ON PROPERTIES OF ANALYTICALLY SOLVABLE FAMILIES OF LOCAL VOLATILITY DIFFUSION MODELS

ON PROPERTIES OF ANALYTICALLY SOLVABLE FAMILIESOF LOCAL VOLATILITY DIFFUSION MODELS

GIUSEPPE CAMPOLIETI AND ROMAN MAKAROV

MATHEMATICS DEPARTMENT, WILFRID LAURIER UNIVERSITY

75 UNIVERSITY AVENUE WEST, WATERLOO, ONTARIO N2L 3C5, CANADA

E-MAIL: [email protected], [email protected]

Abstract. We present some recent developments in the construction and

classification of new analytically solvable one-dimensional diffusion models for

which the transition densities and other quantities that are fundamental to

derivatives pricing are represented in closed form. Our approach is based on

so-called diffusion canonical transformations that allow us to uncover new mul-

tiparameter processes that are mapped onto various simpler diffusions. Using

an asymptotic analysis, we arrive at a rigorous endpoint boundary classifi-

cation as well as a characterization with respect to probability conservation

and the martingale property of the newly constructed diffusions. The ap-

proach is applicable to a fairly general class of nonlinear volatility diffusions.

Specifically, we analyze and classify in detail four main families of driftless

regular diffusion models that arise from the underlying squared Bessel process

(the Bessel family), CIR process (the confluent hypergeometric family), the

Ornstein-Uhlenbeck diffusion (the OU family) and the Jacobi diffusion (hy-

pergeometric family). We show that the Bessel family is a superset of the

constant elasticity of variance (CEV) model without drift. The Bessel family,

in turn, is generalized by the confluent hypergeometric family. For these two

families we find further subfamilies of conservative strict supermartingales and

martingales with an absorbing endpoint. For the new classes of absorbed dif-

fusions we also derive analytically exact first-hitting time densities for paths

hitting an absorbing endpoint boundary, with densities given in terms of gen-

eralized inverse gaussians and extensions. As for the two other new models,

we show that the OU family of processes are conservative strict martingales,

whereas the Jacobi family are non-conservative (absorbed) non-martingales.

Considered as asset price diffusion models, we also show that these models

demonstrate a large range of local volatility shapes including pronounced skew

and smile patterns. A discussion of applications to option pricing concludes

the paper.

0This research was supported by a Discovery Research Grant of the Natural Sciences and

Engineering Research Council of Canada (NSERC).

Date: October 28, 2008.

Key words and phrases. Option pricing, state-price density, first-hitting time, local volatility

models, implied volatility smile, hypergeometric, confluent hypergeometric, Bessel process, CIR

process, OU process, Jacobi process, CEV model.

1

2 Giuseppe Campolieti and Roman Makarov

1. Introduction

In the study of stochastic processes and their applications to finance, geomet-ric Brownian motion (GBM) is the simplest model used for continuous-time assetpricing. The local volatility in GBM is constant, i.e., the volatility or diffusioncoefficient is a linear function of the underlying asset price, as is the drift coef-ficient. For a long period of time the GBM model has stood as one of the fewknown continuous diffusion models which admits exact analytically tractable tran-sition probability density functions (i.e., state-price densities or pricing kernels) andclosed-form pricing formulas for various standard, barrier and lookback European-style options. As is well known, the lognormal transition density for the unrestrictedasset price process (with positive half-line as the state space where the origin andinfinity are natural boundaries) describes a process that conserves probability and,moreover, the discounted (forward) price process obeys the martingale property ina risk-neutral measure. These are desirable probabilistic features for arbitrage-freerisk-neutral asset pricing. However, the lognormal assumption of asset price re-turns disagrees with most empirical evidence. A very notable defect of the GBMmodel is the fact that, by construction, the implied volatility surface is completelyflat while market observed implied volatility surfaces exhibit various pronouncedvolatility smiles and skews. These and other important market observations havespawned the development of several pricing models based on alternative stochasticprocesses.

Among the many alternative models proposed and used for option pricing andother applications in the mathematical finance literature are jump-diffusion mod-els (Anderson and Andreasen 2000, Cont and Tankov 2004), stochastic volatilitymodels (Heston 1993, Hobson and Rogers 1998), and local (i.e., level- or state-dependent) volatility diffusion models (Albanese et al. 2001, Kuznetsov 2004, Al-banese and Campolieti 2005, Campolieti and Makarov 2007; 2008a, Cox 1975, Coxand Ross 1976, Davydov and Linetsky 2001, Dupire 1994, Linetsky 2004a). In therealm of local volatility diffusion models, the constant elasticity of variance (CEV)model of Cox (1975) and Cox and Ross (1976), for instance, has provided a first stepinto the introduction of a nonlinear state-dependent volatility model that exhibitsan implied volatility (half) smile as function of strike. The CEV model has a powerlaw local volatility function with two adjustable parameters and a linear drift func-tion, and admits closed-form pricing formulas for plain vanilla, barrier and lookbackEuropean options (see Cox (1975), Cox and Ross (1976), Davydov and Linetsky(2001), Linetsky (2004b)). Spectral expansions and Laplace transform techniquesare very useful in deriving analytical pricing formulas and transition densities for

On Properties of Analytically Solvable Families 3

the CEV process and other time-homogeneous models with more complex nonlinearlocal volatility functions.

In recent years, a new approach was introduced for uncovering various new fam-ilies of exactly solvable multiparameter local volatility diffusion models (Albaneseet al. 2001, Kuznetsov 2004, Albanese and Campolieti 2005). For these familiesthe transition probability densities, or fundamental solutions to the correspondingKolmogorov PDE subject to appropriate boundary conditions, are obtainable inanalytically closed-form. Moreover, the state-price densities and pricing formulasfor standard European options are readily given in analytically closed-form for alarge class of such models that nest the CEV and other popular models as specialcases. The method is based on combining special measure changes with nonlinearvariable transformations and has been coined as “diffusion canonical transforma-tion” in light of the fact that a diffusion process (e.g., for the asset price) is reduced,or mapped, to another underlying (simpler) diffusion process. As discussed in pastrelated work and in this paper, the Laplace transforms of the transition densitiesfor the simpler underlying processes provide the Green’s functions and hence theso-called generating functions that define the transformations, and such functionsfollow analytically by standard ODE methods. One of the main formulas in ourcanonical diffusion methodology then allows us to simply relate the transition densi-ties for the new family of processes to those for the respective underlying processes.A distinctive simplifying feature of the approach is that the problem of determiningtransition densities for the (more complex) transformed processes of interest is rel-egated to the simpler underlying processes for which exact solutions are tractableby applying standard methods such as spectral expansions.

The types of transition probability densities and hence the types of new processesthat can be generated via the diffusion canonical transformation method depend, ofcourse, on the choice of the underlying process and the analytical properties of itsallowable transition densities subject to appropriate endpoint boundary conditions.Some important and fundamental questions concerning the probabilistic characterof the new families of regular processes have so far not been addressed in any rigor-ous and complete fashion. Specifically, the characteristics referred to here are themartingale property, probability conservation (i.e., absorption or nonabsorption)of the regular process defined on its natural domain, and the endpoint boundaryclassification. One of the main contributions of this paper is to provide a rigorousframework for such a characterization, i.e., we solve the problem of classifying thefamilies of transformed processes that arise via diffusion canonical transformationsin terms of these probabilistic properties which are highly relevant to financial ap-plications, namely derivatives pricing. Under quite general assumptions for thediffusion and drift coefficients of the underlying process, Lemma 4.1 (and 4.2) gives


the complete boundary classification and Theorems 4.3 and 4.4 give necessary andsufficient conditions for probability conservation and the martingale property. Asa consequence of applying the Laplace transform, the theorems provide easy-to-implement limit (asymptotic) conditions. As an additional result of Theorem 4.3,we obtain a formula for computing the first-hitting (or exit) time density for a pro-cess that admits absorption at an endpoint. The formula is easily implemented asa special endpoint limit involving the transition density, and the speed and scaledensities, of the underlying process. This leads to new families of closed-form ex-pressions for first-hitting time densities for all processes that map into analyticallysolvable underlying processes.

The diffusion canonical transformation approach is applicable to quite generalchoices of underlying processes, however we single out four main families of time-homogeneous driftless processes among the classes of analytically solvable modelsthat have applications in finance. These are specified by multiparameter nonlin-ear local volatility functions and are named as Bessel, confluent hypergeometric,Ornstein-Uhlenbeck and the hypergeometric (Jacobi) family. These processes arisevia the diffusion canonical transformation method by respectively choosing the stan-dard squared Bessel process, CIR process, Ornstein-Uhlenbeck process and Jacobiprocess as underlying diffusions. We provide a detailed classification of the rele-vant probabilistic properties of these processes as well as an analysis of their localvolatility profiles. We show that, depending on the choice of values of the modelparameters, these families further subdivide into subfamilies of either bounded orunbounded (with positive half-line as state space) processes. Among these, someprocesses are strictly conservative (i.e. conserve probability) and others are non-conservative, and in the zero drift coefficient case they are either martingales, orstrict supermartingales or strict submartingales.

For the driftless Bessel family, we identify a dual set of 3-parameter unboundedprocesses that are shown to share the same probabilistic properties as, and yet gen-eralize, the known respective nonconservative martingale and conservative strictsupermartingale CEV models. The confluent hypergeometric family further gener-alizes the Bessel family and also consists of a bounded and a dual set of unbounded(4-parameter) processes obeying either the nonconservative martingale or conser-vative strict supermartingale property. Interestingly, the local volatility profiles forthe confluent hypergeometric family consist of various pronounced smiles of vary-ing skewness as one varies the model parameters. The driftless OU family formsa separate set of processes with a bounded (BOU) subfamily and an unbounded(UOU) subfamily. For all choices of model parameters we prove that the BOU andUOU subfamilies are strictly conservative and are martingales. The OU family alsogenerates local volatility profiles with varied pronounced smiles and skews. The


newly discovered UOU models are hence among the few known pure diffusion statedependent models that are strictly conservative, analytically solvable local volatilitysmile models, with highly nonlinear volatility functions and that are strict martin-gales for all choices of model parameters. These features are of obvious importanceto no-arbitrage option pricing and model calibration. For the hypergeometric Ja-cobi family of transformed diffusions we simply prove that the processes are allnonconservative non-martingales.

The paper is organized as follows. In Section 2 we state the standard Green’sfunction and diffusion kernel methodology and provide the basic background for-malism and general assumptions that define the underlying diffusion processes. InSection 3 we present the diffusion canonical transformation technique for generat-ing transformed driftless diffusions. Section 4 presents the stochastic analysis forthe transformed diffusions, i.e. Lemmas 4.1 and 4.2 and Theorems 4.3 and 4.4.Section 4 also presents a general formula for computing first-hitting (exit) timedensities across endpoint boundaries of the transformed processes generated by dif-fusion canonical transformations. Section 5 applies the theory of Section 4 to theprobabilistic classification and analysis of the four above-named family of diffusionmodels. Section 6 discusses the inclusion of drift into state dependent diffusionsby applying additional scale and time changes to the processes. We also demon-strate how to price standard European options and provide some computed impliedvolatility surfaces. Some conclusions are drawn in Section 7.

2. Green’s function methodology for probability kernels

Consider a one-dimensional time-homogeneous diffusion process {Xt}t≥0 whoseregular state space is some open interval I = (l, r) ⊆ R with endpoints l and r,−∞ ≤ l < r ≤ ∞, and which obeys the stochastic differential equation (SDE)

(2.1) dXt = λ(Xt)dt + ν(Xt)dWt, Xt=0 = X0 ∈ I,

where {Wt}t≥0 is the standard Brownian motion (Wiener) process.

Assumption 1. The diffusion coefficient ν(x) is assumed to be strictly positive andλ′(x), ν′(x), ν′′(x) are assumed to be continuous on I.

Let u(x, x0, t) be a probability kernel for the process (2.1) specified by the con-ditional probability for any Borel subset B ⊂ I:

P{Xt+τ ∈ B|Xτ = x0} =∫

B

u(x, x0, t) dx , x0 ∈ (l, r), t, τ > 0.

This kernel is, in fact, a transition probability density function (p.d.f.) for thediffusion {Xt}t≥0.


