Page 1
THE PRICING OF THE OPTION IMPLICITLY GRANTED
BY THE ITALIAN TREASURY TO THE PRIMARY DEALERS
IN THE RESERVED AUCTION REOPENING
Chiara Coluzzi
Università di Roma “Tor Vergata”
Preliminary and Incomplete
May 2007
Abstract
The Italian Government finances its public deficit by issuing debt securities, as of October 2005 they account for 86% of public debt. The efficiency of the issuance mechanism of Government Securities is crucial and it entails a correct pricing of the issued bond and notes. The aim of this paper is to address a particular feature of the Italian auction placement mechanism, that has received scant attention by the academic literature. Modeling the short term interest rate as a square root process and I provide a pricing of the option in the framework of the Cox – Ingersoll – Ross model. It turns out that the option helps explaining the mispricing occurring between the primary and secondary market.
1
Page 2
Introduction
One of the most important functions of the Italian Treasury is to provide the financing of the
public debt at the lowest possible cost for a given level of risk, making a correct pricing of GS
extremely important. To this aim, the efficiency of the issuance mechanism of GS is crucial,
giving great prominence to the study of market conditions, security design and placement
techniques. The present research particularly refers to the third issue.
The Italian GS primary market avails itself of a primary dealership system. The Ministry of
Economy and Finance (MEF) chooses a group of intermediaries, called Specialists1, among
the Primary Dealers of the MTS. The status of Specialist implies obligations and privileges,
e.g. among the former, Specialists should buy at least 3% of the auction, on the other hand,
among the latter, they have exclusive access to the reserved reopenings. Thus, the relationship
between the issuer and the primary dealers should be taken into account when considering
primary and secondary markets prices. A visible effect of such a relationship on prices is the
so called mispricing, i.e. the prices at which Treasury notes, bills and bonds are sold in
auction are higher (overpricing, see Brandolini 2004) or lower (underpricing, see Drudi and
Massa 1997, Scalia 1997 and Pacini 2005,) than the when-issued or secondary market prices.
Since Specialists get almost the entire auction, the sign of mispricing presumably depends on
the above-mentioned obligations and privileges. To this aim, the quantitative measurement of
such obligations and privileges, in terms of cost and profits, is necessary in order to make a
complete assessment of the whole placement mechanism. The option implicit in the
reopenings reserved to the Specialists is one of the main privileges they have. Indeed, the
reopening allows Specialists to buy predetermined additional quantities of GS at the price
settled at the auction. The application deadline is fixed at 15.30 p.m. of the business day
following the auction. The goal of this research is to provide a pricing of such an option as a
call written by MEF with a strike price equal to the stop-out price of the auction. Moreover, in
recent years Specialists complain about the existence of overpricing that generates money
losses in their balance sheet, a part of the explication is in their aggressive behaviour during
auction in order to obtain the right to participate to other special issues.2 This study could be
able to explain another part of the overpricing as the cost of the option embedded in the
1 D.P.R. n. 398, 30/12/2003. 2 MEF ranks Specialists according to the quantity they are awarded in the auctions. The higher ranked have the privilege to participate in other particular and highly remunerative operations on the primary market.
2
Page 3
reopening procedures. The option is written both on medium and long term bonds and on 6-
months BOTs, while the former securities are auctioned through a marginal auction, the latter
entail a competitive bidding. This research is focused on medium and long term bonds, hence
on BTPs (Buoni Poliennali del Tesoro) and CTZs (Certificati del Tesoro Zero Coupon). The
analysis covers the quarters 2004:1-2006:1. The source of the data is the database MTS Time
Series.
The rest of the paper is organized as follows: Section I is aimed to determine the
characteristics of the implicit option and then to chose a pricing model; in Section II the
model is calibrated to data and the option priced; Section III relates the results to the
mispricing phenomenon; Section IV concludes.
Section I In the mid 90’s the reserved reopening was introduced as a privilege for the Specialists
participating the ordinary auction. Since its introduction, the feature that has been changing in
the years is the time of expiration. For instance, before June 27th 2000 the deadline for bidding
at the ordinary auction, and then the option’s starting time, was at 1.00pm. Moreover, since
January 15th 2005 the deadline to submitting bids for the reopening changed from 12.00pm to
3.30pm. The higher panel of Table 1.1 contains the number of reopenings and the option’s
exercise rate from July 2003 to October 2006 for a selected kind of securities. The lower
panel breaks down the sample according to the last change on the option’s expiration. Total
exercise means that the complete amount offered in the reserved reopening has been allotted
to the Specialists. In the higher panel this quantity goes from about 23% of the CCT to the
60% of the 10 year BTP. The partial exercise is always smaller and goes from 10% in the 10
year BTP to about 30% in the CTZ. The change in the option’s expiration seems to have not
significantly affected the rate of exercise.
