show it as
CplTi (t) = δD(t, Ti+1 )ETi+1 [(LTi (Ti ) − K)+ |F (t)]= δD(t, Ti+1 )ETi+1 [(LTi (Ti ) − K 0 )+ |F (t)],0 ≤ t ≤ Ti .

(52)

Using the same calculation as the Black-Scholes Formula, we obtain
CplTi (t) = δD(t, Ti+1 ) LTi (t)Φ(d(t, Ti )) − K 0 Φ(d(t, Ti ) − ν(t, Ti )) ,(53)

where LTi (t), Φ(d(t, Ti )), K 0 , d(t, Ti ), and ν(t, Ti ) are defined as follows:1
LTi (t) := LTi (t) + ,
δ
Z d
1
2
Φ(d(t, Ti )) := √e−x /2 dx,
2π −∞1
K 0 := K + ,δln LTi (t)/K 0
ν(t, Ti )d(t, Ti ) :=+,ν(t, Ti )2σν(t, Ti ) := p[1 − 2e−β(Ti+1 −Ti ) + e−2β(Ti+1 −Ti ) − e−2β(Ti −t)32β+2e−β(Ti+1 +Ti −2t) − e−2β(Ti+1 −t) ]1/2 .

(54)
(55)
(56)
(57)

(58)

Similarly, by the risk neutral evaluation method, the fair (no–arbitrage) price F ll Ti (t) of
i–floorlet at time t (∈ [0, Ti ]) is given by:∗ δ[K − LTi (Ti )]+F llTi (t) = E
F (t) , 0 ≤ t ≤ Ti ,(59)
B(t, Ti+1 )so that
F llTi (t) = δD(t, Ti+1 )ETi+1 [(K 0 − LTi (Ti ))+ |F (t)],0 ≤ t ≤ Ti .

(60)

To calculate the theoretical price of the floorlet, we derive the parity relation of the caplet
and the floorlet. From the relation9(x − K)+ − (K − x)+ = x − K,(61)

we take the expectation of both sides under the forward neutral probability P Ti and multiply both sides by δD(t, Ti+1 ). Then we have
LHS =
=
RHS =
=
=

δD(t, Ti+1 )ETi+1 [(LTi (Ti ) − K)+ |F (t)] − δD(t, Ti+1 )ETi+1 [(K − LTi (Ti ))+ |F (t)]CplTi (t) − F llTi (t),
(62)
Ti+1
δD(t, Ti+1 )E [(LTi (Ti ) − K)|F (t)]δD(t, Ti+1 )ETi+1 [(LTi (Ti ) − K 0 )|F (t)]δD(t, Ti+1 )(LTi (t) − K 0 ).
(63)

Using the parity relation
CplTi (t) − F llTi (t) = δD(t, Ti+1 )(LTi (t) − K 0 ),(64)

we can calculate the floorlet price as follows:
F llTi (t) =
=
=
=

7

CplTi (t) − δD(t, Ti+1 )(LTi (t) − K 0 )δD(t, Ti+1 ){LTi (t)Φ(d(t, Ti )) − K 0 Φ(d(t, Ti ) − ν(t, Ti )) − (LTi (t) − K 0 )}
δD(t, Ti+1 ){−LTi (t)(1 − Φ(d(t, Ti ))) + K 0 (1 − Φ(d(t, Ti ) − ν(t, Ti ))}δD(t, Ti+1 ){−LTi (t)Φ(−d(t, Ti )) + K 0 Φ(ν(t, Ti ) − d(t, Ti ))}.
(65)

Calibration

From the results of Section 6 unknown parameters α(·), β, σ could be estimated as follows.
In the Hull–White model by solving the theoretical value formula of the forward rate:∂∂a(0, T ) + r(0) b(0, T )
∂T
∂T
Z T
σ 2 −βT∂∂2= − 2 (e− 1) +
α(s) b(s, T )ds + r(0) b(0, T ),
2β
∂T
∂T
0

f (0, T ; r(0)) =0 ≤ T ≤ T ∗,(66)

with respect to the function α(·), we have




∂
σ 2 −βT
σ 2 −βT22α(T ) =f (0, T ) + 2 (e− 1) − β f (0, T ) + 2 (e− 1) ,∂T
2β
2β0 ≤ T ≤ T ∗.
(67)
If we let the observed forward rate curve at current time 0 as
fmkt (0, T ),0 ≤ T ≤ T ∗,(68)

then the next includes unknown parameters β and σ, only:




