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: delbaen@math.ethz.ch

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: sirakawa@me.titech.ac.jp

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 : delbaen@math.ethz.ch 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 : sirakawa@me.titech.ac.jp

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 Martingale Measure, Arbitrage.

1 Introduction

We consider a security market model of two assets, bond and stock. Let (Ω, (F_{t})_{0≤t}, P) be a filtered probability space satisfying the usual condition and let W_{t} be a Wiener process under P. We suppose that F_{t} is generated by {W_{u}; 0 ≤ u ≤ t}, the bond price B_{t} increases with constant riskless return r > 0 and stock price process S_{t} has independent increments with constant elasticity of variance. That is :

dB_{t} = rB_{t}dt, B_{0} = 1,  (1.1)
dS_{t} = μS_{t}dt + σS_{t}^{ρ}dW_{t}, S_{0} = 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 S_{t}. 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 [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.

†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.

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) by X_{t}^{(δ)}. It follows the stochastic differential equation

dX_{t}^{(δ)} = 2\sqrt{|X_{t}^{(δ)}|}dW_{t} + δdt,  (2.1)

starting with X_{0}^{(δ)} = s^{\frac{2}{2−δ}}. Let ζ be the first passage time of 0 for the process X_{t}^{(δ)}, i.e.,

ζ = inf{t > 0 ; X_{t}^{(δ)} = 0}.  (2.2)

For the parameters ν > 0 and δ < 2, we consider a deterministic time change defined by

τ_{t}^{(δ,ν)} = \frac{σ^{2}}{2ν(2 − δ)} (1 − exp \{−\frac{2νt}{2 − δ} \}) .  (2.3)

The process Y_{t}^{(δ,ν)} is defined as

Y_{t}^{(δ,ν)} = exp\{νt\} (X_{τ_{t}^{(δ,ν)}∧ζ}^{(δ)})^{1− \frac{1}{2} δ} .  (2.4)

From Itô’s lemma, we can “easily” check that Y_{t}^{(δ,ν)} follows

dY_{t}^{(δ,ν)} = \{ \begin{matrix}νY_{t}^{(δ,ν)}dt + σ(Y_{t}^{(δ,ν)})^{\frac{1−δ}{2−δ}} dW_{t}^{(δ,ν)}, & if τ_{t}^{(δ,ν)} ≤ ζ, \\ 0, & if τ_{t}^{(δ,ν)} > ζ, \end{matrix}  (2.5)

where W_{t}^{(δ,ν)} is the Wiener process defined by

W_{t}^{(δ,ν)} = ∫_{0}^{τ_{t}^{(δ,ν)}} \frac{2 − δ}{\sqrt{σ^{2} − 2μ(2 − δ)v}} dW_{v}.  (2.6)

Let δ_{ρ} = \frac{1−2ρ}{1−ρ} which means \frac{1−δ_{ρ}}{2−δ_{ρ}} = ρ. Since ρ ∈ (0, 1), we have δ_{ρ} ∈ (−∞, 1) and hence

\frac{2 − δ}{2ν} log (\frac{σ^{2}}{σ^{2} − 2ν(2 − δ)ζ} ) on the set \{ζ < \frac{σ^{2}}{2ν(2 − δ)}\}

also plays a role as first passage time of the point 0 for the process Y_{t}^{(δ_{ρ},μ)}.

Remark 2.1 Before continuing the analysis of the relation between the Bessel square processes and the CEV model, we make the remark that for δ < 1 and for τ_{t}^{(δ,ν)} > 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 0 ≤ t < +∞ only uses the part of the squared Bessel process X up to time \frac{σ^{2}}{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

ζ^{S} = \frac{1}{2μ(1 − ρ)} log (\frac{σ^{2}(1 − ρ)}{σ^{2}(1 − ρ) − 2μζ} ) on the set \{ζ < \frac{σ^{2}(1 − ρ)}{2μ}\}

and by

ζ^{S} = +∞ on the set \{ζ ≥ \frac{σ^{2}(1 − ρ)}{2μ}\} .

