CHAPTER 4
Multi-Period Model
Assumption.
(1) Trading time t = 0, 1, 2, ..., T ;
(2) The situation at time t = T : sample space Ω;
(3) Information at time t: Ft , for t = 0, 1, 2, ..., T ;
(4) The price is adapted to (Ft ), i.e., the price at time t is Ft -measurable.
dddddddddd: t = 0, 1, 2, 3, Ω = {ω1 , ω2 , ..., ω8 }. The price is given in
Figure 4.1.
w1
w2
w3
w4
w5
w6
w7
t=0
t=1
t=2
Figure 4.1
91
t=3
w8
92
4. MULTI-PERIOD MODEL
The filtration in this model
F0 = {∅, Ω},
F1 = σ({ω1 , ω2 , ω3 , ω4 }, {ω5 , ω6 , ω7 , ω8 }),
F2 = σ({ω1 , ω2 }, {ω3 , ω4 }, {ω5 , ω6 }, {ω7 , ω8 }),
F3 = σ({ω1 }, {ω2 }, {ω3 }, {ω4 }, {ω5 }, {ω6 }, {ω7 }, {ω8 }).
4.1. The market model
Definition 4.1. A market model is a pair (S̄, F), where F = (Ft )t=0,1,...,T is a filtration
1
N T
and
⎛ S̄ ⎞= (S̄t )t=0,1,...,T is an (Ft )-adapted stochastic process, S̄t = (Bt , St , ..., St ) =
⎜ Bt ⎟
⎠ is the security price at time t.
⎝
St
Definition 4.2. A stochastic process (Xt )t=0,1,...,T is called previsible (or predictable)
if X0 is F0 -measurable and Xt is Ft−1 -measurable for all t = 1, 2, ..., T .
T
N +1
-valued
Definition 4.3. A trading strategy (h̄t ) = ((h0t , h1t , ..., hN
t ) )t=0,1,...,T is an R
predictable process, where hit is the number of the shares of ith asset between the time
t − 1 (after the trading) and the time t (before the trading).
Figure 4.2 dddddddd S̄t , dddd h̄t ddd t ddddd. dddddd
ddddddd trading strategy h̄ dddddd t d total value.
Remark 4.4. The total value of the portfolio h̄ at time t − 1 (after the trading) is
given by
h̄t · S̄t−1 =
h0t Bt−1
+
N
i=1
i
hit St−1
.
4.1. THE MARKET MODEL
_
_
_
h0 h1
_
h2
_
h3
1
2
3
S0
S1
S2
S3
_
_
_
_
ht
0
t-1
_
_
St-1
93
_
ht+1
t
t+1
St
St+1
_
hT
T-1
_
_
T_
ST-1 ST
Figure 4.2. Strategy strategies, stock prices and trading period
At time t, before the trading, the value of the portfolio h̄ has changed to
h̄t · S̄t =
h0t Bt
+
N
hit Sti .
i=1
Definition 4.5. A trading strategy h̄ is called self-financing if
h̄t · S̄t = h̄t+1 · S̄t ,
for t = 0, 1, 2, ..., T − 1.
Self-financing dddddddddddd, ddddddddddd.
Remark 4.6.
(1) If h̄ is self-financing, then
h̄t+1 · S̄t+1 − h̄t · S̄t = h̄t+1 · (S̄t+1 − S̄t ).
This implies that the losses and gains resulting from the asset price fluctuations
are the only source of variations of the portfolio values.
(2) If h̄ is self-financing, then the value of the portfolio h̄ at time t is given by
h̄t · S̄t =
t
+
h̄k · (S̄k − S̄k−1 ),
h̄0 · S̄0
k=1
initial endowment
for t = 0, 1, ..., T.
Remark 4.7. Often it is assumed that Bt plays the role of (locally) riskless asset. We
may take
B0 = 1
and
spot rate (dddd) rt ≥ 0.
94
4. MULTI-PERIOD MODEL
Thus,
Bt =
t
(1 + rk ).
k=1
This means that Bt > 0 P-a.s. for all t.