Table 1.1 Options’ exercise rate by security
3
Page 4
7/14/03 - 10/31/06# reopenings total
exercisepartial exercise tot
CTZ 36 47.22% 29.41% 76.63%CCT* 13 23.08% 33.33% 56.41%BTP 3y 32 46.88% 26.67% 73.54%BTP 5y 32 46.88% 13.33% 60.21%BTP 10y 33 60.61% 10.00% 70.61%total 146* 12/29/04 - 2/16/06
# reopenings total exercise
partial exercise tot
before 1/15/05 53 43.40% 21.74% 65.14%after 1/15/05 93 44.09% 19.51% 63.60%total 146
Overall the participation in reserved reopenings is always above 70% in the most traded
securities. Add table-quantity
The first task to address in order to understand the features of the option is to focus on the
reopening mechanism. In particular, we have to figure out the option type, the strike price, the
expiration date, whether it is European or American and the quantity of the underling asset.
The reopening allows Specialists to buy a predetermined quantity of the auctioned bond at the
price settled at the auction. By definition we can describe the option as a plain vanilla call
option where the strike price should equal the stop-out price. Anyway, we have to consider
that the marginal price includes a placement fee that the Italian Ministry of Finance pays to
the Bank of Italy. The Specialists are reimbursed since they can not apply subscription fees to
their final clients. The fee is equal to twenty basis points for the 10 year-BTP, thirty for the 3-
year BTP and forty for the other maturities. Then the strike price of the option is simply the
difference between the marginal price and this fee. The option’s life starts at 11.00am of the
auction day and it expires at 3.30pm of the following business day. Hence this is a one-day
option. Even if, in principle, each Specialist is able to exercise the option whenever, just short
selling the underling and then placing a bid for the reopening, I model the option as a
European one. In fact, the exercise is convenient as soon as the price on the secondary market
for the underling asset exceeds the strike price. Anyway, this approximation is not severe
since the short life of the option. The settlement day is the same both for the first auction and
for the supplementary placements according to the calendar set at the beginning of the year.
4
Page 5
This is of particular interest when considering the strategies that can be implemented by the
Specialists using the quantity they are entitled when participating at the reopenings. The
reopenings are set up for a maximum amount equal to 25% of the amount offered in the first
tranche of every new securities and to 10% for the following placements. Each Specialist has
the right to a minimum share of the total amount issued. This fraction equals the sum of the
quantities awarded in the last three auctions for the same security, excluding the reopenings,
divided by the quantity allotted to all the Specialists in the same auctions. Only those
Specialists who took part in the first auction are allowed to participate to the reserved
reopenings. Bids are satisfied by first assigning to each Specialist the lesser between the
amount requested by the specialist and the amount rightfully due to him. Should one or more
Specialists present bids inferior to those rightfully due3, or not present any bid at all, the
difference is allocated to dealers who presented bids greater than those rightfully due. This
explains why, in general, it is optimal for the Specialists to bid for a quantity above the
minimum one. Now we know the main option’s characteristics and we can turn to the choice
of the pricing model.
The tools developed for derivative asset pricing are an example of theories and methods
originally developed by physicists in order to solve problems in economics, usually those
including uncertainties or stochastic elements and nonlinear dynamics. The option is
implicitly written on bonds, hence the spot rate dynamics are the source of uncetainty. These
dynamics may be modeled like the ones of a particle in a fluid and then using stochastic
differential equations of the kind
tttt dwrtdtrtdr ),(),( σµ +=
where wt is a standard Brownian motion. Given the choice of the drift and the diffusion
coefficients a model for the spot rate dynamics can be worked out.
So far only Brandolini (2004) attempted to price the option with the Black and Scholes
formula. Although for some classes of derivatives B&S’s assumption of constant risk free rate
can be maintained, for instance for options on stock, in general this assumption may not be
correct. In particular, when dealing with interest rate derivatives, movements in interest rate
are the motivation for the existence of such instruments. Bond options give to the holder the
right to buy (call options) or sell (put options) a bond Pt at a fixed strike price K. Since the
3 Ex art. 12 of the Issuance Decree. See for instance http://www.dt.tesoro.it/ENGLISH-VE/Public-Deb/Italian-Go/Medium-Lon/index.htm.