We can summarize the previous discussion in the following:

Theorem 2.2

{S_{t}; 0 ≤ t} =^{law} {Y_{t}^{(δ_{ρ},μ)}; 0 ≤ t}  (2.7)

where =^{law} means equivalence in law under the original probability measure P. □

Let us consider the risk neutral evaluation when B_{t} is used as a numéraire. Define the process U_{t} by

U_{t} = \frac{S_{t}}{B_{t}} = exp\{−rt\}S_{t}.  (2.8)

From Itô’s lemma, U_{t} follows

dU_{t} = σ exp{−(1 − ρ)rt}U_{t}^{ρ}dW̃_{t}  (2.9)

where

W̃_{t} = W_{t} + ∫_{0}^{t} θS_{v}^{1−ρ}dv  (2.10)

and θ = \frac{μ−r}{σ}. Using the process W̃_{t}, the stock price process follows

dS_{t} = rS_{t}dt + σS_{t}^{ρ}dW̃_{t}.  (2.11)

By Cameron-Martin-Maruyama-Girsanov’s theorem [9, p.191], we shall consider the unique equivalent measure change candidate for P |_{F_{T ∧ζ^{S}}} defined by the Radon-Nikodym derivative

η_{T} = exp \{−θ ∫_{0}^{T} S_{t}^{1−ρ}dW_{t} − \frac{θ^{2}}{2} ∫_{0}^{T} S_{t}^{2(1−ρ)}dt\} = exp \{−θ ∫_{0}^{T ∧ζ^{S}} S_{t}^{1−ρ}dW_{t} − \frac{θ^{2}}{2} ∫_{0}^{T ∧ζ^{S}} S_{t}^{2(1−ρ)}dt\} ,

(2.12)

under which {U_{t}; 0 ≤ t ≤ T } would be martingale.

Theorem 2.3 For all T < ∞, E[η_{T}] = 1 and hence there exists a unique P - equivalent martingale measure P̃ on F_{T} defined by

P̃[A] = E[1_{A}η_{T}] for A ∈ F_{T} .  (2.13)

Proof. There are (at least) two ways to prove the theorem. One way is to analyse the Novikov condition for the local martingale

η_{u} = exp \{−θ ∫_{0}^{u∧ζ^{S}} S_{t}^{1−ρ}dW_{t} − \frac{θ^{2}}{2} ∫_{0}^{u∧ζ^{S}} S_{t}^{2(1−ρ)}dt\} .

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[η_{T}] = 1 also fortimes T that are bigger than frac1(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:

dS_{t} = μS_{t} dt + σS_{t}^{ρ}dW_{t}.

The second equation is the equation obtained when replacing μ by r, i.e.

dS_{t} = rS_{t} dt + σS_{t}^{ρ}dW_{t}.

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 F_{t}. 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 dS_{t} = μS_{t} dt + σS_{t}^{ρ}dW_{t}. In the same way we can find a P̃− Brownian motion W̃ so that dS_{t} = rS_{t} dt + σS_{t}^{ρ}dW̃_{t}. The passage from P to P̃ is not difficult. If we define the stopping times

τ_{n} = inf {u | ∫_{0}^{u} S_{t}^{2(1−ρ)} dt ≥ n} ,

we have that on the σ−algebra F_{τ_{n}} both measures, P and P̃, are equivalent. Furthermore the density of P̃ with respect to P is given by the random variable

η_{τ_{n}} = exp \{−θ ∫_{0}^{τ_{n}} S_{t}^{1−ρ}dW_{t} − \frac{θ^{2}}{2} ∫_{0}^{τ_{n}} S_{t}^{2(1−ρ)}dt\} .

Since P −almost surely we have ∫_{0}^{T} S_{t}^{2(1−ρ)} dt < ∞, we get that for n → +∞, necessarily τ_{n} → +∞. This implies that P ≪ P̃.

Conversely the same reasoning applies to the measure P̃ and using that P̃−almost surely we have ∫_{0}^{T} S_{t}^{2(1−ρ)} dt < ∞, we get that P̃ is absolutely continuous with respect to P.