Since the drift and volatility coefficients are assumed to have no explicit timedependence, the kernel u = u(x, x0, t) satisfies the time-homogeneous forward Kol-mogorov equation

(2.2)∂u

∂t=

12

∂2

∂x2

(ν2(x)u

)− ∂

∂x(λ(x)u) , Lxu

and the corresponding backward Kolmogorov equation:

(2.3)∂u

∂t=

12ν2(x0)

∂2u

∂x20

+ λ(x0)∂u

∂x0, Gx0u ,

subject to the Dirac delta function initial condition u(x, x0, 0+) = δ(x− x0). Herewe have defined the Fokker-Planck differential operator Lx , which acts on thevariable x, and its formal Lagrange adjoint Gx0 acting on x0. The operator G is theinfinitesimal generator of the X-diffusion {Xt}t≥0.

The X-diffusion has two basic characteristics: the speed measure m(dx) andscale function S(x) (see, e.g., Borodin and Salminen (2002)). For any X-diffusionconsidered here these characteristics are absolutely continuous w.r.t. the Lebesguemeasure and have smooth derivatives. In other words, m(dx) = m(x)dx holds, andthe speed and scale density functions m(x) and s(x) = S′(x) are defined as follows:

(2.4) s(x) = exp(−

∫ x 2λ(z)ν2(z)

dz

)and m(x) =

2ν2(x)s(x)

.

The functions s(x) and m(x) are hence continuous and strictly positive on I. Thedifferential operators Gx and Lx , acting on any smooth function f , can be re-writtenin compact form:

Gxf(x) =1

m(x)

(f ′(x)s(x)

)′, Lxf(x) =

(1

s(x)

(f(x)m(x)

)′)′

.

Throughout this paper we conveniently define the inner product of two functionsf , g with m on a closed interval [a, b] as

(2.5) (f, g)[a,b] ,∫ b

a

f(x)g(x)m(x)dx

and (f, g)(a,b] , limε→a+

(f, g)[ε,b], (f, g)[a,b) , limε→b−

(f, g)[a,ε], ||f ||2(a,b) , (f, f)(a,b).

For our purposes, we need to construct two linearly independent solutions of thefollowing ordinary differential equation (ODE)

(2.6) Gxϕ(x) = sϕ(x), s ∈ C, x ∈ I.

There exist two linearly independent solutions ϕ+s and ϕ−s such that for real s =

ρ > 0 the solutions ϕ+ρ (x) and ϕ−ρ (x) are correspondingly increasing and decreasing

functions of x (see Borodin and Salminen (2002), Karlin and Taylor (1981), Mandl(1968), Rogers and Williams (2000)). These solutions are convex on I.


The linearly independent functions ϕ+s and ϕ−s are called fundamental solutions

of (2.6). Their Wronskian can be computed as follows:

(2.7) W [ϕ−s (x), ϕ+s (x)] , ϕ−s (x)

dϕ+s (x)dx

− dϕ−s (x)dx

ϕ+s (x) = wss(x) ,

where ws is a constant w.r.t. x (but may depend on s) so that wρ > 0 for positivereal s = ρ > 0.

The functions ϕ+s and ϕ−s can be characterized as the unique (up to a multiplica-

tive constant) solutions that pose certain boundary conditions at the endpoints l

and r. In this paper, we consider endpoints l and r as non-exit boundaries for theX-diffusion (see Feller’s classification in Borodin and Salminen (2002), Karlin andTaylor (1981)). Hence the diffusion {Xt}t≥0 conserves probability on I.

Generally, the functions ϕ±ρ (x), ρ > 0, satisfy the asymptotic relations

(2.8) limx→l+

ϕ+ρ (x)/ϕ−ρ (x) = 0 and lim

x→r−ϕ+

ρ (x)/ϕ−ρ (x) = ∞.

Moreover, the square integrability conditions

(2.9) (ϕ+ρ , ϕ+

ρ )(l,x] < ∞ and (ϕ−ρ , ϕ−ρ )[x,r) < ∞

generally hold true for ρ > 0 with x ∈ I as any interior point.Here we consider a standard Green’s function framework for finding solutions for

the X-diffusion transition p.d.f. u(x, x0, t) subject to homogeneous boundary con-ditions. Consider the Green’s function G(x, x0, s) defined via the Laplace transformw.r.t. time t:

(2.10) G(x, x0, s) , L[u(x, x0, t)

](s) ,

∫ ∞

0

e−stu(x, x0, t)dt , s ∈ C .

Consequently, a solution u(x, x0, t) that satisfies the same homogeneous boundaryconditions as G follows by Laplace inversion, as given by the Bromwich integral:u(x, x0, t) = L−1

[G(x, x0, s)

](t) = 1

2πi

∫ c+i∞c−i∞ estG(x, x0, s)ds.

The Green’s function methodology is based on (generally singular) Sturm-Liou-ville theory of linear ordinary differential equations (Davies 2002, Duffy 2001).Here we shall specifically focus on the theory of Green’s functions arising from theKolmogorov equations.Assumption 2.We shall assume throughout that any density u is absolutely integrablewith respect to t on any interval 0 ≤ t ≤ T , T > 0, and that G(x, x0, s), as givenby the integral (2.10), exists for some real value s = c. Moreover, the operators Land G commute with the Laplace transform.By Laplace transforming (2.3), then G solves (Gx0 − s)G = δ(x − x0). Similarly,the adjoint equation (Lx − s)G = δ(x− x0) follows from (2.2)

Then from complex analysis it readily follows that G(x, x0, s) is an analytic func-tion on the complex s-half-plane where Re s > c, for any given x, x0 ∈ I. We note


that the minimum admissible value of c is also called the abscissa of convergence.While the Laplace integral defined by equation (2.10) may only converge in thehalf-plane with Re s greater than the abscissa of convergence, the resulting func-tion G is analytically continued into the remainder of the complex s-plane once itssingularity structure has been revealed.

The Green’s function G is then written in terms of the functions ϕ+s , ϕ−s , and

m(x) in the standard form (Borodin and Salminen 2002):

(2.11) G(x, x0, s) = w−1s m(x)ϕ+

s (x<)ϕ−s (x>),

where x< = min{x, x0} and x> = max{x, x0}. The boundary conditions for G areset by an appropriate choice of the two solutions ϕ+

s and ϕ−s . In general cases, theyare not necessarily the two fundamental solutions introduced above, but any twosuitable linearly independent solutions to (2.6). Hence if either endpoint is a regularboundary, different Green’s function can be constructed, so the equations (2.2) and(2.3) admit multiple solutions u. For applications to regular killing at interior pointssee, e.g., Campolieti (2008), Linetsky (2004a). A more general discussion on theexistence and classification of the possible types of admissible Green’s functions iscontained in (Campolieti and Makarov 2008b).

3. Diffusion canonical transformations

Consider an F -diffusion process {Ft}t≥0 obeying the driftless SDE

(3.1) dFt = σ(Ft)dWt,

where {Wt}t≥0 is a standard Brownian motion w.r.t. some appropriate probabilitymeasure. In financial terms, Ft may denote the price at calendar time t of someasset, such as the forward or discounted price of a stock under some forward mea-sure. {Ft}t≥0 can also represent other processes such as a credit rating. The statedependent volatility function σ(F ) is generally nonlinear.

The process {Ft}t≥0 is a regular diffusion on an open real intervalD = (F (l), F (r)).A transition p.d.f. U(F, F0, t) for the F -diffusion {Ft}t≥0 is a fundamental solutionto the time-homogeneous Kolmogorov equations, i.e. the forward equation is

(3.2)∂U

∂t=

12

∂2

∂F 2

(σ2(F )U

),

accompanied by the Dirac delta initial condition U(F, F0, 0+) = δ(F − F0). Wenote that in general situations (3.2) may have multiple solutions subject to someboundary conditions. Solutions with different probabilistic properties can arisewhen imposing certain boundary conditions at the endpoints of D.

In Section 5, we consider several families of state dependent volatility modelsderived from an underlying X-diffusion that can be solved analytically. We shall


refer to this construction as the ”diffusion canonical transformation” methodology.The method is in one sense a reduction approach that essentially reduces the morecomplex state dependent F -diffusion problem into a simpler underlying X-diffusionprocess. From another viewpoint, the simpler underlying X-diffusion is used togenerate new analytically tractable families of transformed F -diffusions. One ofthe basic ideas of the approach is to start by considering X-diffusions that areanalytically tractable by the use of more standard methods, e.g. for which Green’sfunction methods can be used to arrive at the probability kernels.

As is shown in (Albanese et al. 2001, Kuznetsov 2004, Albanese and Campolieti2005), by considering a family of so-called diffusion canonical invertible transfor-mations x = X(F ) such that ∣∣∣∣

dX

dF

∣∣∣∣ =ν(x)σ(F )

with volatility function given by

(3.3) σ(F ) = σ(F(x)) =σ0ν(x) exp

(− 2∫ x λ(z)

ν2(z)dz)

u2ρ(x)

=σ0ν(x)s(x)

u2ρ(x)

,

where σ0 > 0 is an arbitrary positive constant, then the F -diffusion transition p.d.f.U is related to that of an underlying X-diffusion as follows:

(3.4) U(F, F0, t) =ν(X(F ))

σ(F )uρ(X(F ))uρ(X(F0))

e−ρtu (X(F ), X(F0), t) .

The generating function u satisfies equation (2.6) with s = ρ,

(3.5)12ν2(x)

d2

dx2uρ(x) + λ(x)

d

dxuρ(x)− ρuρ(x) = 0 ,

where the parameter ρ is an arbitrary positive real constant. The map F = F(x) isgiven by

(3.6) F(x) = F ± σ0

∫ x

x

s(z)u2

ρ(z)dz ,

with F = F(x) and x ∈ I as arbitrary constants. The ± factor allows for twopossible branches of either monotonically increasing or decreasing maps (+ signbranch for monotonically increasing or − sign branch for monotonically decreasing).

Throughout what follows, the generating function u is strictly positive and isgenerally a linear combination of the fundamental solutions ϕ±ρ (x):

(3.7) uρ(x) = q1ϕ+ρ (x) + q2ϕ

−ρ (x),

with real parameters q1, q2 ≥ 0 where at least one of them is strictly positive.Any F -diffusion is given by a monotonic map {Ft = F(X(ρ)

t )}t≥0, where {X(ρ)t }t≥0

is a regular diffusion on I and defined by the generator

(3.8) G(ρ)f(x) , 12ν2(x)

d2f(x)dx2

+(

λ(x) + ν2(x)u′ρ(x)uρ(x)

)df(x)dx

,


where u′ρ(x) ≡ duρ(x)dx

, and having the transition p.d.f.

(3.9) u(ρ)(x, x0, t) = e−ρt uρ(x)uρ(x0)

u(x, x0, t) .

The speed and scale densities of the X(ρ)-diffusion process are

(3.10) mρ(x) = u2ρ(x)m(x), sρ(x) =

s(x)u2

ρ(x).

To obtain all the possible maps, we employ the following indefinite integral iden-tity which follows simply from the Wronskian of any two linearly independent func-tions ϕ+

ρ and ϕ−ρ :(3.11)

∫W [ϕ+

ρ , ϕ−ρ ](x)

(q1ϕ+ρ (x) + q2ϕ

−ρ (x))2

dx =

ϕ−ρ (x)/q1

q1ϕ+ρ (x) + q2ϕ

−ρ (x)

+ const, (q1 6= 0),

−ϕ+ρ (x)/q2

q1ϕ+ρ (x) + q2ϕ

−ρ (x)

+ const, (q2 6= 0).

Employing this identity in equation (3.6) while using expression (2.7) and (3.7), wearrive at a general form for any map F(x) as a quotient:

(3.12) F(x) =c1ϕ

+ρ (x) + c2ϕ

−ρ (x)

q1ϕ+ρ (x) + q2ϕ

−ρ (x)

.

The parameters c1, c2 ∈ R are given in terms of q1, q2 and some other arbitraryreal constant C. In particular, two sets of monotonically increasing or decreasingmaps arise where either c1 = q1C, c2 = q2C ± σ0w

−1ρ /q1 (for maps with q1 6= 0) or

c1 = q1C ±σ0w−1ρ /q2, c2 = q2C (for maps with q2 6= 0). For any choice of map, the

absolute value of the determinant of the matrix of coefficients in (3.12) is

|D| = |q2c1 − q1c2| = σ0/wρ 6= 0 .

The Jacobian of this transformation is given by

F′(x) =DW [ϕ−ρ , ϕ+

ρ ](x)

(q1ϕ+ρ (x) + q2ϕ

−ρ (x))2

=D wρs(x)

u2ρ(x)

= Dwρsρ(x) ,

Thus, we recover the expression (3.3) for the volatility function associated with thetransformation (3.12):

σ(F ) = σ(F(x)) = |F′(x)| ν(x) = σ0ν(x)sρ(x) =σ0ν(x)s(x)

u2ρ(x)

.

