VALUATION OF NONCENTRAL CHI-SQUARED DISTRIBUTION

FOR OPTIONS ON A ZERO COUPON BONDS

UNDER CIR DIFFUSION*

Manuela Larguinho+

Department of Mathematics, ISCAC

José Carlos Dias

Finance Research Center (FRC/ISCTE) and Department of Finance, ISCAC

Carlos A. Braumann

Centro de Investigação em Matemática e Aplicações, Universidade de Évora

Área temática: B) Valuation and Finance (Valoración y Finanzas)

Keywords: Option pricing, CIR model, Greeks, statistical methods

*Larguinho and Braumann acknowledges the financial support from the Centro de Investigação em Matemática e Aplicações (CIMA) financed by the Fundação para a Ciência e Tecnologia (FCT) and Dias gratefully acknowledges the financial support from FCT grant number PTDC/EGE-ECO/099255/2008.

+Corresponding author: Department of Mathematics, ISCAC, Quinta Agrícola, Bencanta, 3040-216 Coimbra, Portugal. Tel: +351 239 802185. Fax: +351 239 445445.

198b

1

VALUATION OF NONCENTRAL CHI-SQUARED DISTRIBUTION

FOR OPTIONS ON A ZERO COUPON BONDS

UNDER CIR DIFFUSION

Abstract

Pricing options and evaluating Greeks under the Cox-Ingersoll-Ross (CIR) model

require the computation of the noncentral chi-square distribution function. In this article,

we compare the performance in terms of accuracy and computational time of

alternative methods for computing such probability distributions against an externally

tested benchmark. All methods are generally accurate over a wide range of parameters

that are frequently needed for pricing options, though they all present relevant

differences in terms of running times. Finally, we present closed-form solutions for

computing some possible Greeks measures under the CIR model for zero coupon

bond option pricing model.

Resumen

Las opciones de precios y la evaluación de las letras griegas bajo el modelo Cox-

Ingersoll-Ross (CIR) requiere el cálculo de la función de distribución de chi-cuadrado

no central. En este artículo, se comparan los resultados en términos de precisión y

tiempo de cálculo de los métodos alternativos para el cálculo de distribuciones de

probabilidad contra un benchmark externamente probado. En general todos los

métodos son exactos sobre la amplia gama de parámetros de frecuencia, se necesitan

que se las opciones de precios, aunque presente todos ellos diferencias relevantes en

cuanto a tiempo de funcionamiento. Finalmente, se presentan las soluciones de forma

cerrada para el cálculo de algunas letras griegas bajo el modelo CIR para las

opciones sobre bonos cupón cero.

2

1. Introduction

The CIR model is general single-factor model equilibrium approach developed

by Cox et al. (1985), and has been a benchmark for a many years because of

its analytical tractability and the fact that the short rate is always positive,

contrary to the well-known Vasicek (1977) model.

The CIR model is used to price zero-coupon bonds, coupon bonds and to price

options on these bonds. This model has been extensively studied and has lead

to generalizations in several directions. For instance Jamshidian (1995),

Maghsoodi (1996), and Brigo and Mercurio (2001) make extensions to this

model and obtain closed-form solutions to the zero-coupon bond options.

Bacinello et al. (1996) use CIR diffusion to the valuation of sinking-funds bonds,

and Mallier and Alobaidi (2004) consider ?xed-for-?oating interest rate swaps

under the assumption that the interest rates are given by the mean-reverting

CIR model.

To compute options prices under the CIR process typically involves the use of

the noncentral chi-square distributions function. There exists an extensive

literature devoted to the efficient of this distribution function. In this article we

will examine the methods proposed by Schroder (1989), Ding (1992), and

Benton and Krishnamoorthy (2003). The noncentral chi-square distribution

function can also be computed using methods based on series of incomplete

gamma series, which will be used as our benchmark. For certain ranges of

parameter values, some of the alternative representations available can be

more computationally efficient than the incomplete gamma functions.

Moreover, for some parameter configurations the use of analytic

approximations (e.g., Sankaran (1963), Fraser et al. (1998) and Penev and

Raykov (2000)) may be preferable. Hence, it is important to compare the

performance of alternative methods for computing noncentral chi-square

distributions for a large set of parameter values.