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

η_{u} = exp \{−θ ∫_{0}^{u∧ζ^{S}} S_{t}^{1−ρ}dW_{t} − \frac{θ^{2}}{2} ∫_{0}^{u∧ζ^{S}} S_{t}^{2(1−ρ)}dt\} .

is not only a local martingale but is a strictly positive martingale. Therefore necessarily E[η_{T}] = 1 and η_{T} > 0. □

3 Arbitrage Free Option Pricing

By Theorem 2.3, there always exists an P -equivalent measure P̃ on F_{T} so that W̃_{t} becomes a Wiener process under the measure P̃. By construction, the discounted price process U_{t} is a local martingale under the P - equivalent measure.

From the reasoning in section 2, we immediately deduce the following Corollary 3.1

{S_{t}; 0 ≤ t} under the measure P̃ =^{law} {Y_{t}^{(δ_{ρ},r)}; 0 ≤ t} under the measure P  (3.1)

□

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

X_{t}^{(δ)} =^{law} t · V^{(δ, \frac{x}{t})}, x ≥ 0, t > 0,  (3.2)

where X_{0}^{(δ)} = x and V^{(a,b)} means the noncentral χ^{2} random variable with a (≥ 0) degrees of freedom and noncentrality parameter b ≥ 0. That is, the density of V^{(a,b)} is given by

f(v; a, b) = \frac{1}{2^{\frac{a}{2}}} exp \{−\frac{1}{2}(b + v)\} v^{\frac{a}{2} −1} ∑_{n=0}^{∞} (\frac{b}{4})^{n} \frac{v^{n}}{n! Γ(\frac{1}{2}a + n)}.  (3.3)

Proof. Consider the Laplace transform of V^{(a,b)},

E[exp{−λV^{(a,b)}}]
= ∫_{v≥0} \frac{1}{2^{\frac{a}{2}}} exp \{−\frac{1}{2}(b + (1 + 2λ)v)\} v^{\frac{a}{2} −1} ∑_{n=0}^{∞} (\frac{b}{4})^{n} \frac{v^{n}}{n! Γ(\frac{1}{2}a + n)}dv

= \frac{exp \{−\frac{λ}{1+2λ}b\}}{(1 + 2λ)^{\frac{δ}{2}}} ∫_{v≥0} \frac{1}{2^{\frac{a}{2}}} exp \{−\frac{1}{2} (\frac{b}{1 + 2λ} + v)\} v^{\frac{a}{2} −1} ∑_{n=0}^{∞} (\frac{b}{4(1 + 2λ)})^{n} \frac{v^{n}}{n! Γ(\frac{1}{2}a + n)}dv
= \frac{exp\{−\frac{λ}{1+2λ}b\}}{(1 + 2λ)^{\frac{b}{2}}}.

On the other hand, the Laplace transform of X_{t}^{(δ)} is given by (see [10, p.411] ),

E[exp\{λX_{t}^{(δ)}\}] = \frac{exp\{−\frac{λt}{1+2λt} \frac{x}{t}\}}{(1 + 2λt)^{\frac{δ}{2}}} = E[exp\{−λtV^{(δ, \frac{x}{t})}\}].

Since the Laplace transforms for both random variables are equal, we have (3.2). □

Yor [11, (2.c)] derived the following relationship between the squared Bessel processes X_{t}^{(δ)} and X_{t}^{(4−δ)}.

Lemma 3.3 Let δ ∈ (−∞, 2) and ϕ be a function such that

lim_{x↓0} E [ϕ (X_{t}^{(4−δ)}) (X_{t}^{(4−δ)})^{\frac{δ}{2} −1} |X_{0}^{(4−δ)} = x] < ∞.  (3.4)

Then for any and x > 0,

E[ϕ(X_{t}^{(δ)})1_{\{ζ>t\}}|X_{0}^{(δ)} = x] = x^{1− \frac{δ}{2}} · E [ϕ (X_{t}^{(4−δ)}) (X_{t}^{(4−δ)})^{\frac{δ}{2} −1} |X_{0}^{(4−δ)} = x] . □  (3.5)

The martingale property of U_{t} under P̃ follows from Corollary 3.1 and Lemma 3.3. That is, let ϕ(x) = x^{1− \frac{δρ}{2}} , then