5
Page 6
bond price depends on the current and future spot rates, bond options will be sensitive to
movements in the underling interest rate, say rt. Assuming a constant interest rate would mean
that Pt is completely predictable and as a consequence its volatility should equal zero, if this is
the case there would have been no reasoning for bond options. Since we depart from the B&S
environment the analysis can turn on the choice of an alternative pricing model. The selection
of a proper model involves a number of considerations. With respect to the arbitrage
condition we adopt for the pricing, there are two approaches that can be followed, the
classical and the Heat Jarrow and Morton (1992) ones. In the classical approach to fixed
income a risk-adjusted model for spot rate under the risk neutral martingale measure is
worked out from the price of arbitrage-free bonds. This approach requires to estimate drift and
volatilities of the spot rate dynamics, moreover the spot rate is assumed to be Markovian and
consequently drt depends only on rt. The fundamental theorem of asset pricing places no
restrictions on what the drift should be since the spot rate is not the price of an asset. The
advantage of this method is that one is able to price interest sensitive instruments without
having a look to the markets for these securities, therefore it would be helpful in determining
the price of the implicit option that by definition is not traded. Furthermore it takes the
advantage of the liquidity of the Italian GS market. However, the traditional approach has
several disadvantages. Because the drift of the instantaneous spot rate under the risk-neutral
martingale measure is not directly observable, it is almost impossible to verify the consistency
of the assumed dynamics with time series data. While it is possible to construct multi-factor
spot rate models in order to obtain a less than perfect correlation across spot (or forward) rates
of different maturities, it is difficult to calibrate exactly the correlations across the factors in
order to obtain a desired (empirical) correlation matrix across rates. No known short-rate
model is consistent with the Black formulas used by the market to quote prices for caps/floors
and swaptions and known spot-rate model can ensure an exact fit to a set of caps/floors or
swaption quotes. Yet, such an exact fit is very desirable for exotic derivatives traders who
hedge their trades using vanilla derivatives.
The Heath-Jarrow-Morton approach (HJM), on the other end, exploits the arbitrage relation
between forward rates and bond prices to impose restrictions on the dynamics of
instantaneous forward rates directly. By doing this it eliminates the need to model the
expected rate of change of rt, still volatilities remain to be estimated. The HJM approach was
later applied to discrete-tenor forward rates in a series of papers that started with the work of
Brace-Gatarek-Musiela (1997). Models based on assumptions regarding the evolution of
6
Page 7
discrete-tenor forward rates are known as BGM models, market models or LIBOR market
models. The modern approach has several advantages over the classical approach. Because
the volatilities of forward rates are the same under the martingale measure and under the true
(historical) probability measures, the consistency of the assumed dynamics for forward rates
with time series data can be easily verified. An exact fit of the initial term structure is
obtained by construction, since the initial forward curve is exogenous to the model. Since
assumptions are made directly on the volatilities of forward rates, it is easy to calibrate a
model in order to obtain a desired (empirical) correlation matrix across forward rates. The
"standard" market model leads to lognormally-distributed forward rates and is consistent with
the Black formulas used to quote prices for caps/floors. In addition, it closely approximates
the Black formulas used to quote prices for swaptions. Moreover, it is very easy to calibrate
the "standard" market model to obtain an exact fit of a set of at the money cap/floor prices.
One disadvantage of the modern pricing approach is that arbitrary specifications of the
forward rate volatilities will in general lead to non-Markovian dynamics, thus requiring the
simulation of a large number of state variables.