Based on these relations and (3.12), three general cases (or subfamilies) arise asfollows.

(i) q1 = 0, q2 > 0, uρ(x) = q2ϕ−ρ (x): |F(l+)| < ∞ and |F(r−)| = ∞ with interval

D = (F(l+),∞) or (−∞, F(l+)) for increasing or decreasing map, respectively.


(ii) q1 > 0, q2 = 0, uρ(x) = q1ϕ+ρ (x): |F(l+)| = ∞ and |F(r−)| < ∞ with

interval D = (−∞,F(r−)) or (F(r−),∞) for increasing or decreasing map,respectively.

(iii) q1 > 0 and q2 > 0, uρ(x) = q1ϕ+ρ (x)+q2ϕ

−ρ (x): |F(l+)| < ∞ and |F(r−)| < ∞

with interval D = (F (l), F (r)) with F (l) = min{F(l+),F(r−)} and F (r) =max{F(l+), F(r−)}.

Note that for cases (i) or (ii), where one of the parameters q1 or q2 is taken as zero,the F -diffusion is defined on an unbounded (semi-infinite) interval. In case (iii) theF -diffusion is defined on a bounded interval.

4. Stochastic Analysis of F -Diffusions

4.1. Boundary Classification of the F -Diffusions. In this subsection we derivesimple conditions for classifying the endpoints l and r of any regular X(ρ)-diffusion,ρ > 0, defined by generator G(ρ) in (3.8) for the three general families (i)–(iii).

Fix any x < y, x, y ∈ (l, r), then the respective scale and speed measure functionsfor a diffusion {X(ρ)

t }t≥0 are defined by

(4.1) Sρ[x, y] ,∫ y

x

sρ(z)dz , Sρ(l, y] , limx→l+

Sρ[x, y] , Sρ[x, r) , limy→r−

Sρ[x, y]

and

(4.2) Mρ[x, y] ,∫ y

x

mρ(z)dz , Mρ(l, y] , limx→l+

Mρ[x, y] , Mρ[x, r) , limy→r−

Mρ[x, y]

with densities sρ and mρ given in (3.10). Based on these measures, we define therelevant functionals for the boundaries l and r of the X(ρ)-diffusions:

Σρ(l) ,∫ x

l

Sρ(l, z]mρ(z) dz , Σρ(r) ,∫ r

x

Sρ[z, r) mρ(z) dz ,(4.3)

Nρ(l) ,∫ x

l

Sρ[z, x]mρ(z) dz , Nρ(r) ,∫ r

x

Sρ[x, z] mρ(z) dz .(4.4)

Let e ∈ {l, r}, then by the well-known classification theory for regular diffusions(e.g., see Karlin and Taylor (1981)) we have that the boundary e is

regular if Σρ(e) < ∞ and Nρ(e) < ∞;exit if Σρ(e) < ∞ and Nρ(e) = ∞;entrance if Σρ(e) = ∞ and Nρ(e) < ∞;natural if Σρ(e) = ∞ and Nρ(e) = ∞.The following lemma summarizes the classification of the boundary endpoints

for all three families (i)-(iii).

Lemma 4.1. The above three families (i)-(iii) of regular diffusions {X(ρ)t }t≥0 on

(l, r) with generator G(ρ), ρ > 0, defined by (3.7) - (3.8) have the following boundaryclassification:


(i) q1 = 0, q2 > 0: The boundary l is attracting natural if (ϕ+ρ , ϕ−ρ )(l,x] = ∞, exit

if (ϕ+ρ , ϕ−ρ )(l,x] < ∞ and (ϕ−ρ , ϕ−ρ )(l,x] = ∞, and is otherwise regular when

(ϕ−ρ , ϕ−ρ )(l,x] < ∞. The boundary r is non-attracting natural if (ϕ+ρ , ϕ−ρ )[x,r) =

∞ and is otherwise entrance.(ii) q1 > 0, q2 = 0: The boundary r is attracting natural if (ϕ+

ρ , ϕ−ρ )[x,r) = ∞, exitif (ϕ+

ρ , ϕ−ρ )[x,r) < ∞ and (ϕ+ρ , ϕ+

ρ )[x,r) = ∞, and is otherwise regular when(ϕ+

ρ , ϕ+ρ )[x,r) < ∞. The boundary l is non-attracting natural if (ϕ+

ρ , ϕ−ρ )(l,x] =∞ and is otherwise entrance.

(iii) q1 > 0, q2 > 0: The boundary l has the same classification as in (i) and r hasthe same classification as in (ii).

Proof. See the appendix ¤

Lemma 4.2. The boundary classification for an F -diffusion {Ft = F(X(ρ)t )}t≥0 is

equivalent to the corresponding X(ρ)-diffusion {X(ρ)t }t≥0.


4.2. Conditions for Absorption and Computation of First-Hitting (Exit)

Time Distribution. To determine whether a one-dimensional diffusion is ab-sorbed at an endpoint of its state space, one can use the (Feller) boundary classifi-cation of the previous subsection. However, the approach presented below operateswith transition probability densities for the process. Moreover, as an importantconsequence, this also leads us to limit formulas for computing the first passageor exit time densities of the transformed F -diffusions. The related problem ofdetermining whether the transition p.d.f. integrates to unity throughout time issolved by using the Green’s function methodology. In particular, we derive simplenecessary and sufficient boundary conditions on the fundamental solutions ϕ± forensuring non-absorption, i.e. probability conservation on the regular state space.

Consider an F -diffusion started at any F0 ∈ D ≡ (F (l), F (r)) with transitionp.d.f. U . The probability of absorption of the process before time t is given by

(4.5) P (t; F0) = 1−∫ F (r)

F (l)U(F, F0, t)dF = 1−

∫ r

l

u(ρ)(x, x0, t)dx ,

with x0 = X(F0). The rate of absorption, R(F (l), F (r)|F0, t) = − ∂∂tP (t;F0), i.e.

the rate of change of probability that the process has value Ft ∈ D at time t, maybe calculated as follows. Changing variables in (4.5), differentiating w.r.t. t andusing the forward Kolmogorov equation in the right hand side followed then by


integrating gives

R(F (l), F (r)|F0, t) =∫ r

l

∂

∂tu(ρ)(x, x0, t)dx =

∫ r

l

L(ρ)x u(ρ)(x, x0, t)dx(4.6)

=1

sρ(x)∂

∂x

(u(ρ)(x, x0, t)

mρ(x)

) ∣∣∣∣x=r−

x=l+

.(4.7)

Clearly, if no absorption occurs over time, then R(F (l), F (r)|F0, t) ≡ 0. Forexample, zero limits in (4.7) are satisfied automatically if we deal with unattainableboundaries. On the other hand, the kernel U integrates to unity (w.r.t. F ) in thelimit t → 0+ because of the initial condition U(F, F0, 0+) = δ(F − F0). Therefore,the condition R ≡ 0, if satisfied, guarantees that the kernel U integrates to unityfor t ∈ (0,∞), and hence the absorption probability P (t;F0) is identically zero. Inthis paper we shall say that a diffusion is conservative if probability is conserved(R ≡ 0); otherwise the process is said to be nonconservative.

Theorem 4.3. An F -diffusion {Ft = F(X(ρ)t )}t≥0 with map F in (3.12) conserves

probability if and only if the limits

(4.8) limx→r−

W [uρ, ϕ−ρ+s](x)

s(x)= 0 and lim

x→l+

W [uρ, ϕ+ρ+s](x)

s(x)= 0

hold when Re s ≥ c for some real constant c.


By Theorem 4.3, probability conservation for a regular F -diffusion is then, inpractice, readily verified via the asymptotic forms of ϕ± and the scale density s forthe corresponding X-diffusion.

Since P (τ ;F0) gives the c.d.f. for absorption up to time τ ≥ 0, then the first-exit time p.d.f., denoted by p(τ ; F0), for the F -diffusion started at F0 ∈ D andhitting either of the boundaries F (l) and F (r) (and being absorbed) follows simplyby differentiating w.r.t. τ . From equation (4.7) we immediately obtain

(4.9) p(τ ;F0) =∂

∂τP (τ ; F0) = − 1

sρ(x)∂

∂x

(u(ρ)(x, X(F0), τ)

mρ(x)

) ∣∣∣∣x=r−

x=l+

.

Note that if neither boundary is absorbing then p(τ ;F0) ≡ 0. If only one endpointis absorbing then (4.9) obviously reduces to just one limit involving the absorbingendpoint, and in this case p(τ ; F0) represents a first-hitting time density.

4.3. Martingale property. In this subsection we extend the above method andapply the Laplace transform to verify the property of conserving the mathematicalexpectation, i.e. the martingale property since we deal here with driftless diffu-sions. A diffusion process {Ft}t≥0 is a martingale, if for any t ≥ 0, the expectationE[|Ft|] < ∞, and the conditional expectation of FT given Ft satisfies E[FT |Ft] = Ft


for all 0 ≤ t ≤ T . Within the theory of diffusion processes, solutions to drift-less SDEs (i.e., with zero drift function) often possess the martingale property(Øksendal 2000). In what follows, we shall see that this martingale property doesnot necessarily hold true for certain families of F -diffusions discussed in this paper.In fact, besides martingale diffusions, in some cases we also obtain subfamilies thatare either strict supermartingales or strict submartingales.

By analogy with the previous subsection let us consider the rate of change of theexpected value of an F -diffusion conditioned on the process starting at F0 ∈ D:

(4.10) M(F (l), F (r)|F0, t) =∂

∂tE[Ft|Ft=0 = F0]

Clearly, a time-homogeneous diffusion with E[|Ft|] < ∞ is a martingale if and onlyif

M(F (l), F (r)|F0, t) = 0 for all F0 ∈ D and t > 0.

If we arbitrarily fix F0 ∈ D, then the trivial initial value problem E′(t) = 0,E(0) = F0 for E(t) , E[Ft|Ft=0] = F0] has only the constant solution E(t) ≡ F0.Note that the F -diffusion is a strict submartingale if E[Ft|Ft=0 = F0] > F0 and isa strict supermartingale if E[Ft|Ft=0 = F0] < F0. Such cases can be verified byanalyzing the sign of the rate M .

By changing variables F → x = X(F ), using the forward Kolmogorov equationfor u(ρ), and then using integration by parts we obtain

M(F (l), F (r)|F0, t) =∂

∂t

∫ F (r)

F (l)F U(F, F0, t)dF =

∫ r

l

F(x)∂

∂tu(ρ)(x, x0, t)dx

=∫ r

l

F(x)L(ρ)x u(ρ)(x, x0, t)dx(4.11)

=[

F(x)sρ(x)

∂

∂x

(u(ρ)(x, x0, t)

mρ(x)

)− (±σ0)

u(ρ)(x, x0, t)mρ(x)

]x=r−

x=l+

,(4.12)

where we define x0 = X(F0) and use F′(x)/sρ(x) = +σ0 (or −σ0) for an increasing(or decreasing) map.

Theorem 4.4. An F -diffusion {Ft = F(X(ρ)t )}t≥0 with map F in (3.12) and satis-

fying E[|Ft|] < ∞ is a martingale if and only if both limits

(4.13) limx→l+

[F(x)

W [uρ(x), ϕ+ρ+s(x)]

s(x)− (±σ0)

ϕ+ρ+s(x)uρ(x)

]= 0,

(4.14) limx→r−

[F(x)

W [uρ(x), ϕ−ρ+s(x)]s(x)

− (±σ0)ϕ−ρ+s(x)uρ(x)

]= 0,

hold when Re s ≥ c for some real constant c.

Proof. See the appendix. ¤


5. Solvable Diffusions

5.1. Bessel family. In this subsection, we specifically consider models arising froman underlying λ0-dimensional squared Bessel (SQB) process (Xt)t≥0 ∈ I ≡ (0,∞)obeying the SDE

(5.1) dXt = λ0dt + ν0

√XtdWt

with constant ν0 > 0 and λ0 > 0. The diffusion scale and speed densities are

(5.2) s(SQB)(x) = x−µ−1 and m(SQB)(x) =2ν20

xµ,

where µ , 2λ0ν20− 1.