Ẽ[U_{t}|U_{0} = u] = Ẽ [\binom{}{X̃_{τ_{t}^{(δρ,r)}∧ζ̃}^{(δ_{ρ})}}^{1− \frac{δρ}{2}} |X̃_{0}^{(δ_{ρ})} = u^{\frac{2}{2−δρ}} ]

= Ẽ [\binom{}{X̃_{τ_{t}^{(δρ,r)}}^{(δ_{ρ})}}^{1− \frac{δρ}{2}} 1_{\{ζ̃≥t\}} |X̃_{0}^{(δ_{ρ})} = u^{\frac{2}{2−δρ}} ]  (3.6)

= (u^{\frac{2}{2−δρ}} )^{1− \frac{δρ}{2}} Ẽ [\binom{}{X̃_{τ_{t}^{(δρ,r)}}^{(4−δ_{ρ})}}^{1− \frac{δρ}{2}} \binom{}{X̃_{τ_{t}^{(δρ,r)}}^{(4−δ_{ρ})}}^{\frac{δρ}{2} −1} |X̃_{0}^{(δ_{ρ})} = u^{\frac{2}{2−δρ}} ]
= u.

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 S_{T} under the original measure P (see also Cox [1]).

Theorem 3.5

P [S_{T} ≤ x|S_{0} = s] = 1 − ∑_{n=1}^{∞} g(n + λ, z)G(n, w)  (3.7)

where

\{ \begin{matrix}λ & = & \frac{1}{2(1−ρ)} \\ z & = & \frac{s^{2(1−ρ)}}{2τ_{T}^{(δρ,μ)}} = \frac{2μλ exp\{\frac{μT}{λ} \}s^{\frac{1}{λ}}}{σ^{2}(exp\{ \frac{μT}{λ} \}−1)} \\ w & = & \frac{(exp\{−μT \}x)^{2(1−ρ)}}{2τ_{T}^{(δρ,μ)}} = \frac{2μλx^{\frac{1}{λ}}}{σ^{2}(exp\{ \frac{μT}{λ} \}−1)} \\ g(u, v) & = & \frac{v^{u−1}}{Γ(u)} exp\{−v\} \\ G(u, v) & = & ∫_{w≥v} g(u, w)dw. \end{matrix}  (3.8)

Proof. From Theorem 2.2 and Lemma 3.3,

P [S_{T} ≥ x|S_{0} = s]
= P [Y_{T}^{(δ_{ρ},μ)} ≥ x |Y_{0}^{(δ_{ρ},μ)} = s]
= P [X_{τ_{T}^{(δρ,μ)}∧ζ}^{(δ_{ρ})} ≥ (exp\{−μT \}x)^{\frac{2}{2−δρ}} |X_{0}^{(δ_{ρ})} = s^{\frac{2}{2−δρ}} ]  (3.9)
= E [1 \{X_{τ_{T}^{(δρ,μ)}}^{(δ_{ρ})} ≥ (exp\{−μT \}x)^{\frac{2}{2−δρ}} \} 1\{ζ ≥ T \} |X_{0}^{(δ_{ρ})} = s^{\frac{2}{2−δρ}} ]
= s · E [1 \{X_{τ_{T}^{(δρ,μ)}}^{(4−δ_{ρ})} ≥ (exp\{−μT \}x)^{\frac{2}{2−δρ}} \} (X_{τ_{T}^{(δρ,μ)}}^{(4−δ_{ρ})})^{\frac{δρ}{2} −1} |X_{0}^{(4−δ_{ρ})} = s^{\frac{2}{2−δρ}} ]

And from Lemma 3.2,

E [(X_{τ_{T}^{(δρ,μ)}}^{(4−δ_{ρ})})^{\frac{δρ}{2} −1} 1 \{X_{τ_{T}^{(δρ,μ)}}^{(4−δ_{ρ})} ≥ (exp\{−μT \}x)^{\frac{2}{2−δρ}} \} |X_{0}^{(4−δ_{ρ})} = s^{\frac{2}{2−δρ}} ]