∂
σ 2 −βT
σ 2 −βT22αmkt (T ) :=fmkt (0, T ) + 2 (e− 1) − β fmkt (0, T ) + 2 (e− 1) ,∂T
2β
2β0 ≤ T ≤ T ∗.
(69)
10

Accordingly, the remaining unknown parameters β and σ could be determined, for example, by the minimizers of the following criterion function:X
C(β, σ) := w1|D(t, Ti )mkt − D(t, Ti )mdl |2
i

+w2

X

|CplTi (t)mkt − CplTi (t)mdl|2+w3

X|F llTi (t)mkt − F llTi (t)mdl |2 ,

i

i

(70)

where D(t, Ti )mkt , CplTi (t)mkt , F llTi (t)mkt : the observed prices of i–th discount bond,caplet, and floorlet, respectively; D(t, Ti )mdl , CplTi (t)mdl , F llTi (t)mdl : the theoretical prices
of i–th discount bond, caplet, and floorlet, respectively; w1 , w2 , w3 (∈ R+ ) (w1 +w2 +w3 =1): weighting coefficients. We can take weighting coefficients in other ways like taking
summation after we put weighting coefficients on each difference between the market
data and the theoretical price of each i-th instrument. We can also use other financial
instruments for this calibration.
The real processes of financial asset prices (bond, simple interest derivatives, etc.)
and financial variables (yield, forward rate, LIBOR, etc.) are governed by the original
market probability measure P (Observation Measure). However, in order to price exotic
interest rate derivatives, we need the probability laws of the above financial assets and/or
variables under the risk neutral probability measure P∗ . These measures are not directly
nor obviously related but only partially related. Roughly speaking, we can not fully infer
the probability laws under the risk neutral probability measure P∗ from the historical
data of the above financial assets and variables by utilizing various tools of statistical
data analyses. Instead, we must learn from the observed market prices of related financial
assets, in which information about the probability laws under the risk neutral probability
measure P∗ are implicitly involved, that is a calibration. In Financial Engineering, various
peculiar methodologies have been developed and accumulated.

8

Construction of the Trinomial Tree for Chooser
Flexible Cap Pricing

Using a trinomial tree of the short rate r(t) that follows the Hull–White model, we construct the trinomial tree of the short rate and the chooser flexible cap price in the discrete
time setting under the risk neutral probability P∗ in this section. We can also construct
the trees under the forward neutral probabilities PTN and PTi as the same way. We construct the short rate trinomial tree based on Hull–White (1994). For the simplicity of thecalculation, we set ∆t = δ = Ti+1 − Ti as constant for all i. (i, j) represents a node t = i∆t
and r = j∆r. We define D(i, i + 1, j) as the (i + 1)∆t-bond price, L(i, i + 1, j) as the
LIBOR with the payment time (i + 1)∆t, and ri,j as the short rate value at the node (i, j).

11In the case of the Hull–White model from (25) and (26) we haveZ
Z Ti+1σ 2 Ti+12
a(Ti , Ti+1 ) = −{b(s, Ti+1 )} ds +α(s)b(s, Ti+1 )ds
2 Ti
Ti
Z
Z Ti+1σ 2 Ti+12∼{b(s, Ti+1 )} ds + αib(s, Ti+1 )ds
= −
2 Ti
Ti
2
1 −2β(Ti+1 −Ti )
3 
σ2 
e
−
= − 2 (Ti+1 − Ti ) + e−β(Ti+1 −Ti ) −
2β
β
2β
2β


1
+αi 2 β(Ti+1 − Ti ) + e−β(Ti+1 −Ti ) − 1β=: a(i, i + 1),
1 − e−β(Ti+1 −Ti )=: b(i, i + 1).
b(Ti , Ti+1 ) =
β

(71)
(72)

