Necessary and Sufficient Conditions
for No Static Arbitrage among European Calls ∗
Laurent Cousot
Courant Institute
New York University
251 Mercer Street
New York, NY, 10012, USA
[email protected]
First Version: September 15, 2004†.
This version: October 28, 2004.
Abstract
Under the assumption of the absence of arbitrage, European call prices on a given asset must satisfy
well-known inequalities, which have been described in the landmark paper Merton (1973). If we further
assume that there is no interest rate volatility and that the underlying pays continuously deterministic
dividends, cross maturity inequalities must also be satisfied by the call prices.
We show that there exists an arbitrage free model, which is consistent with the call prices, if these inequalities are satisfied. Furthermore, we describe an algorithm to obtain a realistic calibrated martingale
Markov chain model, using the notions of entropy and of copula.
JEL classification: C51, G13.
AMS classification codes: 60J10, 60J20, 60J22.
Key words: option pricing, arbitrage, Markov chains, entropy, copula.
∗ I am grateful to Peter Carr, Bruno Dupire, and more generally to the Bloomberg Quantitative Finance Research Team for
all their insights. I am also grateful to Brian Barber for useful comments. All errors are of course my own.
† Extended version of a paper registered at SSRN, 09/21/2004, Abstract ID 594141.
1
1
Introduction
In trying to understand the dynamics of a given underlying, the financial markets only tell part of the story.
For example, options quotes yield some information about the marginal distributions at specific future
times. However a complete specification of the possible future behavior of the underlying remains largely
undetermined.
Consequently, to bet on an arbitrage strategy based on a particular market structure can be very risky as
explained in the enlightening binomial-tree based example of Carr, Geman, Madan and Yor (2003) (henceforth CGMY). It makes more sense to restrict oneself to static strategies in which the future behavior of the
underlying is less relevant. Nevertheless, it is not sufficient to make the notion of arbitrage independent of
the real ”statistical measure”.
Indeed, classical static arbitrage strategies involving vertical (or call) spreads, butterfly spreads and
calendar spreads do not depend on the ”statistical measure”. But is it possible to claim that the elimination
of these strategies is sufficient to prevent static arbitrage without specifying a ”statistical measure”? The
answer is no. Indeed, if the price of a call is zero and the probability that the underlying will be greater than
the strike at maturity is positive, then to buy the call is an arbitrage strategy. Likewise, if the price of a call
is positive and the probability that the underlying will be greater than or equal to the strike at maturity is
zero, then to sell the call is an arbitrage strategy.
In this paper, we will show that these necessary conditions for the absence of static arbitrage are also
sufficient for the existence of a market model which is consistent with the call quotes and further, is (static)
arbitrage free. We generalize the results of Carr and Madan (2004) by allowing the underlying to be a
deterministic dividend paying stock, the short interest rate to be a deterministic function of time, and the
grid of the available strikes to be finite and not rectangular. Furthermore, we describe an algorithm to
construct a more realistic calibrated martingale Markov chain model.
The structure of this paper is given as follows. Section 2 describes the assumptions we make and introduces some calendar versions of the classical vertical and butterfly spreads; Section 3 describes a way to
construct a marginal distribution consistent with call quotes corresponding to a given maturity; Section 4
uses this construction to exhibit a martingale measure, which prevents arbitrage; Section 5 recalls briefly why
the conditions are necessary for the absence of static arbitrage; Section 6 explains how to construct a more
realistic calibrated Markov chain model; Section 7 explains how to relax some of the starting assumptions;
Section 8 concludes.
2
Assumptions and definitions
We assume that we have at our disposal a finite set of call quotes on a given non-dividend paying stock. The
maturities, indexed in increasing order, will be denoted by (Ti )16i6m . For a given maturity Ti , ni quotes
are available corresponding to the strikes K1i < ... < Kji < ... < Kni i and will be denoted by (Cji )16j6ni . We
also assume that there is no interest rate. For each maturity, we will add a quote corresponding to a call
struck at 0. By convention, K0i = 0 and C0i = S0 - the initial stock price - by arbitrage.
Now we will extend the classical definitions of vertical, calendar and butterfly spreads because the grid
of available strikes is not rectangular. The calendar version of a vertical (resp. butterfly) spread must be
thought of as a linear combination of calendar spreads and a classical vertical (resp. butterfly) spread.
Definition 2.1 (Vertical Spreads). ∀i ∈ [1, m],
V Sji ≡
i
Cj−1
− Cji
i
Kji − Kj−1
1 6 j 6 ni
V S0i ≡ 1
Definition 2.2 (Calendar (Vertical) Spreads). ∀i1 , i2 ∈ [1, m] s.t. i1 6 i2 , ∀j1 ∈ [0, ni1 ], ∀j2 ∈ [0, ni2 ]
s.t. Kji11 > Kji22
,i2
CV Sji11 ,j
≡ Cji22 − Cji11
2
2
Definition 2.3 ((Calendar) Butterfly Spreads). ∀i, i1 , i2 ∈ [1, m] s.t. i 6 i1 and i 6 i2 , ∀j ∈
[0, ni ], ∀j1 ∈ [0, ni1 ], ∀j2 ∈ [0, ni2 ] s.t. Kji11 < Kji < Kji22
,i
CV Sji11 ,j
i,i1 ,i2
CBSj,j
≡
1 ,j2
3
Kji11 − Kji
−
i,i2
CV Sj,j
2
Kji − Kji22
Construction of discrete risk-neutral marginal distributions
Lemma 3.1 (Discrete Distribution). If N ∈ N∗ , k0 = 0 < ... < kj < ... < kN are N + 1 real numbers and
PC : R+ → R is a function, which is continuous, convex, linear and decreasing on each interval [kj ; kj+1 ]
with 0 6 j 6 N − 1 (with a slope greater than or equal to -1 on [k0 ; k1 ]) and zero after kN then the following
distribution
µ=
N
X
qj δkj
j=0
with
PC (k0 ) − PC (k1 )
k1 − k0
PC (kj−1 ) − PC (kj ) PC (kj ) − PC (kj+1 )
−
, 16j 6N −1
qj =
kj − kj−1
kj+1 − kj
PC (kN −1 )
qN =
kN − kN −1
q0 = 1 −
is such that:
g(K) ≡
for all K ∈ R+ .