Notation 4.8. The assumption “Bt > 0 for all t” allows us the use B as a “numéraire”
(dd). Define the discounted price process
Xt0 =
Bt
= 1,
Bt
Sti
,
Bt
⎛
0
⎜ Xt
⎜
⎜ 1
⎜ Xt
= ⎜
⎜ .
⎜ ..
⎜
⎝
XtN
Xti =
X̄t
Definition 4.9.
t = 0, 1, ..., T, i = 1, 2, ..., N,
⎞
⎟ ⎛
⎞
⎟
⎟
⎟ ⎜ 1 ⎟
⎟=⎝
⎠.
⎟
⎟
Xt
⎟
⎠
(1) The discounted value process V = (Vt )t=0,1,...,T associated
with a trading strategy h̄ is given by
Vt (h̄) := h̄t · X̄t ,
for t = 0, 1, 2, ..., T.
(2) The gain process G = (Gt )t=0,1,...,T associated with h̄ is given by
G0 := 0,
Gt :=
t
h̄k · (X̄k − X̄k−1 ),
for t = 1, 2, ..., T.
k=1
Remark 4.10. By Notation 4.8,
Vt = h̄t · X̄t =
h̄t · S̄t
,
Bt
This implies that Vt is the portfolio value at the end of the tth trading period expressed
in units of the numéraire asset.
4.1. THE MARKET MODEL
95
Proposition 4.11. For a trading strategy h̄, the following conditions are equivalent.
(1) h̄ is self-financing;
(2) h̄t · X̄t = h̄t+1 · X̄t , for t = 0, 1, 2, ..., T − 1;
t
(3) Vt = V0 + Gt = h̄0 · X̄0 +
hk · (Xk − Xk−1 ) for all t.
k=1
Proof. “(1) ⇐⇒ (2)”: h̄ is self-financing, by Definition 4.5, if and only if
h̄t · S̄t = h̄t+1 · S̄t ,
for all t.
This is equivalent to
h̄t · X̄t = h̄t ·
=
1
S̄t
=
(h̄t · S̄t )
Bt
Bt
1
(h̄t+1 · S̄t ) = h̄t+1 · X̄t .
Bt
“(2) ⇐⇒ (3)”: The equivalent relation holds due to the equation
Vt+1 − Vt = h̄t+1 · X̄t+1 − h̄t · X̄t = h̄t+1 · (X̄t+1 − X̄t )
= ht+1 · (Xt+1 − Xt )
for all t = 0, 1, ..., T − 1. Taking summation, we get the desired result.
Remark 4.12. By Proposition 4.11(2), we know that if h̄ is self-financing, the trading
strategy of the bond is given by
h0t+1 − h0t = −(ht+1 − ht ) · Xt ,
for t = 0, 1, 2, ..., T − 1.
96
4. MULTI-PERIOD MODEL
4.2. Arbitrage opportunities
Definition 4.13.
(1) A self-financing trading strategy is called an arbitrage opportunity
if its value process satisfies
(i) V0 ≤ 0,
(ii) VT ≥ 0 P-a.s., and P(VT > 0) > 0.
(2) A market model does not allow for arbitrage opportunities is called arbitrage-free.
dddddddd, dddd one-period model dd arbitrage opportunity ddd
dd. dddddddddddd terminal time T ddd. ddddddd t d? d
dddddd t ddd t ddddddd? dddddddddddd arbitrage
opportunity ddddddddddd t ddddd one-period model dd arbitrage
opportunity dd.
Proposition 4.14. The market model admits an arbitrage opportunity if and only if
there exists t ∈ {1, 2, ..., T } and a trading strategy h ∈ Ft−1 such that
h · (Xt − Xt−1 ) ≥ 0,
P − a.s.
(4.1)
and
P(h · (Xt − Xt−1 ) > 0) > 0.
