Article
Reference
Path dependent options on yields in the affine term structure model
LEBLANC, Boris, SCAILLET, Olivier
LEBLANC, Boris, SCAILLET, Olivier. Path dependent options on yields in the affine term structure model. Finance and stochastics , 1998, vol. 2, no. 4, p. 349-367
DOI : 10.1007/s007800050045
Available at:
http://archive-ouverte.unige.ch/unige:41793
Disclaimer: layout of this document may differ from the published version.
1 / 1
c Springer-Verlag 1998
Path dependent options on yields in the affine term structure model
Boris Leblanc1, Olivier Scaillet2,?
1 Banque Nationale de Paris, Universit´e Paris VII and CREST Laboratoire de Finance Assurance, Bˆatiment Malakoff 2 – Timbre J320, 15 Boulevard Gabriel P´eri, F-92245 Malakoff Cedex, France
2 Institut d’Administration et de Gestion and D´epartement des Sciences Economiques, Universit´e Catholique de Louvain, 3 place Montesquieu, B-1348 Louvain-la-Neuve, Belgique
Abstract. We give analytical pricing formulae for path dependent options on yields in the framework of the affine term structure model. More precisely, Eu- ropean call options such as the arithmetic average call, the call on maximum and the lookback call are examined. For the two last options approximate formulae using the law of hitting times of an Ornstein-Uhlenbeck process are proposed.
Numerical implementation is also briefly discussed and results are given in the case of the arithmetic average option.
Key words: Term structure, path dependent options, affine model, hitting time, Laplace transform
JEL classification: E43, G13
Mathematics Subject Classification (1991): 60E10, 60G17, 60J70, 65U05
1 Introduction
Nowadays, banks and other financial intermediaries are aware that their liquidity needs and their risk-return preferences could be more effectively dealt with by actively managing their assets and liabilities. Often, interest rate derivative se- curities such as bond or yield options are used to influence their risk exposure.
?We would like to thank the editor Dieter Sondermann and an anonymous referee for construc- tive criticism. We also thank Marc Yor for his many helpful advices and Dilip Madan for useful suggestions. We have received fruitful comments from Christine Demol, Paul-Henri Heenen, Chris- tian Gouri´eroux, participants at AFFI meeting Bordeaux 95, ESWC Tokyo 95, the Workshop on the Mathematics of Finance Montreal 96, and seminar participants at ESSEC, CREST, HEC and CORE.
All remaining errors are our own. The second author gratefully acknowledges financial support from ICM, Belgium, and by the grant “Actions de Recherche Concert´ee” 93/98-162 from the Ministry of Scientific Research (Belgian French Speaking Community).
Manuscript received: September 1996; final version received: October 1997
For example, bond put options or yield call options can be bought to protect a portfolio in case of rising interest rates. Besides, sophisticated credit loans often involve hidden interest rate options.
In this paper, our aim is to price some of these interest rate options namely European path dependent options on yields. Options on yields have already been considered by Longstaff (1990) in the European case and by Chesney et al.
(1993) in the American one. Here we are mainly concerned with three types of path dependent options : arithmetic average call option (or Asian options), call on maximum and lookback call. For these options we give analytical pricing formulae.
Such path dependent options have been priced in the literature about options on stocks in the Black-Scholes (1973) world where stock prices are supposed to follow a Geometric Brownian Motion.
Analytical solutions for options based on the arithmetic average of stock prices have been proposed by Geman and Yor (1992, 1993). This exact approach has also been applied to the pricing of insurance futures contracts by Cummins and Geman (1993). Along this analytical approach, a number of other approaches coexist in the literature such as numerical approaches or price approximations : Carverhill and Clewlow (1990); Kemna and Vorst (1990); Ruttiens (1990); Turn- bull and Wakeman (1991); Levy (1992); Bouaziz et al. (1994); Rogers and Shi (1995).
Others path dependent options on stock prices such as the call on maximum and the lookback call have been priced by : Goldman et al. (1979a,b); Merton (1973); Conze and Viswanathan (1991).
