No. 96-03
A Note of Option Pricing for
Constant Elasticity of Variance Model
by
Freddy Delbaen
Department of Mathematics, Eidegenössische Technische Hochschule Zürich
CH-8092 Zurich, Switzerland
E-mail: [email protected]
and
Hiroshi Shirakawa
Department of Industrial Engineering and Management, Tokyo Institute of Technology
2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japan
E-mail: [email protected]
A Note of Option Pricing for Constant Elasticity of Variance Model
Freddy Delbaen
Department of Mathematics, Eidgenössische Technische Hochschule Zürich
CH-8092 Zürich, Switzerland
E-mail : [email protected]
and
Hiroshi Shirakawa†
Department of Industrial Engineering and Management, Tokyo Institute of Technology
2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japan
E-mail : [email protected]
Abstract: We study the arbitrage free option pricing problem for constant elasticity of variance (CEV) model.
To treat the stochastic aspect of the CEV model, we direct attention to the relationship between the CEV model
and squared Bessel processes. Then we show the existence of a unique equivalent martingale measure and derive
the Cox’s arbitrage free option pricing formula through the properties of squared Bessel processes. Finally we
show that the CEV model admits arbitrage opportunities when it is conditioned to be strictly positive.
Keywords:
Constant Elasticity of Variance Model, Squared Bessel Process, Option Pricing, Equivalent Mar-
tingale Measure, Arbitrage.
1
Introduction
We consider a security market model of two assets, bond and stock. Let (Ω, (Ft )0≤t , P )
probability space satisfying the usual condition and let Wt be a Wiener process under P .
that Ft is generated by {Wu ; 0 ≤ u ≤ t}, the bond price Bt increases with constant riskless
and stock price process St has independent increments with constant elasticity of variance.
dBt
dSt
= rBt dt,
B0 = 1,
= µSt dt +
σStρ dWt ,
be a filtered
We suppose
return r > 0
That is :
(1.1)
S0 = s ≥ 0,
(1.2)
where ρ means the elasticity of variance. This model was first considered by Cox [1] where it was called
the constant elasticity of variance model (hereafter abbreviated as the CEV model).
We require the following conditions for the parameters in (1.2).
(C.1)
0 < ρ < 1.
Under Condition (C.1), the point 0 is an attainable state. As soon as the process S reaches zero. we keep
it equal to zero. The reader can verify that the process so defined, still satisfies the stochastic differential
equation (1.2). In doing so the point 0 becomes the absorbing state for the stock price process St . A
straightforward analysis shows that when we want the equation (1.2) to be satisfied, this is the only way
to treat the point 0.
The object of this paper is to study the existence of a unique equivalent martingale measure of the
CEV model and to derive arbitrage free option pricing formula through the probabilistic analysis. Cox
† Part of the research was done while the second auther was visiting the Vrije Universiteit Brussel, Departement Wiskunde.
He wishes to extend his deep thanks for their hospitality. Also we appreciate Marc Yor for his helpful comments. This
research is partly supported by the Dai-ichi Life Insurance Co. and the Toyo Trust and Banking Co. Ltd.
1
[1] has already investigated the transition probability of the stock price process for the CEV model and
obtained the arbitrage free call option pricing formula through the risk neutral evaluation [2]. However
it was not shown whether there is a unique equivalent martingale measure for the CEV model. The risk
neutral method is a convenient approach to derive the arbitrage free option prices, but it requires only
the local arbitrage free property and that is not equivalent to the arbitrage free property [4, 5]. Here we
direct attention to the relationship between the CEV model and squared Bessel process which enables
us to study the CEV model through the basic properties of squared Bessel process. Then we prove the
existence of a unique equivalent martingale measure and derive the law of stock price process for the
CEV model. Furthermore we show that the CEV model admits arbitrage opportunities when the stock