= (τ_{T}^{(δ_{ρ},μ)})^{\frac{δρ}{2} −1} E [(V^{(4−δ_{ρ},2z)})^{\frac{δρ}{2} −1} 1\{V^{(4−δ_{ρ},2z)} ≥ 2w\}]
= \frac{exp\{−z\}}{(2τ_{T}^{(δ_{ρ},μ)}z)^{1− \frac{δρ}{2}}} ∑_{n=0}^{∞} \frac{z^{n+1− \frac{δρ}{2}}}{Γ(n + 2 − \frac{δ_{ρ}}{2} )} ∫_{v≥w} \frac{v^{n} exp\{−v\}}{n!}dv

= \frac{1}{s} ∑_{n=1}^{∞} g(n + λ, z)G(n, w).  (3.10)

Thus from (3.9) and (3.10), (3.7) is obtained. □

As the special case of Theorem 3.5, we can derive the probability that X_{t}^{(δ)} hits the point 0 for the dimension δ ∈ (−∞, 2).

Corollary 3.6 For the squared Bessel process X_{t}^{(δ)} with dimension δ ∈ (−∞, 2), we have

P [X_{u}^{(δ)} hits the point 0 during 0 ≤ u ≤ t|X_{0}^{(δ)} = x]
= 1 − (\frac{x}{2t})^{1− \frac{δ}{2}} ∑_{n=1}^{∞} \frac{(\frac{x}{2t})n−1}{Γ(n + 1 − \frac{δ}{2})} exp \{−\frac{x}{2t}\} , x ≥ 0.  (3.11)

Proof. Let s = 0 (w = 0) and S = x^{1− \frac{δρ}{2}} in (3.7). Then G(n, w) = 1 and hence

P [S_{T} = 0|S_{0} = x^{1− \frac{δρ}{2}} ] = 1 − ∑_{n=1}^{∞} g(n + λ, z) = P [ζ ≤ τ_{T}^{(δ_{ρ},μ)}|X_{0}^{(δ)} = x]  (3.12)

Substituting τ_{T}^{(δ_{ρ},μ)} = t, z = \frac{x}{2t} and δ_{ρ} = δ in (3.12), we get (3.11). □

Remark 3.7 It is well known that when δ ≥ 2, the point 0 is polar and hence the probability that X_{t}^{(δ)} hits the point 0 is 0 (see [10, p.415]).

Here note that the martingale measure P̃ is not unique on F_{T}. However it is unique on F_{τ}. Hence if the payoff of the option depends only on the stock price process S_{t}, 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 C_{T} depends only on the stock price at the maturity S_{T}. That is, C_{T} is given by C(S_{T}) for some function C(·). Then we can derive an arbitrage free option pricing formula.

Theorem 3.8 Let the initial stock price S_{0} = s and C_{0}(s) be the arbitrage free price for the option at time 0 with the terminal payoff C(S_{T}). Then

C_{0}(s) = s · Ẽ [exp\{−rT \}C (exp\{rT \} \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})}}^{1− \frac{δρ}{2}} ) \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})}}^{\frac{δρ}{2} −1} |X̃_{0} = s^{2(1−ρ)} ]

+ exp\{−rT \}C(0) (1 − s (\frac{1}{2τ_{T}^{(δ_{ρ},r)}} )^{1− \frac{δρ}{2}} ∑_{n=1}^{∞} \frac{(\frac{s^{2(1−ρ)}}{2τ_{T}^{(δρ,r)}} )n−1}{Γ(n + 1 − \frac{δ_{ρ}}{2} )} exp \{−\frac{s^{2(1−ρ)}}{2τ_{T}^{(δ_{ρ},r)}} \}) .