In order to answer our pricing problem, we consider the affine class of one factor term structure models with time invariant parameter studied in a more general framework by Brown and Schaefer (1991); Frachot and Lesne (1993a,b) and Duffie and Kan (1996). Particular cases are the Cox et al. (CIR) (1985) and Vasicek (1977) models. In these models, the yield is an affine function of the instantaneous interest rate and this feature allows to use extensively the distribution properties of the instantaneous interest rate (see Leblanc and Scaillet (1995) for applications to the pricing of forward and futures contracts and Scaillet (1996) to the valuation of compound and exchange options).
Section 2 briefly reviews the continuous time setting underlying the pricing.
In Sect. 3 analytical formulae for European path dependent options on yields are derived. We examine arithmetic average options, options on maximum and lookback options and we discuss the problems of the numerical implementation of these formulae. Some pratical results for the European arithmetic average call option are presented. Section 4 concludes. All proofs and technical details are gathered in appendices.
2 The framework
We first introduce some notations. We denote by B (t,t +τ) the price at date t of a discount bond of maturity t +τ, i.e. the price of the asset delivering one
monetary unit at time t +τ (τ is independent of t ). The yield corresponding to this bond namely the yield at date t with time to maturityτ is defined by :
Y (t,t +τ) =−1
τlog B (t,t +τ).
Here we focus on models belonging to the affine class of one factor term structure models with time invariant parameter (see Brown and Schaefer 1991;
Frachot and Lesne 1993a,b; Duffie and Kan 1996). In these models the yield Y (t,t +τ) is an affine function of the instantaneous interest rate rt :
Y (t,t +τ) = 1
τ[A(τ)rt+ b(τ)].
Under the risk neutral probabilityQ, the instantaneous interest rate rt satisfies the SDE :
drt = (φ−λrt)dt +p
αrt+βd ˜Wt,
where ˜Wt is a standard Brownian Motion. The initial value of the process is denoted r0.
This model is called the one factor affine term structure model or the extended CIR model since takingβ= 0 leads to the CIR (1985) model. The Vasicek (1977) model is obtained by takingα= 0.
Table 1 in which the following notations are adopted : µ = λ
r 1 + 2α
λ2, k = µ−λ
µ+λ, φ˜ = φ+λβ α,
gives the different forms of the coefficients A(τ) and b(τ) (these coefficients are easily deduced from solving a PDE, see e.g. Duffie and Kan 1996).
3 Path dependent options on yields
In this paper we aim to price at date 0 assets which deliver at date T a cash-flow depending on the evolution between 0 and T of the yield with time to maturityτ:
hT = hT(Y (t,t +τ)) = hT(1
τ[A(τ)rt+ b(τ)]), t ∈[0,T ].
Due to the affine structure of the model, this is thus equivalent to valuate path- dependent options on the instantaneous interest rate.
Using the standard framework of Artzner and Delbaen (1989); Heath et al.
(1992); and El Karoui et al. (1993) for term structure modelling, we have that the price of such assets may be computed from :
Table 1. Forms of A(τ) and b(τ) in the different models
A(τ) b(τ)
Affine model 1 + k
µ
1−e−µτ 1 + ke−µτ
2 ˜φ α log
1 + ke−µτ 1 + k
+
φ˜(µ−λ) α −β
α
τ+β αA(τ) CIR model
1 + k µ
1−e−µτ 1 + ke−µτ
2φ α log
1 + ke−µτ 1 + k
+φ(µ−λ) α τ
Vasicek model 1−e−λτ
λ
2λφ−β 2λ2
τ−1−e−λτ λ
+ β 4λ
1−e−λτ λ
2
H (hT,0,T ) = ErQ0[e− RT
0 rsds
hT], (1)
where ErQ0 denotes the conditional expectation under the risk-neutral probability Q.
From equation (1), the price of the discount bond B (0,T ) is given by : B (0,T ) = H (1,0,T ) = ErQ0[e−
RT 0 rsds
].