Then the bond price D(i, i + 1, j) and the LIBOR L(i, i + 1, j) are represented respectivelyasD(i, i + 1, j) = e−a(i,i+1)−b(i,i+1)ri,j ,(73)
1
1
L(i, i + 1, j) = − +.
δ δD(i, i + 1, j)(74)

The bank account B(Ti , Ti+1 ) with r(t) transiting from a node (i, j) to a node (i + 1, j 0 )can be approximated utilizing the trapezoidal rule asn Z Ti+1onr + roi,ji+1,j 0B(Ti , Ti+1 ) = exp
r(s)ds ∼
exp
δ
.
=
2
Ti
(75)
We define pu (i, j), pm (i, j), pd (i, j) as the transition probabilities from the node (i, j) toup, same, and down states at t = (i + 1)∆t respectively, W (i, j, l) as the chooser flexible
cap price at the node (i, j) with the exercise opportunity l, and W (i + 1, j + 1, l), W (i +1, j, l), W (i + 1, j − 1, l) as the chooser flexible cap prices at each state, j + 1, j, andj − 1, at time (i + 1)∆t with above corresponding transition probabilities from (i, j), andBu (i, j), Bm (i, j), Bd (i, j) as the bank account values at each state, j + 1, j, and j − 1, attime (i + 1)∆t with above corresponding transition probabilities from (i, j). The chooser
flexible cap price with the exercisable number l(1 ≤ l < N − i) at (i, j) , W (i, j, l), can bederived asn
W (i + 1, j + 1, l − 1)
W (i, j, l) = max D(i, i + 1, j)δ(L(i, i + 1, j) − K)+ + pu (i, j)
Bu (i, j)
W (i + 1, j, l − 1)
W (i + 1, j − 1, l − 1)+pm (i, j)+ pd (i, j),
Bm (i, j)
Bd (i, j)
W (i + 1, j + 1, l)
W (i + 1, j, l)W (i + 1, j − 1, l) opu (i, j)+ pm (i, j)+ pd (i, j).
Bu (i, j)
Bm (i, j)
Bd (i, j)(76)
By the backward induction from the terminal condition, we can finally calculate the current
chooser flexible cap price at (0, 0).
129Numerical Examples

9.1

Construction of the Trinomial Tree of the Short Rate andthe Chooser Flexible Cap

Following Hull and White (1994), we construct the trinomial tree of the short rate in
Figure 1 which fits the term structure with a gently increasing initial yield curve, where
we set the parameters as β = 0.3 and σ = 0.01. We also set the time interval δ = 0.25
for both of the short rate and the spot LIBOR in this example. But we can set different
time intervals for the tree construction of each rate. For example, we can set the shortertime interval for the short rate tree than the spot LIBOR. We make a programming forthe calculation with MATLAB. We also construct the trinomial tree with K = 0.7 andl = 1 in Figure 2 and l = 2 in Figure 3.

9.2

Comparative Statics

In this subsection, we discuss how each parameter and an initial condition affect the priceof the chooser flexible cap. The Table 1 shows the chooser flexible cap price calculatedwith only one parameter changed where other parameters and initial conditions are fixed
on the bench mark values, β=0.3, R : increasing 0.0025 in each period starting from 0.1,
σ=0.01, N =8, T =2, δ=0.25, K=0.8, and l=2.
The result shows that the chooser flexible cap price is larger as l(Exercise opportunity)
is bigger. This is because the payoff increases as the option holder has more exercise
opportunities. The chooser flexible cap price is larger as β(Parameter of HW model) isbigger. The value of β affects the level and speed of the mean reversion of the short rate.
The larger β is, the smaller mean reversion level is. This keeps the values of the short rateand LIBOR small, and causes the small chooser flexible cap price. The chooser flexible
cap price is larger as σ(Parameter of HW model) is bigger. The reason is the same as theresult of the simple Black-Scholes formula. As σ is bigger, the value of the instruments,
with which we can hedge the floating interest rate risk, is highly evaluated by buyers of
the option. The chooser flexible cap price is smaller as K(Exercise rate) is bigger. This
result is caused by the payoff function, δ(LTi (Ti ) − K)+ , at each period. As the valueof K increases, the payment value gets smaller. The chooser flexible cap price is larger
as T (Option maturity) is bigger. Because we can replicate the chooser flexible cap withthe short maturity by the longer one, the longer one is at least more expensive than the
shorter one. Finally, we examine six patterns of the initial yield curve such as increasing
curves and decreasing curves at different yield value levels. Among the six initial yield
curve examples, the price of the case of R5(the increasing, 0.01 at each period, curvefrom 0.1) is most expensive, and the prices of decreasing and flat yield curve are zeroin this example. But the prices with other parameter set cases do not necessarily resultin the same. Although the increasing initial yield curves cause the bigger values of the
spot LIBOR than the decreasing one, the bigger values cause not only the bigger valuesof payments but also the bigger values of the discount bonds, which are the discountvalues used on the backward calculation. As we calculate the values of the chooser flexible
cap backwardly, the chooser flexible cap values are discounted more in the case of theincreasing case than the decreasing case. As the result, the price differences between theincreasing and decreasing cases get smaller and smaller on the backward calculation. In
some cases, the option price with the decreasing yield curve may be more expensive than
the increasing case.
13