price is conditioned to be strictly positive. This is the advantage of our approach since the standard risk
neutral argument is useless to study the existence of arbitrage for the conditioned the CEV model. The
analysis also shows that one cannot discard the possibility that a CEV process will hit 0.
The paper is organized as follows. In Section 2, we show the relationship between the CEV model
and squared Bessel process. Then we prove the existence of a unique equivalent martingale measure. In
Section 3, we study the law of the risk neutral stock price process and derive the arbitrage free option
pricing formula. Finally in Section 4, we show the existence of arbitrage opportunities for the CEV model
when the stock price is conditioned to be strictly positive.
2
Weak Solution by Squared Bessel Process
First we shall represent the weak solution of the stochastic differential equation (1.2) by a squared
Bessel process with time and state changes. We denote the δ-dimensional squared Bessel process (δ ∈ R)
(δ)
by Xt . It follows the stochastic differential equation
(δ)
(δ)
dXt = 2 |Xt |dWt + δdt,
(2.1)
(δ)
starting with X0
2
(δ)
= s 2−δ . Let ζ be the first passage time of 0 for the process Xt , i.e.,
(δ)
ζ = inf{t > 0 ; Xt
= 0}.
(2.2)
For the parameters ν > 0 and δ < 2, we consider a deterministic time change defined by
σ2
2νt
(δ,ν)
τt
=
1 − exp −
.
2ν(2 − δ)
2−δ
(δ,ν)
The process Yt
(2.3)
is defined as
(δ,ν)
Yt
= exp{νt} X
1− 12 δ
(δ)
(δ,ν)
τt
∧ζ
.
(2.4)
(δ,ν)
From Itô’s lemma, we can “easily” check that Yt
follows
(δ,ν)
(δ,ν) 1−δ
(δ,ν)
(δ,ν)
dt + σ(Yt
) 2−δ dWt
, if τt
≤ ζ,
νYt
(δ,ν)
dYt
=
(δ,ν)
0,
if τt
> ζ,
(δ,ν)
where Wt
(2.5)
is the Wiener process defined by
(δ,ν)
Wt
=
0
(δ,ν)
τt
2−δ
σ2
2
− 2µ(2 − δ)v
dWv .
(2.6)
1−2ρ
1−ρ
Let δρ =
which means
1−δρ
2−δρ
2−δ
log
2ν
= ρ. Since ρ ∈ (0, 1), we have δρ ∈ (−∞, 1) and hence
σ2
σ 2 − 2ν(2 − δ)ζ
σ2
ζ<
2ν(2 − δ)
on the set
(δρ ,µ)
also plays a role as first passage time of the point 0 for the process Yt
.
Remark 2.1 Before continuing the analysis of the relation between the Bessel square processes and the
(δ,ν)
CEV model, we make the remark that for δ < 1 and for τt
> ζ, the process Y still satisfies, in a
trivial way, the first equation of 2.5. The solution of the stochastic differential equation 1.2 (except for
the substitution of µ by ν is therefore given by the transformation 2.4. It follows that the process S for
σ2
0 ≤ t < +∞ only uses the part of the squared Bessel process X up to time 2−δ
= σ 2 (1 − ρ). Also it
follows that there are two kinds of trajectories for the process S. The first kind consists of the trajectories
absorbed at 0. The second kind consists of the trajectories that (at least for µ > 0) will converge to +∞
when t → +∞. (Just for the information of the reader we add that for µ ≤ 0 the above analysis does not
apply and that all trajectories will be absorbed by 0.) The passage time through zero is given by
1
σ 2 (1 − ρ)
σ 2 (1 − ρ)
ζS =
log
on
the
set
ζ
<
2µ(1 − ρ)
σ 2 (1 − ρ) − 2µζ
2µ
and by
ζ≥
ζ S = +∞ on the set
σ 2 (1 − ρ)
2µ
.
We can summarize the previous discussion in the following:
Theorem 2.2
law
(δρ ,µ)
{St ; 0 ≤ t} = {Yt
; 0 ≤ t}
(2.7)
law
where = means equivalence in law under the original probability measure P .
2
Let us consider the risk neutral evaluation when Bt is used as a numéraire. Define the process Ut by
Ut =
St
= exp{−rt}St .
Bt
(2.8)
From Itô’s lemma, Ut follows
dUt = σ exp{−(1 − ρ)rt}Utρ dW̃t
where
(2.9)
t
θSv1−ρ dv
W̃t = Wt +
(2.10)
0
and θ =
µ−r
σ .
Using the process W̃t , the stock price process follows
dSt = rSt dt + σStρ dW̃t .
(2.11)
By Cameron-Martin-Maruyama-Girsanov’s theorem [9, p.191], we shall consider the unique equivalent
measure change candidate for P |FT ∧ζS defined by the Radon-Nikodym derivative
ηT = exp −θ
0
T
St1−ρ dWt
θ2
−
2
T
2(1−ρ)
St
dt
= exp −θ
0
0
T ∧ζ S
St1−ρ dWt
θ2
−
2
T ∧ζ S
2(1−ρ)
St
dt
,
0
(2.12)
under which {Ut ; 0 ≤ t ≤ T } would be martingale.
Theorem 2.3 For all T < ∞, E[ηT ] = 1 and hence there exists a unique P - equivalent martingale
measure P̃ on FT defined by
P̃ [A] = E[1A ηT ] for A ∈ FT .
(2.13)
3
Proof.
There are (at least) two ways to prove the theorem. One way is to analyse the Novikov
condition for the local martingale