From this expression, we may determine which distribution the random variable XT =RT
0 rudu follows under the risk neutral probabilityQ. We denote f the pdf of XT.
Property 1 : Let XT =RT
0 rudu where rt is solution of the SDE : drt = (φ−λrt)dt +p
αrt+βd ˜Wt,
and starts from r0. The Laplace transform of the distribution of XT is given by : ErQ0[e−aXT] = B0(0,T ), (2) where B0(0,T ) = ErQ0
0
[e− RT
0 ru0du
] is the price of a discount bond with maturity T depending on the evolution of the instantaneous interest rate rt0 starting from r00 = ar0and satifying :
drt0 = (φ0−λrt0)dt +p
α0rt0+β0d ˜Wt, (3) withφ0= aφ, α0= aαandβ0 = a2β.
Proof. See Appendix 1.
The discount bond price B0(0,T ) is easily derived using Table 1 and the reparametrisation induced by (3), and from equation (2), we see that it is equal to a function of a :
F (a) = Z ∞
0
e−axf (x )dx.
This Laplace transform can be inverted by numerical procedures in order to get the pdf of XT (see 3.4 below for a review of available methods).
Moreover if an explicit expression for the Laplace transform ErQ
0[e−aXT] is known it is possible to write an expression for ErQ0[(XT −K )+], and for ErQ
0[e−XT(XT −K )+], as a one dimensional integral involving this known ex- pression. Indeed we have :
e−aXT = Z
IR
(K −XT)+a2e−aKdK, (4) which leads to :
ErQ0[e−aXT] = Z
IRErQ0[(K−XT)+]a2e−aKdK. (5) Therefore by inverting (5) and using :
ErQ0[(XT−K )+] = ErQ0[(K−XT)+] + ErQ0[XT]−K,
we get the first stated result (let us remark that we use (K −XT)+in (4) instead of (XT−K )+ for integrability conditions). Finally after multiplying both sides of equation (4) by e−XT, similar computations give the second result.
Let us now turn to the option pricing. We first consider arithmetic average call on yields (Asian options) before studying the call on maximum and the lookback call.
3.1 European arithmetic average option
The price of a European call option on the arithmetic average of yields with time to maturityτ from time 0 to time T and strike price K is given by :
C (1 T
Z T 0
Y (u,u +τ)du,0,T,K ) = ErQ0[e− RT
0 rsds
(1 T
Z T 0
Y (u,u +τ)du−K )+]. From the affine form of the yield : Y (T,T +τ) = 1
τ[A(τ)rT+ b(τ)], we deduce : C (1
T Z T
0
Y (u,u +τ)du,0,T,K ) = A(τ) τ C (1
T Z T
0
rudu,0,T,k ),
with C (1 T
Z T 0
rudu,0,T,k ), the price of an option on the average of the instan- taneous interest rate with strike price k = τK−b(τ)
A(τ) . Hence the price of the
yield option may be rewritten in terms of the price of an instantaneous interest rate option with a modified strike price.
Thanks to Property 1 after inversion of (2) the option price is computable since it writes :
C (1 T
Z T 0
rudu,0,T,k ) = Z ∞
Tk
e−x(1
Tx−k )f (x )dx. (6) Let us remark that if k is negative, the integration has only to be performed from 0 to ∞ in the CIR and affine models (the density is defined on positive values) which means that the option will be exercised with probability one.
3.2 European option on maximum
The price at date 0 of a European call option on maximum of maturity T and strike price K is given by :
C (sup
[0,T ]
Y (u,u +τ),0,T,K ) = ErQ0[e− RT
0 rsds
(sup
[0,T ]
Y (u,u +τ)−K )+]. Again we deduce from the affine structure of the model an expression involving an instantaneous interest rate option :
C (sup
[0,T ]
Y (u,u +τ),0,T,K ) = A(τ) τ C (sup
[0,T ]
ru,0,T,k ).
In order to compute this price, we need the following property (see Ito and McKean (1974) p. 145 for related results).