Brigo and Mercurio (2001) provide a deep analysis of these issues. The choice of the model
may require also to set the number of risk factors and the eventual inclusion of a jump
component in the interest rate process. However, before the model is chosen one has to
consider also the market data that would be available to calibrate its parameters. Given the
data available to me the model I chose is the one-factor time-homogeneous version of the Cox
– Ingersoll –Ross model (CIR). The general equilibrium approach developed by Cox,
Ingersoll and Ross (1985) led to the introduction of a square root term in the diffusion
coefficient of the instantaneous short-rate dynamics proposed by Vasicek in 1977. The
resulting model has been a benchmark for many years and it remains useful because of its
analytical tractability and the fact that, contrary to the Vasicek (1977) one, the spot rate is
always positive. Moreover, it provides closed form solutions for both the price of bonds and
of plain vanilla options written on bonds. Under the risk neutral martingale measure the
formulation of the model is:
( ) dwrdtrdr σθκ +−=
where κ is the speed of adjustment of the interest rate towards its long-run average θ, rσ is
the volatility of changes in the instantaneous interest rate and dw is a standardized Wiener
7
Page 8
process. Moreover if 0 < κ < 1 the process is mean-reverting, this means that the interest rate
converges to its long-run value. It can be proved that the CIR model possesses an affine term
structure, that is the price of a pure discount bond with residual maturity τ = T-t can be
written as:
( ) ( ) ( )rTtGeTtFTtrP ,,,, −=
where
( ) ( )3
1
2
12
1
1,
φ
τφ
τφ
φφφ
+−
=e
eTtF
and
( ) ( ) 12 11,
1
1
φφ τφ
τφ
+−−
=eeTtG
Subject to the boundary condition
P(r,T,T)=1
This means that the price of a pure discount bond with residual maturity τ is a function of the
state variable r and of the three parameters φ1, φ2 and φ3 where:
( ) 221 2σλκφ ++=
21
2φλκ
φ++
=
232σκθφ =
and –λ is as usual the market price of risk. Given P(r,t,T) the whole term structure of interest
rates R(r,t,T) can be worked out using the following relationship:
8
Page 9
( ) ( )[ ]tT
TtrPTtrR−
−=
,,ln,, .
In particular, we have that:
( ) rTtrRRtT
==→
,,lim0
( ) ( ) 321,,lim φφφ −==∞→∞ TtrRR
T
and
( )22212 φφφσ −= .
If we substitute the value of for θ we can gain economic intuition on the three parameter.
In fact, now they can be written as:
∞R
λκφ −=1
κφ =2
λθφ −=3 .
The CIR model is easy to implement and this property is very important for a financial
institution. For this reason, the CIR model is often used for practical purposes. As our aim is
to price an option that is part of the Specialists’ privileges and that they may want to price,
then it is useful to start with instruments already used by banks and managers.
Of the Italian GS trading on a given date there are both coupon and zero coupon bonds, this
must be taken into account when calibrating the CIR model. Moreover, the quoted prices are
clean prices, meaning they do no incorporate the coupon that has been maturing from the last
payment date. Hence we have to work out the accrued interest in order to obtain the dirty
price or cum-coupon price. The CIR model is estimated using the one-stage approach
developed by Brown and Dybvig (1986). This assumes that if we consider a coupon bond that
entitles the holder to a vector of remaining payments, cf, to be received in a vector of dates, d,
the value of such bond at time t is equal to
9
Page 10
( ) ( )∑ >=
td iii
dtrPcfdcftV ,,,,* .
Hence a zero coupon bond can be represented as a bond with a single payment at its maturity
date. To estimate the parameters of the model we make the assumption that the bond price V
at time t deviate by the model price V* by an error term, εt:
( ) ( ) tdcftVdcftV ε+= ,,,, * .
While Brown and Dybvig assume that the error term is zero-mean and independent and
identically distributed as a Normal, I follow Barone, Cuoco and Zautiek (1989) and assume
that is increasing in the bond’s duration. Indeed, it is natural to assume that a pricing error is
smaller the closer is the bond’s maturity date. This implies that in order to make the error term
homoskedastic both sides of the previous equations are divided by the square root of the
product between the modified duration and the cum-coupon price of the bond. The resulting
model is of the kind
( ) uXPY += βτ ,
where X is the cash flows’ matrix and β is the vector of the parameters.
Section II The model is calibrated on daily data on Italian GS for the quarters 2004:1 – 2006:1 for a total
of 579 trading days. These data come from MTS Time Series database. MTS Time Series
package contains both high frequency tick data and daily data. In particular, I use daily trade
and quote information taken at 5.00 p.m. Central European Time (CET) and an identifying
cross-sectional file providing bond descriptions. For each day I have the closing mid price of
all the bonds and notes traded that day, the time to maturity, the coupon payments, the day
counting market conventions. Moreover the database includes the accrued interest and the
modified duration. When the last two quantities were missing they have been worked out.
Data cover a wide range of maturities, from one day up to February 1st, 2037 and are ordered
by quote date and then by maturity date.
10
Page 11
The CIR model has been estimated in many different ways [For estimations using non-linear
regression techniques, see S. Brown and P. Dybvig (1986) and R. Brown and S. Schaefer
(1988). For an estimation based on the method of moments, see M. Gibbons and K.
Ramaswamy (1986). An alternative method, known as the two-stage method, consists in first
estimating the parameters κ, µ and σ of process followed by the instantaneous interest rate,
using the time series of a short-term rate, and then λ on the basis of equation (3): see A.