The main purpose of this article is to provide comparative results in terms of

accuracy and computation time of existing alternative algorithms for computing

the noncentral chi-square distribution function to be used for option pricing and

hedging under the CIR model. Although in the article of the Larguinho et al.

(2010) have done an exhaustive analysis of the noncentral chi-square

distribution function, will be important to do a new study since the nature of the

parameters values is very different for zero coupon bond options under CIR

model.

3

All tested methods are generally accurate over a wide range of parameters that

are frequently needed for pricing options on zero-coupon bonds, though they all

present relevant differences in terms of running times. The iterative procedure

of Benton and Krishnamoorthy (2003) has better performance in terms of

accuracy, while the iterative procedure of Ding (1992) is the most efficient in

terms of computation time needed for determining zero-coupon bond options

under the CIR diffusion, but is about three times slower than the benchmark

gamma series method.

The theoretical contribution of this paper is the derivation of closed-form

solutions for computing some possible Greeks of European-type zero-coupon

bond options under the CIR model that to our knowledge are not known in the

finance literature.

The structure of the paper is organized as follows. Section 2 outlines the

noncentral chi-square; Section 3 gives a brief review of CIR model and zero-

coupon bond options under this model and the closed-form solutions for the

Greeks. Section 4 compares the alternative methods in terms of speed and

accuracy and Section 5 concludes.

2. Alternative Methods for Computing the Noncentral Chi-Square

Distribution

2.1. The Noncentral Chi-Square Distribution

If Z1, Z2, ..., Zv are independent unit normal random variables, and d1, d2, ..., dv

are constants, then

is the noncentral chi-square distribution with ? degrees of freedom and non

centrality parameter and is denoted as . When for all j,

then Y is distributed as the central chi-square distribution with v degrees of

freedom, and is denoted as .

Hereafter, is the probability density function (pdf) of a

noncentral chi-square distribution , and is the probability

density function of a central chi-square distribution . Likewise,

4

is the cumulative distribution function of , and

is the cumulative distribution function of .

The cumulative distribution functions of , is given by (see, for

instance, Johnson et al. (1995, Equation 29.3) or Abramowitz and Stegun

(1972, Equation 26.4.25)):

(1)

while for and where is the central chi-square

distribution function as is given by Abramowitz and Stegun (1972, Equation

26.4.1). This definition express the , for , as a weighted sum of

central chi-square probabilities with weights equal to the probabilities of a

Poisson distribution with expected value ?/2.

The probability density function of can, similarly, be expressed as a mixture

of central chi-square probability density functions (see, for instance, Johnson et

al. (1995, Equation 29.4)):

, (2)

where is the modified Bessel function of the first kind of order q, as defined

by Abramowitz and Stegun (1972, Equation 9.6.10):

.

Using equation (2) we may also express the function F (w; ?, ?) as integral

representations:

. (3)

5

2.2. Alternative Methods

It is well-known that the function F(w; v +2n, 0) is related to the so-called

incomplete gamma functions (see, for instance, Abramowitz and Stegun (1972,

Equation 26.4.19)). Hence, we may express noncentral chi-square distribution

function (3) using series of incomplete gamma functions as follows:

(4)

with ?(m, t) and G(m, t) being, respectively, the incomplete gamma function and

the complementary incomplete gamma function as defined by Abramowitz and

Stegun (1972, Equations 6.5.2 and 6.5.3), and where G(m) is the Euler gamma

function, as defined by Abramowitz and Stegun (1972, Equation 6.1.1).

The gamma series method has been applied by Fraser et al. (1998) as a

benchmark for computing exact probabilities to be compared with several

alternative methods for approximating the noncentral chi-square distribution

function, and by Dyrting (2004) for computing the noncentral chi-square

distribution function to be used under Cox et al. (1985) diffusion processes.

Carr and Linetsky (2006) also use the gamma series approach but for

computing option prices under a jump-to-default CEV framework.