(3.13)

Proof. By the general theorem for the complete market asset pricing [7, 8], the unique arbitrage free price C_{0}(S) is given by

C_{0}(s) = Ẽ[exp{−rT }C(S_{T})|S_{0} = s]
= Ẽ [exp\{−rT \}C (exp\{rT \} \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(δ_{ρ})}})^{1− \frac{δρ}{2}} |X̃_{0}^{(δ_{ρ})} = s^{\frac{2}{2−δρ}} ]

= Ẽ [exp\{−rT \}C (exp\{rT \} \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(δ_{ρ})}}^{1− \frac{δρ}{2}} ) 1_{\{ζ>τ_{T}^{(δρ,r)}\}} |X̃_{0}^{(δ_{ρ})} = s^{\frac{2}{2−δρ}} ]
+ exp\{−rT \}C(0)P̃[ζ̃ ≤ τ_{T}^{(δ_{ρ},r)}|X̃_{0}^{(δ_{ρ})} = s^{\frac{2}{2−δρ}}].  (3.14)

From Lemma 3.3,

Ẽ [exp\{−rT \}C (exp\{rT \} \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(δ_{ρ})}}^{1− \frac{δρ}{2}} ) 1_{\{ζ̃>τ_{T}^{(δρ,r)}\}} |X̃_{0} = s^{\frac{2}{2−δρ}} ]  (3.15)

= s · Ẽ [exp\{−rT \}C (exp\{rT \} \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})}}^{1− \frac{δρ}{2}} ) \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})}}^{\frac{δρ}{2} −1} |X̃_{0} = s^{\frac{2}{2−δρ}} ] .

Also from Corollary 3.6,

P̃[ζ̃ ≤ τ_{T}^{(δ_{ρ},r)}|X̃_{0} = s^{\frac{2}{2−δρ}}]

= 1 − s (\frac{1}{2τ_{T}^{(δ_{ρ},r)}} )^{1− \frac{δρ}{2}} ∑_{n=1}^{∞} \frac{(\frac{s^{\frac{2}{2−δρ}}}{2τ_{T}^{(δρ,r)}} )n−1}{Γ(n + 1 − \frac{δ_{ρ}}{2} )} exp \{−\frac{s^{\frac{2}{2−δρ}}}{2τ_{T}^{(δ_{ρ},r)}} \} .  (3.16)

Then from (3.14) through (3.16) and \frac{2}{2−δ_{ρ}} = 2(1 − ρ), we have (3.13). □

Call Option Case : Consider the arbitrage free pricing for the call option C_{T} = (S_{T} − K)^{+} with the nonnegative exercise price K. From Theorem 3.8, the arbitrage free price of C_{T} is given by

C_{0}(s)
= s · Ẽ [exp\{−rT \} (exp\{rT \} \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})}}^{1− \frac{δρ}{2}} − K)^{+} \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})}}^{\frac{δρ}{2} −1} |X̃_{0} = s^{\frac{2}{2−δρ}} ]
= s · P̃ [X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})} ≥ (exp\{−rT \}K)^{\frac{2}{2−δρ}} |X̃_{0}^{(4−δ_{ρ})} = S^{\frac{2}{2−δρ}} ]  (3.17)

− exp\{−rT \}Ks · Ẽ [\binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})}}^{\frac{δρ}{2} −1} 1_{\{X̃_{τ{(δρ,r) \atop T}}^{(4−δρ)} ≥(exp\{−rT \}K)^{\frac{2}{2−δρ}} \}} |X̃_{0}^{(4−δ_{ρ})} = s^{\frac{2}{2−δρ}} ] .

From Lemma 3.2,

P̃ [X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})} ≥ (exp\{−rT \}K)^{\frac{2}{2−δρ}} |X̃_{0}^{(4−δ_{ρ})} = s^{\frac{2}{2−δρ}} ]
= P [V^{(4−δ_{ρ},2z′)} ≥ 2w′]
= exp\{−z′\} ∑_{n=0}^{∞} \frac{z^{′n}}{n!} ∫_{v≥w′} \frac{v^{n+1− \frac{δρ}{2}} exp\{−v\}}{Γ(n + 2 − \frac{δ_{ρ}}{2} )}dv
= ∑_{n=1}^{∞} g(n, z′)G(n + λ, w′)  (3.18)

where λ, g(x, y), G(x, y) are given by (3.8) and z′, w′ are defined by

\{ \begin{matrix}z′ & = & \frac{s^{2(1−ρ)}}{2τ_{T}^{(δρ,r)}} = \frac{2rλ exp\{\frac{rT}{λ} \}s^{\frac{1}{λ}}}{σ^{2}(exp\{ \frac{rT}{λ} \}−1)} \\ w′ & = & \frac{(exp\{−rT \}K)^{2(1−ρ)}}{2τ_{T}^{(δρ,r)}} = \frac{2rλK^{\frac{1}{λ}}}{σ^{2}(exp\{ \frac{rT}{λ} \}−1)}. \end{matrix}  (3.19)

