and the expected rate of return = EP (R)
; r = EP (R) ; EQ(R) = ;CovP (z R)
since EQ(R) = r: If z is attainable, i.e. z = PN=1 ndn, then, n
(1.21)
z=(
N X n=1
n pn )(1 + Rz )
where Rz is the rate of return of portfolio z. We have
EP (Rz ) ; r = ;(
and The
Conversely, Suppose that none of the associated one period models admits arbitrage but the price process p(t) t = 0 1 T admits arbitrage. Let (t) t = 0 1 T ;1 be an arbitrage strategy. Thus,
c(t) =
N X n=1
( n(t ; 1) ; n (t)pn(t) 0
for t = 0 1 T and c(s)
3.2 RiskNeutral Probability Measures
We now always assume that the price process p(t) t = 0 1 T does not admit arbitrage. We will show in this section that under the noarbitrage assumption, there is a probability measure on ( F ) such that the present v
where Q(s Fti) = Q(s !) for ! 2 Fti and s t. It is well de ned since Q(s) s = 1 all are Ftmeasurable. Proceeding in this manner yields
J X j =1
t
(3.5)
Q(!j ) =
=
m0 X k=1
Q(1 F1k ) = 1:
T XY
Furthermore, for any Fti 2 Ft
Q(Fti) =
=
X
!j 2Fti t Y s=1
Q(!
From we have
EQ(an(t) jFt;1) = an(t ; 1) D0(Ftj;1 )Q(Ftj;1 ) = (1 + r)p(Ftj;1):
Thus, none of the one period models admits arbitrage, neither does the multiperiod model. The probability measure Q under which the present value processes are martingales is
Since any consumption process c = (c(t) t = 0 1 2 trading strategy (t) t = 0 1 2 T ; 1 such that
T ), is attainable, we have a
(3.15)
c(t) =
for t = 1 2
N X
T . Noarbitrage implies that the price of this consumption process is N X (c) = c(0) + (3.16) n (
TN X X h EQ( n(t ; 1)pn(t ; 1) EQ( n (t + 1)pn(t) i EQ(a) = c(0) + ; (1 + r)t;1 (1 + r)t t=1 n=1 N X = c(0) + (3.18) n (1)pn (0) = (c): n=1
In particular, if we let
8 > 1 ! 2 Fti < i (! ) = Ft > 0 otherwise :
(3.19)
be the ArrowDebreu security which pays
regular intervals. The stock price S (t) follows the random walk model described in Section 2.2. Hence, PrfS (t) = uS (t ; ) jS (t ; )g = q PrfS (t) = dS (t ; ) jS (t ; )g = 1 ; q 0 < q < 1 (3.21) for t = T : The interest rate of the riskless bond at each
F01
QQ QQ Q
: 1 XXXX z X 2 F11 F2 3PPPPP : PPP F23 q PXXX XXX z
F21 X
QQ
Q F12 s QPP
1F2 PPP
4

: PPP F25 q PXXX XXX z
Figure 2.1: Tree structure of a lattice model Thus, ij ; ij;1 is the number of sets in Ft split from Ftj;1 . Summarizing the discussio
(t) such that
c(t) =
N X
for t = 1 2 T . A market is complete if every consumption process is attainable. A self nancing trading strategy is a trading strategy such that
N X
n=1
( n (t ; 1) ; n (t)pn(t)
(2.20)
for t = 1 2 T ; 1. In other words, under a s
observe from X are in the algebra generated by events fX xg for all real numbers x. We call the algebra generated by fX xg x 2 R the Borel algebra with respect to X and denote it as BX . Hence, BX represents all the information that can be obtained fro
Thus, Bt t = 0 1 form a ltration on ( F P ), called the Borel or natural ltration with respect to X (t) t = 0 1 : This ltration contains exact information obtained from X (t) t = 0 1 and Bt is the exact information obtained from X (t) t = 0 1 up to time t
can be used to model securities. Let Y1 Y2 Yk be a sequence of independent, identically distributed(iid) Bernoulli random variables de ned on a probability space ( F P ). First, let us assume that for a given h > 0, 8 >h < Yk = > (2.1) : ;h and Pr(Yk = h)
m=0 1 t. We can also easily compute its mean and variance. E (X (t) = tE (Y1) = 0 2 V ar(X (t) = tV ar(Y1) = th2 = t h :
Moreover, there is a recursive relation among P (x t) t = 0 . It follows from (2.5) (2.6)
P (x t + ) = Pr(Xt = x ; Yt+1) = E (Pr(Xt =
1. Adjust the probabilities of the up movement and the down movement. Let Pr(Yk = h) = q Pr(Yk = ;h) = 1 ; q: To satisfy equations in (2.9), we must have