While this method is accurate over a wide range of parameters, the number of

terms that must be summed increases with the non centrality parameter ?. To

avoid the infinite sum of the series we use the stopping rule as proposed by

Knusel and Bablok (1996) which allows the specification of a given error

tolerance by the user.

There have been several alternative proposals for evaluating expression (7) -

see, for instance, Farebrother (1987), Posten (1989), Schroder (1989), Ding

(1992), Knusel and Bablok (1996), Benton and Krishnamoorthy (2003), and

Dyrting (2004) - all of which involve partial summation of the series. For certain

ranges of parameter values, some of the alternative representations available

are more computationally efficient than the series of incomplete gamma

functions. Hence, it is important to evaluate the speed and accuracy of each

method for computing the noncentral chi-square distribution as well as for

option pricing and dynamic hedging purposes.

For the numerical analysis of this article we will concentrate the discussion on

Schroder (1989) and Ding (1992) methods since both are commonly used in the

finance literature. The algorithm provided by Schroder (1989) has been

6

subsequently used by Davydov and Linetsky (2001). The popular book on

derivatives of Hull (2008) suggests the use of the Ding (1992) procedure. We

will also use the suggested approach of Benton and Krishnamoorthy (2003),

since it is argued by the authors that their algorithm is more computationally

efficient than the one suggested by Ding (1992). A detailed explanation of how

to compute the noncentral chi-square distribution function using these three

algorithms can be seen in Larguinho et al. (2010).

The cumulative distribution function of the noncentral chi-square distribution

with degrees of freedom v > 0 and a noncentrality parameter ? = 0 is usually

expressed as an infinite weighted sum of central chi-square cumulative

distribution functions. For numerical evaluation purposes this infinite sum is

being approximated by a finite sum. For large values of the noncentrality

parameter, the sum converges slowly. To overcome this issue, a number of

approximations have been proposed in the literature.

In this article, we will consider the approximation method of Sankaran (1963)

which is well-known in the finance literature due to Schroder (1989) who

recommends its use for large values of w and ?. In addition, two more recent

approximations, namely Fraser et al. (1998) and Penev and Raykov (2000), will

be also considered since both of them are commonly referenced by the statistic

literature as accurate methods for approximating the noncentral chi-square

distribution.

All these analytical approximations map the argument w, and parameters ?, and

? to a new variable z that is approximately normally distributed. Thus

F (w; v, ?) = N (z),

where N (z) is the standard normal distribution.

3. The CIR Option Price

3.1. CIR Diffusion

Under the risk-neutral measure Q, Cox et al. (1985) modeled the evolution of

the interest rate, rt, by the stochastic differential equation (SDE):

, (5)

where is a standard Brownian motion under Q, ?, ? and s are positive

constants representing reversion rate, asymptotic rate and volatility

parameters, respectively, and ? is the market risk. The condition 2?? > s2 has

to be imposed to ensure that the interest rate remains positive.

7

The interest rate behavior implied by this structure has the following empirically

relevant properties:

i. Does not allow negative interest rates;

ii. If the interest rate reaches zero, it can subsequently become positive;

iii. The absolute variance of the interest rate increases when the interest

rate itself increases;

iv. There is a steady state distribution for the interest rate.

Following Cox et al. (1985), the price of a general interest rate claim F (r, t) with

cash flow rate C (r, t) satisfies the following partial differential equation

. (6)

The price of a zero coupon bond with maturity at S, Z (r, t, S), satisfies the

equation (6) with C(r, t) = 0 subject to the boundary condition Z(r, S, S) = 1 and

is given by

Z (r, t, S) = A(t, S)e- B(t,S)r

(7) where

,

,

.

Denote by ZCcall(r, t, T , S, X ) the price at time t of a European call option with

maturity T > t, strike price X, written on a zero coupon bond maturity at S > T

and with the instantaneous rate at time t given by rt . X is restricted to be less

than A(T, S) the maximum possible bond price at time T, since otherwise the

option would never be exercised and would be worthless. The option price will

follow the basic valuation equation with terminal condition

,

and is given by

, (8)

8

where

F(.; ?, ?) is the noncentral chi-square distribution function with ? degrees of

freedom and non-centrality parameter ? and, r* is the critical rate below which

exercise will occur, this is, X = Z (r*, T, S).