Ananthanarayanan and E. Schwartz (1980). For an application of the two-stage method to the
Italian market, see E. Barone and R. Cesari (1986)]. Here I chose to calibrate the model to the
available data using a non-linear least squares method. This is a numerical method that find
the minimum of a function in a recursive way, starting from a parameter vector arbitrary
chosen. From a geometrical point of view, this method can be thought as a path that, from a
starting point, chooses the way that brings to the nearest deep valley down. Anyway, it could
be possible that the nearest deep valley is not the deepest valley, that is the non-linear least
squares method could bring only to a local minimum down. So, the choice of the starting
point is very delicate. In general we do not know, neither approximately, where the global
minimum lies. This means that in general we can not choose a reasonable starting point for
the parameter vector sufficiently close to the minimum. A rough but efficient method to
overcome this problem is suggested by Torosantucci Uboldi (200?). They propose to build a
net of starting points, calibrate the model using n set of starting points from the grid, and then
to chose the parameters that ensure both a reasonable expectation of the market, and the
mathematical hypothesis of the CIR model. If more than one set of such parameters satisfies
these requirements they chose the set that minimize their score function. The main drawback
of this method is the long time necessary to give a solution. For this reason they suggest to
restrict the number of iterations either by applying this procedure on few distant days, or to
apply it just on the first day and then use that day’s local minimum as starting point of the
following days calibration. The problem with this method is that if at day t a strange and
particular market situation occurs it is reasonable to obtain an anomalous minimum parameter
vector.
Given that per each day I have about 60 observations, it should not be problematic to find the
global minimum independently by the chosen starting point. I then used a procedure that is
close to Torosantucci and Uboldi’s (henceforth TU), robust as well but less time consuming.
First I fixed an interval for each element of the parameter’s vector. According to TU the
intervals should be large enough to include the global minimum. In order to limit the
11
Page 12
dimension of the intervals I chose them in such a way that they include the values found in
previous calibrations of the CIR model on Italian data (add reference). Then I made a draw of
the first starting point from the uniform distributions functions built on the chosen intervals,
and I calibrated the CIR model for all the days in the sample using the selected starting point.
I stored the result and then made a second draw and another calibration. At the end of the
second simulation I compared the resulting minima with the previous ones. Then I saved the
best ones coming from this comparison into a separate file, where for best selection criterion I
used, accordingly with the non-linear least square method, the sum of square errors. I repeated
this procedure 100 times. The result was a file containing 101 different values for each
parameter on each day and a file including the time series of the parameters minimizing the
function. This procedure allows each set of parameters to come from a different set of starting
points.4 In order to check for the stability of the obtained parameters, I also counted the
number of revisions on parameters coming from the 101 simulations. The resulting
parameters are summarized by year in the following table.
Table 2.1 Parameters’ statistics
years # obs. mean sd min max2004 258 0.2186 0.0348 0.1326 0.35122005 257 0.1779 0.0287 0.0791 0.25002006 64 0.2386 0.0529 0.1664 0.4154
2004 258 0.2186 0.0348 0.1326 0.35122005 257 0.1779 0.0287 0.0791 0.25002006 64 0.2386 0.0529 0.1664 0.4154
2004 258 4.24 1.63 3.23 13.872005 257 5.52 3.35 3.69 13.902006 64 5.66 3.46 3.36 13.78
2004 258 0.0180 0.0016 0.0140 0.02542005 257 0.0195 0.0023 0.0161 0.02792006 64 0.0259 0.0013 0.0238 0.0283
φ 3
r
φ 1
φ 2
4 I ended up with ??? starting points for the 579 days in the sample.
12
Page 13
In Table 2.2 are reported statistics about the number of parameters’ revisions generated by the
101 simulations. On average 2005 and the beginning of 2006 exhibit a higher number of
revisions.
Table2.2 Number of revisions
years # obs. mean sd min max2004 258 5.01 0.88 2 82005 257 5.96 0.84 4 102006 64 6 1.49 3 10
number of revisions
Further information about parameters’ stability can be inferred from the following figures. In
particular, Figure 2.1 shows the time series of the parameters. Although I did not exclude any
day from the sample, the resulting parameters seems to be pretty stable. This holds in
particular for φ1, φ2 and r. The parameter φ3 shows many peaks, some of them coincide with
auction and reopenings days (Add analysis). This path is particularly unstable in the last part
of the sample. Figure 2.2 shows the time series of the diffusion coefficient and of the long
term interest rate respectively. The long term rate’s peak is probably due to liquidity problems
since it happens on December 24th 2004. The last two figures shows the time series evolution
of the volatility and of the sum of square errors generated by the model.
