NBER WORKING PAPER SERIES
TRANSFORM ANALYSIS AND
ASSET PRICING FOR AFFINE
JUMP-DIFFUSIONS
Darrell Duffie
Jun Pan
Kenneth Singleton
Working Paper 7105
http://www.nber.org/papers/w7105
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
April 1999
We are grateful for extensive discussions with Jun Liu; conversations with Jean Jacod, Monika Piazzesi,
Philip Protter, and Ruth Williams; and support from the Financial Research Initiative, The Stanford
Program in Finance, and the Gifford Fong Associates Fund, at the Graduate School of Business, Stanford
University. The views expressed herein are those of the authors and do not necessarily reflect the views
of the National Bureau of Economic Research.
©
1999
by Darrell Duffie, Jun Pan, and Kenneth Singleton. All rights reserved. Short sections of text, not
to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including
©
notice,
is given to the source.
Transform Analysis and Asset Pricing for
Affine Jump-Diffusions
Darrell Duffie, Jun Pan, and Kenneth Singleton
NBER Working Paper No. 7105
April 1999
JELNo. Gi
ABSTRACT
In the setting of "affine" jump-diffusion state processes, this paper provides an analytical
treatment of a class of transforms, including various Laplace and Fourier transforms as special cases,
that allow an analytical treatment of a range of valuation and econometric problems. Example
applications include fixed-income pricing models, with a role for intensity-based models of default,
as well as a wide range of option-pricing applications. An illustrative example examines the
implications of stochastic volatility and jumps for option valuation. This example highlights the
impact on option 'smirks' of the joint distribution ofjumps in volatility and jumps in the underlying
asset price, through both amplitude as well as jump timing.
Darrell Duffie
Jun Pan
Graduate School of Business
Graduate School of Business
Stanford University
Stanford University
Stanford, CA 93405
Stanford, CA 93405
duffie@stanford.edu
junpan@stanford.edu
Kenneth Singleton
Graduate School of Business
Stanford University
Stanford, CA 93405
and NBER
kenneths@future.stanford.edu
1 Introduction
In valuing financial securities in an arbitrage-free environment, one inevitably
faces a trade-off between the analytical and computational tractability of
pricing and estimation, and the complexity of the probability model for the
state vector X. In the light of this trade-off, academics and practitioners
alike have found it convenient to impose sufficient structure on the condi-
tional distribution of X to give closed- or nearly closed-form expressions for
securities prices. An assumption that has proved to be particularly fruitful
in developing tractable, dynamic asset pricing models is that X follows an
affine jump-diffusion (AJD), which is, roughly speaking, a jump-diffusion
process for which the drift vector, "instantaneous" covariance matrix, and
jump intensities all have affine dependence on the state vector. Prominent
among AJD models in the term-structure literature are the Gaussian and
square-root diffusion models of Vasicek [1977] and Cox, Ingersoll, and Ross
[1985]. In the case of option pricing, there is a substantial literature building
on the particular affine stochastic-volatility model for currency and equity
prices proposed by Heston [1993].
This paper synthesizes and significantly extends the extant literature on
affine asset pricing models by deriving a closed-form expression for an "ex-
tended transform" of an AJD process X, and then showing that this trans-
form leads to analytically tractable pricing relations for a wide variety of
valuation problems. More precisely, fixing the current date t and a future
payoff date T, suppose that the stochastic "discount rate" R(X), for com-
puting present values of future cash flows, is an affine function of X. Also,
consider the generalized terminal payoff function (vo + v1 XT) euX(T) of XT,
where
v0 is scalar and the n elements of each of the v1 and u are scalars.
These scalars may be real, or more generally, complex. We derive a closed-
form expression for the transform
E (exp (_
1T
R(X8, s) ds) (v0 + v1 XT) eT),
(1.1)
where E denotes expectation conditioned on the history of X up to t. Then,
using this transform, we show that the tractability offered by extant, special-
ized affine pricing models extends to the entire family of AJDs. Additionally,
by selectively choosing the payoff (v0+v1 .XT) e'(T), we significantly extend
the set of pricing problems (security payoffs) that can be tractably addressed
2
with X following an AJD. To motivate the usefulness of our extended trans-
form in theoretical and empirical analyses of affine models, we briefly outline
three applications.
