Pricing and Hedging Path-Dependent
Options Under the CEV Process
Dmitry Davydov • Vadim Linetsky
Equities Quantitative Strategies, UBS Warburg, 677 Washington Boulevard,
Stamford, Connecticut 06901
Department of Industrial Engineering and Management Sciences,
McCormick School of Engineering and Applied Sciences, Northwestern University,
2145 Sheridan Road, Evanston, Illinois 60208
dmitry.davydov@ubsw.com • linetsky@iems.nwu.edu
M
uch of the work on path-dependent options assumes that the underlying asset price
follows geometric Brownian motion with constant volatility. This paper uses a more
general assumption for the asset price process that provides a better fit to the empirical
observations. We use the so-called constant elasticity of variance (CEV) diffusion model where
the volatility is a function of the underlying asset price. We derive analytical formulae for
the prices of important types of path-dependent options under this assumption. We demon-
strate that the prices of options, which depend on extrema, such as barrier and lookback
options, can be much more sensitive to the specification of the underlying price process than
standard call and put options and show that a financial institution that uses the standard
geometric Brownian motion assumption is exposed to significant pricing and hedging errors
when dealing in path-dependent options.
(Path-Dependent Options; Barrier Options; Lookback Options; Diffusion Processes; CEV Model;
Generalized Bessel Process; Radial Ornstein-Uhlenbeck Process)
1. Introduction
A standard option gives its owner the right to buy
(or sell) some asset in the future for a fixed price. The
fixed price is known as the strike price. Call options
confer the right to buy the asset, while put options
confer the right to sell the asset. Path-dependent
options represent extensions of this concept. For
example, a lookback call option confers the right to buy
an asset at its minimum price over some time period.
A barrier option resembles a standard option except
that the payoff also depends on whether or not the
asset price crosses a certain barrier level during the
option’s life. Lookback options and barrier options
are two of the most popular types of path-dependent
options.
Most of the academic literature on path-dependent
options follow the lead set by Black and Scholes (1973)
and assume that the underlying asset price follows
geometric Brownian motion with constant volatility.
This implies that the future asset prices are log-
normaly distributed and leads to tractable analytical
formulae.
1
However, the evidence indicates that this
distributional assumption is not rich enough to cap-
ture the empirical observations. If the true asset price
process was geometric Brownian motion with con-
stant volatility, then the Black-Scholes formula could
be used to find out this volatility by equating the
model price of a standard option to its market price.
The volatility thus obtained is known as the implied
1
Merton (1973) derives a closed-form pricing formula for down-
and-out call options. Rubinstein and Reiner (1991) extend Merton’s
result to other types of barrier options. Goldman et al. (1979),
Goldman et al. (1979), and Conze and Vishwanathan (1991)
provide closed-form pricing formulae for lookback options.
0025-1909/01/4707/0949$5.00
1526-5501 electronic ISSN
Management Science © 2001 INFORMS
Vol. 47, No. 7, July 2001 pp. 949–965
DAVYDOV AND LINETSKY
Path-Dependent Options Under the CEV Process
volatility of the option. Empirically, we find that the
implied volatilities computed from market prices of
options with different strike prices are not constant
but vary with strike price. This variation is observed
across a wide range of markets and underlying assets
and is known as the implied volatility smile or frown
depending on its shape. The lognormal assumption
with constant volatility does not capture this effect.
There are different ways of extending the basic
model to incorporate this feature. The present paper
uses the constant elasticity of variance (CEV) diffusion
to model asset prices. This process, first introduced
to finance by Cox (1975), is capable of reproducing
the volatility smile observed in the empirical data.
Our paper uses this model to examine the pricing and
hedging of lookback and barrier options.
The contributions of the present paper are two-fold.
First, we derive solutions for barrier and lookback
option prices under the CEV process in closed
form. More generally, we first derive closed-form
expressions for barrier and lookback option prices
under a general time-homogeneous, one-dimensional
diffusion in terms of the two independent solutions
of the stationary Black-Scholes differential equation
with state-dependent volatility. Specializing to the
CEV process, we then derive closed-form expressions
for the stationary solutions and compute barrier and
lookback option prices under the CEV process.
Second, we use the closed-form pricing formulae to
carry out a comparative statics analysis. We demon-
strate that barrier and lookback option prices and
hedge ratios under the CEV process can deviate
dramatically from the lognormal values. Therefore,
substantial model risk exposure exists for finan-
cial institutions making markets in path-dependent
options. Strikingly, we find that deltas of up-and-
out calls, down-and-out puts, double knock-out, and
lookback options can have different signs under the
CEV and lognormal specifications. In these cases,
delta hedging with a misspecified model can produce
worse results than no hedging at all. Finally, to further
assess model risk inherent in writing these contracts,
we carry out a dynamic hedging simulation experi-
ment that quantifies the impact of model misspecifi-
cation on the outcome of dynamic hedging strategies.