i

0

1

2

3

4

j
3
0.14100.15270.12730.13842

0.11370.1242

1

0

0.1
0.11

0.09630.1070-1

0.11000.12120.10140.1126
0.09270.1040-2-30.10500.11560.11870.1298ri,jLTi (Ti )0.13230.14410.12370.13540.11500.12680.10640.1182
0.09770.10960.08900.10110.60010.7036
0.59150.69380.58280.68410.57410.67430.56550.6646
0.55680.65480.54820.645150.65340.77220.64480.7622
0.63610.75230.62740.74240.61880.73250.61010.72270.60150.71286
0.67960.80620.67100.7962
0.66230.78620.6536
0.77620.64500.76630.63630.75640.62770.74647
0.69280.82340.68410.81340.67550.8034
0.66680.79340.6582
0.78340.64950.77340.64080.7635Figure 1: The Trinomial Tree of the Short Rate and the Spot LIBOR. We construct thetree following Hull–White (1994) based on the parameter set, β = 0.3, R : increasing
0.0025 in each period starting from 0.1, σ = 0.01, N = 8, T = 2, δ = 0.25, K = 0.7, andl = 2. The upper nodal values, ri,j , are the short rates and lower values, LTi (Ti ), are thespot LIBORs at each node (i, j).

14

i

0

1

2

3

4

j
3
0.0053
X
0.0040X

2

0.0032
X

1

0
-1
-2

0.0026
X

0.0029X
0.0026
X

0.0036
X
0.0032
X
0.0028X
0.0025X-3
W (i, j, 1)
Exercise(E) or Not(X)0.0047
X
0.0041
X
0.0035
X
0.0031
X
0.0027
X
0.0024X

0.0078E
0.0069E
0.0059E
0.0050E
0.0042
X
0.0037
X
0.0032
X

5

60.0221E

0.0151E

0.0201E

0.0131E

0.0180E

0.0110E
0.0089E

0.0160E
0.0139E

0.0075E

0.0118E

0.0065E
0.0055E

0.0098E

70.0256E
0.0236E
0.0215E
0.0195E
0.0174E
0.0154E
0.0133
E

Exercise

Figure 2: The Trinomial Tree of Chooser Flexible Cap Values with the Exercise Opportunity 1. The tree is built calculating the chooser flexible cap backwardly based on the
LIBOR tree of the Figure 1. The upper nodal values, W (i, j, 1), are the chooser flexible cap prices with the exercise opportunity 1 at each node (i, j). The lower rows showwhether the option holder exercises the i-caplet, E, at each node or not, X. Furthermore,
the surrounded area with the thick line represents the exercise nodes.