Here we specifically consider models arising from the SQB process with λ0 > ν20/2

(hence µ > 0). The boundary l = 0 is entrance, and r = ∞ is an attracting naturalboundary. The transition density of the SQB process is given by

(5.3) u(SQB)(x, x0, t) =(

x

x0

)µ2 e−2(x+x0)/ν2

0 t

ν20 t/2

Iµ

(4√

xx0/ν20 t

), and x, x0, t > 0,

and it satisfies probability conservation:∫∞0

u(SQB)(x, x0, t)dx = 1, x0, t > 0.

The pair of fundamental solutions is

(5.4) ϕ+ρ (x) = x−µ/2Iµ(2

√2ρx/ν0) and ϕ−ρ (x) = x−µ/2Kµ(2

√2ρx/ν0) ,

where Iµ(z) and Kµ(z) are the modified Bessel functions (of order µ) of the first andsecond kind, respectively (for definitions and properties see Abramowitz and Stegun(1972)). These solutions satisfies the Wronskian relation (2.7) where wρ = 1/2.

Proposition 5.1. The Bessel family of F -diffusions {Ft = F(X(ρ)t ), t ≥ 0} with

map F in (3.12) generated by the SQB process has the following properties.

(i) q1 = 0, q2 > 0 : The boundary F(0+) is exit when µ ≥ 1 and is regular killingwhen 0 < µ < 1; the boundary F(∞) is non-attracting natural. The process Ft

is nonconservative. It is a martingale when c2 = 0, is a strict supermartingalewhen c2 > 0 and is a strict submartingale when c2 < 0.

(ii) q1 > 0, q2 = 0 : The boundary F(0+) is entrance, the boundary F(∞) is at-tracting natural. The process Ft is conservative. It is a strict supermartingalewhen c2 > 0 and is a strict submartingale when c2 < 0.

(iii) q1 > 0, q2 > 0 : The boundary F(0+) is exit when µ ≥ 1 and is regular killingwhen 0 < µ < 1; the boundary F(∞) is attracting natural. The process Ft isnonconservative. It is a martingale when c2 = 0, is a strict supermartingalewhen c2 > 0 and is a strict submartingale when c2 < 0.



5.1.1. Bessel-I and Bessel-K subfamilies. By considering the two maps wherein oneof the parameters q1 or q2 is zero (for fixed nonzero the other), the above Besselfamily gives rise to two separate 3−parameter subfamilies (and for our purposes wesimply set F = 0):

(5.5) F(x) =

aIKµ(2

√2ρx/ν0)

Iµ(2√

2ρx/ν0), Bessel I-subfamily for q2 = 0,

aKIµ(2

√2ρx/ν0)

Kµ(2√

2ρx/ν0), Bessel K-subfamily for q1 = 0,

where aI , aK , ρ, and µ are independently adjustable positive parameters. Thetwo functions F(x) (and the respective inverses x = X(F )) in (5.5) now map x ∈(0,∞) and F ∈ (0,∞) into one another, where F(x) is monotonic with dF(x)/dx =±σ(F(x))/(ν0

√x) (with respective + (or −) sign for K- (or I-) subfamily). The

transformation (5.5) hence leads to a pair of subfamilies of processes Ft ∈ (0,∞)with respective volatility functions

(5.6) σ(I,K)(F ) =

aI ν0/2√X(F )I2

µ(2√

2ρX(F )/ν0), I-subfamily,

aK ν0/2√X(F )K2

µ(2√

2ρX(F )/ν0), K-subfamily.

Despite the similarity between these two subfamilies, they possess essentially dif-ferent properties. Of interest are the qualitative differences in the local volatilityfunctions σloc(F ) ≡ σ(F )/F for both models. Let σ

(K)loc and σ

(I )loc denote the re-

spective local volatility functions of the K- and I-subfamilies. From (5.5) and (5.6),and the large and small argument asymptotics of the modified Bessel functions, weobtain the local volatility asymptotes:

(5.7) σ(K)loc (F ) ∼

µ√

2ρ

[2 aK

Γ(µ)Γ(µ + 1)

] 12µ

F−12µ , as F → 0+ ,

2√

2ρ , as F →∞ ,

and

(5.8) σ(I )loc(F ) ∼

2√

2ρ , as F → 0+ ,

µ√

2ρ

[2a−1

I

Γ(µ)Γ(µ + 1)

] 12µ

F12µ , as F →∞ .

Figure 1 shows typical plots of σ(K)loc and σ

(I )loc. The Bessel K-subfamily hence ap-

proaches a constant local volatility in the limit of large asset prices (i.e. approachesa lognormal model), whereas for small asset values the local volatility exhibits apower law singularity. On the other hand, the Bessel I-subfamily approaches aconstant local volatility for small asset values, with unbounded power law in theopposing large asset value limit.


A pair of exact transition densities for the two models on the regular statespace (F (l), F (r)) = (0,∞) are obtained by combining equation (3.4) (uρ(x) =x−µ/2Iµ(2

√2ρx/ν0) for I- and uρ(x) = x−µ/2Kµ(2

√2ρx/ν0) for K-subfamily) with

equations (5.3) and (5.6) giving:

(5.9) U (I,K)(F, F0, t) =

x e−ρt−2(x+x0)/ν20 t

aI ν20 t/4

I3µ(2

√2ρx/ν0)

Iµ(2√

2ρx0/ν0)Iµ

(4√

xx0

ν20 t

),

x e−ρt−2(x+x0)/ν20 t

aK ν20 t/4

K3µ(2

√2ρx/ν0)

Kµ(2√

2ρx0/ν0)Iµ

(4√

xx0

ν20 t

),

where x = X(F ), x0 = X(F0) are given by the respective inverses of (5.5).By Proposition 5.1 for case (ii), the origin 0 = F(∞) is attracting natural and

∞ = F(0+) is entrance. Since c2 > 0 in F(x), the I-subfamily of processes, withtransition p.d.f. given by the top expression in (5.9), are positive conservativestrict supermartingales for all choices of positive model parameters aI , ρ, µ. If Ft

represents a forward price or discounted asset price process under an assumed risk-neutral measure, then the I-subfamily is a candidate for a so-called “bubble market”model for which Ft is a strict supermartingale, and where the usual textbook resultsof equivalent martingale arbitrage-free pricing do not strictly hold. For discussionson some of the financial implications of using strict supermartingale models foroption pricing we refer the reader to Cox and Hobson (2005) and references therein.

The K-subfamily with transition p.d.f. given as the bottom expression in (5.9)has quite different properties, stated as follows. By Proposition 5.1 for case (i), theorigin F(0+) = 0 is absorbing (exit for µ ≥ 1 and regular killing for 0 < µ < 1)and F(∞) = ∞ is non-attracting natural. Since c2 = 0 in F(x), the K-subfamily ofprocesses are all martingales. This is in direct contrast to the strict supermartingaleproperty that holds for the I-subfamily. The density, p(K)(τ ; F0), for the first-hittingtime down at the origin (i.e. first exit time at 0) for a Bessel-K process started atF0 > 0 is readily derived by computing the limits in equation (4.9) (where theupper limit x →∞ is zero), giving the generalized inverse Gaussian distribution:

(5.10) p(K)(τ ; F0) =

(2X(F0)/ρν2

0

)µ/2

2 Kµ

(2√

2ρX(F0)/ν0

) τ−µ−1e−ρτ−2X(F0)/ν20τ , τ > 0.

An interesting property of the above Bessel subfamilies is that they form a super-set of the CEV models with zero drift coefficient (see Cox (1975)), which formallyobey the SDE:

(5.11) dFt = δF β+1t dWt, t ≥ 0, F0 > 0.

That is, the CEV volatility and the respective transition and first-hitting timedensities are recovered in special limiting cases as ρ → 0+ (for details see Campolietiand Makarov (2007) and Campolieti (2008) for more generally killed F -diffusions).


5.1.2. Bessel-IK subfamily. For the case when both q1 and q2 are strictly positive(we refer to this as the IK-subfamily), the map F(x) in (3.12) is:

(5.12) F(x) =c1Iµ(

√8ρx/ν2

0) + c2Kµ(√

8ρx/ν20)

q1Iµ(√

8ρx/ν20) + q2Kµ(

√8ρx/ν2

0),

where |c1q2− c2q1| = σ0/wρ = 2σ0. According to (3.3) the volatility function takesthe general form

(5.13) σ(IK)(F ) =σ0 ν0

√X(F )

[q1Iµ

(√8ρX(F )/ν2

0

)+ q2Kµ

(√8ρX(F )/ν2

0

)]2 .

By the boundary conditions in (2.8), which follow from the small and largeargument asymptotics of the modified Bessel functions, the interval x ∈ (0,∞)is mapped onto a bounded interval (F (l), F (r)) where F (l) = min{c1/q1, c2/q2}and F (r) = max{c1/q1, c2/q2}. From equation (3.4) an exact expression for thetransition p.d.f. of such F -diffusions on the bounded interval (F (l), F (r)) takes theform

U(F, F0, t) =x e−ρt−2(x+x0)/ν2

0 t

σ0ν20 t/2

[q1Iµ

(√8ρxν20

)+ q2Kµ

(√8ρxν20

)]3[q1Iµ

(√8ρx0ν20

)+ q2Kµ

(√8ρx0ν20

)]Iµ

(4√

xx0

ν20 t

)

where x = X(F ), x0 = X(F0) are given by the inverse of (5.12). We note that thisdensity reduces to the respective densities in (5.9) (and (5.12) to the respective mapsin (5.5)) in the respective cases q2 = 0 or q1 = 0 where the constants aI = 2σ0/q2

1

and aK = 2σ0/q22 are defined.

By case (iii) of Proposition 5.1, the IK-subfamily of diffusions with above tran-sition p.d.f. are all nonconservative with endpoint F(0+) = c2/q2 as an absorbingboundary and the other endpoint F(∞) = c1/q1 as attracting natural. Moreover,the subset of all such diffusions for which F(0+) = 0 (i.e. c2 = 0) are martingales.

Since the endpoint F(0+) is an absorbing boundary, we can obtain the first-hitting (exit) time at this boundary for a Bessel-IK process started at F0 > 0. Thisdensity, denoted by p(IK)(τ ; F0), is obtained in closed-form via equation (4.9), aswas done to derive (5.10), and is proportional to the generalized inverse Gaussiandistribution in (5.10):

(5.14) p(IK)(τ ;F0) = P (IK)p(K)(τ ; F0) .

Here P (IK) =q2Kµ

(2√

2ρX(F0)/ν0

)

q1Iµ

(2√

2ρX(F0)/ν0

)+ q2Kµ

(2√

2ρX(F0)/ν0

) is the probability that

the process will ever be absorbed at the boundary F(0+). Since P (IK) < 1, thedensity p(IK) does not integrate to unity over all time τ > 0 (unless q1 = 0 whichrecovers the K-subfamily). This means that there is a nonzero probability that the


process is never absorbed, which is of course consistent with the fact that the otherboundary F(∞) is attracting natural.

5.2. Confluent CIR family. Consider the Cox-Ingersoll-Ross (CIR) process (Coxet al. 1985) (Xt)t≥0 ∈ I ≡ (0,∞) with SDE:

(5.15) dXt = (λ0 − λ1Xt)dt + ν0

√XtdWt ,

where λ0 > 0, λ1 > 0, ν0 > 0. The endpoint x = ∞ is a natural boundary (seeBorodin and Salminen (2002)). Throughout, we shall take µ , 2λ0

ν20− 1 > 0 so

that the origin is an entrance boundary of the CIR process. The scale and speeddensities are simply related to those for the SQB process:

(5.16) s(CIR)(x) = x−µ−1eκx and m(CIR)(x) =2ν20

xµe−κx,

where κ , 2λ1ν20

> 0. In generating F -diffusions of the so-called confluent CIRfamily we specifically consider the underlying X-diffusion as the CIR process withtransition p.d.f.

(5.17) u(CIR)(x, x0, t) = cteλ1t

(xeλ1t

x0

)µ2

e−ct(xeλ1t+x0)Iµ

(2ct

√x0xeλ1t

),

where ct , κ/(eλ1t − 1). By using a time and scale transformation, this density isreduced to that in (5.3) for the SQB model:

u(CIR)(x, x0, t) = eλ1tu(SQB)

(eλ1tx, x0,

eλ1t − 1λ1

).

Driftless diffusions with SDE of the form (3.1) for the CIR family are obtainedby applying the diffusion canonical transformation methodology of Section 3. Thegenerating function uρ(x) solving (3.5) is generally given by (3.7) as a linear com-bination of the functions

(5.18) ϕ+ρ (x) = M (υ, µ + 1, κx) and ϕ−ρ (x) = U (υ, µ + 1, κx)

where υ , ρ/λ1 > 0. M(a, b, z) and U(a, b, z) are confluent hypergeometric func-tions, i.e., the standard Kummer functions (see Abramowitz and Stegun (1972)),which are strictly increasing and decreasing positive functions, respectively. Theysatisfy the Wronskian relation (2.7) where wρ = (Γ(µ + 1)/Γ(υ))κ−µ.