Z
∞
0
(x − K)+ dµ = PC (K)
(1)
Proof. The non-negativity of q0 is ensured by the condition on the slope of PC on [k0 ; k1 ]. Further qj > 0
(1 6 j 6 N − 1) due to the convexity of PC . Finally qN is non-negative because PC is also non-negative.
Moreover,
N
X
qk = q0 +
k=0
PC (k0 ) − PC (k1 ) PC (kN −1 ) − PC (kN )
−
+ qN
k1 − k0
kN − kN −1
PC (k0 ) − PC (k1 ) PC (k0 ) − PC (k1 ) PC (kN −1 ) − PC (kN )
PC (kN −1 )
+
−
+
k1 − k0
k1 − k0
kN − kN −1
kN − kN −1
PC (kN )
=1+
kN − kN −1
=1
=1−
We now show that the distribution results in a linear call price function g between two nodes [kj ; kj+1 ],
0 6 j 6 N − 1. ∀K ∈ (kj ; kj+1 ], we have:
g(K) =
N
X
k=0
=
qk (kk − K)+
N
X
j+16k6N

=
N
X
qk (kk − K)
j+16k6N


qk kk  − K 
3
N
X
j+16k6N

qk 
Figure 1: Sketch of a possible discrete distribution.
Moreover g is also right-continuous in kj . Indeed,
g(K) −→+
K→kj
N
X
j+16k6N
qk (kk − kj ) =
N
X
j6k6N
qk (kk − kj ) = g(kj )
Clearly g is zero on [kN , +∞). Now if we prove that the slopes are equal on each segment [kj ; kj+1 ],
0 6 j 6 N − 1, we will be able to conclude that the functions g and PC are equal. Accordingly,
g(kj+1 ) − g(kj ) =
N
X
j+16k6N
qk (kk − kj+1 ) −
= (kj − kj+1 )
= (kj − kj+1 )
N
X
N
X
j+16k6N
qk (kk − kj )
qk
j+16k6N
PC (kj ) − PC (kj+1 )
kj+1 − kj
= PC (kj+1 ) − PC (kj )
In the next section, we will use this lemma to construct marginal distributions. The quoted strikes will be
among the kj and their corresponding prices will be given by the function PC . Consequently the distribution
µ will be a risk-neutral marginal distribution for the given maturity.
4
4.1
Sufficient conditions for the existence of a martingale measure
Outline of the proof
If we are able to construct marginal distributions, that are consistent with the call quotes and the initial stock
price, non-decreasing in the convex order (henceforth NDCO) and have the same mean, then we can conclude
that there exists a martingale consistent with all these marginal distributions due to Kellerer theorem.
4
Theorem 4.1 (Kellerer theorem (1972)). Let (µt )t∈[0,T ] be a family of probability measures on (R, B(R))
with first moment, such that for s < t µt dominates µs in the convex order, i.e. for each convex function φ
R → R µt -integrable for each t ∈ [0, T ], we must have:
Z
Z
φdµt >
φdµs
R
R
Then there exists a Markov process (Mt )t∈[0,T ] with these marginals for which it is a submartingale. Furthermore if the means are independent of t then (Mt )t∈[0,T ] is a martingale.
Proof. See Kellerer (1972).
4.2
Construction of marginal distributions that are NDCO
Proposition 4.2. Let us assume that:
• ∀i ∈ [1, m],
Cji > 0,
V
V
Sji
Sji
0 6 j 6 ni
∈ [0, 1]
>0
1 6 j 6 ni
i
>0
if 1 6 j 6 ni and Cj−1
• ∀i1 , i2 ∈ [1, m] s.t. i1 < i2 , ∀j1 ∈ [0, ni1 ], ∀j2 ∈ [0, ni2 ]
,i2
CV Sji11 ,j
> 0 if Kji11 > Kji22
2
,i2
CV Sji11 ,j
> 0 if Kji11 > Kji22 and Cji11 > 0
2
• ∀i, i1 , i2 ∈ [1, m] s.t. i 6 i1 and i 6 i2 , ∀j ∈ [0, ni ], ∀j1 ∈ [0, ni1 ], ∀j2 ∈ [0, ni2 ] s.t. Kji11 < Kji < Kji22
i,i1 ,i2
>0
CBSj,j
1 ,j2
then there exists m risk-neutral distributions (φj )16j6m corresponding to the different maturities. Let us
define φ0 ≡ δS0 . (φj )06j6m are NDCO and their means are all equal to S0 .
Before proving this proposition, we would like to give some insight on the required conditions. If a
martingale is consistent with the quoted call prices, then the general call price function would be nonincreasing and convex in the strike, as well as non-decreasing in the maturity. For example, we can easily
conclude that a situation as the one described in Table 1 is not compatible with the existence of a pricing
martingale. Indeed if there was a pricing martingale, then x > 30 and x 6 20, which is impossible.
Maturity \ Strike
1 month
3 months
80
x
20
100
30
—
Table 1
Another more complicated example is given by Table 2.
Maturity \ Strike
1 month
3 months
6 months
80
—
x1
40
Table 2
5
100
37
x2
—
120
—
30
—
Here the condition on the calendar butterfly spread is violated and this can be explained as follows: if there
was a pricing martingale, then we would have:
40 > x1 (The price is non-decreasing with maturity.)
x1 + 30
> x2 (The price is convex with respect to the strike.)
2
x2 > 37 (The price is non-decreasing with maturity.)
> 37. This last example allows us to understand why the strike
and this would result in 35 = 40+30
2
corresponding to the nearest maturity is always between the two other strikes in the definition of the calendar
butterfly spread. The same kind of conclusions could not be drawn in a situation such as the one described
in Table 3.
Maturity \ Strike
80 100 120
1 month
40
—
—
3 months
—
—
30
6 months
—
37
—
Table 3
Proof. Lemma 3.1 tells us how to associate in a one-to-one way a distribution with a Ti -call price function
if the latter respects the given conditions. Consequently, to construct the marginal distributions, we will
construct call price functions which have the right properties and which go through the call quotes at Ti .