15

i

0

1

2

3

4

j
3
0.0092X
0.0073X

20.0059
X

1

0
-1
-2

0.0049X

0.0054X
0.0049X

0.0067
X
0.0060X
0.0053
X
0.0046
X

0.0084X
0.0076
X
0.0067
X
0.0058X
0.0050X-3
W (i, j, 2)
Exercise(E) or Not(X)0.0043
X

0.0137E
0.0123E
0.0109E
0.0095E
0.0081E
0.0066E
0.0057X

5

60.0262E
0.0233E
0.0203E
0.0174E
0.0143E
0.0113E
0.0096E

0.0348E
0.0319E
0.0290E
0.0260E
0.0231E
0.0201E
0.0171E

70.0256E
0.0236E
0.0215E
0.0195E
0.0174E
0.0154E
0.0133
E

Exercise

Figure 3: The Trinomial Tree of Chooser Flexible Cap Values with the Exercise Opportunity 2. The tree is built calculating the chooser flexible cap backwardly based on the
LIBOR tree of the Figure 1. The upper nodal values, W (i, j, 2), are the chooser flexible cap prices with the exercise opportunity 2 at each node (i, j). The lower rows showwhether the option holder exercises the i-caplet, E, at each node or not, X. Furthermore,
the surrounded area with the thick line represents the exercise nodes.

16

Table 1. Comparative Staticsa
l=1

l=2

l=3

l = 4 Change of the Pricel=20.0703 0.1222

l=40.0703 0.1222 0.1591 0.1859

β = 0.3

0.0703 0.1222

β = 0.6

0

l bigger ⇒ price bigger

β bigger ⇒ price smaller0

σ = 0.01

0.0703 0.1222

σ = 0.5

18.7

K = 0.80.0703 0.1222

K = 0.1

63.0

σ bigger ⇒ price bigger

26.8
K bigger ⇒ price smaller

101.2

T = 2, N = 8, δ = 0.25

0.03645 0.03949

T = 4, N = 16, δ = 0.25

0.0703 0.1222

T (orN ) bigger ⇒ price bigger

R0: gently increasing from 0.1 0.0703 0.1222

flat and decreasing ⇒ price 0

R1: flat at 0.1

00R5 ⇒ price biggest

R2: decreasing from 0.14

00R3: increasing from 0.06

0.0954 0.1771

R4: decreasing from 0.1

00

R5: increasing from 0.1

0.6

1.1

a

The Table 1 shows the chooser flexible cap prices (the table values ∗10−3 ) calculated with only one
parameter changed where other parameters and initial conditions are fixed on the bench mark values,l (Exercise opportunity)= 2, β (Parameter of HW model)= 0.3, σ (Parameter of HW model)= 0.01,
K (Exercise rate)= 0.8, T (Option maturity)= 2, N (Number of time periods)= 8, δ (Time interval)=
0.25, and R (Initial Yield Curve): increasing 0.0025 in each period starting from 0.1. The column of”Change of the Price” shows the change of the chooser flexible cap price when each corresponding
parameter is changed. R0 is the gently increasing, 0.005 at each period, initial yield curve from 0.1.
R1 is the flat curve at 0.1. R2 is the decreasing, 0.01 at each period, curve from 0.14. R3 is theincreasing, 0.01 at each period, curve from 0.06. R4 is the decreasing, 0.01 at each period, curve from
0.1. R5 is the increasing, 0.01 at each period, curve from 0.1.

17

10

Conclusion

In this paper we propose the pricing method of the chooser flexible cap. We have mainly
three contributions in this paper. Firstly, we utilize the dynamic programming approachwith the short rate model, in particular the Hull–White model. Secondly, deriving thetheoretical prices of bond, the caplet and the floorlet, we use the theoretical prices for thecalibration. Thirdly, we show the numerical examples and discuss comparative statics.
The future plan of the research is to price the chooser flexible cap after the calibration
thorough real market data. Furthermore we should verify if prices derived by the model
suit the chooser flexible cap prices of the real market. We can analytically prove theresults of comparative statics shown in Subsection 9.2. Some of them are under way in
our future research paper. One factor model like this paper has only one driving Brownian
motion and it implies perfect correlation of all forward rates with different maturity dates.
Furthermore, by working with the one factor short rate as the model primitive, it isdifficult to allow for a precise fit to processes of quoted instruments (bond, caplet andfloorlet) because of low degrees of freedom. We can extend the one factor model of thispaper to a multi factor model.

References
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18