The price of a European put option can be found by the put-call parity relation

(9)

3.2. Some Greeks

In this section we determine some sensitivity measures, commonly referred as

“Greek letters” or simply “Greeks”. These measures are vital tools for risk

management and they all represent sensitivity measures of the option price to a

small change of a given parameter. These new formulae are important for

practitioners since closed-form solutions, when available, are generally

preferable to simulations methods because their computational speed

advantage. In addition, the existence of analytical solutions allows that they can

be coded in any desired computer language such as Matlab, FORTRAN, R, or

C. In the following we give the analytical expressions for the Greek letters for

zero coupon bonds options under the CIR diffusion process.

3.2.1. Rho or Interest Rate Delta

1. Call Rho

where

9

2. Put Rho

Using put-call parity we have

3.2.2. Theta

1. Call Theta

,

where

,

.

2. Put Theta

Using put-call parity we have

3.2.3. Eta or Strike Delta

1. Call Eta

10

2. Put Eta

Using put-call parity we have

4. Computational Results

This section aims to present computational comparisons of the alternative

methods of computing the noncentral chi-square distribution function for pricing

and hedging European options on zero coupon bonds under the CIR diffusion.

Similarly to the study conducted by Larguinho et al. (2010), we examine this

CIR option pricing model using alternative combinations of input values over a

wide range parameter space. In this paper we consider that the reversion rate

can assume the values of ? = {0.35, 0.65}, the asymptotic rate, ?, is equal to

0.08, the interest rate can assume the values of r = {0.01, 0.02, ..., 0.15}, the

volatility parameter is assumed to have the following values: s = {0.04, 0.1,

0.16}. We let the market risk to be ? = {- 0.1, 0}. We use alternatives maturities

for options of T = {2, 5}, and for bonds of S = {10, 15}. The striking price of

each option contract can assume values of X = {0.25, 0.30}. These

combinations generate a set of 2, 880 probabilities distributions and 1, 440

unique contracts of zero coupon bond options.

All the calculations in this article were made using Mathematica 7.0 running on

a Pentium IV (2.53 GhZ) personal computer. Option prices and Greeks are

computed using each of the alternative algorithms for approximating the

complementary noncentral chi-square distribution. We have truncated all the

series with an error tolerance of 1E- 10. All values are rounded to four decimal

places. In order to understand the computational speed of the alternative

algorithms, we have computed the CPU times for all the algorithms using the

function Timing [.] available in Mathematica. Since the CPU time for a single

evaluation is very small, we have computed the CPU time for multiple

computations.

11

4.1. Benchmark Selection

The noncentral chi-square distribution function F(w; v, ?) require values for w, ?

, and ?. For option pricing and hedging under the CIR model w can assume

values of x1 or x2 and ? can assume values of b1 or b2. Table 1 shows the

maximum, minimum, and mean values for w, ? , and ? for the following set of

parameters used in the benchmark selection: ? = {0.15, 0.25, ..., 0.85}, ? =

{0.03, 0.06, ..., 0.15}, r = {0.01, 0.02, ..., 0.15}, s = {0.03, 0.05, ..., 0.15}, and ?

= {- 0.1, 0}. We also consider the next two set of parameters: for one bond

maturity of 2, i.e., S = 2, we have T = {1, 1.5, 1.75}, and in this case the strike

price are X = {0.90, 0.95}. For one bond maturity of S = 10 we consider T = {3,

5, 7}, and in this situation the strike price are X = {0.25, 0.35}. These

combinations of parameters produce 98, 280 probabilities1.

Table 1: Maximum, min imum, and mean values for w, ?, and ?.