Finally we will consider the corresponding distributions as candidates.
The distributions must be NDCO. Notably the corresponding call price functions must be non-decreasing
with maturity. By only considering the quotes at a given maturity, it is possible that the corresponding
constructed distributions will not possess this characteristic (See Figure 2). Nevertheless, observe that if we
add the points corresponding to quotes of longer maturities to the quotes of maturity Ti , the frontier of their
convex hull1 seems to have all of the features we are looking for.
Figure 2: Frontier at Ti1 .
To ensure the monotonicity of the distributions, we must also properly specify the call price function to
the right of the greatest quoted strike - if the corresponding price is not zero. Let us determine the maximum
of the non-zero slopes between two points, which could be consecutive on the frontier a priori.
1 In
the definition of the convex hull we should restrict the slopes of the linear functions to be non-positive.
6
We adopt the following notation:
!
)
(
2
\
CV Sji11 ,i
,j2
i1
i2
∗
R−
M axSlope ≡ max
− i1
|1 6 i1 , i2 6 m, 0 6 j1 6 ni1 , 0 6 j2 6 ni2 s.t. Kj1 > Kj2
Kj1 − Kji22
If Cni i > 0, we will add a point (Kni i +1 , 0) such that the slope between (Kni i , Cni i ) and (Kni i +1 , 0) is equal
to M asSlope/2. By convention, if Cni i = 0 then Kni i +1 ≡ Kni i + 1.
∀i ∈ [1, m], ∀j ∈ [0, ni ],
Aij ≡ (Kji , Cji )
Kni i +1 ≡ Kni i − 2Cni i /M axSlope
Ai ≡
n[
i +1
Aini +1 ≡ (Kni i +1 , 0)
Aij
B i ≡ Convex hull of
j=0
m
[
Ai
j=i
We will show that the frontier of B i corresponds to a risk-neutral distribution for the calls of maturity
Ti and furthermore that these distributions are NDCO.
It is obvious that F r(B i ), the frontier of B i , is continuous, piece-wise linear, convex and that the nodes
are among the Kjk (with corresponding value Cjk ), i 6 k 6 m, 0 6 j 6 nk + 1.
Let us show that the slope of the frontier on the first segment is greater than or equal to -1. The value of
the frontier in K0i = 0 is clearly C0i = S0 . Let us denote by Kji11 the smallest non-zero node of the frontier.
Cji11 − C0i
Kji11
=
=
Cji11 − C0i1
Kji11
j1
1 X
i1
−V Ski1 (Kki1 − Kk−1
)
Kji11 k=1
>−
>−
j1
V S0i1 X i1
i1
)
(K − Kk−1
Kji11 k=1 k
i1
, 0 6 j < ni1 by convexity.)
(V Sji1 > V Sj+1
Kji11 − K0i1
Kji11
> −1
We now show that the frontier is zero after a given value. Since (Kni i +1 , 0) is a point in the convex hull
B and the linear functions involved in the definition of the convex hull have non-positive slopes, the frontier
is zero on [Kni i +1 , 0). Denote by K∗i the smallest strike which corresponds to a node of the frontier whose
call value is zero.
Let us examine the monotonicity of F r(B i ). We consider two consecutive nodes of the frontier Kji11 and
Kji22 with Kji11 < Kji22 . Since we know that the frontier is flat at the right of K∗i , we only need to prove that
the segment [Aij11 ; Aij22 ] is decreasing if Kji22 6 K∗i .
i
• If Cji22 = 0, then by definition of K∗i , we have Kji22 = K∗i . Because Kji11 < Kji22 = K∗i , Cji11 > 0, i.e. the
slope is negative.
• If Cji22 > 0 and i2 6 i1 , then the condition on the (calendar) vertical spreads shows the monotonicity.
6 ni2 + 1 since the two call prices are positive).
(i1 6= ni1 + 1 and i2 =
• If Cji22 > 0 and i2 > i1 , [Aij11 ; Aij22 ] is below the segment [Aij11 ; Aij11 +1 ] because Aij22 is on the frontier.
(Remember that i1 6= ni1 + 1). Since [Aij11 ; Aij11 +1 ] is decreasing (because of the condition on the
vertical spreads if j1 < ni1 and the definition of M axSlope if j1 = ni1 ), the segment [Aij11 ; Aij22 ] must
be decreasing as well.
7
We now show that the frontier goes through the points Aij for 0 6 j 6 ni . For j = 0, it is obvious since
all the Ai0 correspond to the same point. Let us consider a point Aij that does not belong to the frontier
(1 6 j 6 ni ). Denote by Kji11 (resp. Kji22 ) the greatest (resp. smallest) node of the frontier which is less
(resp. greater) than Kji . (Kji22 exists since Kni i +1 is on the frontier). Recall that i 6 i1 and i 6 i2 .
• If j1 6 ni1 and j2 6 ni2 , then the non-negativity of the (calendar) butterfly spreads contradicts the
fact that Aij does not belong to the frontier.
• If j1 6 ni1 and j2 = ni2 + 1, then we have the following two cases:
– If Cni2i2 = 0, then Ain2i2 is also on the frontier. Since Ain1i1 and Ain2i
2 +1
Cji11
are consecutive on the
frontier,
= 0. The condition on the (calendar) vertical spreads between the points Aij11 and
i
Aj imposes that Cji 6 0. Combined with the non-negativity condition on call quotes, this ensures
us that the point Aij lies in the flat frontier, which is a contradiction.
– If Cni2i2 > 0, then the slope of the segment [Aij11 ; Ain2i
[Ain2i2 ; Ain2i
2
2 +1
] must be less than the slope of the segment
i1
i2
+1 ], since Aj1 and Ani2 are two consecutive nodes on the frontier. Its slope is therefore
greater than M axSlope/2. But the slope of the segment [Aij11 ; Aij ] is greater than the slope of
[Aij11 ; Ain2i +1 ], since Aij does not belong to the frontier. Consequently, the slope of [Aij11 ; Aij ] is
2
greater than M axSlope/2. We can exclude the case where Cji = 0, since in this case Aij always
belong to the frontier. If Cji > 0, then the slope of [Aij11 ; Aij ] is strictly negative because of the
condition on the (calendar) vertical spread and must therefore be less than or equal to M axSlope,
which is a contradiction.