S
u∧ζ S
θ2 u∧ζ 2(1−ρ)
1−ρ
ηu = exp −θ
St dWt −
St
dt .
2 0
0
This is not completely possible and only works for (1 − ρ)µT ≤ 1. A repeated application of the Markov
property allows to extend the equality E[ηT ] = 1 also fortimes T that are bigger than f rac1(1 − ρ)µ. The
calculations are not difficult but are tedious. We will proceed in another way using a more qualitative
approach. We first consider two stochastic differential equations. The first equation is the original
equation 1.2:
dSt = µSt dt + σStρ dWt .
The second equation is the equation obtained when replacing µ by r, i.e.
dSt = rSt dt + σStρ dWt .
The law of the first process, a measure on the space C[0, T ] of continuous functions on the interval [0, T ],
is denoted by P , the law of the second process is denoted by P̃ . The coordinate process is denoted by S, it
generates a filtration denoted by Ft . This notation is not really misleading, it is inspired by transporting
the processes defined on Ω to the canonical space C[0, T ].
Since we supposed that the process S generates the filtration (up to time ζ S ) we can find a P −Brownian
motion W so that on the space C[0, T ] we have dSt = µSt dt + σStρ dWt . In the same way we can find a
P̃ − Brownian motion W̃ so that dSt = rSt dt + σStρ dW̃t . The passage from P to P̃ is not difficult. If we
define the stopping times
u
τn = inf u |
2(1−ρ)
St
dt ≥ n ,
0
we have that on the σ−algebra Fτn both measures, P and P̃ , are equivalent. Furthermore the density of
P̃ with respect to P is given by the random variable
τn
θ2 τn 2(1−ρ)
ητn = exp −θ
St1−ρ dWt −
St
dt .
2 0
0
T 2(1−ρ)
Since P −almost surely we have 0 St
dt < ∞, we get that for n → +∞, necessarily τn → +∞. This
implies that P P̃ .
Conversely the same reasoning applies to the measure P̃ and using that P̃ −almost surely we have
T 2(1−ρ)
S
dt < ∞, we get that P̃ is absolutely continuous with respect to P .
t
0
As a result, we find that both measures on C[0, T ] are equivalent. Transporting back to Ω and using
the fact that also on Ω, the filtration is generated by S, we can conclude that on Ω, the density process
S
u∧ζ S
θ2 u∧ζ 2(1−ρ)
1−ρ
ηu = exp −θ
St dWt −
St
dt .
2 0
0
is not only a local martingale but is a strictly positive martingale. Therefore necessarily E[ηT ] = 1 and
ηT > 0. 2
3
Arbitrage Free Option Pricing
By Theorem 2.3, there always exists an P -equivalent measure P̃ on FT so that W̃t becomes a Wiener
process under the measure P̃ . By construction, the discounted price process Ut is a local martingale
under the P - equivalent measure.
From the reasoning in section 2, we immediately deduce the following
4
Corollary 3.1
law
(δρ ,r)
{St ; 0 ≤ t} under the measure P̃ = {Yt
; 0 ≤ t} under the measure P
(3.1)
2
Before giving explicit formulas we first need to state some important results for the squared Bessel
process.
Lemma 3.2 For any δ ∈ [0, ∞), we have
(δ) law
Xt
x
= t · V (δ, t ) ,
x ≥ 0,
t > 0,
(3.2)
(δ)
where X0 = x and V (a,b) means the noncentral χ2 random variable with a (≥ 0) degrees of freedom and
noncentrality parameter b ≥ 0. That is, the density of V (a,b) is given by
∞ n
a
1
b
1
vn
f (v; a, b) = a exp − (b + v) v 2 −1
.
(3.3)
2
4
22
n! Γ( 12 a + n)
n=0
Proof.
Consider the Laplace transform of V (a,b) ,
E[exp{−λV (a,b) }]
∞ n
a
b
1
vn
1
2 −1
=
(b
+
(1
+
2λ)v)
v
dv
exp
−
a
2
4
n! Γ( 12 a + n)
v≥0 2 2
n=0
λ
n
∞ exp − 1+2λ
b a
b
1
vn
1
b
−1
2
=
−
+v
v
dv
a exp
δ
2 1 + 2λ
4(1 + 2λ)
n! Γ( 12 a + n)
(1 + 2λ) 2
v≥0 2 2
n=0
=
λ
b}
exp{− 1+2λ
b
(1 + 2λ) 2
.
(δ)
On the other hand, the Laplace transform of Xt
(δ)
E[exp{λXt }] =
is given by (see [10, p.411] ),
λt x
exp{− 1+2λt
t}
(1 + 2λt)
δ
2
x
= E[exp{−λtV (δ, t ) }].
Since the Laplace transforms for both random variables are equal, we have (3.2).
2
(δ)
Yor [11, (2.c)] derived the following relationship between the squared Bessel processes Xt
(4−δ)
and Xt
Lemma 3.3 Let δ ∈ (−∞, 2) and φ be a function such that
δ −1 (4−δ)
(4−δ)
(4−δ) 2
lim E φ Xt
Xt
= x < ∞.
X0
(3.4)
x↓0
Then for any and x > 0,
(δ)
(δ)
E[φ(Xt )1{ζ>t} |X0
1− δ2
= x] = x
δ −1 (4−δ)
(4−δ)
(4−δ) 2
· E φ Xt
=x .
Xt
X0
.
2
(3.5)
The martingale property of Ut under P̃ follows from Corollary 3.1 and Lemma 3.3. That is, let
δρ
φ(x) = x1− 2 , then