Parameter Maximum Minimum Mean w 4,647.8959 0.3476 275.0822 ? 566.6667 2.1302 53.3679 ? 649.6682 0.0034 30.0593

For benchmark selection we consider the same strategies as used by

Larguinho et al. (2010). Our benchmark is the noncentral chi-square distribution

F(w; ?, ?) expressed as a gamma series as given by Equation (4). For instance,

Fraser et al. (1998) uses the gamma series method as a benchmark for

computing exact probabilities to be compared with several alternative methods

for approximating the noncentral chi-square distribution function. In this

benchmark selection we compare the gamma series method for computing the

noncentral chi-square distribution function F(w; ?, ?) based on equation (4),

with a pre-defined error tolerance of 1E- 10, against four external benchmarks

based on the Mathematica built-in function (with the call

CDF[NoncentralChiSquareDistribution[? ,?],w]), the integral representation

method based on equation (3) and using the NIntegrate [.] function available in

Mathematica, the Matlab built-in-function (with the call ncx2cdf(w,? ,?)), and the

1 We obtained these probabilities by computing the values of F (x1 ; ?, b1 ), for this set of parameters.

12

R built- in-function (with the call pchisq(w,? ,?)). The Table 2 reports the results

obtained.

Table 2: Benchmark selection.

Methods MaxAE RMSE k1 k2

GS vs CDF of Mathematica 1.29E- 04 4.13E- 07 79 1,769 GS vs Integral Representation 1.29E- 04 4.13E- 07 81 20,541 GS vs CDF of Matlab 6.46E- 11 1.16E- 11 0 0 GS vs CDF of R 6.45E- 11 1.16E- 11 0 0

This table compares the gamma series method for computing the noncentral

chi-square distribution function F(w; ?, ?) based on equation (4), with a pre-

defined error tolerance of 1E- 10, against four external benchmarks. The

MaxAE, RMSE, k1, and k2 denote, respectively, the maximum absolute error,

the root mean absolute error, the number of times the absolute difference

between the two methods exceeds 1E- 07, and the number of times a

computed probability is greater than 1.

Two test statistics obtained from computing these noncentral chi-square

probabilities are shown. The first statistic, MaxAE, is the maximum absolute

error, while the second, RMSE, is the root mean squared error. The results

show that the MaxAE and RMSE are higher for the comparison between the GS

vs CDF of Mathematica and the GS vs Integral representation, though the

number k1 is small in relative terms (in both cases, it represents about 0.08%

of the 98,280 computed probabilities). However, the number k2 is slightly higher

for the CDF of Mathematica2 (about 1.80% of computed probabilities computed)

and much higher for the Integral Representation method3 (about 20.90% of

computed probabilities).

The results comparing the GS vs CDF of Matlab and GS vs CDF of R show that

the corresponding differences are smaller and very similar (never exceeds

1E- 07). Under the selected wide parameter space we have not obtained any

probability value greater than 1 either in gamma series method, Matlab or R. In

summary, the results show that the gamma series method is an appropriate

choice for a benchmark. 2 This means care must be taken if one wants to use the CDF built-in-function of Mathematica for computing the noncentral chi-square distribution function. 3 It should be noted that here we have used the NIntegrate [.] function that is available in Mathematica which, to the authors knowledge, chooses the “best” numerical integration method for each particular case.

13

4.2. Noncentral Chi-Square Distribution and Zero-Coupon Bond Options

using Alternative Methods

Now we want to evaluate the differences in approximations of noncentral chi-

square probabilities F (w; ?, ?) for the iterative procedures of Schroder (1989)

(S89), Ding (1992) (D92) and Benton and Krishnamoorthy (2003) (BK03), and

the analytic approximations of Sankaran (1963) (S63), Fraser et al. (1998)

(FWW98) and Penev and Raykov (2000) (PR00a and PR00b) compared

against the benchmark based on the gamma series approach, and examine

these differences in options prices. We will concentrate our analysis on call

options, but the same line of reasoning applies also for put options.

Table 3 reports such comparison results using the following set of parameters:

? = {0.35, 0.65}, ? = 0.08, s = {0.04, 0.10, 0.16}, r = {0.01, 0.02, ..., 0.15}, ? =

{- 0.1, 0.0}, X = {0.25, 0.30}, T = {2, 5}, and S = {10, 15}. The MaxAE, MaxRE,

RMSE, MeanAE, and k1 denote, respectively, the maximum absolute error, the

maximum relative error, the root mean absolute error, the mean absolute error,

and the number of times the absolute difference between the two methods

exceeds 1E- 07. The second rightmost column of the table reports the CPU

time for computing 1,000 times the 2,880 probabilities4.