• If j1 = ni1 + 1, then Cji22 = 0 because the corresponding point belongs to the frontier, which is flat at
the right of Kni1i +1 . We have the following two cases:
1
– If
= 0, the condition on the (calendar) vertical spread between the points Aij11 and Aij imposes
that Cji 6 0. Combined with the non-negativity condition on call quotes, this ensures that the
point Aij lies in the flat frontier, which is a contradiction.
Cni1i1
– If Cni1i1 > 0, then the slope of the segment [Ain11 ; Aij ] is greater than the slope of [Ain11 ; Ain11 +1 ],
which is given by M axSlope/2. But this slope is strictly negative because of the condition on
the vertical (calendar) spread (Cji > 0 since Aij does not belong to the frontier). Consequently, it
must be less than or equal to M axSlope. This is a contradiction
We have proved that F r(B i ) satisfies all the conditions of Lemma 3.1. Therefore we can associate to it
a distribution µi satisfying (1). Moreover this distribution is risk-neutral for the calls of maturity Ti since
F r(B i ) goes through the points Aij . We still have to prove that these distributions are NDCO and that their
mean is constant over time.
For a given strike, the prices are non-decreasing with the maturity since B i+1 ⊆ B i implies that F r(B i ) 6
F r(B i+1 ), 1 6 i < m. The call price function corresponding to the distribution φ0 ≡ δS0 starts at S0 , has
a slope of -1 until S0 and is zero after. Consequently, it is less than the one corresponding to φ1 , which
starts at S0 , has a starting slope greater than or equal to -1 and is convex. The prices of European put
options are also non-decreasing with maturity because of put-call parity. Since any convex function can be
approximated by linear combinations with positive weights of put and call functions, the distributions are
NDCO.
Finally the means of the different distributions are constant over time because all the calls struck at 0
have the same price.
5
Necessity of the conditions
The conditions of Proposition 4.2 are of course necessary for the existence of a pricing martingale but they
are also necessary for the absence of static arbitrage.
8
First of all, let us be precise in what we mean by static strategies. At the initial time, one is allowed
to take long and short positions in the options and in the stock. Furthermore, as explained in CGMY
(2003), one has the possibility to short the stock for a given future period of time if the stock is greater
than a specified value at the beginning of the period. Under the assumption of no static arbitrage and of a
frictionless market, it is costless. In other words, at the initial time, one can purchase at zero cost, a security
whose payoff at time T2 (> T1 > 0) is:
1{ST1 >K} (ST1 − ST2 )
The inequalities of Proposition 4.2 are classic necessary conditions and will not be detailed in this paper.
We just consider, as an example, the case of a (vertical) calendar spread constructed from the call of strike
K1 , maturity T1 and price C1 , and the call of strike K2 , maturity T2 and price C2 (K1 > K2 and T1 < T2 ).
The strategy is to sell the call of maturity T1 , buy the call of maturity T2 and short one share if the stock
at T1 is greater than K1 . The price of this strategy is of course C2 − C1 and its payoff at T2 is:
−(ST1 − K1 )+ + 1{ST1 >K1 } (ST1 − ST2 ) + (ST2 − K2 )+ > 1{ST1 >K1 } (K1 − K2 )
We remark that the payoff of this strategy is always non-negative. Consequently, its price should also be
non-negative, i.e. C1 6 C2 . Moreover, if the probability that ST1 > K1 is positive and K1 > K2 , then
the payoff is non-negative and positive with positive probability. Therefore, in this case, the price of this
strategy should be positive and we notice that the condition P (ST1 > K1 ) > 0 is equivalent to C1 > 0, since
we assumed no static arbitrage.
6
Construction of a more intuitive and realistic model
The previous construction of the pricing martingale offers little intuition and may result in an unrealistic
model for the stock. In this section, we will construct a martingale Markov chain model, which will be
calibrated to the quoted call prices. First we will describe an algorithm based on entropy maximization to
construct more realistic, NDCO, marginal distributions. Once these distributions are constructed, we will be
able to claim the existence of martingale transition matrices due to the Sherman-Stein-Blackwell theorem,
as explained for example in Davis (2004). To choose realistic transition matrices, we will once again describe
an algorithm based on the notion of entropy and of Brownian copula.
6.1
Construction of marginal distributions
The marginal distributions we have constructed may be quite unrealistic. Indeed, the number of possible
states for the stock may be decreasing with maturity. We will describe here, an algorithm allowing to
construct more realistic discrete marginal distributions.
Let (ki )06i6N be a finite subset of R+ containing all the (Kji )16i6m,06j6ni +1 . This set will be the support
of the marginal distributions.
In the proof of Proposition 4.2, we have already constructed some marginal distributions having the
required features and (ki )06i6N as a support. To choose a more realistic set of marginal distributions, we
will use the notion of entropy, which is appealing for several reasons. One reason is that the less information a
discrete distribution contains, the higher the entropy - the maximum being reached for a uniform distribution.
A direct consequence is that the entropy maximization will distribute the masses among all the (ki ) in a
more uniform manner.
Let us denote by P the set of the call price functions satisfying the assumptions of Lemma 3.1 and by Q
the set of the corresponding distributions, whose support is consequently (ki ). Let us denote by PCi the call
price function corresponding to the maturity Ti constructed in the proof of Proposition 4.2. The new call
1
price function, PC,new
, corresponding to the maturity T1 , is obtained by maximizing the entropy2 :
max
qC ∈Q
2 By
−
N
X
!
qC (ki ) log(qC (ki ))
i=0
convention 0 × log (0) = 0 in the definition of entropy.
9
s.t. 0 6 PC 6 PC1 and ∀j ∈ [0, n1 ], PC (Kj1 ) = PC1 (Kj1 )
Now, let us assume that we have constructed the new marginal distributions until the maturity Ti−1 .
i
PC,new
is obtained by maximizing:
max
qC ∈Q
−
N
X
!
qC (ki ) log(qC (ki ))
i=0
i−1
s.t. PC,new
6 PC 6 PCi and ∀j ∈ [0, ni ], PC (Kji ) = PCi (Kji )
We remark that the new distributions are risk neutral since:
i
∀i ∈ [1, m], ∀j ∈ [0, ni ], PC,new
(Kji ) = PCi (Kji ) = Cji
Moreover these distributions are NDCO by construction. A numerical example is given in Appendix 1.