h(2q ; 1) =
which yields
4h2 q(1 ; q) =
2
1h h = 2 + 2 2 q = 2 1 + 1 + 1= 2 The corresponding recursi
are only two states of economy over the next time interval: the upstate and the downstate. The probablities of the upstate and the downstate are q and 1 ; q, respectively. The return of the security over the next time interval is u when the upstate is att
There are T + 1 consumption dates separated in regular intervals. Without loss of generality, we assume these dates are t = 0 1 T . Tradings take place only at t=0 1 T ; 1. There are a nite number of states of economy = f!1 !2 with the probability at stat
Since at time t the securities are priced based on the information available up to time t, the price process (p(0) p(1) p(T ) is a stochastic process on ( F Ft P ). We further assume that one of these securities, say, the rst security, is a riskfree bond
01 B T C usdT ;sqs (1 ; qu )T ;s = (1 + r);T S (0) @A u log(K=S (0);T log d s s log(u=d) 01 X B T C qs (1 ; qu)T ;s: ; (1 + r);T K @ Au log(K=S (0);T log d s s log(u=d) X
Let Then, from (3.22)
qu = 1uqur : + qd = (1 ; qu) = 1dqdr : +
(3.24) (3.25)
The pri
This formula is the well known option pricing formula of Ross, Cox and Rubinstein 5]. The delta is B (d T qd). Other Greeks can be calculated easily. It also reveals that to replicate a European call, the strategy is to form a portfolio long in stock and
process e = (e(0) e(T ) and the price system p. Mathematically, a budget set is an a ne space of RJ +1. A consumption process is said to be attainable if its terminal consumption can be expressed as the payo of a portfolio, i.e.
c(T ) =
N X
It is easy to
Similarly, the payo of a European put option with strike price K then is maxfK ; d 0g:
Example 1.1 Consider two securities with payo d1 = (1 2 4)0 d2 = (2 0 1)0 respec
tively. Since the number of the states is 3 and the number of securities is 2, the mar
d(A B ) is the distance between A and B de ned by d(A B ) = inf fkx ; yk for any x 2 A and y 2 B g:
Then, there exists a z 2 H and a scalar h such that for any x 2 A x z > h, and for any y 2 B y z < h: See Appendix B for a proof. It is easy to see that th
Summerizing the above arguments, we conclude that
Theorem 1.2 The price system does not admit arbitrage if and only if there is a positive
vector such that
D0 = p:
(1.6)
Let us now consider the case that one of these securities is a riskless bond, say the
1.4 Valuation
We now denote the time0 price of a consumption process c = (c(0) c(T ) by (c). Then, no arbitrage implies that for any attainable comsumption process with c(T ) = PN=1 ndn, n (c) = c(0) +
N X n pn :
This formula itself is trivial but it rep
For each j ,
= 1 EQ( j ) = 1 Q(!j ): (1.11) 1+r 1+r In other words, the riskneutral probability for each state is actually the accumulated value of the corresponding state price at the riskfree rate.
j
Example 1.2 Consider an economy with only two states
On the other hand, suppose that a portfolio S + B , where is the number of shares of the stock and B is the bond value, gives the same payo as (Cu Cd)0. We then have
S u + B (1 + r) = Cu S d + B (1 + r) = Cd
(1.13) (1.14)
C uCd ;dC Thus, = S(uu;Cdd) B = (
Thus, they together are also su cient conditions for a consistent noarbitrage price system for all consumption processes. The correspondance ! Q is an onetoone correspondance since a linear functional is uniquely determined by its values on a basis whi
There are N primitive securities. The nth security has price pn at time 0 and terminal payo 0 1 B dn(!1) C B B dn(!2) C C B B.C dn = B . C B.C C B C @ A dn(!J ) Thus, we have a price system
p = (p1 p2
pN ) 0
where 0 denotes the corresponding transpose, a