1.1 Affine, Defaultable Term Structure Models
There is a large literature on the term structure of default-free bond yields
that presumes that the state vector underlying interest rate movements fol-
lows an AJD (see, e.g., Dai and Singleton [1999] and the references therein).
Assuming that the instantaneous riskless short-term rate Tt is an affine func-
tion of an n-dimensional AJD process X (that is Tt =
Po
+ ,oi X) ]Duffie
and Kan [1996] show that the (T —
t)-period zero-coupon bond price,
E (exp (JT))
(1.2)
is known in closed form, where expectations are computed under the risk-
neutral measure.1
Recently, considerable attention has been focused on extending these
models to allow for the possibility of default in order to price corporate
bonds and other credit-sensitive instruments.2 To illustrate the new pricing
issues that may arise with the possibility of default, suppose that default
is governed by a stochastic intensity A and that, upon default, the holder
recovers a constant fraction w of face value. Then, from results in Lando
[1998], the price of a (T —
t)-period
zero-coupon bond is given under techni-
cal integrability conditions by
E (exp (_
f(rs
+ A8) ds)) + w f
E
(s
exp
(
f8( +
A) du)) ds.
(1.3)
The first term in (1.3) is the value of a claim that pays $1 contingent on
survival to maturity T, while the second term is the value of the claim that
pays w at date s should the issuer default at that date, and nothing otherwise.
Both the first term and, for each s, the expectation in the second term
can be computed in closed form using our extended transform. Specifically,
1The entire class of affine term structure models is obtained as the special case of (1.1)
found by setting R =
r,
u =
0,
vo =
1,
and v1 =
0.
2See, for example, Jarrow, Lando, and Turnbull [1997] and Duffie and Singleton [1999].
3
assuming that both rt and
are affine in an AJD process X, the first
expectation in (1.3) is the special case of (1.1) that is obtained by letting
R(X, t) =
Tt
+ )t, u =
0,
v0
1 and v1 =
0.
Similarly, each expectation in
(1.3) of the form E (A8 exp (—
f8 r,
+ )' du)) is obtained as a special case
of (1.1) by setting u =
0,
R(X, t) =
Tt
+ At, and v0 + v1 X =
A.
Thus,
using our extended transform, the pricing of defaultable zero-coupon bonds
with constant fractional recovery of par reduces to the computation of a
one-dimensional integral of a known function. Similar reasoning can be used
to derive closed-form expressions for zero prices in environments where the
default arrival intensity is affine in X, and there is "gapping" risk associated
with unpredictable transitions to different credit categories (see Lando [1998]
for the case of w =
0).
A different application of the extended transform is pursued by Piazzesi
[1998] who extends the AJD model in order to treat term-structure models
with releases of macro-economic information and with central-bank interest-
rate targeting. She considers jumps at both random and at deterministic
times, and allows for an intensity process and interest-rate process that have
linear-quadratic dependence on the underlying state vector, extending the
basic results of this paper.
1.2 Estimation of Affine Asset Pricing Models
Another useful implication of (1.1) is that, by setting R =
0,
v0 =
1,
and
v1 =
0,
we obtain a closed-form expression for the conditional characteristic
function of XT given X, defined by
X, t, T) E (eT X) .
Because
knowledge of is equivalent to knowledge of the joint conditional density
function of XT, this result is useful in estimation and all other applications
involving the transition densities of an AJD.
For instance, Singleton [1998] exploits knowledge of to derive maximum
likelihood estimators for AJDS based on the conditional density of X1 given
X, obtained by Fourier inversion of
f(Xt+iXt;7) —
N
f
e_1(u,Xt,t,t
+ 1) du.
(1.4)
(2ir) a
Das [1998] exploits (1.4) for the specific case of a Poisson-Gaussian AJD to
compute method-of-moments estimators of a model of interest rates.
Method-of-moments estimators can also be constructed directly in terms
4