(4.2)
Proof. “=⇒”: Suppose that the market model admits an arbitrage opportunity, then
there exists an arbitrage opportunity k̄ = (k0 , k) with value function V . Let
t := min{k : Vk ≥ 0 P − a.s. and P(Vk > 0) > 0}.
Then t ≤ T . By assumption, we know that either Vt−1 = 0 P-a.s or P(Vt−1 < 0) > 0.
4.2. ARBITRAGE OPPORTUNITIES
97
(i) Vt−1 = 0, P-a.s: Since
Vt = Vt − Vt−1 = kt · (Xt − Xt−1 )
P − a.s.
due to Proposition 4.11(3). Let h = kt , by the definition of t
h · (Xt − Xt−1 ) ≥ 0
P − a.s.
and
P(h · (Xt − Xt−1 ) > 0) > 0.
(ii) P(Vt−1 < 0) > 0: Let
h = kt I{Vt−1 <0} .
Then h is Ft−1 -measurable, and
h · (Xt − Xt−1 ) = kt · (Xt − Xt−1 )I{Vt−1 <0} = (Vt − Vt−1 )I{Vt−1 <0}
≥ −Vt−1 I{Vt−1 <0} .
Clearly, h · (Xt − Xt−1 ) satisfies both of (4.1) and (4.2).
“⇐=”: For t and h defined as in (4.1) and (4.2), define an N -dimensional predictable
process (ks ) by
ks :=
⎧
⎪
⎨ k,
if s ≤ t,
⎪
⎩ 0,
otherwise.
Define ks0 by Remark 4.12. Then k̄ is uniquely determined as a self-financing trading
strategy. Since
VT = h · (Xt − Xt−1 )
by Proposition 4.11(3), we see that k̄ is an arbitrage opportunity.
98
4. MULTI-PERIOD MODEL
B1(w1)=B1(w2)=1.1
S1(w1)=S1(w2)=12
B2(w1)=1.2
S2(w1)=14
B2(w2)=1.2
S2(w2)=8
B2(w3)=1.2
S2(w3)=12
B0(wi)=1
S0(wi)=10
for all i
B1(w3)=B1(w4)=1.1
S1(w3)=S1(w4)=10
B2(w4)=1.2
S2(w4)=9
Figure 4.3
Example 4.15. Let Ω = {ω1 , ω2 , ω3 , ω4 } and t = 0, 1, 2. The price at each time and
each states is given in Figure 4.3.
Then this market model is arbitrage-free.
ddddddddddd. dddddddd no arbitrage, ddddddd no
arbitrage. ddddddddd arbitrage opportunities, ddd model dd arbitrage
opportunity.
Exercise
(1) (24 points) Consider a market model given as in Figure 4.4. Show that this market
model is not arbitrage-free and find an arbitrage opportunity in this model.
4.3. Martingale measures
Definition 4.16.
(1) A probability measure Q on (Ω, F) is called a martingale
measure if the discounted price process X is a (N -dimensional) Q-martingale,
4.3. MARTINGALE MEASURES
99
(2, 30)
(1,14)
(2,0)
(2, 8)
(1, 12)
(1,10)
(2, 16)
(1,10)
(2, 60)
(1, 12)
(2,12)
(2, 12)
(1,8)
(1,6)
(2,4)
Figure 4.4. Market model in Exercise (1)
i.e., for 0 ≤ s < t ≤ T , i = 1, 2, ..., N ,
EQ [Xti ] < ∞,
and
Xsi = EQ [Xti |Fs ].
(2) A martingale measure P∗ is called an equivalent martingale measure if P∗ ∼ P.
(3) Denote P the collection of all equivalent martingale measures.
Example 4.17. As shown in Example 4.15. The probability at each period is given
in Figure 4.5.
100
4. MULTI-PERIOD MODEL
28/33
1/2
5/33
7/11
1/2
4/11
Figure 4.5
ddddd equivalent martingale measures ddddd (dd Chapter 3 dddd
d), ddddddd. Then the equivalent martingale measure Q is given by
1
2
1
Q({ω2 }) =
2
1
Q({ω3 }) =
2
1
Q({ω4 }) =
2
Q({ω1 }) =
28
33
5
·
33
7
·
11
4
·
11
·
14
,
33
5
= ,
66
7
= ,
22
2
= .