Proposition 5.2. The confluent hypergeometric family of F -diffusions {Ft =F(X(ρ)

t ), t ≥ 0} with map F in (3.12) generated by the CIR process has the sameproperties as the Bessel family of F -diffusions, as stated in Proposition 5.1.



By equating either one or none of the coefficients q1 and q2 to zero, we obtainthree separate processes: the confluent M-subfamily with q2 = 0, the U -subfamilywith q1 = 0, and the mixed MU-subfamily when both q1 and q2 are nonzero.We remark that these subfamilies generalize the Bessel I-, K-, and IK-subfamilies,respectively, which are obtained in the limit λ1 → 0+. Moreover, they share thestochastic properties of the Bessel subfamilies, as stated in Proposition 5.2.

5.2.1. Confluent-M subfamily. The generating function is

(5.19) uρ(x) = q1M(υ, µ + 1, κx).

This corresponds to q1 6= 0, q2 = 0, c1 = 0 and c2 = σ0/(q1wρ) in (3.12). The mapF(x) = F(M)(x) takes the form:

(5.20) F(M)(x) = aMU(υ, µ + 1, κx)M(υ, µ + 1, κx)

,

where aM = c2/q1 is an arbitrary positive constant. Hence F(M) is monotonicallydecreasing and maps x ∈ (0,∞) onto F ∈ (0,∞). This transformation leads to asubfamily of processes (Ft)t≥0 with volatility function

(5.21) σ(M)(F(x)) =aM ν0 wρ eκx

xµ+1/2M 2(υ, µ + 1, κx).

Applying the large and small argument asymptotics of the confluent hypergeometricfunctions to (5.20) and (5.21), we obtain the asymptotic forms of the local volatilityσloc(F ) = σ(F )/F :

(5.22) σ(M)loc (F ) ∼

ν0

√−κ ln

(µ F

aMΓ(υ)/Γ(µ)

), as F → 0+,

ν0µ√

κ

((Γ(υ)/Γ(µ)) F

aM

) 12µ

, as F →∞.

The local volatility has a pronounced smile-like pattern as observed in Figure 2.For large values of F , σ

(M)loc exhibits an unbounded power law, as observed for the

Bessel I-subfamily. As F approaches zero the local volatility increases and tends to∞ but slower than any power function.

The processes defined by the above confluent-M subfamily are all positive con-servative strict supermartingales. The exact transition density defining the aboveM-subfamily is obtained simply by substituting (5.21), (5.19) and (5.17) into (3.4):

(5.23) U (M)(F, F0, t) =e−ρt−κxxµ+1M 3(υ, µ + 1, κx)

aM wρM (υ, µ + 1, κx0)u(CIR)(x, x0, t)

where x = X(F ), x0 = X(F0) are given by the inverse of (5.20).


5.2.2. Confluent-U subfamily. The generating function is

(5.24) uρ(x) = q2U(υ, µ + 1, κx).

This corresponds to q2 6= 0, q1 = 0, c2 = 0 and c1 = σ0/(q2wρ) in (3.12). The mapF takes the form:

(5.25) F(U)(x) = aUM(υ, µ + 1, κx)U(υ, µ + 1, κx)

,

where aU = c1/q2 is an arbitrary positive constant. Hence F(U) is monotonicallyincreasing and maps x ∈ (0,∞) onto F ∈ (0,∞). This transformation leads to asubfamily of F -diffusions with volatility function

(5.26) σ(U)(F(x)) =aU ν0 wρ eκx

xµ+1/2 U 2(υ, µ + 1, κx).

By proceeding in a similar fashion as in the above M-subfamily and applying thelarge and small argument asymptotics of M and U to (5.25) and (5.26), we obtainthe asymptotic forms of the local volatility:

(5.27) σ(U)loc (F ) ∼

ν0µ√

κ

(F

aUΓ(υ)/Γ(µ)

)− 12µ

as F → 0+,

ν0

√κ ln

((Γ(υ)/Γ(µ))F

µaU

)as F →∞.

As seen in Figure 2, the local volatility curves generated by the U-subfamily alsohave a pronounced smile-like pattern, but with different sloping features for largerand smaller values of F , as compared to that for the M-subfamily. For small valuesof F , σ

(M)loc exhibits an unbounded power law like that for the Bessel K-subfamily.

As F → ∞, the local volatility increases and approaches infinity but slower thanany power function.

The processes defined by the above confluent-U subfamily are all nonconserva-tive martingales. The exact transition density for the above U-subfamily (i.e., theanalogue of (5.23)) is:

(5.28) U (U)(F, F0, t) =e−ρt−κxxµ+1U 3(υ, µ + 1, κx)

aU wρU (υ, µ + 1, κx0)u(CIR)(x, x0, t)

where x = X(F ), x0 = X(F0) are given by inverting (5.25).By Proposition 5.2, the lower endpoint F(0+) = 0 is an absorbing boundary and

the upper endpoint F(∞) = ∞ is non-attracting natural. The exact first-hitting(exit) time density, p(U)(τ ; F0), for a confluent-U process started at F0 > 0 andhitting the origin is readily obtained by combining equations (3.10),(5.16), (5.17)and (5.24) into (4.9). Then, by using the small and large argument asymptotics ofthe Bessel I and Kummer function U we arrive at:

(5.29) p(U)(τ ;F0) =λ1e

−ρτ−cτ x0(1− e−λ1τ

)−µ−1

U(υ, µ + 1, κx0)Γ(υ), τ > 0


where x0 = X(F0) is obtained by inverting (5.25), i.e. F(U)(x0) = F0. Since theupper boundary is non-attracting natural, then this density must integrate to unity

over all τ > 0. This is shown by making a time change T (τ) , e−λ1τ

1− e−λ1τ, with

inverse τ(t) =1λ1

ln(

1 + TT

), which maps τ ∈ (0,∞) onto T ∈ (0,∞) and is a

strictly decreasing function. The density (5.29) is then rewritten as follows:

p(U)(τ ;F0) = |T ′(τ)| e−κx0T (τ)(T (τ))υ−1(1 + T (τ))µ−υ

U(υ, µ + 1, κx0)Γ(υ)= |T ′(τ)| p(U)

T (T (τ); x0) .

The latter density as a function of T = T (τ) integrates to unity as is readily verifiedby the integral representation of the Kummer U function:

Γ(a)U(a, b, z) =

∞∫

0

e−ztta−1(1 + t)b−a−1 dt .

We note that the expression for the density in (5.29) also directly recovers theformula in (5.10) in the limit λ1 → 0+. This follows by applying the knownformula (see 13.3.3 of Abramowitz and Stegun (1972))

lima→∞

Γ(1 + a− b)U(a, b, z/a) = 2z(1−b)/2Kb−1(2√

z)

with a = ρ/λ1, b = µ + 1, z = 2ρx0/ν20 .

5.2.3. Confluent-MU subfamily. The generating function is

uρ(x) = q1M(υ, µ + 1, κx) + q2U(υ, µ + 1, κx).

This corresponds to q1, q2 > 0 with |c1q2 − c2q1| = σ0/wρ in (3.12). The map F

takes the form:

(5.30) F(x) =c1M(υ, µ + 1, κx) + c2U(υ, µ + 1, κx)q1M(υ, µ + 1, κx) + q2U(υ, µ + 1, κx)

.

This function maps x ∈ (0,∞) onto a bounded interval F ∈ (F (l), F (r)) withF (l) = min{ c1

q1, c2

q2}, F (r) = max{ c1

q1, c2

q2}, since F(0+) = c2

q2, F(∞) = c1

q1. This

transformation leads to a subfamily of F -diffusions with volatility function

(5.31) σ(MU)(F(x)) =a ν0 wρ eκx

xµ+1/2(q1M(υ, µ + 1, κx) + q2U(υ, µ + 1, κx))2,

where a = |c1q2 − c2q1| = σ0/wρ is an arbitrary positive constant.By case (iii) of Proposition 5.2, the bounded processes of the MU-subfamily

are nonconservative. The endpoint F(0+) is an absorbing boundary and F(∞) isattracting natural. Moreover, the subset of all such diffusions for which F(0+) = 0(i.e. c2 = 0) are martingales. Substituting the above generating function uρ and(5.31) into (3.4) gives the exact closed-form transition density for such processes inanalogy with (5.23) and (5.28).


Contrary to the confluent-U subfamily, the first-hitting time density for the aboveMU-subfamily of processes to hit F(0+) does not integrate to one since now F(∞)is attracting. Employing the above asymptotic relations into (4.9), the exact closed-form density is found to be:

(5.32) p(MU)(τ ;F0) = P (MU) p(U)(τ ; F0) ,

where P (MU) =q2U(υ, µ + 1, κx0)

q1M(υ, µ + 1, κx0) + q2U(υ, µ + 1, κx0)is the probability that

the process will ever be absorbed at endpoint F(0+). Here the process starts atF0 = F(MU)(x0). We note that P (MU) < 1, while P (MU) → 1 as q1 → 0+ recoversthe U-subfamily.

As noted above, the confluent hypergeometric family generalizes the Bessel fam-ily which is recovered as a special limiting case as λ1 → 0+ . As is readily observedfor the U-subfamily, the map (5.25), volatility function (5.31), and first-hitting timedensities (5.29) and (5.32) are all reduced to F(K) of (5.5), σ(K) of (5.6), p(K) andp(IK) given by (5.10) and (5.14), respectively, as λ1 → 0+. Thus the M-, U-, andMU-subfamilies generalize the respective I, K-, and IK-subfamilies.

5.3. Ornstein-Uhlenbeck family of conservative martingales. The Ornstein-Uhlenbeck (OU) process solves the SDE

(5.33) dXt = (λ0 − λ1Xt)dt + ν0dWt.

In what follows we consider diffusion families that are generated by an underlyingOrnstein-Uhlenbeck process with state space I ≡ (−∞,∞). Without loss in gener-ality we will henceforth set λ0 = 0. Otherwise we can consider the shifted processYt = Xt− λ0

λ1and the formulae follow by simply shifting x → x− λ0

λ1, x0 → x0− λ0

λ1.

The speed and scale densities are as follows:

s(OU)(x) = eκx2/2, m(OU)(x) = (2/ν20)e−κx2/2.

As in the previous subsection, we define the positive constants κ , 2λ1ν20

, υ , ρλ1

where we assume λ1 > 0. Both boundaries l = −∞ and r = ∞ are natural for allchoices of parameters. The OU process has the transition p.d.f.

(5.34) u(OU)(x, x0, t) =√

κ

2π(1− e−2λ1t)exp

(−κ(x− x0e

−λ1t)2

2(1− e−2λ1t)

).

A pair of fundamental solutions on the entire real line is given by

(5.35) ϕ−ρ (x) = eκx2/4 D−υ(√

κx) and ϕ+ρ (x) = ϕ−ρ (−x)

where D−υ(x) is Whittaker’s parabolic cylinder function (see Abramowitz and Ste-gun (1972) for definitions and properties). The Wronskian constant in equation


(2.7) is easily shown to be wρ =√

2κπΓ(υ) . The parabolic cylinder functions are also

related to the confluent hypergeometric functions. The integral representation

D−υ(±√κx) =e−κx2/4

Γ(υ)

∫ ∞

0

e∓√

κ xs− 12 s2

sυ−1ds

shows that ϕ−ρ (x) and ϕ+ρ (x) given by (5.35) are positive and respectively mono-

tonically decreasing and increasing on (−∞,∞). Based on the above properties,the OU process generates two separate new subfamilies of F -diffusions as is shownin detail below. One subfamily is unbounded over the half-line while the other isbounded. However, both subfamilies are characterized by the following result.

Proposition 5.3. The family of F -diffusions {Ft = F(X(ρ)t ), t ≥ 0} with map F in

(3.12) generated by the Ornstein-Uhlenbeck process are all conservative martingalesand have the following boundary classification.

(i) q1 = 0, q2 > 0: F(−∞) is attracting natural, F(∞) is non-attracting natural.(ii) q1 > 0, q2 = 0: F(−∞) is non-attracting natural, F (∞) is attracting natural.(iii) q1 > 0, q2 > 0: F(−∞) and F (∞) are both attracting natural.