6.2
Construction of martingale transition matrices
Now that we have constructed risk neutral marginal distributions, which are NDCO, we will focus on the
construction of martingale transition matrices. The Sherman-Stein-Blackwell theorem ensures the existence
of such matrices. This theorem is a generalization to the N-dimensional case of the Hardy-Littlewood-Polya
theorem (1929) and has been generalized by Strassen (1965) to probability measures in RN . The Kellerer
theorem (1972), which we stated in section 4.1 is a continuous time version of the latter.
Theorem 6.1 (Sherman-Stein-Blackwell theorem). If X = (a1 , ..., an ) is a finite set in RN , and q 1
and q 2 are probability measures on X such that
n
X
φ(ai )q 1 (ai ) 6
i=1
n
X
φ(ai )q 2 (ai )
i=1
for every continuous convex function φ defined on the convex hull of X, then there is a martingale transition
matrix (qi,j )16i,j6n such that:
n
X
qi,j > 0
for 1 6 i, j 6 n
qi,j = 1
for 1 6 j 6 n
i=1
n
X
qi,j q 1 (ai ) = q 2 (aj )
for 1 6 j 6 n
i=1
n
X
qi,j aj = ai
for 1 6 i 6 n
i=1
Proof. See Davis (2004) for Strassen’s proof restricted to the finite case.
Now that the existence of the transition matrices is established, we would like to have at our disposal an
extra criterion to choose realistic ones. Once again we will minimize under constraints the Kullback-Leibler
distance of the joint distributions:
!
i
X
Jα,β
i i,prior
i
D(J |J
)=
Jα,β log
i,prior
Jα,β
16α,β6N
10
i,prior
i
)16α,β6N is a
(where Jα,β
≡ P (STi = kα , STi+1 = kβ ) is the joint distribution of (STi , STi+1 ) and (Jα,β
given prior joint distribution), under the following constraints
i
Jα,β
>0
X
i
Jα,β
= q i (α)
1 6 α 6 N (Law of total probability)
16β6N
X
i
Jα,β
= q i+1 (β)
1 6 β 6 N (Chapman-Kolmogorov equations)
16α6N
X
i
kβ Jα,β
= q i (α)kα
1 6 α 6 N (Martingale conditions)
16β6N
where the Ti −marginal distribution obtained in Section 6.1 is denoted by (q i (α))16α6N .
As explained in Cover and Thomas (1991) or in Avellaneda et al (2000), this problem can be reduced to
the unconstrained maximization problem in (λ, µ, ν) of:

W (λ, µ, ν) = − log 
X
16α,β6N

i
Jα,β
eλα +να kβ +µβ  +
X
q i (α) (λα + να kα ) +
16α6N
X
µβ q i+1 (β)
16β6N
Indeed, if (λ0 , µ0 , ν 0 ) is a critical point of W , the joint distribution defined by:
0
i,0
Jα,β
=P
0
0
i,prior λα +να kβ +µβ
Jα,β
e
0
16α,β6N
0
0
i eλα +να kβ +µβ
Jα,β
i,prior
is a solution of the initial problem, provided that Jα,β
> 0.
We now need to specify a prior joint distribution. One way is to discretize the Brownian copula as
explained in Carr and Cousot (2004).
Recall that once the marginal distributions are specified, the knowledge of the copula of (STi , STi+1 ),
CSTi ,STi+1 , and of the joint distribution are equivalent (See Darsow, Nguyen and Olsen (1992)). Indeed,
under
of bilinear
interpolation, the copula is uniquely determined by its values at the points
P the assumption
Pβ
α
i
i+1
(b) :
a=1 q (a),
b=1 q
i
Cα,β
≡ CSTi ,STi+1
α
X
i
q (a),
a=1
β
X
q
i+1
(b)
b=1
!
=
β
α X
X
i
Ja,b
a=1 b=1
Conversely, if the copula is given, the joint distribution is uniquely specified by:
i
i
i
i
i
− Cα−1,β
− Cα,β−1
+ Cα−1,β−1
= Cα,β
Jα,β
for 1 6 α, β 6 N .
We would like the joint distribution to imply a realistic copula. Consequently, one way is to choose as a
prior joint distribution the one implied by the Brownian copula characterized in Darsow et al (1992):
!
p
√
Z x
Ti+1 Φ−1 (y) − Ti Φ−1 (u)
p
CBTi ,BTi+1 (x, y) =
Φ
du
Ti+1 − Ti
0
where Φ is the cumulative normal distribution.
B
Cα,β
≡ CBTi ,BTi+1
α
X
a=1
i,prior
B
B
Jα,β
= Cα,β
− Cα−1,β
−
i
q (a),
β
X
2
!
q (b)
b=1
B
B
Cα,β−1 + Cα−1,β−1
i,prior
We remark that this so defined joint distribution (Jα,β
) has the right marginal distributions, but
unfortunately does not infer a martingale. That is why we need to minimize the Kullback-Leibler distance.
A numerical example is given in Appendix 2.
11
7
Generalization to the case of a deterministic dividend paying
stock in presence of deterministic interest rates
In this section we generalize the results of Section 4 by assuming that the stock pays continuously a deterministic dividend q and that the short interest rate is a deterministic
function r. By convention, we will add
R Ti
a call struck at 0 for each maturity Ti : K0i = 0 and C0i = S0 e− 0 qu du with S0 the initial stock price.