13
Page 15
Given the calibrated parameters it is possible to price the option. Let c(r,t,T,s,K) be the price
of a call option with strike price K and maturity T, written on a pure discount bond with
15
Page 16
maturity s (s > T > t). The price of the option, obtained solving the problem (Insert
PDE+boundary cond) is
( ) ( ) ( ) ( ) ( )2222
1112 ,,,,,,,,,,,,, ncdfdTtrKPncdfdstrPKcfsTtrc χχ −=
where is the non-central chi-square distribution function valued at point d, with
df degree of freedom and non-centrality parameter nc. These parameters are specified both by
Cox, Ingersoll and Ross (1985) and Jamshidian (1990). In what follows I will refer to the CIR
(1985) ones:
( ncdfd ,,2χ )
( )[ ]sTGrd ,2 *1 ++= ψϕ
( )ψϕ += *2 2rd
321 2φ== dfdf
( )
( )sTGrenc
tT
,2 12
1 ++=
−
ψϕϕ φ
( )
ψϕϕ φ
+=
−tTrenc12
22
( )[ ]1212
1
−= −tTeφσ
φϕ
222
σφ
ψ =
( )
( )sTGK
sTF
r,
,ln*
=
Where r* is the critical interest rate below which the option will be exercised, and it is
obtained solving P(r*,T, s) = K with respect to r*. The option formulation tells us that the
replicating strategy is simply to buy ( )1112 ,, ncdfdχ pure discounts bonds with maturity s and
sell pure discount bonds with maturity T. Since the options to be
considered here are written on coupon bonds, we have to take it into account. In particular
with one-factor model it can be proved that the option written on a coupon bond is equivalent
to a portfolio of options on pure discount bonds (Jamshidian 1989, Longstaff 1990):
( 2222 ,, ncdfdKχ )
16
Page 17
( ) ( )∑=
=m
qjjjj KsTtrccfKcfsTtrc 0,,,,,,,,,, .
Hence, given the formula for the price of an option on pure discount bonds c, we obtain
( ) ( ) ( ) ( ) ( )jjj
m
qjjjjjjj ncdfdTtrPKncdfdstrPcfKcfsTtrc ,2,2,2
2,1,1,1
2 ,,,,,,,,,,,,, χχ∑=
−=
where cfj is the bond’s payments at time sj (T < sq ≤ sj ≤ sm), r* is the solution of
and K(∑=
=m
qjjj strPcfK ,,* ) j the solution of ( )jj strPK ,,*= .
In order to work out the value of the option I use the parameters calibrated the closing prices
of the day before the ordinary auction.5 The results of the option pricing for the reopenings
occurred during the year 2004 are reported in the following table.
Table 2.3 Option Pricing Results
5 These values are the most recent available before the auction.
17
Page 18
auction date bond description option
pricestrike price
reserved reopening
dateσ σ r^0.5
long term rate φ1 φ2 φ3 r
1/14/2004 BTP 15/01/07 2,75% 0.0085 99.72 1/15/2004 0.0805 0.0106 0.0600 0.236 0.222 4.104 0.017441/14/2004 BTP 15/09/08 3,50% 1.0446 100.61 1/15/2004 0.0805 0.0106 0.0600 0.236 0.222 4.104 0.017441/27/2004 CTZ 03-31/08/05 24M 0.0008 96.39 1/28/2004 0.0822 0.0106 0.0605 0.242 0.227 4.066 0.016501/29/2004 BTP 15/01/07 2,75% 0.0035 99.57 1/30/2004 0.0899 0.0123 0.0607 0.244 0.226 3.390 0.018761/29/2004 BTP 01/08/14 4,25% 1.6802 98.8 1/30/2004 0.0899 0.0123 0.0607 0.244 0.226 3.390 0.018762/12/2004 BTP 15/09/08 3,50% 1.0996 100.96 2/13/2004 0.0840 0.0108 0.0603 0.243 0.228 3.892 0.016552/24/2004 CTZ 03-31/08/05 24M 0* 96.72 2/25/2004 0.0829 0.0106 0.0612 0.228 0.212 3.766 0.016502/26/2004 BTP 15/01/07 2,75% 0.0033 100.27 2/27/2004 0.0735 0.0100 0.0625 