5.3.1. Unbounded Ornstein-Uhlenbeck (UOU) subfamily. Due to the reflection sym-metry ϕ+

ρ (x) = ϕ−ρ (−x) the subfamilies, arising when either q1 or q2 is zero, coin-cide. For definiteness sake, we let q1 = 0 and q2 = 1. So the generating function isuρ(x) = ϕ−ρ (x), and the map (3.12) takes the form F = F(UOU):

(5.36) F(UOU)(x) = aϕ+

ρ (x)

ϕ−ρ (x)= a

D−υ(−√κx)D−υ(

√κx)

,

where a is an arbitrary positive constant. The monotonically increasing functionF(UOU) maps x ∈ (−∞,∞) onto F ∈ (0,∞), with F(0) = a. This transformationleads to a subfamily of F -diffusions with volatility function

(5.37) σ(UOU)(F(x)) =a ν0 wρ eκx2/2

(ϕ−ρ (x))2=

a ν0 wρ

(D−υ(√

κx))2.

As |x| → ∞, the local volatility function σ(UOU)loc (F(x)) , σ(UOU)(F(x))

F(x) is asymp-

totically equivalent to a linear function in x: σ(UOU)loc (F(x)) ∼ κν0 |x|. By inverting

(5.36) in the limit |x| → ∞, then as a function of F the local volatility has theasymptotics:

(5.38) σ(UOU)loc (F ) ∼

ν0

√−2κ ln F

C1as F → 0+,

ν0

√2κ ln F

C2as F →∞,


with some positive constants C1,2. The local volatility exhibits a pronounced smilepattern as is shown in Figure 3. The exact transition density for this UOU-subfamilyis:

(5.39) U (UOU)(F, F0, t) =e−ρt+κ(x2−x2

0)/4

awρ

D 3−υ(

√κx)

D−υ(√

κx0)u(OU)(x, x0, t)

where F, F0 > 0, t > 0, and x = X(F ), x0 = X(F0) are given by inverting (5.36).

5.3.2. Bounded Ornstein-Uhlenbeck (BOU) subfamily. When both q1 and q2 arenon-zero, the state space is a bounded interval. Here we focus specifically on thecase where q1 = q2 = 1 and c2 = 0 in (3.12) although the analysis for other choicesof parameters follows similarly. The generating function is uρ(x) = ϕ+

ρ (x) + ϕ−ρ (x)so that F = F(BOU):

(5.40) F(BOU)(x) =b ϕ+

ρ (x)

ϕ+ρ (x) + ϕ−ρ (x)

=bD−υ(−√κx)

D−υ(−√κx) + D−υ(√

κx),

where b > 0 is a constant. This monotonically increasing function maps x ∈(−∞,∞) onto F ∈ (0, b) with F(0) = b/2 and this leads to a subfamily of F -diffusions with volatility function

(5.41) σ(BOU)(F(x)) =b ν0 wρ eκx2/2

(ϕ+ρ (x) + ϕ−ρ (x))2

=b ν0 wρ

(D−υ(−√κx) + D−υ(√

κx))2.

The local volatility exhibits a mixed smile-skewed behavior. From (5.40) and(5.41), and the asymptotic formulas for the parabolic cylinder functions, we havethe limiting expressions for the local volatility:

(5.42) σ(BOU)loc (F ) ∼

ν0

√−2κ ln F

B1as F → 0+,

B3∆F(− 2

κ ln ∆FB2

)1−υ as F → b−,

where ∆F , |F − b| and B1,2,3 are positive constants. As is seen from (5.42) andFigure 3, the local volatility function is skewed to the left and is zero at F = b−.

5.4. Hypergeometric Jacobi family. The Jacobi diffusion solves the SDE

(5.43) dXt = (λ0 − λ1Xt)dt + ν0

√Xt(A−Xt)dWt.

The state space is the finite interval I = (0, A). The speed and scale densities are

(5.44) s(J)(x) = x−β−1(A− x)−α−1, m(J)(x) =2ν20

xβ(A− x)α,

where α = 2λ1ν20− 2λ0

ν20A

− 1, β = 2λ0ν20A

− 1. Assuming α > 0 and β > 0 whenboth l = 0 and r = A are natural boundaries of the Jacobi process, then thetransition probability density function u(J)(x, x0, t) admits a spectral expansion inthe Jacobi orthogonal polynomials (see Karlin and Taylor (1981)). Here we shallonly analyze the Jacobi family in terms of whether or not the non-absorption andmartingale properties are satisfied. However, we note that such a closed-form series


representation for u(J)(x, x0, t), and hence for U (J)(F, F0, t), can be derived by usingthe residue theorem of complex analysis while evaluating the Bromwich integral.fundamental solutions ϕ+

ρ and ϕ−ρ to the ODE

ν20

2x(A− x)

d2ϕ(x)dx2

+ (λ0 − λ1x)dϕ(x)

dx= ρϕ(x), x ∈ (0, A).

The pair of fundamental solutions consists of the hypergeometric functions:

(5.45) ϕ+ρ (x) = 2F1(α1, α2; β + 1; x/A), ϕ−ρ (x) = 2F1(α1, α2; α + 1; 1− x/A),

where the parameters satisfy the following system of equations:

(5.46)

{α1 + α2 = α + β + 1 = 2λ1

ν20− 1,

α1α2 = 2ρν20.

The Wronskian factor in (2.7) is ws = ν20

2s Γ(α + 1)Γ(β + 1)Aα+β+1.The map in (3.12) gives rise to either bounded or unbounded F -diffusions. For

the case q1 > 0 and q2 > 0 the process is bounded where F(0+) = c2/q2, F(A−) =c1/q1 with F(x) as increasing when c1/q1 > c2/q2 and decreasing otherwise. Theother two cases, (i) q1 = 0, q2 > 0, c1 6= 0 and (ii) q2 = 0, q1 > 0, c2 6= 0, correspondto dual sets of unbounded processes. In case (i) Ft ∈ (c2/q2,∞) or (−∞, c2/q2) andin case (ii) Ft ∈ (c1/q1,∞) or (−∞, c1/q1), depending on whether the map F(x) isincreasing or decreasing.

Proposition 5.4. For any choice of parameters in (5.43) such that α > 0 andβ > 0, the F -diffusions {Ft = F(X(ρ)

t ), t ≥ 0} with map F in (3.12) generated bythe hypergeometric Jacobi process are nonconservative non-martingales and havethe following boundary classification.

(i) q1 = 0, q2 > 0: The boundary F(0+) is exit when β ≥ 1 and is regular killingwhen 0 < β < 1; the boundary F(A−) is entrance.

(ii) q1 > 0, q2 = 0: The boundary F(0+) is entrance; the boundary F(A−) is exitwhen α ≥ 1 and is regular killing when 0 < α < 1.

(iii) q1 > 0, q2 > 0: The boundary F(0+) has the same classification as in (i) andF(A−) has the same classification as in (ii).


6. Diffusions with Drift

6.1. Adding the affine drift. So far we have considered F -diffusions that solvea driftless SDE (3.1) with nonlinear diffusion coefficient σ(F ). For the specialcase of the CEV process with σ(F ) = δF β+1, it is known that the process St =ertFτ(t) defined by a scale and time change from the process Fτ , where τ(t) =


(e2rβt − 1)/(2rβ) for generally nonzero drift parameter r, simply introduces a driftin the SDE dSt = rStdt + δSβ+1

t dWt (for example, see Goldenberg (1991)). Wenow study the applicability of this type of transformation for generally nonlineardiffusions with SDE (3.1) and show that the CEV diffusion (i.e., the power lawvolatility function) is the only diffusion model for which this approach leads to atime-homogeneous process with linear drift.

Consider then a transformed process St = ertFτ(t) with some strictly increasingdifferentiable function τ(t) such that τ(0) = 0 and τ ′(0) = 1. From the Kol-mogorov PDE for generally nonzero drift parameter r, one readily derives p.d.f.sas Ur(S, S0, t) = e−rtU(e−rtS, S0, τ(t)), where U solves (3.2). The transition p.d.f.Ur satisfies the forward Kolmogorov equation:

∂Ur

∂t(S, S0, t) =

12

∂2

∂S2

(e2rtτ ′(t)σ2(e−rtS)Ur(S, S0, t)

)− ∂

∂S

(rSUr(S, S0, t)

).

As is seen, the process {St}t≥0 solves a time-inhomogeneous SDE

(6.1) dSt = rStdt + σ(t, St)dWt, where σ(t, S) = ert√

τ ′(t)σ(e−rtS)

is the new diffusion coefficient. Of interest is when this diffusion coefficient isindependent of t. Differentiating σ2 w.r.t. t and equating the resulting derivativeto zero gives: ∂σ2

∂t (t, S) =(2r + τ ′′(t)

τ ′(t) − 2r e−rtSσ′(e−rtS)σ(e−rtS)

)σ2(t, S) = 0. Hence the

equality holds if the function τ solves the ODE

(6.2) τ ′′(t) + 2r

(1− e−rtSσ′(e−rtS)

σ(e−rtS)

)τ ′(t) = 0

along with the imposed initial conditions: τ(0) = 0 and τ ′(0) = 1. To ensure thata solution τ is independent of S, the derivative w.r.t. S of the last term in thebrackets has to be zero: ∂

∂S

(e−rtSσ′(e−rtS)

σ(e−rtS)

)= 0. Hence the diffusion coefficient σ

has to solve(

Fσ′(F )σ(F )

)′= 0. The only solution to this ODE is a power function:

σ(F ) = AFB , where A and B are constants. Then the solution to (6.2) is

(6.3) τ(t) =e2r(B−1)t − 12r(B − 1)

.

Thus the only model, for which the scale and time change described above leads toa time-inhomogeneous diffusion, is the CEV model with diffusion coefficient σ(F ) =δF β+1. Moreover, the transformation in this case is also volatility preserving, i.e.,σ(t, S) = σ(S) = δSβ+1.

In general cases, when the diffusion coefficient is not of the form σ(F ) = AFB ,

one cannot entirely eliminate the time dependence. Yet, if desired one can choose avalue for B so as to minimize the time dependence of σ(t, S) in certain domains ofS based on the asymptotic form (or approximation) of σ(S) by a power function.For example, for the Bessel family of models we can use the asymptotics in (5.7) or(5.8). In this case we guarantee the time-homogeneity of σ in a neighborhood of the


corresponding endpoint of the underlying asset price, either as S tends to zero orbecomes large. The same idea can be used for the other nonlinear volatility models(see Figure 4). Qualitatively, one observes that the overall shape of σ(t, S) w.r.t.the asset price S does not vary substantially as t is varied within the models that wehave presented in this paper. However, and perhaps interestingly, the introductionof the drift due to the interest rate adds a slight time dependence in the volatilityfunction which can be made relatively more pronounced in certain regions of theasset price.

Another approach to introducing an affine drift is based on changing the mappingfunction F. This new approach further generalizes the diffusion canonical transfor-mation method discussed in this paper. It is shown in (Campolieti and Makarov2008c) how a modification of (3.12) allows us to construct newly solvable familiesof time-homogeneous F -diffusions with an affine drift. Applications of such affinediffusions are presented in (Campolieti 2008).

6.2. Pricing vanilla European options. Consider a positive asset price process{St}t≥0 modeled as a diffusion according to (6.1). Assume an economy with con-stant domestic interest rate r. Choosing the money-market account gt = ert asnumeraire, we introduce the risk-neutral pricing measure Q = Q(g) and Wt = WQ

t

– a standard Brownian motion under Q. The risk-neutral transition p.d.f. isUr(S, S0, t) = e−rtU(e−rtS, S0, τ(t)) for the asset price process in the measure Q.From the previous section, we observe that the process St is reduced to the forwardprice process Fτ solving (3.1) and having the transition p.d.f. U(F, F0, τ) withSt = ertFτ(t). The time change τ(t) can be taken as in (6.3).

As usual, a standard European-style option with time-to-maturity T > 0 isdefined by its payoff function Λ(ST ). The value of a European-style option is givenby the discounted risk-neutral expectation

V (S0, T ) = e−rTEQ[Λ(ST ) | St=0 = S0] = EQ[Λr,T (Fτ(T )) | Fτ=0 = F0]

where Λr,T (F ) , e−rT Λ(erT F ) is an effective payoff function. The option valuationcan hence be reduced to an integral in the forward price space while using the exactforward price transition p.d.f. U(F, F0, τ(T )):

V (S0, T ) = e−rT

∫ ∞

0

Ur(S, S0, T )Λ(S) dS =∫ ∞

0

U(F, F0, τ(T ))Λr,T (F ) dF .