Property 2 : Let rt be a solution of the SDE :
drt=µ(rt)dt +σ(rt)d ˜Wt, (7) starting from r0. Let Ta = inf{t ; rt ≥ a} and let V be a bounded function on [0,a] such thatAV =γV , where :
A= 1
2σ2(r )d2
dr2 +µ(r )d dr is the infinitesimal generator associated to (7).
Then, we have :
ErQ0[e−γTa1ITa<∞] = V (r0)
V (a). (8)
Proof. See Appendix 2.
This property gives the Laplace transform of the distribution of a hitting time of a diffusion characterized by its infinitesimal generator and so allows one to recover the density of a hitting time by inverting equation (8).
Let us now examine the affine model.
Property 3 : Let rt be a solution of the SDE : drt = (φ−λrt)dt +p
αrt+βd ˜Wt, (9) starting from r0.
Then, we have :
ErQ0[e−γTa] =M (γλ,(φ+λβα)α2,2λ(r0+αβ/α)) M (γλ,(φ+λβα)α2,2λαa) .
Proof. See Appendix 3.
This property characterizes the distribution of the hitting time of level a by the process rt involved in the affine model by means of a Laplace transform.
We are now able to give an analytical pricing formula for the European call on maximum which takes the form of a Laplace transform. Indeed rewriting:
(sup
[0,T ]
ru −k )+ = Z ∞
k
1supI [0,T ]ru>vdv, we deduce :
Z ∞
0
e−aT¯ C (sup
[0,T ]
ru,0,T,k )dT = ErQ0[ Z ∞
k
dv Z ∞
0
dT e−aT¯ − RT
0 rsds
1supI [0,T ]ru>v]
= ErQ
0[ Z ∞
k
dv Z ∞
Tv
dT e−aT¯ − RT
0 rsds
]
= ErQ
0[ Z ∞
k
dv Z ∞
Tv
dT e−a(T¯ −Tv)−aT¯ v− RTv
0 rsds−RT Tvrsds
]
= Z ∞
k
dvErQ0[e−aT¯ v− RTv
0 rsds
ErQ0[ Z ∞
Tv
dT e−a(T¯ −Tv)− RT
Tvrsds
|Tv]]
= Z ∞
k
dvErQ
0[e−aT¯ v− RTv
0 rsds
]EvQ[ Z ∞
0
dTe−aT¯ − RT
0 rsds
],
where Tv= inf{t ; rt > v}. The last equality is due to the strong Markov property and time homogeneity of the instantaneous interest rate process. The new starting point of the instantaneous interest rate is v in the second expectation. It still remains to compute the two expectations.
For the first one, we can use the following change of probability measure :
dQ∗ dQ = e−
RTv
0 θ√rsd ˜Ws−12
RTv 0 θ2rsds
= e−√1α RTv
0 θ[drs−(φ+λβα−λrs)ds]−12
RTv 0 θ2rsds
. The instantaneous interest rate rt satisfies under the probability Q∗ :
drt = (φ+ λβ
α −(λ+θ√
α)rt)dt +√αrtd ˜Wt∗, (10) where ˜Wt∗ is a standard Brownian Motion under Q∗. By choosing θ such that
√λθα+ θ22 = 1, we obtain :
ErQ0[e−aT¯ v− RTv
0 rsds
] = e
θ(v−r0)√ α ErQ0∗[e
−( ¯a + (φ+λβα)θ
√α )Tv ], which can be computed thanks to Property 4 with γ= ¯a +(φ+√λβα)θ
α , a = v, and the adequate reparametrization implied by (10).
The second expectation is equal to : Z ∞
0
dTEvQ[e−aT¯ − RT
0 rsds
] = Z ∞
0
dTe−aT¯ Bv(0,T ),
where Bv(0,T ) is the discount bond price corresponding to an instantaneous interest rate process starting from the initial valuev.