0.190 0.175 4.044 0.018682/26/2004 BTP 01/08/14 4,25% 0.6690 99.95 2/27/2004 0.0735 0.0100 0.0625 0.190 0.175 4.044 0.018683/11/2004 BTP 15/09/08 3,50% 1.2604 101.97 3/12/2004 0.0781 0.0102 0.0601 0.202 0.185 3.643 0.017043/29/2004 CTZ 04-28/04/06 24M 0.0021 95.56 3/30/2004 0.0810 0.0096 0.0611 0.208 0.190 3.547 0.014023/30/2004 BTP 15/01/07 2,75% 0.0471 100.75 3/31/2004 0.0742 0.0099 0.0635 0.171 0.153 3.543 0.017803/30/2004 BTP 01/08/14 4,25% 1.2904 100.95 3/31/2004 0.0742 0.0099 0.0635 0.171 0.153 3.543 0.017804/13/2004 BTP 15/04/09 3,00% 1.5231 98.32 4/14/2004 0.0824 0.0103 0.0602 0.237 0.222 3.937 0.015564/13/2004 BTP 01/08/34 5,00% 2.8229 97.375 4/14/2004 0.0824 0.0103 0.0602 0.237 0.222 3.937 0.015564/27/2004 CTZ 04-28/04/06 24M 0.0054 95.1 4/28/2004 0.0836 0.0109 0.0601 0.248 0.233 4.006 0.016894/29/2004 BTP 15/01/07 2,75% 0.3616 99.76 4/30/2004 0.0507 0.0067 0.0600 0.277 0.272 12.721 0.017414/29/2004 BTP 01/08/14 4,25% 5.6604 98.92 4/30/2004 0.0507 0.0067 0.0600 0.277 0.272 12.721 0.017415/13/2004 BTP 15/04/09 3,00% 0.3832 97.02 5/14/2004 0.0826 0.0115 0.0613 0.240 0.224 4.029 0.019505/13/2004 BTP 01/08/34 5,00% 1.0345 96.75 5/14/2004 0.0826 0.0115 0.0613 0.240 0.224 4.029 0.019505/26/2004 CTZ 04-28/04/06 24M 0.0068 95.09 5/27/2004 0.0858 0.0115 0.0606 0.263 0.249 4.089 0.017915/28/2004 BTP 01/06/07 3,00% 1.4104 100.07 5/31/2004 0.0839 0.0110 0.0606 0.251 0.236 4.070 0.017315/28/2004 BTP 01/08/14 4,25% 1.5790 98.67 5/31/2004 0.0839 0.0110 0.0606 0.251 0.236 4.070 0.017316/14/2004 BTP 15/04/09 3,00% 0.6552 96.57 6/16/2004 0.0873 0.0123 0.0594 0.277 0.263 4.103 0.019976/25/2004 CTZ 04-28/04/06 24M 0.0415 95.26 6/28/2004 0.0857 0.0117 0.0590 0.267 0.253 4.063 0.018596/28/2004 BTP 01/06/07 3,00% 0.2736 99.77 6/30/2004 0.0872 0.0119 0.0590 0.274 0.259 4.019 0.018536/28/2004 BTP 01/08/14 4,25% 1.7795 98.47 6/30/2004 0.0872 0.0119 0.0590 0.274 0.259 4.019 0.018537/13/2004 BTP 15/04/09 3,00% 0.7297 97.68 7/14/2004 0.0844 0.0115 0.0585 0.262 0.247 4.059 0.018447/27/2004 CTZ 04-31/07/06 24M 0.0490 94.64 7/28/2004 0.0884 0.0120 0.0579 0.284 0.269 3.987 0.018277/29/2004 BTP 01/06/07 3,00% 0.2485 99.86 7/30/2004 0.0990 0.0131 0.0576 0.352 0.338 3.971 0.017467/29/2004 BTP 01/08/14 4,25% 0.8061 98.68 7/30/2004 0.0990 0.0131 0.0576 0.352 0.338 3.971 0.017468/26/2004 CTZ 04-31/07/06 24M 0.0151 95.25 8/27/2004 0.0808 0.0108 0.0580 0.241 0.226 4.016 0.017748/30/2004 BTP 01/06/07 3,00% 0.3014 100.52 8/31/2004 0.0733 0.0107 0.0595 0.188 0.173 3.832 0.021448/30/2004 BTP 01/02/15 4,25% 0.8085 100.03 8/31/2004 0.0733 0.0107 0.0595 0.188 0.173 3.832 0.021449/15/2004 BTP 15/04/09 3,00% 0.9823 98.6 9/16/2004 0.0783 0.0110 0.0568 0.236 0.222 4.119 0.019759/15/2004 BTP 01/08/34 5,00% 13.5940 102.15 9/16/2004 0.0783 0.0110 0.0568 0.236 0.222 4.119 0.019759/27/2004 CTZ 04-31/07/06 24M 0.0187 95.43 9/28/2004 0.0786 0.0108 0.0561 0.234 0.220 3.999 0.018809/29/2004 BTP 01/06/07 3,00% 0.7484 100.52 9/30/2004 0.0683 0.0100 0.0580 0.176 0.162 4.019 0.021609/29/2004 BTP 01/02/15 4,25% 1.4605 100.98 9/30/2004 0.0683 0.0100 0.0580 0.176 0.162 4.019 0.0216010/14/2004 BTP 15/04/09 3,00% 0.0033 99.47 10/15/2004 0.0763 0.0104 0.0562 0.224 0.210 4.066 0.0186310/28/2004 BTP 01/02/15 4,25% 0.5952 101.79 10/29/2004 0.0405 0.0057 0.0577 0.201 0.197 13.870 0.0196612/28/2004 CTZ 04-31/07/06 24M 0.0081 96.36 12/29/2004 0.0668 0.0092 0.0524 0.187 0.174 4.097 0.01911
