No Arbitrage Criteria
for Exponential Lévy Models
A.V. Selivanov
Moscow State University
Financial Mathematics
3 ”columns” (Z. Bodie, R.C. Merton ”Finance”):
Pricing
Resource
Allocation
Utility
maximization
Equilibrium
Arbitrage
Pricing Theory
Risk
Management
Concept of No Arbitrage

Practical:

Mathematical:
air
A  L0  {0}, A – set of incomes
No Free Lunch (NFL)
(J.M. Harrison, D.M. Kreps 1979)
No Arbitrage condition
in discrete time
(, F ,P), ( S n ) nN0 – any sequence, Fn   ( S k , k  n)
M  {Q ~ P : S is an ( Fn , Q) - martingale }
Fundamental Theorem of Asset Pricing:
NFL  M  
• J.M. Harrison, S.R. Pliska 1981 – finite 
• R.C. Dalang, A. Morton, W. Willinger 1990 –
general case
No Arbitrage conditions
in continuous time

No Free Lunch with Vanishing Risk
(NFLVR)
F. Delbaen, W. Schachermayer 1994

No Generalized Arbitrage (NGA)
A.S. Cherny 2004
Definition of sigma-martingales
The definition is given by T. Goll and J. Kallsen
A semimartingale M is a sigma-martingale if
there exist predictable sets ( Dn )nsuch
that
1
• Dn  Dn 1 ,
•



Dn    [0, T ] or  [0, )
n 1
I Dn dSt is a uniformly integrable
0
martingale for any n
Sigma-martingales and
local martingales
sigma-martingales
local martingales
positive
sigma-martingales
Classes of Martingale Measures
(, F ,P), S – any process, Ft   ( Su , u  t ), t  0
M   {Q ~ P : S is an ( Ft , Q) -  - martingale }
M  {Q ~ P : S is an ( Ft , Q) - martingale }
UM  {Q ~ P : S is an ( Ft , Q) - uniformly
integrable martingale}
Fundamental Theorem
of Asset Pricing
no arbitrage condition
finite time
horizon
infinite time
horizon
M  
M  
M 
UM  
F. Delbaen, W. Schachermayer
NFLVR 
S – semimartingale
A.S. Cherny
NGA 
S – any positive process
absence
of arbitrage
existence of certain
martingale measure
completeness
of the model
uniqueness
of the measure
Models under consideration

exponential Lévy model:
St  e

Lt
(1)
time-changed exponential Lévy model:
St  e
( L  )t
e
L t
(2)
L – nonzero Lévy process
 – independent increasing non-constant process
Black-Scholes and Merton models

Black-Scholes model
 Bt   t
St  e

B – Brownian motion,  ,   0
Merton model
 t   t
St  e
 – Poisson process,
,   0
Theorem for models
with finite time horizon
statement
M
NFLVR
  
MNGA
 
Let M   . Then
| M |  1 
| M | 1 
condition for
model (1)
condition for
model (2)
process S is not monotone
Black-Scholes or
Black-Scholes or Merton model;
Merton model
 is deterministic
and continuous
Theorem for model (1)
with infinite time horizon
statement
condition for model (1)
M   
NFLVR
either the process S is a P-martingale,
or E S1  1 and the jumps of S are not
bounded from above
UM GA

always
Let M   . Then
| M |  1 
Black-Scholes or Merton model
Theorem for model (2)
with infinite time horizon
Suppose that     P-a.s.
Then always
UM  GA.
.
An example: NFLVR and GA
St  e
Bt  t / 2
, t0
NFLVR is satisfied;
NGA is not satisfied
Strategy:
H t  I(t   )
X   0.5
  inf{t  0 : St  0.5}
Conclusions
We have obtained:
 the criteria for the NFLVR and the NGA
conditions for models with finite time horizon;
for these models
NFLVR  NGA

the criteria for the NFLVR and the NGA
conditions for models without time change and
with infinite time horizon; for these models the
NGA is never satisfied, while the NFLVR is
satisfied in certain cases