Observe that similar computations could be made for any one factor term structure model since Property 2 is not particular to the affine model. In the case of the Ornstein-Uhlenbeck process it is also possible to use another approach based on the law of hitting times of this process in order to approximate the option price. Indeed if we replace the stochastic discount factor by a deterministic one we get the following approximation :
C (sup
[0,T ]
Y (u,u +τ),0,T,K ) ≈ ErQ
0[e−r0T(sup
[0,T ]
Y (u,u +τ)−K )+]
= e−r0T Z ∞
0
(A(τ)
τ x +b(τ)
τ −K )+dP (sup
[0,T ]
ru >k ),
with : k =τK−b(τ) A(τ) .
The computation of P (sup[0,T ]ru >k ) is given in Appendix 4.
3.3 European lookback option
The price of a lookback call is given by : C (Y (T,T +τ),0,T, inf
[0,T ]Y (u,u +τ)) = ErQ0[e− RT
0 rsds
(Y (T,T +τ)
−inf
[0,T ]Y (u,u +τ))],
from which we get :
C (Y (T,T +τ),0,T, inf
[0,T ]Y (u,u +τ)) = A(τ)
τ C (rT,0,T, inf
[0,T ]ru). For the lookback call, we can separate the price in two expectations :
C (rT,0,T,inf
[0,T ]ru) = ErQ0[e− RT
0 rsds
rT]−ErQ0[e− RT
0 rsds
[0inf,T ]ru].
The first expectation follows from a standard computation (see e.g. Leblanc and Scaillet 1995) and is equal to :
B (0,T )LT(4φ λ +ζT), with :
LT = α 4µ
(1−e−µT)(1 + k ) (1 + ke−µT) = α
4A(T ), ζT = 4(r0+β/α)µ
α
(1 + k ) (1 + ke−µT)(eµT−1).
For the second expectation we may proceed as for the call on maximum : Z ∞
0
dTe−aT¯ ErQ0[e− RT
0 rsds
inf
[0,T ]ru]
= Z ∞
0
dvErQ0[e−aT¯ v− RTv
0 rsds
] EvQ[
Z ∞
0
dTe−aT¯ − RT
0 rsds
],
where Tv = inf{t ; rt ≤ v}. The first expectation can be computed with the same change of probability as in the maximum case but here using successively Properties 2 and 3 with Ta = inf{t ; rt≤a}and assuming V bounded on [a,∞[.
Since we need a function V bounded on [a,∞[, we have to retain Kummer’s function U instead of M (see Abramowitz and Stegun (1965), Eq. 13.1.4 and 13.1.8). It has to be noted that an approximation similar to the one proposed for the call on maximum can also be derived in the case of the Ornstein-Uhlenbeck process.
3.4 Numerical implementation
In this section, we discuss some problems linked to the numerical inversion of the Laplace transform. We examine in detail the European arithmetic average call where equation (2) has to be inverted in order to give the density involved in the pricing formula.
As quoted by Davies and Martin (1979), “Laplace transform inversion is still an art more than a science”. Indeed, Laplace transform inversion is known to be
unstable (see also Bellman et al. 1966, p. 17-19) and to be an ill-posed problem in the sense of Hadamard (see Fioravanti et al. 1984) and no clear cut solution to this problem has already been found. So, following the recommendations of Davies and Martin (1979), we have tried several methods. For each methods, we have verified that the inverse Laplace transform is a density function by checking visually the form of the density (the form must be closed to a noncentral chi-square distribution) and by integrating the function using a truncated Gauss- Legendre quadrature (equation 25.4.30 of Abramowitz and Stegun 1965) with 96 points. This same rule has been used in computing the integral (6) which gives the option price. The truncation points have been chosen on the ground of the form of the density function. Of course more precision could be obtained by dividing the contour of the integral into segments around the mode of the distribution.