1− δ2ρ 2
(δρ )
(δ )
Ẽ[Ut |U0 = u] = Ẽ  X̃ (δρρ ,r)
X̃0 = u 2−δρ 
τt
∧ζ̃


1− δ2ρ
2
(δ )
(δ )
= Ẽ  X̃ (δρρ ,r)
1{ζ̃≥t} X̃0 ρ = u 2−δρ 
τt
=
u
2
2−δρ
1− δ2ρ


1− δ2ρ δ2ρ −1 2
(δρ )
(4−δρ )
(4−δρ )
Ẽ  X̃ (δρ ,r)
X̃ (δρ ,r)
X̃0 = u 2−δρ 
τt
τt
= u.
5
(3.6)
Remark 3.4 The martingale property of the process U can also be proved by using more structural
methods. However since we need the density functions later on, we did not insist on such a presentation.
Furthermore by Theorem 2.2 and Lemma 3.2, 3.3, we can derive the probability distribution of ST under
the original measure P (see also Cox [1]).
Theorem 3.5
P [ST ≤ x|S0 = s] = 1 −
∞
g(n + λ, z)G(n, w)
(3.7)
n=1











where
λ
=
1
2(1−ρ)
z
=
s2(1−ρ)
(δ ,µ)
2τT ρ
1
=
λ
2µλ exp{ µT
λ }s
σ 2 (exp{ µT
λ
2(1−ρ)
}−1)
1
(exp{−µT }x)
w =
=
(δ ,µ)


2τT ρ


u−1


g(u, v) = vΓ(u) exp{−v}



 G(u, v) = g(u, w)dw.
w≥v
Proof.
(3.8)
2µλx λ
σ 2 (exp{ µT
λ
}−1)
From Theorem 2.2 and Lemma 3.3,
P [ST ≥ x|S0 = s]
(δ ,µ)
(δ ,µ)
= P YT ρ ≥ x Y0 ρ = s
2
2
(δ )
(δ )
= P X (δρρ ,µ) ≥ (exp{−µT }x) 2−δρ X0 ρ = s 2−δρ
τT
∧ζ
2
2
(δ )
(δ )
= E 1 X (δρρ ,µ) ≥ (exp{−µT }x) 2−δρ 1{ζ ≥ T } X0 ρ = s 2−δρ
τT