Our results indicate that it is much more important
to have an accurate model specification for pricing
and hedging barrier and lookback options than for
standard options.
This paper is organized as follows. Section 2
introduces barrier and lookback options. Section 3
focuses on general valuation results. We derive a
closed-form expression for the Laplace transform of
a barrier option price in time to expiration under
a time-homogeneous, one-dimensional diffusion. The
formula involves two independent solutions of the
stationary Black-Scholes differential equation with
state-dependent volatility. Similarly, closed-form for-
mulae for lookback options are given in terms of
stationary solutions. In §4, we specialize to the CEV
diffusion and derive closed-form expressions for the
stationary solutions in terms of Whittaker and Bessel
functions. In §5, we compute barrier and lookback
option prices and sensitivities, conduct a compara-
tive statics analysis, and carry out a dynamic hedg-
ing experiment to quantify model risk due to model
misspecification faced by financial institutions mak-
ing markets in path-dependent options. Section 6 con-
cludes the paper. Proofs are collected in Appendix A.
Appendix B discusses the case of positive elasticity.
Appendix C contains explicit expressions for the inte-
grals entering the barrier option pricing formulae.
Appendix D discusses Laplace transform inversion.
2. The Contracts: Barrier and
Lookback Options
Barrier options are probably the oldest path-dependent
options. Snyder (1969) describes down-and-out stock
options as “limited risk special options.” Merton
(1973) derives a closed-form pricing formula for
down-and-out calls under the lognormal assumption.
A down-and-out call is identical to a European
2
call
with the additional provision that the contract is can-
celed (knocked out) if the underlying asset price hits
a prespecified lower barrier level. The contract may
also specify a cash rebate to be received by the option
holder if cancellation occurs. The rebate is received
when the knock-out barrier is first reached.
2
European options can only be exercised on the expiration date,
while American options can be exercised on any business day
through and including expiration.
950 Management Science/Vol. 47, No. 7, July 2001
DAVYDOV AND LINETSKY
Path-Dependent Options Under the CEV Process
An up-and-out call is the same, except the contract is
canceled when the underlying asset price first reaches
a prespecified upper barrier level. A cash rebate may
also be received when the barrier is first reached.
Unlike down-and-out calls which are canceled out-
of-the-money (the lower barrier is placed below the
strike price), up-and-out calls are canceled in-the-
money (the upper barrier is placed above the strike).
Generally, contracts that are canceled in-the-money
are called reverse knock-out options.
Down-and-out and up-and-out puts are similar
modifications of European put options. Knock-in
options are complementary to the knock-out options:
They pay off at expiration if and only if the under-
lying asset price does reach the prespecified barrier.
The combination of otherwise identical in and out
options is equivalent to the corresponding stan-
dard European option. Rubinstein and Reiner (1991)
derive closed-form pricing formulae for all eight
types of single-barrier options under the lognormal
assumption.
Double-barrier (double knock-out) options are canceled
when the underlying asset first reaches either the
upper or the lower barrier. A rebate may also be
received at that time. Different representations of
the closed-form pricing formula for double barrier
options under the lognormal assumption are obtained
by Kunitomo and Ikeda (1992), Geman and Yor (1995),
Pelsser (2000), and Schroder (2000).
A capped call is an up-and-out call with the cash
rebate equal to the difference between the upper bar-
rier (cap) and the strike price. It combines a European
exercise feature and an automatic exercise feature. The
automatic exercise is triggered when the index value
first exceeds the cap (Broadie and Detemple 1995).
Barrier options are one of the most popular types
of path-dependent options traded over-the-counter on
stocks, stock indexes, currencies, commodities, and
interest rates. Derman and Kani (1996) offer a detailed
discussion of their investment, hedging, and trading
applications. There are several reasons to use bar-
rier options rather than standard options. First, bar-
rier options may more closely match investor beliefs
about the future behavior of the asset. By buying a
barrier option, one can eliminate paying for those
scenarios one feels are unlikely. Second, barrier option
premiums are generally lower than those of standard
options because an additional condition has to be met
for the option holder to receive the payoff (e.g., the
lower barrier not reached for down-and-out options).
The premium reduction can be substantial, especially
when volatility is high.