We have examined eight option prices corresponding to the CIR model and the choice of parameter values given in Table 2. We have taken r0= 0.10. Hence options are out of the money since Y (0,10) = 0.099905,Y (0,10) = 0.0917731 for α = 0.0002, α = 0.02 and Y (0,0.25) = 0.0999998,Y (0,0.25) = 0.0999799 for α= 0.0002, α = 0.02. The results given below provide some guidelines for the selection of one particular method, bearing in mind that in practice on the financial markets a trade-off has to be made between speed and accuracy. The accuracy needed does not exceed in general 10−4 (the bid ask spread).
Table 2. Parameters for Asian options in the CIR model
Case φ λ α T τ K
1 0.02 0.2 0.0002 1 10 0.1
2 0.02 0.2 0.02 1 10 0.1
3 0.02 0.2 0.0002 0.25 10 0.1
4 0.02 0.2 0.02 0.25 10 0.1
5 0.02 0.2 0.0002 1 0.25 0.1
6 0.02 0.2 0.02 1 0.25 0.1
7 0.02 0.2 0.0002 0.25 0.25 0.1
8 0.02 0.2 0.02 0.25 0.25 0.1
The problem of the inversion of the Laplace transform is to evaluate : f (x ) = 1
2πi Z
B
eaxF (a)da, (11)
where B is the usual Bromwich contour which is conventionally defined from c−i∞to c + i∞with c chosen in order to have all the singularities of F (a) to the left of the contour. Moreover c belongs to the region of the complex a-plane Re(a)>c0for which the function F (a) is analytic. Alternative forms of (11) are given by :
f (x ) = 1 2πi
Z ∞
−∞
e(c+ia)xF (c + ia)da
= 2ecx π
Z ∞
0
Re[F (c + iu)] cos(ux )du,
where Re[F (.)] denotes the real part of F (.). The first methods are based on the computation of a sample. They are respectively due to ter Haar (1951); Schapery (1962); Widder (1964) and Stehfest (1970). The first two of them give cumulative density functions which exceed one. The method of Widder (1964) corresponds to the following equation :
f (x )≈ (−1)nnn+1(n!)−1x−(n+1)F(n)(n
x), (12)
where F(n)(a) denotes the nth derivative of F (a) with respect to a. This method gives densities (the resulting function integrates to one) with a form of a very peaked noncentral chi-square distribution or the distribution of a stopping time (as illustrated by Fig. 1) and seems to give option prices which make sense as can be seen from Table 3.
Fig. 1. pdf : method of Widder (1964) (n = 2, T = 1,α= 0.0002)
Table 3. Asian options in the CIR model
Case n = 1 n = 2 n = 3 n = 5
1 0.0365451 0.0249558 0.0182874 0.0082519 2 0.0292038 0.0189691 0.0133078 0.0121406 3 0.0807969 0.0381072 0.0240933 0.0145258 4 0.0678984 0.0300541 0.0177808 0.0095593 5 0.0826462 0.0564752 0.041413 0.0275278 6 0.0817052 0.0563472 0.0419683 0.0288775 7 0.18264 0.0862042 0.0545448 0.0329291 8 0.182206 0.0863378 0.0549048 0.0335003
The method of Stehfest (1970) gives negative values for some points. We have tested other methods such as the method of Weeks (1966) based on La- guerre polynomials or the methods of Dubner and Abate (1968) and Silverberg (1970) based on Fourier series. They all report negative values for some points and the form obtained for the density depends very much on the choice of the parameters characterising the method. We have not used the methods of Piessens and Branders (1971) and Piessens (1972) which are based respectively on La- guerre polynomials and Chebyshev polynomials since they require the knowledge
of b such that lim
a→∞
F (a)
a−b = 1. Indeed in our case the behavior of F (a) when a goes to∞is given by :
alim→∞
F (a) exp(−√
a
√2α(r0+τφ)
α )
= exp(λr0 α +2φ
α log 2 +τφλ α ). This equation is of the form :
F (a)'Ke−d√a. (13)
and gives the behavior of the inverse Laplace transform f (x ) when x goes to 0.
Indeed Laplace transforms and their inverses share the property :
alim→∞
F (a)
G(a) = 1 iff lim
x→0
f (x ) g(x ) = 1.