δ2ρ −1 2
2
(4−δρ )
(4−δρ )
(4−δρ )
= s · E 1 X (δρ ,µ)
X (δρ ,µ)
≥ (exp{−µT }x) 2−δρ
= s 2−δρ 
X0
τT
(3.9)
τT
And from Lemma 3.2,


δ2ρ −1 2
2
(4−δ )
(4−δρ )
(4−δρ )
E  X (δρ ,µ)
1 X (δρ ,µ)
≥ (exp{−µT }x) 2−δρ X0 ρ = s 2−δρ 
τT
=
(δ ,µ)
τT ρ
τT
δ2ρ −1
E
V
(4−δρ ,2z)
δ2ρ −1
1{V
(4−δρ ,2z)
≥ 2w}
=
ρ
∞
exp{−z}
z n+1− 2
v n exp{−v}
dv
δρ
1− δ2ρ
n!
(δρ ,µ)
n=0 Γ(n + 2 − 2 ) v≥w
2τT
z
=
∞
1
g(n + λ, z)G(n, w).
s n=1
δ
Thus from (3.9) and (3.10), (3.7) is obtained.
(3.10)
2
(δ)
As the special case of Theorem 3.5, we can derive the probability that Xt
dimension δ ∈ (−∞, 2).
(δ)
Corollary 3.6 For the squared Bessel process Xt
hits the point 0 for the
with dimension δ ∈ (−∞, 2), we have
(δ)
P [Xu(δ) hits the point 0 during 0 ≤ u ≤ t|X0 = x]
x !n−1
∞
x 1− δ2 x
2t
= 1−
exp
−
, x ≥ 0.
2t
2t
Γ(n + 1 − 2δ )
n=1
6
(3.11)
Proof.
Let s = 0 (w = 0) and S = x1−
δρ
2
P [ST = 0|S0 = x1−
δρ
2
in (3.7). Then G(n, w) = 1 and hence
]=1−
∞
(δρ ,µ)
g(n + λ, z) = P [ζ ≤ τT
(δ)
|X0
= x]
(3.12)
n=1
(δρ ,µ)
Substituting τT
= t, z =
x
2t
2
and δρ = δ in (3.12), we get (3.11).
(δ)
Remark 3.7 It is well known that when δ ≥ 2, the point 0 is polar and hence the probability that Xt
hits the point 0 is 0 (see [10, p.415]).
Here note that the martingale measure P̃ is not unique on FT . However it is unique on Fτ . Hence
if the payoff of the option depends only on the stock price process St , we can derive the unique option
price to duplicate the terminal payoff [7, 8]. That is, the unique no arbitrage price of the option is given
by its discounted expected value under the equivalent martingale measure P̃ . Hereafter we shall consider
the arbitrage free pricing for the option whose payoff CT depends only on the stock price at the maturity
ST . That is, CT is given by C(ST ) for some function C(·). Then we can derive an arbitrage free option
pricing formula.
Theorem 3.8 Let the initial stock price S0 = s and C0 (s) be the arbitrage free price for the option at
time 0 with the terminal payoff C(ST ). Then




1− δ2ρ δ2ρ −1 (4−δρ )
ρ)
 X̃ (4−δ
C0 (s) = s · Ẽ exp{−rT }C exp{rT } X̃ (δρ ,r)
X̃0 = s2(1−ρ) 
(δρ ,r)
τT



+ exp{−rT }C(0) 1 − s

'
τT
(1− δ2ρ
1
(δ ,r)
2τT ρ

∞
s2(1−ρ) 

exp
−
.
δρ
(δρ ,r) 
2τT
n=1 Γ(n + 1 − 2 )
s2(1−ρ)
(δ ,r)
2τT ρ
n−1
(3.13)
Proof. By the general theorem for the complete market asset pricing [7, 8], the unique arbitrage free
price C0 (S) is given by
C0 (s)
= Ẽ[exp{−rT }C(ST )|S0 = s]