Table 3: Differences in approximations of noncentral chi-square probabilities

F(w; ?, ?) for each method compared against a benchmark based on the

gamma series approach.

Methods MaxAE MaxRE RMSE MeanAE CPU k1

S89 3.79E-

10

4.19E-

01

1.21E-

10

9.03E-

11

9,773.37 0

D92 9.60E-

11

1.83E-

11

6.20E-

11

5.94E-

11

9,085.95 0

BK03 4.22E-

11

1.71E-

11

4.29E-

12

1.31E-

12

103,152.7

0

0

S63 1.53E-

03

2.87E-

01

2.35E-

04

7.93E-

05

463.45 1,378

FWW98 2.95E-

01

4.16E-

01

1.42E-

02

1.96E-

03

407.15 1,282

4 The CPU time for the gamma series method is 3,303.23 seconds.

14

PR00a 1.46E-

01

2.55E-

01

4.55E-

03

2.56E-

04

1,101.77 718

PR00b 1.46E-

01

2.55E-

01

4.54E-

03

2.59E-

04

1,065.50 853

When comparing the iterative procedures based on S89, D92 and BK03

methods with benchmark we found that all are accurate for determining

noncentral chi-square probabilities that are needed for computing zero-coupon

bond options prices. However, the differences in terms of computational time

play a key role for the tradeoff between and accuracy. Of the three alternatives

methods discussed, the iterative procedure of D92 is the most efficient in terms

of running time, but for this set of parameters the D92 is about three times

slower than the benchmark gamma series (GS). In terms of accuracy the BK03

has a best performance but is the less efficient in terms of speed.

These results agree with we know about the running time needed for computing

the noncentral chi-square distribution F(w; ?, ?), which increases when

noncentrality, ?, is large. For our parameters set the ? is always less than 200,

this is, the ? is a moderate value, not a large value. For this reason we think that

is not convenient to use approximations methods.

The Table 4 analyzes the impact of these competing methods for pricing of the

call options under the CIR diffusion.

Table 4: Differences in call option prices using each alternative method for

computing the noncentral chi-square distribution compared against a

benchmark based on the gamma series approach.

Methods MaxAE MaxRE RMSE MeanAE CPU K3

S89 1.22E-

10

5.24E+0

1

2.60E-

11

1.63E-

11

9,796.68 0

D92 3.96E-

11

3.43E-

02

1.25E-

11

1.00E-

11

9,013.49 0

BK03 6.97E-

12

4.76E-

05

6.62E-

13

1.66E-

13

106,413.0

0

0

S63 1.54E-

04

1.50E+0

0

1.61E-

05

5.25E-

06

483.57 0

FWW98 1.06E-

03

3.05E-

01

3.79E-

05

7.32E-

06

424.23 0

15

PR00a 3.80E-

02

2.45E+0

0

1.51E-

03

1.06E-

04

1,114.11 7

PR00b 3.79E-

02

2.45E+0

0

1.51E-

03

1.06E-

04

1,084.18 7

k3 denote the number of times the absolute difference between the two

methods exceeds $0.01.

The results of Table 4 highlight that the iterative procedure of S89, D92 and

BK03 are all accurate for computing zero-coupon bonds options prices under

the CIR model, though the iterative procedure of D92 is still the most efficient in

terms of computation time needed for determining options prices (but it is again

about three times slower than the benchmark, since the time taken by GS

method to calculate these prices was 3,340.48 seconds) and the BK03 is still

the more accurate.

Again, the analytic approximations run quickly but have a reasonable accuracy

when all possible values of w and ? are considered since the S63 and F98

approximations returns a value of k3 = 0. For this set of parameters is not

problematic to use the S63 approximation, once the use of this approach does

not leads to large errors in zero-coupon bond options prices, since all absolute

errors are less than 0.001.