The inverse Laplace transform of function (13) when K = 1 is known to be a density (density of the stopping time of a Brownian Motion) equal to :
f (x ) = d 2√
πx3e−d 24x. (14)
Since the density function derived with the Widder method bears some resem- blance with the form of the density (14), we have tried to fit with a non linear least square procedure F (a) by function (13) taking K = 1 and d as a parame- ter. We have also tried to fit the Laplace transform of the noncentral chi-square distribution. Unfortunately, both procedures give very poor fits.
In order to check if the order of magnitude of Asian option prices in the CIR model is comparable with similar options, we compute Asian and Euro- pean option prices given by the Vasicek model. We choose in the Vasicek model the same parameter values for the drift and we take as diffusion parameter val- ues : β = 0.00002 and β = 0.002 (the stationary distributions under Q of the instantaneous interest rate for both models have then the same first and sec- ond moments). These option prices are easily computed since in the case of the Ornstein-Uhlenbeck process we have that rT under QT and RT
0 rsds under Q follow Gaussian processes given by :
rT ∼N
φA(T ) + e−λTr0−β
λ(A(T )−1−e−2λT
2λ ), β1−e−2λT 2λ
, Z T
0
rsds ∼N φ
λT + (r0−φ
λ)A(T ), β λ2T − β
λ2A(T )− β 2λA(T )2
. The yields for this model are Y (0,10) = 0.0999048,Y (0,10) = 0.09044811 for β = 0.00002, β = 0.002 and Y (0,0.25) = 0.0999998,Y (0,0.25) = 0.0999799 for β = 0.00002, β = 0.002. We also compute European option prices in the CIR model. Table 4 gives the option prices for the different cases. By comparing the results of Table 3 and Table 4, we can remark that the prices obtained for Asian options in the CIR model are probably too high (the prices of the European
Table 4. Asian and European options in Vasicek and CIR models Case Asian (V.) European (V.) European (CIR)
1 0.000331815 0.00059001 0.000589524 2 0.000872459 0.00284672 0.00282785 3 0.000169869 0.00032182 0.000322069 4 0.000088411 0.00074882 0.000813155 5 0.000847218 0.00142596 0.0014257
6 0.0081878 0.013931 0.01378892
7 0.000480616 0.000831179 0.000827467 8 0.00478025 0.00824066 0.00822438
options for the two models are rather close). The prices of Table 3 also seem to decrease when the derivative order in equation (12) increases and therefore they likely constitute an upper bound.
More recently Abate and Whitt (1992) have proposed a method especially designed for highly decreasing function F (a). The inverse Laplace transform is given by :
f (x ) = lim
n−→∞fn(x ), with :
fn(x ) =eA/2 x
1
2Re[F ( A 2x)] +
Xn
k =1
(−1)kRe[F (A + i 2kπ 2x )]
! .
We have taken A = 40 and n = 1000 and the results of the inversion are shown in Table 5. As it is seen the results are very satisfactory : the density integrates to one, the bond prices computed with the analytical formula B (0,T ) and the numerical procedureR∞
0 e−xf (x )dx are very close and the option prices are more sensible.
Table 5. Inversion of Abate and Whitt and prices of Asian options in the CIR model Case R∞
0 f (x )dx R∞
0 e−xf (x )dx B (0,T ) Asian (CIR)
1 0.99997 0.90425 0.90509 0.000332
2 1.00000 0.90509 0.90509 0.000949
3 0.98546 0.97514 0.975315 0.000170
4 1.00000 0.97531 0.975315 0.000120
5 0.99997 0.90462 0.90509 0.000842
6 1.00000 0.90509 0.90509 0.008131
7 0.9819 0.97329 0.975315 0.00049
8 1.00000 0.975315 0.975315 0.008131
4 Concluding remarks
We have proposed some analytical forms for different path dependent securities on yields in the framework of the affine term structure model. In the case of
the European arithmetic average option a density has to be recovered by the inversion of a Laplace transform and for the other path dependent options, prices themselves are expressed under the form of Laplace transforms. Numerical in- version procedures of the Laplace transform thus have to be performed. We have discussed the problems linked with such procedures in the case of the numerical inversion of the density involved in pricing Asian options on yields and we hope that we have brought some elements of answers. However further research is still requested in this area. Beside these numerical points, the result concerning the density of hitting times of the Ornstein-Uhlenbeck process can lead to new theoretical developments in the Vasicek model.