For a vanilla European call option with Λ(S) = (S −K)+,K > 0 as strike price,the effective payoff is given by Λr,T (F ) = (F − e−rT K)+, which retains the sameform as Λ with strike replaced by discounted strike. The above option value can berecast as an expectation integral that is more readily computed by switching to thex-space variables. This also mainly avoids the inversion implicit in the mapping


from x to F = F(x), giving:

V (S0, T ) =∫ r

l

u(ρ)(x,X(S0), τ(T ))Λr,T (F(x)) dx .(6.4)

We observe that the option value is generally reduced to a closed-form integral overspecial functions (i.e., integrands involving the Bessel, or confluent hypergeometricfunctions, etc.). For some types of options, the option value can be expressed as aclosed-form rapidly convergent eigenfunction expansion. The European call optionvalue using (6.4) is also readily computed by numerical integration. For the specialfamilies considered in this paper we used Mathematica1 where the modified Besseland confluent functions are available as built-in functions. which contains numer-ous routines for computing special functions, and a suitable numerical quadratureroutine (e.g., from the NAG2 Fortran Library).

We now present typical implied Black-Scholes volatility surfaces computed forsome of the models presented in this paper. Since the models support smiles andskewed shapes for the local volatility, these same features are also expected toarise within the implied volatility surfaces. For illustration, we restrict ourselves totwo interesting models – the Ornstein-Uhlenbeck UOU model and the confluent-Usubfamily. Figures 5 and 6 shows typical implied volatility surfaces. As is seenfrom the plots, the implied volatility against the strike price of the option exhibitsvarious smile and skewed shapes. In fact the models support a rather wide range ofshapes that also tend to flatten at large values of time to maturity. These featuresare desirable for realistic model calibration to observed option market data. Forother applications in pricing exotic options under these models, and under someextensions of these models with drift, we refer to Campolieti and Makarov (2007;2008a), Campolieti (2008).

7. Conclusion

By applying the diffusion canonical transformation method, this paper has de-veloped new families of exactly solvable multiparameter nonlinear local volatilitydiffusion models (“F -diffusions”) for asset price processes and other financial quan-tities. In particular, analytically exact expressions for transition densities wereobtained for various families of F -diffusions in terms of known special functions.Subfamilies of these models encompass the driftless CEV and other models as spe-cial cases, and have been shown to exhibit a wide range of implied volatility surfaceswith pronounced smiles and skews.

This paper has also developed the boundary classification for all families of F -diffusions. In addition, the paper presents two simple theorems for classifying

1Mathematica is a registered trademark of Wolfram Research Inc.2The Numerical Algorithms Group.


the martingale and probability conservation properties of any regular F -diffusion.Moreover, an easy-to-implement limit formula for computing first-hitting (exit)time densities was derived for any F -diffusion defined on its entire regular statespace with one or two possible absorbing endpoint boundaries. Its implementationlead to new closed-form first-exit time densities for such processes. By employing aspectral expansion approach, the theory of the present paper is also readily extendedto cover F -diffusions with killing at arbitrary levels. This then leads to new familiesof analytically exact results for: the transition densities for killed F -diffusions, thedensities for any first-hitting (or exit) time of any F -diffusion, the densities for theextrema of any F -diffusion, the joint densities of the extrema and the value of anyF -diffusion. As a consequence, various new analytical pricing formulas for exoticoptions (barriers, lookbacks, etc.) are derivable in closed-form. These applicationsare the subject of currently related research (see Campolieti (2008)).

Appendix A. Boundary Classification for the Transformed

Diffusions

A.1. The proof of Lemma 4.1. Note that by (4.3) and (4.4) we equivalentlyhave that e = l is regular if Sρ(e, x] < ∞ and Mρ(e, x] < ∞, exit if Σρ(e) < ∞and Mρ(e, x] = ∞ and entrance if Sρ(e, x] = ∞ and Nρ(e) < ∞, with analogousconditions for the right boundary e = r. Moreover, the left (right) boundary l (r)of the diffusion X(ρ) is attracting if and only if Sρ(l, x] < ∞ (Sρ[x, r) < ∞).

The scale measure in (4.1) is evaluated simply by using equation (3.11). Takingleft and right limits and employing the asymptotic properties in (2.8) gives

(A.1) Sρ(l, x] =

1q2wρ

ϕ+ρ (x)

uρ(x) , q2 > 0,

∞ , q2 = 0,

Sρ[x, r) =

1q1wρ

ϕ−ρ (x)

uρ(x) , q1 > 0,

∞ , q1 = 0.

Hence, for the three general cases we have:

(i) q1 = 0, q2 > 0: l is attracting and r is non-attracting,(ii) q1 > 0, q2 = 0: l is non-attracting and r is attracting,(iii) q1 > 0, q2 > 0: l and r are both attracting.

A boundary e is attainable in finite time if and only if Σρ(e) < ∞. By the abovescale measures, (4.3) gives

Σρ(l) =

1q2wρ

(ϕ+

ρ , uρ

)(l,x]

, q2 > 0,

∞ , q2 = 0,

Σρ(r) =

1q1wρ

(ϕ−ρ , uρ

)[x,r)

, q1 > 0,

∞ , q1 = 0.

Note that throughout we use the notation for the inner product w.r.t. the speedmeasure m (not mρ) as defined in (2.5). Recalling the conditions in (2.9), then forthe three general cases we have:


(i) q1 = 0, q2 > 0: Σρ(l) < ∞ if and only if (ϕ+ρ , ϕ−ρ )(l,x] < ∞; Σρ(r) = ∞.

(ii) q1 > 0, q2 = 0: Σρ(l) = ∞; Σρ(r) < ∞ if and only if (ϕ+ρ , ϕ−ρ )[x,r) < ∞.

(iii) q1 > 0, q2 > 0: Σρ(l) < ∞ if and only if (ϕ+ρ , ϕ−ρ )(l,x] < ∞; Σρ(r) < ∞ if and

only if (ϕ+ρ , ϕ−ρ )[x,r) < ∞.

The asymptotic properties (2.8) obviously imply that (ϕ−ρ , ϕ−ρ )(l,x] < ∞ =⇒(ϕ+

ρ , ϕ−ρ )(l,x] < ∞ and (ϕ+ρ , ϕ+

ρ )[x,r) < ∞ =⇒ (ϕ+ρ , ϕ−ρ )[x,r) < ∞. Since mρ(x) =

u2ρ(x)m(x), and (2.9) holds, then (4.2) immediately gives the following conditions

for the boundedness of the speed of X(ρ)-diffusions at the boundaries:

(i) q1 = 0, q2 > 0: Mρ(l, x] < ∞ if and only if (ϕ−ρ , ϕ−ρ )(l,x] < ∞; Mρ[x, r) < ∞.(ii) q1 > 0, q2 = 0: Mρ[x, r) < ∞ if and only if (ϕ+

ρ , ϕ+ρ )[x,r) < ∞, Mρ(l, x] < ∞.

(iii) q1 > 0, q2 > 0: Mρ(l, x] < ∞ if and only if (ϕ−ρ , ϕ−ρ )(l,x] < ∞; Mρ[x, r) < ∞if and only if (ϕ+

ρ , ϕ+ρ )[x,r) < ∞.

The functionals defined by (4.4) roughly measure the time taken to reach aninterior point x ∈ I of the diffusion X(ρ) started at the respective boundariesl or r. Now, combining Sρ[z, x] = Sρ(l, x] − Sρ(l, z] for q2 > 0 and Sρ[z, x] =Sρ[z, r)− Sρ[x, r) for q1 > 0 with (A.1) into (4.4) gives:

(i) q1 = 0, q2 > 0: Nρ(l) < ∞ if and only if (ϕ−ρ , ϕ−ρ )(l,x] < ∞; Nρ(r) < ∞ if andonly if (ϕ+

ρ , ϕ−ρ )[x,r) < ∞.(ii) q1 > 0, q2 = 0: Nρ(l) < ∞ if and only if (ϕ+

ρ , ϕ−ρ )(l,x] < ∞; Nρ(r) < ∞ if andonly if (ϕ+

ρ , ϕ+ρ )[x,r) < ∞.

(iii) q1 > 0, q2 > 0: Nρ(l) < ∞ if and only if (ϕ−ρ , ϕ−ρ )(l,x] < ∞; Nρ(r) < ∞ if andonly if (ϕ+

ρ , ϕ+ρ )[x,r) < ∞.

By simply combining the conditions in the above analysis for the four functionals,we have therefore proven the lemma.

A.2. The proof of Lemma 4.2. Using the Ito formula for the diffusion Ft =F(X(ρ)

t ) and changing integration variables x = X(z) = F−1(z), the scale densitysF of Ft is recast as sF(F ) = csρ(X(F )) |X′(F )| for some constant c > 0. That is,sF(F )dF = ±csρ(x)dx and the scale measure function for diffusion Ft is

(A.2) S(F)[F1, F2] ,∫ F2

F1

sF(F ) dF = cSρ[x<, x>] .

where F1 = F(x1), F2 = F(x2) ∈ D, F1 < F2, x< = min{x1, x2}, x> = max{x1, x2}for any x1, x2 ∈ (l, r). Now, pick any x ∈ (l, r), with F = F(x) ∈ D. Then, sincesF(F ) = csρ(x)|X′(F )|, we have mF(F )dF = ±(1/c)mρ(x)dx, i.e.,

(A.3) M (F)[F1, F2] ,∫ F2

F1

mF(F ) dF =1cMρ[x<, x>] .

Hence, the scale and speed measures for an F -diffusion are simply constant multiplesof those for the corresponding X(ρ)-diffusion and the boundary classifications are


the same. In particular, define two functionals Σ(F) and N (F) analogous to (4.3) and(4.4) with mρ → mF,Sρ → S(F). Then, Σ(F)(F (e)) = Σρ(e) and N (F)(F (e)) = Nρ(e)where F (e) ∈ {F (l), F (r)}, F (l) = F(l+) (or F(r−)) and F (r) = F(r−) (or F(l+))for increasing (or decreasing) maps.

A.3. The proof of Theorem 4.3. By applying the Laplace transform w.r.t.time t to (4.6), changing order of integration and differentiation, and using (2.11),the Laplace transform of the rate of absorption takes the form:

(A.4) L[R(F (l), F (r)|F0, t)](s) =W

[uρ(x), ϕ+

ρ+s(x<)ϕ−ρ+s(x>)]

wρ+s s(x) uρ(x0)

∣∣∣∣x=r−

x=l+

,

Here we used the identity (f(x)/g(x))′ = W [g, f ](x)/g2(x). If there is no absorption,then the Laplace transform is identically zero. Hence, setting L ≡ 0 in (A.4) andusing x< = x0 (x> = x0) when x → r− (x → l+) gives

(A.5) ϕ+ρ+s(x0) lim

x→r−W [uρ, ϕ

−ρ+s](x)

s(x)− ϕ−ρ+s(x0) lim

x→l+

W [uρ, ϕ+ρ+s](x)

s(x)= 0 .

Since ϕ+ρ+s(x0) and ϕ−ρ+s(x0) are nontrivial linearly independent fundamental so-

lutions to Gx0ϕ(x0) = (ρ + s)ϕ(x0), then (A.5) holds for all x0 ∈ I if and onlyif the limits in (4.8) hold. The converse statement follows since (4.8) =⇒ (A.5)=⇒ L ≡ 0. By Laplace inversion (with Bromwich contour chosen in the half-planeRe s ≥ c) and taking into account the uniqueness of the Laplace inverse we arriveat R(F (l), F (r)|F0, t) ≡ 0, i.e. no absorption. ¤

A.4. The proof of Theorem 4.4. The proof follows similar steps as above forTheorem 4.3. We now take the Laplace transform L = L[M(F (l), F (r)|F0, t)](s) of(4.11) w.r.t. t, change order of integration and differentiation, and use (2.11), (3.9)and x< = x0 (x> = x0) when x → r− (x → l+). Then, setting L = 0 gives

(A.6) ϕ+ρ+s(x0)

(lim

x→r−g−(x)

)− ϕ−ρ+s(x0)(

limx→l+

g+(x))

= 0,

where g±(x) , F(x)sρ(x)

∂∂x

(ϕ±ρ+s(x)

uρ(x)

) − sgn(F′(x))σ0ϕ±ρ+s(x)

uρ(x) , sgn(F′(x)) = ±1. Linearindependence of the pair ϕ±ρ+s(x0) gives that (A.6) holds, for all x0 ∈ I, if and onlyif the two separate limits are zero, i.e. (4.13) and (4.14) hold. As in the proof ofTheorem 4.3, the converse statement again follows by Laplace inversion. ¤


A.5. The proof of Proposition 5.1. First, let us analyze case (i). For the Besselfamily we have the asymptotics:

as x → 0+ : as x → +∞ :

W [uρ,ϕ+ρ+s](x)

s(x) ∼ C(s),W [uρ,ϕ−ρ+s](x)

s(x) ∼ C(s)e−2(√

ρ+s+√

ρ)√

2x/ν0 ,

ϕ+ρ+s(x)

uρ(x) ∼ C(s)xµ,ϕ−ρ+s(x)

uρ(x) ∼ C(s)e−2(√

ρ+s−√ρ)√

2x/ν0 ,

F (x) ∼ c2C + Cxµ F (x) ∼ Ce4√

2ρx/ν0 ;

where here and everywhere below C(s) is a generic nonzero constant that dependson s, C is a generic nonzero constant that is independent of s. Hence, (A.4) givesL[R](s) = C(s) 6≡ 0 for Re s > c, ∀c > 0. For the rate M = M(F (l), F (r)|F0, t) wehave L[M ](s) ≡ 0 for Re s > c for some c > 0 if and only if c2 = 0.