On the other hand from (3.10),

\begin{matrix}Ẽ & [\binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(4−δ_{ρ})}}^{\frac{δρ}{2} −1} 1_{\{X̃_{τ{(δρ,r) \atop T}}^{(4−δρ)} ≥(exp\{−rT \}K)^{\frac{2}{2−δρ}} \}} |X̃_{0}^{(4−δ_{ρ})} = s^{\frac{2}{2−δρ}} ] \\ = \frac{1}{s} ∑_{n=1}^{∞} g(n + λ, z′)G(n, w′). \end{matrix}  (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].

C_{0}(s) = s ∑_{n=1}^{∞} g(n, z′)G(n + λ, w′) − exp{−rT }K ∑_{n=1}^{∞} g(n + λ, z′)G(n, w′),  (3.21)

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 S_{t} 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 [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 ξ_{T} by

ξ_{T} = \frac{exp\{−rT \}S_{T}}{s} =^{l̃aw} \frac{1}{s} \binom{}{X̃_{τ_{T}^{(δρ,r)}∧ζ̃}^{(δ_{ρ})}}^{1− \frac{δρ}{2}} .  (4.1)

From (3.6), we have Ẽ[ξ_{T} |X̃_{0}^{(δ_{ρ})} = s^{2(1−ρ)}] = 1. Hence we can use ξ_{T} as a Radon-Nikodym derivative for an absolutely continuous (not equivalent) measure change from P̃ to another distribution P̂ on F_{T}. Furthermore since ξ_{T} > 0 ⇔ S_{T} > 0, P̂ is equivalent to the conditional distribution P̃[·|S_{T} > 0]. This together with the equivalence of P and P̃ yields the equivalence of P [·|S_{T} > 0] and P̂.

Next we shall consider the Markov process Ŝ_{t} defined by

dŜ_{t} = (rŜ_{t} + σ^{2}Ŝ_{t}^{2ρ−1})dt + σŜ_{t}^{ρ}dW̃_{t},  (4.2)

starting with Ŝ_{0} = S. Then we have the following lemma.

Lemma 4.1

{S_{t}; 0 ≤ t ≤ T } under P̂ =^{law} {Ŝ_{t}; 0 ≤ t ≤ T } under P̃.  (4.3)

Proof. First we shall represent Ŝ_{t} by squared Bessel process with dimension δ′ > 2. For ν > 0 and δ < 2, let

Z_{t}^{(δ,ν)} = exp\{νt\} \binom{}{X̃_{τ_{t}^{(δ,ν)}}^{(4−δ)}}^{1− \frac{δ}{2}} ,

starting with X̃_{0}^{(4−δ)} = s^{\frac{2}{2−δ}}. From Itô’s lemma,

dZ_{t}^{(δ,ν)} = (νZ_{t}^{(δ,ν)} + σ^{2} (Z_{t}^{(δ,ν)})^{− \frac{δ}{2−δ}} ) dt + σ (Z_{t}^{(δ,ν)})^{\frac{1−δ}{2−δ}} dW̃_{t}^{(δ,ν)}

where W̃_{t}^{(δ,ν)} is another Wiener process under P̃ given by (2.6) for W̃_{t}. Since −\frac{δ_{ρ}}{2−δ_{ρ}} = 2ρ − 1 and \frac{1−δ_{ρ}}{2−δ_{ρ}} = ρ, we have

{Ŝ_{t}; 0 ≤ t} =^{l̃aw} {Z̃_{t}^{(δ_{ρ},r)}; 0 ≤ t}.