Based on the closed-form solutions for rho, theta, and eta under CIR option

pricing model, we should also consider how the different methods for computing

the noncentral chi- square distribution affect the computations of the Greeks.

Table 5 shows results for rhos, thetas and etas for the European standard call

and put options under the CIR assumption for different specifications of the

option parameters. In this analysis we will consider the following parameters

values ? = 0.2339, ? = 0.0808, s = 0.0854, ? = 0, taken from empirical work of

Chan et al. (1992), and X = 0.6, T = 4, S = 10, and r = {0.01, 0.02, ..., 0.14,

0.15}. The last five lines of the table report the CPU times for computing 1, 000

times the Greeks of the eight options contracts using the closed - form

solutions based on the gamma series method (CPU time 1), on the iterative

procedures of Schroder (1989), Ding (1992), and Benton and Krishnamoorthy

(2003) (CPU time 2-4, respectively), and via elementary differentiation of the

gamma series method through Mathematica with nmax = 1000 (CPU time 5).

Table 5: Greeks for European-style standard call and put options under the CIR

assumption.

16

European call options European put options Theta Rho Eta Theta Rho Eta

0.01 0.0137 -0.7992 -0.8185 -0.0011 0.0625 0.0524 0.02 0.0113 -0.7364 -0.7765 -0.0012 0.0814 0.0724 0.03 0.0091 -0.6755 -0.7327 -0.0012 0.0979 0.0946 0.04 0.0071 -0.6169 -0.6878 -0.0011 0.1141 0.1185 0.05 0.0053 -0.5608 -0.7023 -0.0009 0.1296 0.1437 0.06 0.0037 -0.5075 -0.5965 -0.0006 0.1441 0.1695 0.07 0.0024 -0.4573 -0.5512 -0.0001 0.1571 0.1954 0.08 0.0012 -0.4104 -0.5067 0.0004 0.1691 0.2210 0.09 0.0002 -0.3664 -0.4634 0.0011 0.1792 0.2458 0.10 -0.0006 -0.3262 -0.4218 0.0019 0.1875 0.2695 0.11 -0.0012 -0.2891 -0.3821 0.0028 0.1940 0.2917 0.12 -0.0017 -0.2553 -0.3445 0.0037 0.1986 0.3122 0.13 -0.0020 -0.2244 -0.3093 0.0047 0.2016 0.3308 0.14 -0.0023 -0.1967 -0.2764 0.0058 0.2028 0.3474 0.15 -0.0024 -0.1717 -0.2460 0.0068 0.2024 0.3620

CPU time 1 35.54 33.10 18.72 39.38 36.86 19.63 CPU time 2 29.20 26.77 15.58 32.40 29.41 16.07 CPU time 3 27.74 25.51 14.95 31.47 28.64 15.34 CPU time 4 1,027.03 1,024.16 514.37 1,009.12 1,007.05 501.68 CPU time 5 61,622.89 26,406.60 15,298.12 60,168.17 26,757.07 14,962.99 Through the analysis of the Table 5, we confirm that the computational time

required for the calculation of Greeks letters is clearly inferior to that obtained

using an elementary differentiation of the gamma series method through

Mathematica with nmax = 1000. However, we note that for this particular set of

values the method proposed by Ding (1992) is slightly faster than the

benchmark gamma series.

We can conclude that the computation time for computing analytic Greeks falls

substantially, which is extremely relevant when one need to design hedging

strategies through time.

5. Conclusions

In this article, we compare the performance of alternative algorithms for

computing the noncentral chi-square distribution function in terms of accuracy

and computation time for evaluating option prices and some Greeks under the

CIR model. We find that the gamma series method and the iterative procedures

of Schroder (1989), Ding (1992), and Benton and Krishnamoorthy (2003) are all

17

accurate over a wide range of parameters, though presenting significant speed

computation differences. For our set of parameters values the benchmark

gamma series method algorithm is clearly the most efficient in terms of

computation time needed for determining zero-coupon bond option prices under

the CIR assumption, followed by the Ding (1992) algorithm. Finally, we present

closed-form solutions for computing some possible Greeks measures under the

CIR option pricing model.

18

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