For case (ii) when q1 > 0 and q2 = 0 the results follow from the asymptotics:

as x → 0+ : as x → +∞ :

W [uρ,ϕ+ρ+s](x)

s(x) ∼ C(s)xµ+1,W [uρ,ϕ−ρ+s](x)

s(x) ∼ C(s)e−2(√

ρ+s−√ρ)√

2x/ν0 ,

ϕ+ρ+s(x)

uρ(x) ∼ C(s),ϕ−ρ+s(x)

uρ(x) ∼ C(s)e−2(√

ρ+s+√

ρ)√

2x/ν0 ,

F (x) ∼ Cx−µ; F (x) ∼ c1C + Ce−4√

2ρx/ν0

Clearly, L[R](s) ≡ 0 for Re s > 0; L[M ](s) = C(s) 6≡ 0 for Re s > c, ∀c > 0. Fromthe asymptotics of u(SQB), ϕ±ρ and their derivatives it follows that the sign of therate M is a negative of the sign of c2. Although we skip the details here, the explicitexpression is easily derived and gives that the rate M is a monotonic function oftime t. The analysis of case (iii) when q1, q2 > 0 is similar to cases (i)-(ii). Theboundary classification for all three cases follows directly from Lemmas 4.1 and 4.2and the following limits: ϕ+

ρ (x)ϕ−ρ (x)m(x) ∼ C and ϕ−ρ (x)ϕ−ρ (x)m(x) ∼ Cx−µ asx → 0+; ϕ+

ρ (x)ϕ−ρ (x)m(x) ∼ Cx−1/2 as x → +∞.

A.6. The proof of Proposition 5.2. The proof is very similar to that of Propo-sition 5.1 and is based on the asymptotics of the Kummer functions.Case (i) when q1 = 0 :

as x → 0+ : as x → +∞ :

W [uρ,ϕ+ρ+s](x)

s(x) ∼ C(s),W [uρ,ϕ−ρ+s](x)

s(x) ∼ C(s)x−2υ−µ− sλ1 e−κx,

ϕ+ρ+s(x)

uρ(x) ∼ C(s)xµ,ϕ−ρ+s(x)

uρ(x) ∼ C(s)x−s

λ1 ,

F (x) ∼ c2C + Cxµ F (x) ∼ Cx2υ−µ−1eκx.


Case (ii) when q2 = 0 :

as x → 0+ : as x → +∞ :

W [uρ,ϕ+ρ+s](x)

s(x) ∼ C(s)xµ+1,W [uρ,ϕ−ρ+s](x)

s(x) ∼ C(s)e−κx,

ϕ+ρ+s(x)

uρ(x) ∼ C(s),ϕ−ρ+s(x)

uρ(x) ∼ C(s)x−2υ+µ+1− sλ1 ,

F (x) ∼ Cx−µ; F (x) ∼ c1C + Cx−2υ+µ+1e−κx

The boundary classification follows directly from Lemmas 4.1 and 4.2 and the fol-lowing limits: ϕ+

ρ (x)ϕ−ρ (x)m(x) ∼ C and ϕ−ρ (x)ϕ−ρ (x)m(x) ∼ Cx−µ as x → 0+;ϕ+

ρ (x)ϕ−ρ (x)m(x) ∼ Cx−1 as x → +∞.

A.7. The proof of Proposition 5.3. Using asymptotics of parabolic cylinder

functions we have:W [ϕ∓ρ+s,ϕ±ρ ](x)

s(x) ∼ C(s)|x|−s/λ1 , as x → ±∞; andW [ϕ∓ρ+s,ϕ∓ρ ](x)

s(x) ∼C(s)|x|−2υ−s/λ1−1e−κx2/2, as x → ±∞. For Re s > 0 the right-hand sides ofthe above asymptotics all tend to zero. Hence the conditions of Theorem 4.3 aresatisfied for all choices of parameters q1, q2, i.e., probability conservation alwaysholds. In general, F(x) given by (3.12) maps the interval x ∈ (−∞,∞) onto eithera finite interval (if both q1 and q2 are nonzero) or a semi-infinite interval (if only q1

or only q2 is zero). If limx→e

F(x) is finite, with either endpoint e ∈ {−∞, +∞}, then

the corresponding limits limx→e

F(x)W [uρ,ϕ±ρ+s](x)

s(x) in (4.13)-(4.14) are automatically

zero. Suppose that F(±∞) = ∞, then F(x) ∼ |x|2υ−1eκx2/2 as x → ±∞, and

clearly F(x)W [uρ,ϕ∓ρ+s](x)

s(x) → 0 for Re s > 0. Finally, the other ratios in (4.13)-(4.14)also all tend to zero for any choice of uρ(x) since:

limx→∞

ϕ−ρ+s(x)

ϕ−ρ (x)= lim

x→−∞ϕ+

ρ+s(x)

ϕ+ρ (x)

= limx→∞

O( |x|−s/λ1) = 0,

limx→∞

ϕ−ρ+s(x)

ϕ+ρ (x)

= limx→−∞

ϕ+ρ+s(x)

ϕ−ρ (x)= lim

x→∞O( |x|1−(2ρ+s)/λ1e−κx2/2) = 0,

for Re s > 0. Hence, the martingale condition in Theorem 4.4 is satisfied.The boundary classification follows by ϕ+

ρ ϕ−ρ m ∼ C |x|−1, as |x| → ∞, ϕ±ρ ϕ±ρ m ∼C |x|2υ−2eκx2/2, as x → ±∞, and ϕ±ρ ϕ±ρ m ∼ C |x|−2υe−κx2/2, as x → ∓∞.


A.8. The proof of Proposition 5.4. We have the following asymptotic forms forϕ+

ρ and ϕ−ρ and their derivatives (see Abramowitz and Stegun (1972)):

ϕ+ρ (x) ∼ 1, as x → 0+, ϕ+

ρ (x) ∼ Γ(β+1)Γ(α)Γ(α1)Γ(α2)

(1− x

A

)−α, as x → A−,

ϕ−ρ (x) ∼ 1, as x → A−, ϕ−ρ (x) ∼ Γ(α+1)Γ(β)Γ(α1)Γ(α2)

(xA

)−β, as x → 0+,

dϕ+ρ

dx (x) ∼ α1α2A(β+1) , as x → 0+,

dϕ+ρ

dx (x) ∼ Γ(β+1)Γ(α+1)AΓ(α1)Γ(α2)

(1− x

A

)−α−1, as x → A−,

dϕ−ρdx (x) ∼ − α1α2

A(α+1) , as x → A−,dϕ−ρdx (x) ∼ −Γ(α+1)Γ(β+1)

AΓ(α1)Γ(α2)

(xA

)−β−1, as x → 0 + .

These relations also hold for complex values when ρ is replaced by ρ + s in theabove relations and in (5.46), where α1α2 = 2(s + ρ)/ν2

0 . By the above relationsit is easy to show that the Laplace transforms L[R] and L[M ] are given by somefunctions of s that are not identically zero for Re s > c with any constant c.

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Figure 1. Local volatility plots for the Bessel-I (left plot) and Bessel-K

subfamilies with ν0 = 2. For the I-subfamily we use (5.5) and (5.6) with

aI = 24.5302, ρ = 0.001, µ = 0.25 (the thinnest line); aI = 17.7323,

ρ = 0.001, µ = 0.5 (the thickest line); and aI = 5.0574, ρ = 0.001,

µ = 1.5. For the K-subfamily we use aK = 78.5398, ρ = 0.005, µ = 0.5

(the thinnest line); aK = 129.4506, ρ = 0.005, µ = 0.125 (the thickest

line); and aK = 362.2907, ρ = 0.00125, µ = 0.25. The parameters are

chosen in such a way that the value of the local volatility functions is

fixed at 0.25 for F = 100.

Figure 2. Local volatility plots for the confluent hypergeometric

subfamilies. The confluent M-subfamily (left plot) and confluent U-

subfamily (right plot) are plotted using (5.21) and (5.26), correspond-

ingly, for the following choices of parameters: aM = aU = 100, µ = 0.25,

λ0 = 0.25, κ = 0.1; υ = 0.025 (the thinnest line); υ = 0.1(the moderate

line); and υ = 0.5 (the thickest line).


Figure 3. Local volatility plots for the Ornstein-Uhlenbeck diffusion

family. The UOU-subfamily (left plot) and BOU-subfamily (right plot)

are plotted using (5.36)–(5.37) and (5.40)–(5.41), correspondingly, with

ρ = 0.001, κ = 10, υ = 0.025 (the thinnest line); ρ = 0.01, κ = 5,

υ = 0.1 (the moderate line); ρ = 0.02, κ = 1, υ = 0.5 (the thickest line);

and a = 100, b = 300.

Figure 4. Local volatility plots σ(t, S)/S for the confluent CIR U-

subfamily. The curves correspond to t = 0 (thinest line), t = 2, t = 4,

and t = 6 (thickest line) with aU = 100, µ = 0.25, λ0 = 0.25, κ = 0.1,

υ = 0.1, and r = 0.05. The left plot is due to the time change (6.3) with

B = 1− 12µ

= −1, where the asymptotics (5.27), as F → 0+, was used

for the choice of B. The right plot corresponds to the time change (6.3)

with B = 1.5.


5075

100125

150175

200

Strike

0

0.25

0.5

0.75

1

Time

00.10.20.30.40.50.6

5075

100125

150175

Strike

5075

100125

150175

200

Strike

0

0.25

0.5

0.75

1

Time

00.10.20.30.40.50.6

5075

100125

150175

Strike

Figure 5. Black-Scholes implied volatility surfaces computed for the

confluent U-subfamily with SDE (6.1) and diffusion coefficient σ(F )

given by (5.26) and (5.25). The left (right) surface corresponds to the

same choice of parameters as that used to plot the local volatility in

Figure 2, which is drawn with the thinnest (moderate) line: aU = 100,

µ = 0.25, λ0 = 0.25, κ = 0.1; υ = 0.01 (the left plot) and υ = 0.1 (the

right plot). The interest rate is r = 0.1, the spot S0 = 100, and the time

change (6.3) with B = 1.5 is used.

5075

100125

150175

200

Strike

0

0.25

0.5

0.75

1

Time

00.10.20.30.40.50.6

5075

100125

150175

Strike

5075

100125

150175

200

Strike

0.1

0.25

0.5

0.75

1

Time

0

0.1

0.2

0.3

0.4

5075

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150175

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Figure 6. Black-Scholes implied volatility surfaces computed for the

Ornstein-Uhlenbeck UOU-subfamily with SDE (6.1) and diffusion coef-

ficient σ(F ) given by (5.37) and (5.36). The left (right) surface corre-

sponds to the same choice of parameters as that used to plot the local

volatility on Figure 3, which is drawn with the thinnest (moderate) line:

a = 100; ρ = 0.0001, κ = 10, υ = 0.005 (the left plot) and ρ = 0.01,

κ = 5, υ = 0.1 (the right plot). The interest rate is r = 0.1, the spot

S0 = 100, and the time change (6.3) with B = 1.5 is used.

ON PROPERTIES OF ANALYTICALLY SOLVABLE FAMILIES OF LOCAL VOLATILITY DIFFUSION MODELS

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