Lookback options are another important example of
path-dependent options. Their payoff depends on
the maximum or minimum underlying asset price
attained during the option’s life. A standard lookback
call gives the option holder the right to buy at the low-
est price recorded during the option’s life. A standard
lookback put gives the right to sell at the highest price
recorded during the option’s life. Lookbacks were
first studied by Goldman, Sosin, and Gatto (1979)
and Goldman, Sosin, and Shepp (1979) who derived
closed-form pricing formulae under the lognormal
assumption. In addition to standard lookback options,
Conze and Vishwanathan (1991) introduce calls on
maximum and puts on minimum. A call on maximum
pays off the difference between the realized maximum
price and some prespecified strike or zero, whichever
is greater. A put on minimum pays off the difference
between the strike and the realized minimum price
or zero, whichever is greater. These options are called
fixed-strike lookbacks. In contrast, the standard lookback
options are also called floating-strike lookbacks, because
the floating terminal underlying price S
T
serves as the
strike price in standard lookback options.
3. General Valuation Results
We first develop some results for a general
one-dimensional diffusion and later specialize to the
CEV assumption. We take an equivalent martingale
measure (risk-neutral probability measure) Q as given
(Duffie 1996). Under Q, we suppose that the asset
price S
t
t ≥ 0 is a time-homogeneous, nonnegative
diffusion process solving the stochastic differential
equation
dS
t
= S
t
dt +S
t
S
t
dB
t
t≥ 0S
0
= S>0 (1)
where B
t
t ≥ 0 is a standard Brownian
motion defined on a filtered probability space
t
t≥0
Q, is a constant ( = r −q, where
r ≥ 0 and q ≥ 0 are the constant risk-free interest
Management Science/Vol. 47, No. 7, July 2001 951
DAVYDOV AND LINETSKY
Path-Dependent Options Under the CEV Process
rate and the constant dividend yield, respectively),
and = S is a given local volatility function, which
is assumed continuous and strictly positive for all
S ∈ 0 . We also assume that the local volatility
function remains bounded as S →. This is sufficient
to insure that infinity is a natural boundary
3
for the
diffusion (1). The boundary behavior at the origin
depends on the growth behavior of S as S → 0. If
S remains bounded as S →0, then zero is a natural
boundary. If S grows as S
−p
with some 0 <p≤1/2
as S → 0, then it is an exit boundary (bankruptcy). If
S grows as S
−p
with some p>1/2asS → 0, then
zero is a regular boundary point, and we specify it as
a killing boundary by adjoining a killing boundary
condition (bankruptcy).
Throughout this paper t denotes the running time
variable. We assume that all options are written at
time t = 0 and expire at time t = T>0. Time remain-
ing to expiration is denoted by = T −t. Suppose
the initial asset price is S and the lower and upper
barrier levels are L and U , L<S<U. Define the
first hitting time of the lower barrier
L
= inft ≥
0; S
t
= L, the first hitting time of the upper bar-
rier
U
= inft ≥ 0S
t
= U, and the first exit time
from an interval between the two barriers
LU
=
inft ≥ 0S
t
L U (by convention, the infimum of
the empty set is infinity). Then a down-and-out call
with strike price K and no rebate has the payoff at
expiration 1
L
>T
S
T
−K
+
, where 1
A
is the indica-
tor function of the event A, and x
+
≡ maxx 0 is the
positive part of x. The up-and-out call has the payoff
1
U
>T
S
T
−K
+
. The double-barrier call has the pay-
off 1
LU
>T
S
T
−K
+
. Rebates are fixed cash amounts
paid at times =
L
,
U
,or
LU
, given ≤ T .
Before attacking the problem of pricing finitely
lived barrier options with rebates and expiration
T<, it is convenient to examine first three more
primitive securities: perpetual claims that pay one
dollar at times
L
,
U
, and
LU
, respectively, and
have no set expiration date (see Ingersoll 1987, p. 371).
The claim that pays one dollar at
LU
can be
decomposed into a combination of two additional
3
The boundary classification of one-dimensional diffusions due to
Feller is described in Karlin and Taylor (1981, Chapter 15) and
Borodin and Salminen (1996, Chapter 2).
claims, the first claim paying one dollar at
L
, given
the lower barrier is reached first (
L
<
U
), and the
second claim paying one dollar at
U
, given the upper
barrier is reached first (
U
<
L
).