Proposition 7.1. If the following conditions are fulfilled
• ∀i ∈ [1, m], ∀j ∈ [0, ni ]
Cji > 0
• ∀i ∈ [1, m], ∀j1 , j2 s.t. 0 6 j1 < j2 6 ni
R Ti
Cji1 − Cji2
∈ [0, e− 0 ru du ]
i
i
Kj2 − Kj1
Cji1 − Cji2
> 0 if Cji1 > 0
Kji2 − Kji1
• ∀i1 , i2 ∈ [1, m] s.t. i1 6 i2 , ∀j1 ∈ [0, ni1 ], ∀j2 ∈ [0, ni2 ]
e
e
R Ti2
qu du
Cji22 > e
R Ti2
qu du
Cji22 > e
0
0
R Ti1
qu du
Cji11
if Kji11 = Kji22 e
−
R Ti2
(ru −qu )du
R Ti1
qu du
Cji11
if Kji11 > Kji22 e
−
R Ti2
(ru −qu )du
0
0
Ti
1
Ti
1
and Cji11 > 0
• ∀i, i1 , i2 ∈ [1, m] s.t. i 6 i1 and i 6 i2 , ∀j ∈ [0, ni ], ∀j1 ∈ [0, ni1 ], ∀j2 ∈ [0, ni2 ] s.t. e−
e−
R Ti
0
(ru −qu )du
e
e−
R Ti1
0
Kji < e−
R Ti
0
qu du
R Ti2
0
(ru −qu )du
Cji − e
(ru −qu )du K i1
j1
R Ti1
0
− e−
qu du
R Ti
0
R Ti1
0
(ru −qu )du
Kji11 <
Kji22
Cji11
(ru −qu )du K i
j
e
−
e−
R Ti
0
R Ti2
0
qu du
Cji22 − e
(ru −qu )du K i
j
− e−
R Ti
0
qu du
R Ti2
0
(ru −qu )du K i2
j2
then there exists a process (St )t>0 starting at the initial stock price, such that (St e−
martingale and further ∀i s.t. 1 6 i 6 m, ∀j s.t. 0 6 j 6 ni
Cji = E[e−
R Ti
0
ru du
Cji
Rt
0
>0
(ru −qu )du
)t>0 is a
(STi − Kji )+ ]
Proof. We will use the results we have proven in the case of a non-dividend paying stock where the interest
rates are null. First let us define ∀i s.t. 1 6 i 6 m, ∀j s.t. 0 6 j 6 ni
Kji ≡ Kji e−
Cji ≡ e
R Ti
0
R Ti
0
qu du
(ru −qu )du
Cji
We can see that Kji and Cji satisfy the hypothesis of Theorem 4.2. Consequently, there exists a continuous
time Markov martingale (Mt ) such that:
Cji = E[(MTi − Kji )+ ]
This can be written as:
Cji = E[e−
R Ti
0
ru du
(MTi e
R Ti
0
Rt
(ru −qu )du
− Kji )+ ]
Let us define the process (St )t>0 as follows: St ≡ Mt e 0 (ru −qu )du . The last property we have to verify is
that this process starts at S0 . The two processesR have the same starting value by definition and the process
Ti
(Mt )t>0 , which is a martingale, starts at C0i = e 0 qu du C0i = S0 .
12
The inequalities of Proposition 7.1 are again necessary for the absence of static arbitrage. Furthermore,
the algorithm described in Section 6 to obtain a realistic Markov chain model can also be generalized easily.
Indeed, it is possible to return to the no interest rate volatility and no dividend case by the following
one-to-one transformation,
Kji ≡ Kji e−
Cji ≡ e
R Ti
0
R Ti
0
qu du
(ru −qu )du
Cji
Then we can apply the algorithm previously described, and finally return to the general case by the inverse
of the above transformation.
8
Conclusion
The elimination of the well-known static strategies involving European calls are in general not sufficient to
prevent static arbitrage: the limitation to static strategies is not sufficient to make the notion of arbitrage
independent of the underlying ”statistical measure”. Nevertheless, we have proved that these necessary
conditions are sufficient for the existence of an arbitrage free model consistent with the call quotes. We
also focused on providing an algorithm explaining how to construct a realistic, arbitrage free, Markov chain
model. We used the notion of entropy as well as the notion of Brownian copula to choose realistic marginal
and joint distributions. An application of this model is for example the pricing of mildly path-dependent
claims, (e.g. globally floored, locally capped, compounding cliquet options), as explained in Carr and Cousot
(2004).
13
References
[1] Avellaneda, M., R. Buff, C. Friedman, N. Grandechamp, L. Krusk and J. Newman, 2000, “Weighted
Monte Carlo: A new technique for calibrating asset pricing models” International Journal of Theoretical
and Applied Finance, 4, 1, 91-119.
[2] Carr, P., H. Geman, D. Madan, M. Yor, ”Stochastic Volatility for Lévy processes”, Mathematical finance,
Vol.13, No.3 (2003), 345-382.
[3] Carr P., L. Cousot, ”Semi-Static Hedging of Path-Dependent Securities”, Preprint, Courant Institute,
New York University (2004).
[4] Carr, P., D. Madan, ”A Note on Sufficient Conditions for No Arbitrage”, Preprint, Courant Institute,
New York University (2004).
[5] Cover, T. and J. Thomas, 1991, Elements of Information Theory, Wiley, New York.
[6] Darsow, W., B. Nguyen and E. Olsen, 1992, “Copulas and Markov Processes”, Illinois Journal of
Mathematics, 36, 4, 600-642.
[7] Davis, M. ”The Range of Traded Option Prices”, Preprint, Imperial College London (2004).
[8] Hardy, J., J. Littlewood, G. Polya, ”Inequalities”, Cambridge University Press (1934).
[9] Kellerer, H., ”Markov Komposition und eine Anwendung of Martingale”, Math. Ann. 98, (1972) 99-122.
[10] Merton, R, ”Theory of rational option pricing”, Bell Journal of Economics and Management Science,
4 (1), (1973) 141-183.
[11] Sherman, S., ”On a theorem of Hardy, Littlewood Ploya and Blackwell”, Proc. Nat. Acad. Sci. USA 37
(1951) 826-831.
[12] Strassen, V., ”The existence of probability measures with given marginals”, Ann. Math. Stat. 36 (1965)
423-439.
14
Appendix 1: Construction of marginal distributions, a numerical
example
Let us consider the following situation: the current price of the non dividend paying stock is S0 = 100 and
the available call quotes are detailed in Table 1. (We assume that there are no interest rates.)