11
=
dddd martingale ddd, ddddd check.
Theorem 4.18. The following conditions are equivalent.
(1) Q is an equivalent martingale measure.
(2) If h̄ = (h0 , hT )T is self-financing and h is bounded, then the value process V of h̄
is a Q-martingale.
(3) If h̄ is self-financing and its value process V satisfies EQ [VT− ] < ∞, then V is a
Q-martingale.
4.3. MARTINGALE MEASURES
101
(4) If h̄ is self-financing and its value process V satisfies VT ≥ 0 Q-a.s., then
EQ [VT ] = V0 .
Proof. (1) =⇒ (2): Let V be the value process of a self-financing trading strategy
h̄ = (h0 , hT )T such that |hk | ≤ C for all k. By Proposition 4.11(3)
t
|Vt | = V0 +
hk (Xk − Xk−1 )
k=1
≤ |V0 | +
t
|hk |(|Xk | + |Xk−1 |) ≤ |V0 | + C
k=1
t
(|Xk | + |Xk−1 |)
k=1
Thus, EQ |Vt | < ∞ for all t, since EQ |Xt | < ∞ for all t. Moreover, for 0 ≤ t ≤ T − 1,
EQ [Vt+1 |Ft ] = EQ [Vt + ht+1 · (Xt+1 − Xt )|Ft ]
= Vt + ht+1 · EQ [Xt+1 − Xt |Ft ]
= Vt + ht+1 · 0 = Vt .
This means that (Vt ) is a Q-martingale.
ddddddd Föllmer and Schied [12] Chapter 4.
dddddd h is bounded, EQ [VT− ] < ∞, d VT ≥ 0 Q-a.s. ddddd? dddd
ddddd double strategies.
Remark 4.19. Consider a gambling. Interest rate r = 0. dddd, ddd $1, dd
ddddddd, $2, dddddddddd (d Figure 4.6).
Trading strategy h00 = −1 (ddddddd $1). dddddddd. ddddd, d
ddd. dddddddddd $2. dddddddd. ddddd, dddd. dd
dddddddd $4. dddddd. ddddd (d Figure 4.7). dddddddd
double strategy.
102
4. MULTI-PERIOD MODEL
2
1/2
1
1/2
0
Figure 4.6. Payoff
h0=1
stop!
win
win
loss
h1=2
loss
stop!
win
stop!
win
h2=4
loss
h3=8
stop!
win
loss
h4=16
stop!
loss
Figure 4.7. The trading strategies
Then V∞ = 1. This is an arbitrage opportunity. dddddddddddd. ddd
dddddddddd double strategy.
Theorem 4.20 (Fundamental Theorem of Asset Pricing). The market model is arbitragefree if and only if the set P of all equivalent martingale measures is non-empty.
Proof. dddddddddddd.
“⇐=” Suppose that there exists an equivalent martingale measure Q. Then by Theorem
4.4. ARBITRAGE-FREE PRICES FOR EUROPEAN CONTINGENT CLAIMS
103
4.18, the value process of any self-financing strategy with VT ≥ 0 satisfies
EQ [VT ] = V0 .
If Q(VT > 0) > 0, then EQ [VT ] > 0. Thus, V0 > 0. This implies that there is no arbitrage
opportunities, i.e., the market model is arbitrage-free.
dddddddddddddd, ddddddddddddd, ddddddddd
dd.
4.4. Arbitrage-free prices for European contingent claims
Definition 4.21.
(1) A non-negative random variable C on (Ω, F, P) is called a
European contingent claim.
(2) A European contingent claim C is called a derivative of the underlying assets
B, S 1 , ..., S N if C is measurable with respect to the σ-algebra generated by the
price process (S̄t )t=0,1,...,T .