This together with Lemma 3.3 yields

P̂[S_{T} ≤ u|S_{0} = s]
= Ê[1_{{S_{T} ≤u}}|S_{0} = s]
= \frac{1}{S} Ẽ [1_{\{X̃_{τ{(δρ,r) \atop T}}^{(δρ)} ≤(exp\{−μT \}u)^{\frac{2}{2−δρ}} \}} \binom{}{X̃_{τ_{T}^{(δρ,r)}}^{(δ_{ρ})}}^{1− \frac{δρ}{2}} 1_{\{ζ̃>τ_{T}^{(δρ,r)}\}} |X̃_{0}^{(δ_{ρ})} = s^{\frac{2}{2−δρ}} ]

= Ẽ [1_{\{X̃_{τ{(δρ,r) \atop T}}^{(4−δρ)} ≤(exp\{−μT \}u)^{\frac{2}{2−δρ}} \}} |X̃_{0}^{(4−δ_{ρ})} = s^{\frac{2}{2−δρ}} ]
= P̃[Ŝ_{T} ≤ u|Ŝ_{0} = s],

for any u ≥ 0. Thus the transition probabilities for the Markov processes ({S_{t}; 0 ≤ t}, P̂) and ({Ŝ_{t}; 0 ≤ t}, P̃) coincide and hence they are equivalent in law. □

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. : {S_{t}; 0 ≤ t ≤ T } under the conditional probability measure P [·|S_{T} > 0], there always exists arbitrage opportunities.

Proof. The probability measures P̂ and P [·|S_{T} > 0] are equivalent. Furthermore from Lemma 4.1, ({S_{t}; 0 ≤ t ≤ T }, P̂) =^{law} ({Ŝ_{t}; 0 ≤ t ≤ T }, P̃). Then it is enough to show that there exists arbitrage opportunities for the process {Ŝ_{t}; 0 ≤ t ≤ T } under P̃. Define the process X̂_{u} by

X̂_{u} = (exp\{−rt_{u}^{(δ_{ρ},r)}\}Ŝ_{t_{u}^{(δρ,r)}})^{\frac{2}{2−δρ}}  (4.4)

where t_{u}^{(δ,ν)} is given by the inverse transformation of 2.3. From Itô’s lemma, X̂_{u} follows

dX̂_{u} = 2\sqrt{X̂_{u}}dŴ_{u} + (4 − δ_{ρ})du  (4.5)

where Ŵ_{u} is an other Wiener process under P̃ defined as :

Ŵ_{u} = ∫_{0}^{t_{u}^{(δ,ν)}} \frac{σ}{2 − δ_{ρ}} exp \{−\frac{rv}{2 − δ_{ρ}} \} dW̃_{v}.  (4.6)

Thus X̂_{u} is a (4 − δ_{ρ})-dimensional squared Bessel process. Since 4 − δ_{ρ} > 2, we have the following result from Theorem 6 of Delbaen-Schachermayer [5] : L_{u} = X̂_{u}^{\frac{δρ}{2} −1} is a local martingale such that L_{u}^{−1} allows arbitrage with respect to the general admissible integrands under 0 interest rate. That is, there exists an admissible integrand φ_{u} such that

V̂′_{T} = ∫_{0}^{τ_{T}^{(δρ,r)}} φ_{u}dL_{u}^{−1} ≥ 0, P̃-a.s.,  (4.7)
P̃[V′_{T} > 0] > 0.  (4.8)

Since L_{u}^{−1} = exp{−rt_{u}^{(δ_{ρ},r)}}Ŝ_{t_{u}^{(δρ,r)}}, we can easily show that

V̂′_{T} = exp{−rT }V̂_{T}  (4.9)

where

V̂_{t} = ∫_{0}^{t} r(V̂_{u} − φ_{τ_{u}^{(δρ,r)}} Ŝ_{u})du + φ_{τ_{u}^{(δρ,r)}}dŜ_{u}, 0 ≤ t ≤ T.  (4.10)

From (4.7) through (4.10), there exists an arbitrage opportunity for Ŝ_{u} under P̃. □

The intuitive explanation for the existence of arbitrage opportunity is as follows. If there exists an equivalent martingale measure P̌ for Ŝ_{t}, it will have the same law as S_{t} under P̃ by Cameron-Martin-Maruyama-Girsanov’s theorem. However this is impossible since P̃[S_{T} = 0] > 0 whereas P̌[Ŝ_{T} = 0] = 0.

References

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[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.