Maturity
3m
6m
1y
Strike
70
80
90
100
110
120
130
70
90
95
100
105
110
130
150
50
90
100
110
140
180
Implied Volatility
28.0169 %
25.1770 %
22.2610 %
19.2448%
16.6954 %
15.8362 %
15.9489 %
27.0431 %
21.5104 %
20.1062 %
18.7155 %
17.4310 %
16.4006 %
15.4949 %
14.4288 %
30.6808 %
20.4677 %
18.1727 %
16.2882 %
15.0552 %
16.1231 %
Table 4: Quotes, which satisfy the hypothesis of Proposition 4.2.
Instead of setting the slope of the segment [Aini , Aini +1 ] to M axSlope/2, we could have set it to α ×
M axSlope with 0 < α < 1 without changing the validity of the proof of Proposition 4.2. Here we choose
α = 0.999 to reduce the value of Kni i +1 .
α = 0.999
M axSlope = 10−5
n1 = 7
n2 = 8
n3 = 6
Kn11 +1 = 251.21212121
Kn22 +1 = 160.10101010
Kn33 +1 = 250.70707071
The (ki ) in this example are the minimal set of all the (Kji )16i6m,16j6ni +1 . In the following tables, the
i
i
approximations up to 10−8 of the distributions qC
and qC,new
are given.
15
kj
0
50
70
80
90
95
100
105
110
120
130
140
150
Kn22 +1
180
Kn33 +1
Kn22 +1
1
qC
(kj )
2.9000 × 10−4
0 × 10−8
1.487000 × 10−2
6.485000 × 10−2
2.0652000 × 10−1
0 × 10−8
3.8499000 × 10−1
0 × 10−8
2.7638000 × 10−1
4.907000 × 10−2
2.97500 × 10−3
0 × 10−8
4.510 × 10−5
9.90 × 10−6
0 × 10−8
0 × 10−8
0 × 10−8
1
qC,new
(kj )
9 × 10−8
1.01468 × 10−3
1.414523 × 10−2
6.485000 × 10−2
1.2573229 × 10−1
1.6157543 × 10−1
2.0761345 × 10−1
1.9317767 × 10−1
1.7979117 × 10−1
4.907000 × 10−2
2.91428 × 10−3
1.1154 × 10−4
4.08 × 10−6
1.0 × 10−7
0 × 10−8
0 × 10−8
0 × 10−8
1
1
Table 5: the distributions qC
and qC,new
.
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
50
100
0
150
0
50
100
150
1
1
Figure 3: Marginal distributions at 3 month maturity, qC
(left) and qC,new
(right).
16
kj
0
50
70
80
90
95
100
105
110
120
130
140
150
Kn22 +1
180
Kn33 +1
Kn22 +1
2
qC
(kj )
1.77200 × 10−3
3.43800 × 10−3
8.934500 × 10−2
0 × 10−8
1.6476500 × 10−1
1.1972000 × 10−1
1.5100000 × 10−1
1.6582000 × 10−1
2.3560500 × 10−1
0 × 10−8
6.681500 × 10−2
0 × 10−8
1.71010 × 10−5
9.90 × 10−6
0 × 10−8
0 × 10−8
0 × 10−8
2
qC,new
(kj )
1.5190 × 10−4
9.10834 × 10−3
4.683731 × 10−2
7.691490 × 10−2
1.2630755 × 10−1
1.1972000 × 10−1
1.5100000 × 10−1
1.6582000 × 10−1
1.9829393 × 10−1
7.462215 × 10−2
2.808188 × 10−2
2.84410 × 10−3
2.8805 × 10−4
9.90 × 10−6
0 × 10−8
0 × 10−8
0 × 10−8
2
2
Table 6: the distributions qC
and qC,new
.
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
50
100
0
150
0
50
100
150
2
2
Figure 4: Marginal distributions at 6 month maturity, qC
(left), qC,new
(right).
17
kj
0
50
70
80
90
95
100
105
110
120
130
140
150
Kn22 +1
180
Kn33 +1
Kn22 +1
3
qC
(kj )
1.77200 × 10−3
8.958550 × 10−2
0 × 10−8
0 × 10−8
2.5834250 × 10−1
0 × 10−8
2.2209600 × 10−1
0 × 10−8
3.3337433 × 10−1
0 × 10−8
0 × 10−8
9.364917 × 10−2
0 × 10−8
0 × 10−8
1.93760 × 10−3
9.90 × 10−6
0 × 10−8
3
qC,new
(kj )
1.77200 × 10−3
3.669584 × 10−2
6.375830 × 10−2
8.404205 × 10−2
1.1077879 × 10−1
1.0530604 × 10−1
1.0010365 × 10−1
1.3640667 × 10−1
1.8587513 × 10−1
9.514572 × 10−2
4.870317 × 10−2
2.493017 × 10−2
5.30849 × 10−3
1.11284 × 10−3
5.125 × 10−5
9.90 × 10−6
0 × 10−8
3
3
Table 7: the distributions qC
and qC,new
.
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
50
100
0
150
0
50
100
150
3
3
Figure 5: Marginal distributions at 1 year maturity, qC
(left), qC,new
(right).
18
We also tried with a thiner grid. The next distributions have been obtained with:
(ki ) = {0, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105,
110, 115, 120, 130, 135, 140, 145, 150, Kn22+1 , 180, Kn33+1 , Kn22 +1 }
kj
0
30
35
40
45
50
55
60
65
70
75
80
85
90
95
1
qC
(kj )
2.9000 × 10−4
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
1.487000 × 10−2
0 × 10−8
6.485000 × 10−2
0 × 10−8
2.0652000 × 10−1
0 × 10−8
1
qC,new
(kj )
6 × 10−8
1.20 × 10−6
2.01 × 10−6
4.34 × 10−6
1.605 × 10−5
6.429 × 10−5
1.9638 × 10−4
6.0568 × 10−4
1.87157 × 10−3
5.78398 × 10−3
1.322888 × 10−2
3.025651 × 10−2
5.595811 × 10−2
1.0350518 × 10−1
1.5007153 × 10−1
1
1
Table 8: the distributions qC
and qC,new
.