1− δ2ρ 2
(δρ )
(δ )
= Ẽ exp{−rT }C exp{rT } X̃ (δρρ ,r)
X̃0 = s 2−δρ 
τT


= Ẽ exp{−rT }C exp{rT } X̃
(δρ ,r)
+ exp{−rT }C(0)P̃ [ζ̃ ≤ τT
From Lemma 3.3,


Ẽ exp{−rT }C exp{rT } X̃

(δρ )
(δρ ,r)
τT


2
(δρ )
1
= s 2−δρ 
(δ ,r) X̃0
{ζ>τ ρ }
(δρ )
(δρ ,r)
τT
(δρ )
|X̃0
1− δ2ρ

1− δ2ρ
T
2
= s 2−δρ ].
(3.14)


2
2−δρ 
1
(δ ,r) X̃0 = s
{ζ̃>τ ρ }
(3.15)
T
= s · Ẽ exp{−rT }C exp{rT } X̃
(4−δρ )
1− δ2ρ
(δρ ,r)
τT


δ2ρ −1 2
ρ)
 X̃ (4−δ
X̃0 = s 2−δρ  .
(δρ ,r)
τT
Also from Corollary 3.6,
(δρ ,r)
P̃ [ζ̃ ≤ τT
'
=
1−s
2
|X̃0 = s 2−δρ ]
1
(1− δ2ρ
n−1
∞
n=1
Γ(n + 1 −
(δρ ,r)
2τT
2
s 2−δρ
(δ ,r)
2τT ρ
7
δρ
2 )
exp −
2
s 2−δρ
(δρ ,r)
2τT
.
(3.16)
= 2(1 − ρ), we have (3.13).
2
2−δρ
Then from (3.14) through (3.16) and
2
Call Option Case : Consider the arbitrage free pricing for the call option CT = (ST − K)+ with the
nonnegative exercise price K. From Theorem 3.8, the arbitrage free price of CT is given by
C0 (s)


= s · Ẽ exp{−rT } exp{rT } X̃
(4−δρ )
1− δ2ρ
(δρ ,r)
τT
+
− K

δ2ρ −1 2
(4−δρ )
X̃ (δρ ,r)
X̃0 = s 2−δρ 
τT
2
2
(4−δρ )
(4−δρ )
2−δρ 2−δρ
= s · P̃ X̃ (δρ ,r) ≥ (exp{−rT }K)
=S
X̃0
τT

δ2ρ −1
(4−δ
)
ρ
− exp{−rT }Ks · Ẽ  X̃ (δρ ,r)
1 (4−δρ )
{X̃
τT
(δρ ,r)
τ
T
(3.17)

≥(exp{−rT }K)
2
2−δρ
}
2
(4−δρ )
= s 2−δρ  .
X̃0
From Lemma 3.2,
2
2
(4−δ )
(4−δρ )
P̃ X̃ (δρ ,r)
≥ (exp{−rT }K) 2−δρ X̃0 ρ = s 2−δρ
τT
= P V (4−δρ ,2z ) ≥ 2w
δρ
∞
z n
v n+1− 2 exp{−v}
= exp{−z }
dv
n! v≥w Γ(n + 2 − δρ )
n=0
2
∞
=
g(n, z )G(n + λ, w )
(3.18)
n=1
where λ, g(x, y), G(x, y) are given by (3.8) and z , w are defined by

1
λ
2(1−ρ)
exp{ rT

λ }s
 z = s (δρ ,r) = 2rλ
rT
2
σ (exp{
}−1)

 w
=
2τT
(exp{−rT }K)2(1−ρ)
On the other hand from (3.10),

δ2ρ −1
(4−δρ )

Ẽ
X̃ (δρ ,r)
1
(δ ,r)
2τT ρ
τT
=
*∞
1
s
n=1
(3.19)
λ
=
1
2rλK λ
σ 2 (exp{ rT
λ }−1)
2
(4−δρ )
≥(exp{−rT }K) 2−δρ
(δρ ,r)
τ
T
{X̃
}
.