Appendix 1: Distribution of XT =RT 0 rudu We have :
ErQ
0[e−a RT
0 rudu
] = ErQ0
0
[e− RT
0 ru0du
] with :
rt0 = art, which leads to :
drt0 = (φ0−λrt0)dt +p
α0rt0+β0d ˜Wt, withφ0 = aφ, α0= aαandβ0= a2β.
Therefore we deduce immediately that : ErQ0
0
[e−a RT
0 rudu
] = B0(0,T ).
where the price of the discount bond B0(0,T ) is obtained by replacing φ,α,β byφ0,α0,β0 in the coefficients of Table 1, and rt by rt0.
Appendix 2: Distribution of a hitting time We have :
d (V (rt)e−γt) = e−γt[V0(rt)drt+1
2V00(rt)σ2(rt)dt−γV (rt)dt ]
= e−γt[AV (rt)dt +σ(rt)V0(rt)d ˜Wt−γV (rt)dt ],
= e−γt[σ(rt)V0(rt)d ˜Wt].
Hence, V (rt∧Ta)e−γ(t∧Ta), where t∧Ta = min(t,Ta), is a martingale and, ErQ0[V (rt∧Ta)e−γ(t∧Ta)] = V (r0).
Decomposing this expectation on the events where Ta is finite and on the events where Ta is infinite and using the boundedness of the function V , we get :
t−→∞lim ErQ
0[V (rt∧Ta)e−γ(t∧Ta)] = ErQ
0[V (rTa)e−γTa1ITa<∞], which ends the proof.
Appendix 3: Distribution of a hitting time in the affine model In the affine model after a change of variable, we have :
A= (φ+λβ
α −λr )d dr +α
2rd2 dr2, and therefore :
(φ+ λβ
α −λrt)V0(rt) +α
2rtV00(rt) =γV (rt).
Now, using the change of variable z = 2λαrt and equations 13.1.1 (Kummer’s equation) and 13.1.11 (complete solution) of Abramowitz and Stegun (1965), we obtain the complete solution :
V (rt) = C1U (γ
λ,(φ+λβ α )2
α,2λrt
α ) + C2M (γ
λ,(φ+ λβ α)2
α,2λrt
α ),
where C1 and C2 are arbitrary constants and M,U are Kummer’s functions.
As in practical cases we have 2φ/α > 1, M is the only bounded solution on [0,a] (see Abramowitz and Stegun (1965) eq. 13.5.5 - 13.5.12) and we get from Property 2 :
ErQ
0[e−γTa1ITa<∞] = M (γλ,(φ+ λβα)α2,2λ(r0α+β/α)) M (γλ,(φ+ λβα)α2,2λαa) . Lettingγ goes to zero, we deduce from the properties of M :
ErQ0[e−γTa1ITa<∞] = ErQ0[I1Ta<∞] = P (Ta <∞) = 1, which leads to the result.
Appendix 4: Computation of P (sup[0,T ]ru > k ) The Ornstein-Uhlenbeck process is given by
drt = (φ−λrt)dt +p βd ˜Wt,
and we denote the hitting time Tk = inf{t ; rt = k}. We adopt the change of time
˜rt = rt
β, which gives :
d ˜rt = ( ˜φ−λ˜˜rt)dt + d ˜˜Wt, (15) with ˜φ=φ/β, ˜λ=λ/β and ˜Tk = Tk/β= inf{t ; ˜rt = k}. Thanks to the Girsanov Theorem, we can pass from the process (15) to a Brownian Motion (see Leblanc 1994) thanks to the change of probability :