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
50
1
qC
(kj )
3.8499000 × 10−1
0 × 10−8
2.7638000 × 10−1
0 × 10−8
4.907000 × 10−2
0 × 10−8
2.97500 × 10−3
0 × 10−8
0 × 10−8
0 × 10−8
4.510 × 10−5
9.90 × 10−6
0 × 10−8
0 × 10−8
0 × 10−8
kj
100
105
110
115
120
125
130
135
140
145
150
Kn22 +1
180
Kn33 +1
Kn22 +1
100
0
150
0
50
100
1
qC,new
(kj )
2.1766617 × 10−1
1.8457612 × 10−1
1.5649467 × 10−1
5.519453 × 10−2
1.946539 × 10−2
4.01469 × 10−3
8.2857 × 10−4
1.5640 × 10−4
3.228 × 10−5
3.58 × 10−6
1.33 × 10−6
4.9 × 10−7
0 × 10−8
0 × 10−8
0 × 10−8
150
1
1
Figure 6: Marginal distributions at 3 month maturity, qC
(left) and qC,new
(right).
19
kj
0
30
35
40
45
50
55
60
65
70
75
80
85
90
95
2
qC
(kj )
1.77200 × 10−3
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
3.43800 × 10−3
0 × 10−8
0 × 10−8
0 × 10−8
8.934500 × 10−2
0 × 10−8
0 × 10−8
0 × 10−8
1.6476500 × 10−1
1.1972000 × 10−1
2
qC,new
(kj )
2.59 × 10−6
8.372 × 10−5
1.6621 × 10−4
3.2501 × 10−4
6.3851 × 10−4
1.25458 × 10−3
2.46148 × 10−3
4.83131 × 10−3
9.48246 × 10−3
1.860712 × 10−2
2.769184 × 10−2
4.120652 × 10−2
6.131944 × 10−2
9.124919 × 10−2
1.1972000 × 10−1
kj
100
105
110
115
120
125
130
135
140
145
150
Kn22 +1
180
Kn33 +1
Kn22 +1
2
2
Table 9: the distributions qC
and qC,new
.
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
50
2
qC
(kj )
1.5100000 × 10−1
1.6582000 × 10−1
2.3560500 × 10−1
0 × 10−8
0 × 10−8
0 × 10−8
6.681500 × 10−2
0 × 10−8
0 × 10−8
0 × 10−8
1.71010 × 10−5
9.90 × 10−6
0 × 10−8
0 × 10−8
0 × 10−8
100
0
150
0
50
2
qC,new
(kj )
1.5100000 × 10−1
1.6582000 × 10−1
1.5304694 × 10−1
7.739197 × 10−2
3.913371 × 10−2
1.978890 × 10−2
1.000554 × 10−2
3.25187 × 10−3
1.05665 × 10−3
3.4284 × 10−4
1.1167 × 10−4
9.90 × 10−6
0 × 10−8
0 × 10−8
0 × 10−8
100
150
2
2
Figure 7: Marginal distributions at 6 month maturity, qC
(left), qC,new
(right).
20
kj
0
30
35
40
45
50
55
60
65
70
75
80
85
90
95
3
qC
(kj )
1.77200 × 10−3
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
8.958550 × 10−2
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
2.5834250 × 10−1
0 × 10−8
3
qC,new
(kj )
3.196 × 10−5
8.5533 × 10−4
1.47926 × 10−3
2.55834 × 10−3
4.42459 × 10−3
7.65221 × 10−3
1.026585 × 10−2
1.377218 × 10−2
1.847611 × 10−2
2.478669 × 10−2
3.325264 × 10−2
4.461017 × 10−2
5.984692 × 10−2
8.028781 × 10−2
9.479990 × 10−2
3
qC
(kj )
2.2209600 × 10−1
0 × 10−8
3.3337433 × 10−1
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
0 × 10−8
9.364917 × 10−2
0 × 10−8
0 × 10−8
0 × 10−8
1.93760 × 10−3
9.90 × 10−6
0 × 10−8
kj
100
105
110
115
120
125
130
135
140
145
150
Kn22 +1
180
Kn33 +1
Kn22 +1
3
3
Table 10: the distributions qC
and qC,new
.
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
100
0
200
0
100
3
qC,new
(kj )
1.1193507 × 10−1
1.2324995 × 10−1
1.3570853 × 10−1
8.655494 × 10−2
5.520478 × 10−2
3.520961 × 10−2
2.245671 × 10−2
1.432289 × 10−2
9.13515 × 10−3
5.16800 × 10−3
2.92367 × 10−3
9.2500 × 10−4
9.585 × 10−5
9.90 × 10−6
0 × 10−8
200
3
3
Figure 8: Marginal distributions at 1 year maturity, qC
(left), qC,new
(right).
21
Appendix 2: Construction of a joint distribution, a numerical example
If the (ki ) are the minimal set of all the (Kji )16i6m,16j6ni +1 , we obtain:
0.07
0.06
0.08
0.05
0.07
0.06
0.04
0.05
0.04
0.03
0.03
0.02
0.02
200
0.01
150
0 200
0.01
100
150
100
50
50
0
Figure 9: Joint distribution of (ST1 , ST2 ).
0.9
0.8
1
0.8
0.7
0.6
0.6
0.4
0.5
0.2
0
0.4
200
0.3
150
0.2
100
200
150
50
100
50
0
Figure 10: The corresponding transition matrix.
22
0.1
The following joint distribution and the corresponding transition matrix have been obtained with:
(ki ) = {0, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105,
110, 115, 120, 130, 135, 140, 145, 150, Kn22+1 , 180, Kn33+1 , Kn22 +1 }
0.045
0.04
0.035
0.05
0.03
0.04
0.025
0.02
0.03
0
0.015
0.02
50
0.01
0.01
100
0.005
150
0
0
20
40
60
80
100
120
140
160
200
180
Figure 11: Joint distribution of (ST1 , ST2 ).
0.9
0.8
0.7
1
0.6
0.8
0.5
0.6
0.4
0.4
0
0.2
50
0
0.3
0.2
100
0
0.1
50
150
100
150
200
Figure 12: The corresponding transition matrix.
23