The first and the second columns contain information about the date of the ordinary auctions
and the characteristic of the bonds auctioned, namely if it is a BTP or a 2-year CTZ, the
maturity date and the coupon rate. The price of the call option written on the described
security is in the third column. The fourth column contains the option’s strike price, the
remaining report the parameters resulting from the calibration of the CIR model on the
closing price of the day before the ordinary auction takes place.
18
Page 19
References
Barone, E., Cuoco, D. and Zautzik, E. 1989, “La struttura dei rendimenti per scadenza
secondo il modello di Cox, Ingersoll e Ross: una verifica empirica”, Banca d’Italia,
Temi di discussione, n. 128.
Black, F., and M. Scholes 1973, “The Pricing of Options and Corporate Liabilities”, Journal
of Political Economy, 637–654.
Brace, A., Gatarek, D. and M. Musiela 1997, "The Market Model of Interest Rate Dynamics",
Mathematical Finance 7, 127-155.
Brandolini, A. 2004,”Valutazione Finanziaria della Riapertura delle Aste dei Titoli di Stato a
Dieci Anni”, Università di Roma Tor Vergata, mimeo.
Brigo, D. and F. Mercurio 2001, Interest Rate Models: Theory and Practice, Springer- Verlag
Berlin Heidelberg
Cox, J., Ingersoll, J. and S. Ross 1985, “A Theory of the Term Structure of Interest Rates”,
Econometrica, 385–408.
Drudi, F. and M. Massa 1997, "Comportamento Strategico sul Mercato Primario e Secondario
dei Titoli di Stato: il Ruolo dell'Informazione Asimmetrica", Bank of Italy, Temi di
Discussione, No. 301.
Heath, D, Jarrow, D. and A. Morton 1992, “Bond Pricing and the Term Structure of Interest
Rates: A New Methodology for Contingent Claims Valuation”, Econometrica, 77–
105.
Hull, J.C. 2002, Options, Futures, and Other Derivatives 5th edition, Prentice-Hall, Upper
Saddle River, New Jersey
Jamshidian, F. 1989, “An Exact Bond Option Formula”, Journal of Finance 1, 205-209.
Longstaff, F. 1993, “The Valuation of Options on Coupon Bonds”, Journal of Banking and
Finance 17, 27-42.
Musiela, M. and M. Rutkowski 1997, Martingale Methods in Financial Modelling, Springer-
Verlag Berlin Heidelberg
Neftci, S. N. 2000, An Introduction to Mathematics of Financial Derivatives, Academic Press.
Pacini, R. 2005, “The Primary Market of Italian Treasury Bonds: an Empirical Study of the
Uniform Price Auction”, University of Rome Tor Vergata, mimeo.
Public Debt Management Office Department of the Treasury, 2000, “The Italian Treasury
Securities Market: Overview and Recent Development”.
19
Page 20
20
Scalia, A. 1997, "Bidder Profitability under Uniform Price Auctions and Systematic
Reopenings: the Case of Italian Treasury Bonds", Bank of Italy, Temi di Discussione,
No. 303.
Decreto 13 maggio 1999, n. 219, ”Disciplina dei mercati all’ingrosso dei titoli di Stato”.
Vasicek, O. A. (1977) “An equilibrium characterization of the term structure", Journal of
Financial Economics, pp. 177-188.