Proposition 1. Suppose the risk-neutral asset price
process is a diffusion (1) and the constant risk-free interest
rate is r>0. Then the prices at time t =0 of the five perpet-
ual claims described above are (the expectation E
S
is with
respect to the risk-neutral measure Q and the subscript S
indicates that the process (1) is starting at S
0
= S):
• One dollar paid at
L
:
E
S
e
−r
L
1
L
<
=
r
S
r
L
S≥ L (2)
• One dollar paid at
L
, given
L
<
U
:
E
S
e
−r
L
1
L
<
U
=
r
S U
r
LU
L≤ S ≤ U (3)
• One dollar paid at
U
:
E
S
e
−r
U
1
U
<
0
=
r
S
r
U
S≤ U (4)
• One dollar paid at
U
, given
U
<
L
:
E
S
e
−r
U
1
U
<
L
=
r
LS
r
LU
L≤ S ≤ U (5)
• One dollar paid at
LU
:
E
S
e
−r
LU
=
r
LS+
r
SU
r
LU
L≤S ≤U (6)
where ( for any 0 <A<B<)
r
AB =
r
A
r
B −
r
A
r
B (7)
The functions
r
S and
r
S can be characterized as the
unique (up to a multiplicative constant) solutions of the
ordinary differential equation (ODE)
4
1
2
2
SS
2
d
2
u
dS
2
+S
du
dS
−ru = 0S∈ 0 (8)
first by demanding that
r
S is increasing in S and
r
S
is decreasing in S, and secondly, if the origin is a regular
boundary point, posing a killing boundary condition:
r
0+ = 0 (9)
4
In our characterization of the functions
r
and
r
, we follow
Borodin and Salminen (1996, pp. 18–19). The subscript r indicates
the dependence on the risk-free rate that enters the ODE (8).
952 Management Science/Vol. 47, No. 7, July 2001
DAVYDOV AND LINETSKY
Path-Dependent Options Under the CEV Process
The functions
r
S and
r
S have the following
properties (Borodin and Salminen 1996, pp. 18–19). If
zero is an exit boundary, then
r
0+ = 0
r
0+<+ (10)
If zero is a natural boundary, then
r
0+ = 0
r
0+ =+ (11)
Because we assume that S is bounded as S →,
+ is a natural boundary and
lim
S→
r
S =+ lim
S→
r
S = 0 (12)
The functions
r
S and
r
S are called fundamen-
tal solutions of the ODE (8). They are linearly inde-
pendent and all solutions can be expressed as their
linear combinations. Moreover, the Wronskian w
r
,
defined by
r
S
d
r
dS
S −
r
S
d
r
dS
S = Sw
r
(13)
is independent of S. Here S is the scale density of
the diffusion (1)
S = exp
−
S
2dx
2
xx
(14)
Proposition 1 expresses the prices of the five claims
in terms of the two fundamental solutions of the
stationary Black-Scholes differential equation with
the local volatility function S (note that the time
derivative term is absent from Equation (8)). For
geometric Brownian motion S
t
= Se
−
2
/2t+B
t
with
constant volatility , the functional form of the
fundamental solutions is (see Ingersoll 1987, p. 372,
and Carr and Picron 1999):
r
S = S
+
r
S = S
−
±
=− ±
2
+
2r
2
(15)
=
2
−
1
2
In §4 we present closed-form expressions for and
for the CEV process.
To value cash rebates included in finitely lived
knock-out option contracts with expiration T<,we
need to value claims that pay one dollar at times =
L
,
U
,or
L U
, given ≤ T . That is, we need to eval-
uate expectations of the form E
S
1
≤T
e
−r
.
Proposition 2. For any >0, the Laplace transform
of the rebate price in time to expiration
5
is equal to 1/
times the price of the corresponding perpetual claim with
the adjusted discount rate r +:
0
e
−T
E
S
1
≤T
e
−r
dT =
1
E
S
e
−r+
(16)
Given the associated perpetual claim value (Proposition 1),
the rebate price is found by inverting this Laplace
transform.
Now we are ready to price terminal payoffs of
finitely lived knock-out options. First, consider the
more difficult case of double-barrier options. We need
to evaluate the discounted expectation (in this paper
we focus on calls; puts can be treated similarly)
e
−rT
E
S
1
LU
>T
S
T
−K
+
(17)
Proposition 3. Let S be the scale density (14) and
S—the speed density
6
of the diffusion (1)
S =
2
2
SS
2
S
(18)
For 0 <K≤A<B< and >0, define
I
KAB =
B
A
Y −K
Y Y dY (19)
J
KAB =
B
A
Y −K
Y Y dY (20)
where
and
are the functions defined in Proposition 1
(with the risk-free rate r replaced with ). Then the Laplace
5
It is sometimes easier to solve for the Laplace transform of an
option price in time to expiration than for the option price itself.
This Laplace transform (premultiplied by ) can be interpreted as
the price of an exponentially stopped option, i.e., an option expiring
at a random independent exponential time (the first jump time of
a Poisson process with intensity and independent of the under-
lying asset price process). Geman and Yor (1993) use this idea
to obtain closed-from solutions for arithmetic Asian options. Carr
(1998) applies this idea to develop analytical approximations to
value American options.
6
See Karlin and Taylor (1981, p. 194) Karatzas and Shreve (1991,
p. 343) and Borodin and Salminen (1996, p. 17) for discussions of
scale and speed densities. Our definition of the speed density coin-
cides with that of Karatzas and Shreve (1991) and Borodin and
Salminen (1996) and differs from Karlin and Taylor (1981) who do
not include two in the definition.
Management Science/Vol. 47, No. 7, July 2001 953