2
(4−δρ )
= s 2−δρ 
X̃0
g(n + λ, z )G(n, w ).
(3.20)
Thus from (3.18) through (3.20), we arrive at the following call option pricing formula which is derived
by Cox [1, 3].
C0 (s) = s
∞
g(n, z )G(n + λ, w ) − exp{−rT }K
n=1
∞
g(n + λ, z )G(n, w ),
(3.21)
n=1
where z , w , g(x, y) and G(x, y) are given by (3.8) and (3.19).
4
Arbitrage for Positive Conditional Process
From Theorem 2.2 and Corollary 3.6, we see that for a CEV model with ρ < 1, the process St is absorbed
at the point 0 with positive probability. In this section, we study the existence of arbitrage opportunities
for the CEV model when it is conditioned to the strictly positive region during the finite time interval
8
[0, T ]. If we presume the risk neutralized stock price process with drift r, it is absorbed in 0 with strictly
positive probability as is the original process. This means that the conventional risk neutral approach
is useless to study this process. However our approach for the CEV model can be applied to show the
existence of arbitrage opportunity for the positive conditional process.
Let us define ξT by
1− δ2ρ
˜ 1
exp{−rT }ST law
(δρ )
ξT =
=
.
(4.1)
X̃ (δρ ,r)
τT
∧ζ̃
s
s
(δ )
From (3.6), we have Ẽ[ξT |X̃0 ρ = s2(1−ρ) ] = 1. Hence we can use ξT as a Radon-Nikodym derivative
for an absolutely continuous (not equivalent) measure change from P̃ to another distribution P̂ on FT .
Furthermore since ξT > 0 ⇔ ST > 0, P̂ is equivalent to the conditional distribution P̃ [·|ST > 0]. This
together with the equivalence of P and P̃ yields the equivalence of P [·|ST > 0] and P̂ .
Next we shall consider the Markov process Ŝt defined by
dŜt = (rŜt + σ 2 Ŝt2ρ−1 )dt + σ Ŝtρ dW̃t ,
(4.2)
starting with Ŝ0 = S. Then we have the following lemma.
Lemma 4.1
law
{St ; 0 ≤ t ≤ T } under P̂ = {Ŝt ; 0 ≤ t ≤ T } under P̃ .
(4.3)
Proof.
First we shall represent Ŝt by squared Bessel process with dimension δ > 2. For ν > 0 and
δ < 2, let
1− δ2
(δ,ν)
(4−δ)
Zt
= exp{νt} X̃ (δ,ν)
,
τt
2
(4−δ)
starting with X̃0
(δ,ν)
where W̃t
1−δρ
2−δρ
= s 2−δ . From Itô’s lemma,
δ − 2−δ
1−δ
(δ,ν)
(δ,ν)
(δ,ν)
(δ,ν) 2−δ
(δ,ν)
dZt
dt + σ Zt
= νZt
+ σ 2 Zt
dW̃t
δ
ρ
is another Wiener process under P̃ given by (2.6) for W̃t . Since − 2−δ
= 2ρ − 1 and
ρ
= ρ, we have
˜
law
(δρ ,r)
{Ŝt ; 0 ≤ t} = {Z̃t
; 0 ≤ t}.
This together with Lemma 3.3 yields
P̂ [ST ≤ u|S0 = s]
= Ê[1{ST ≤u} |S0 = s]

=
1  Ẽ 1
(δ )
S
X̃ ρ
(δρ ,r)
τ
T


= Ẽ 1
≤(exp{−µT }u)
2
2−δρ
2
(4−δρ )
≤(exp{−µT }u) 2−δρ
(δρ ,r)
τ
T
X̃ (δ(δρ ),r)
ρ
τT
1− δ2ρ

2

(δ )
1,ζ̃>τ (δρ ,r) - X̃0 ρ = s 2−δρ 
T

2
X̃ (4−δρ ) = s 2−δρ 

0
X̃
= P̃ [ŜT ≤ u|Ŝ0 = s],
for any u ≥ 0. Thus the transition probabilities for the Markov processes ({St ; 0 ≤ t}, P̂ ) and ({Ŝt ; 0 ≤
t}, P̃ ) coincide and hence they are equivalent in law. 2
Now we can show the following result for the CEV model.
Theorem 4.2 For the positive price process of the CEV model, i.e. : {St ; 0 ≤ t ≤ T } under the
conditional probability measure P [·|ST > 0], there always exists arbitrage opportunities.
9
Proof.
The probability measures P̂ and P [·|ST > 0] are equivalent. Furthermore from Lemma 4.1,
law
({St ; 0 ≤ t ≤ T }, P̂ ) = ({Ŝt ; 0 ≤ t ≤ T }, P̃ ). Then it is enough to show that there exists arbitrage
opportunities for the process {Ŝt ; 0 ≤ t ≤ T } under P̃ . Define the process X̂u by
2
2−δ
ρ
ρ ,r)
X̂u = exp{−rt(δ
}
Ŝ
(δ
,r)
ρ
u
t
(4.4)
u
(δ,ν)
where tu
is given by the inverse transformation of 2.3. From Itô’s lemma, X̂u follows
dX̂u = 2 X̂u dŴu + (4 − δρ )du
(4.5)
where Ŵu is an other Wiener process under P̃ defined as :
t(δ,ν)
u
Ŵu =
0
σ
rv
dW̃v .
exp −
2 − δρ
2 − δρ
(4.6)
Thus X̂u is a (4 − δρ )-dimensional squared Bessel process. Since 4 − δρ > 2, we have the following result
δρ
−1
from Theorem 6 of Delbaen-Schachermayer [5] : Lu = X̂u2
is a local martingale such that L−1
u allows
arbitrage with respect to the general admissible integrands under 0 interest rate. That is, there exists an
admissible integrand ϕu such that
V̂T
=
(δρ ,r)
τT
ϕu dL−1
u ≥ 0, P̃ -a.s.,
(4.7)
0
P̃ [VT > 0] > 0.
(δ ,r)
ρ
Since L−1
u = exp{−rtu
(4.8)
}Ŝt(δρ ,r) , we can easily show that
u
V̂T = exp{−rT }V̂T
where
V̂t =
0
(4.9)
t
r(V̂u − ϕτ (δρ ,r) Ŝu )du + ϕτ (δρ ,r) dŜu , 0 ≤ t ≤ T.
u
(4.10)
u
From (4.7) through (4.10), there exists an arbitrage opportunity for Ŝu under P̃ . 2
The intuitive explanation for the existence of arbitrage opportunity is as follows. If there exists an
equivalent martingale measure P̌ for Ŝt , it will have the same law as St under P̃ by Cameron-MartinMaruyama-Girsanov’s theorem. However this is impossible since P̃ [ST = 0] > 0 whereas P̌ [ŜT = 0] = 0.
References
[1] Cox, J. C. (1996), ”The Constant Elasticity of Variance Option Pricing Model,” The Journal of
Portfolio Management, ?, 15-17.
[2] Cox, J. C. and S. Ross (1975), “The Valuation of Options for Alternative Stochastic Processes,” J.
Financial Economics, 3, 145-166.
[3] Cox, J. C. and M. Rubinstein (1983), “A Survey of Alternative Option Pricing Models,” in Option
Pricing, ed. M. Brenner, Lexington, Mass : D.C. Heath, 3-33.
[4] Delbaen, F. and W. Schachermayer (1994), “A General Version of the Fundamental Theorem of Asset
Pricing,” Mathematische Annalen, 300, 463-520.
[5] Delbaen, F. and W. Schachermayer (1995), “Arbitrage Possibilities in Bessel Processes and their
Relations to Local Martingales,”Probability and Related Fields, 102, 357-366.
10
[6] Harrison, J. M. and D. M. Kreps (1979), “Martingales and Arbitrage in Multiperiod Security Markets,” J. Economic Theory, 20, 381-408.
[7] Harrison, J. M. and S. R. Pliska (1981), “Martingales and Stochastic Integrals in the Theory of
Continuous Trading,” Stochastic Processes and Their Applications, 11, 381-408.
[8] Harrison, J. M. and S. R. Pliska (1983), “A Stochastic Calculus Model of Continuous Time Trading
: Complete Markets,” Stochastic Processes and Their Applications, 13, 313-316.
[9] Karatzas, I and S. E. Shreve (1988), Brownian Motion and Stochastic Calculus, Springer-Verlag.
[10] Revuz, D. and M. Yor (1991), Continuous Martingales and Brownian Motion, Springer-Verlag.
[11] Yor, M. (1993), “On some Exponential Functionals of Brownian Motion,” Advances in Applied
Probability, 24, 509-531.
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