(3) T is called the expiration date or maturity of C.
(4) The discounted value of C is given by
H :=
C
.
BT
H is called the discounted (European) claim associated with C.
From now on, we assume that our market model is arbitrage-free, i.e.,
P = ∅.
104
4. MULTI-PERIOD MODEL
Definition 4.22. A contingent claim C is called attainable (replicable, or redundant)
if there exists a self-financing strategy h̄ such that
C = h̄T · S̄T ,
P − a.s.
h̄ is called a replicating strategy for C.
Remark 4.23. C is attainable if and only if the corresponding discounted claim
H = h̄T · X̄T = VT (h̄) = V0 +
T
ht · (Xt − Xt−1 ).
t=1
We call h̄ a replicating strategy for H and H is attainable.
Theorem 4.24.
(1) Any attainable discounted claim H is integrable with respect
to each equivalent martingale measures, i.e.,
E∗ [H] < ∞,
for all P∗ ∈ P.
(2) For each P∗ ∈ P, the value process of any replicating strategy satisfies
Vt = E∗ [H|Ft ],
P∗ − a.s.
for all t = 0, 1, 2, ..., T .
(3) (Vt )0≤t≤T is a non-negative P∗ -martingale.
dddddddddd. ddddddddddd, dddddddddd. d
Theorem 4.18 dd, dd C ≥ 0, H ≥ 0. ddd VT ≥ 0, d value process (Vt ) ddd
P∗ -martingale. dddddddddd one-period model ddddd.
Definition 4.25.
(1) A real number π H ≥ 0 is called an arbitrage-free price of
a discounted claim H, if there exists an adapted stochastic process X N +1 such
that
4.4. ARBITRAGE-FREE PRICES FOR EUROPEAN CONTINGENT CLAIMS
105
(i) X0N +1 = π H ,
XtN +1 ≥ 0 for all t = 0, 1, 2, ..., T ,
XTN +1 = H;
(ii) the enlarged market model with discounted price process (X 0 , X 1 , ..., X N , X N +1 )
is arbitrage-free.
(2) The set of all arbitrage-free prices of H = Π(H). Moreover, we denote
πinf (H) = inf Π(H)
and
πsup (H) = sup Π(H).
Remark 4.26. The (undiscounted) arbitrage-free price of the contingent claim C :=
BT H is given by
π C := B0 π H .
(4.3)
(4.3) dddddddddddd numéraire ddddd. H = C/BT dddd
d T dddddd numéraire ddd. ddddd 0 dddd π H dddddd
π H = π C /B0 .
Theorem 4.27. The set
Π(H) = {E∗ [H] : P∗ ∈ P and E∗ [H] < ∞}
and in case Π(H) = ∅, we have
πinf (H) = inf
E∗ [H]
∗
P ∈P
Proof.
and
πsup (H) = sup E∗ [H].
P∗ ∈P
(1) By Theorem 4.20, π H is an arbitrage-free price of and only if there
exists an equivalent martingale measure P̃ for the extended market model. This
implies that
Xti = Ẽ[XTi |Ft ],
106
4. MULTI-PERIOD MODEL
for t = 0, 1, 2, ..., T and i = 1, 2, ..., N + 1. In particular, P̃ ∈ P and satisfies
π H = Ẽ[H]. Thus,
Π(H) ⊆ {E∗ [H] : P∗ ∈ P and E∗ [H] < ∞}.
(2) If π H = E∗ [H] for some P∗ ∈ P, define
XtN +1 := E∗ [H|Ft ],
t = 0, 1, 2, ..., T,
then XtN +1 satisfies the condition (i) in Definition 4.25. Moreover, (XtN +1 )t=0,...,T
is a martingale under P∗ . Thus, π H is an arbitrage-free price, which means that
π H ∈ Π(H). Hence,
Π(H) ⊇ {E∗ [H] : P∗ ∈ P and E∗ [H] < ∞}.
Example 4.28.
(1) Consider a European call option
C call = (ST − K)+
with strike price K and maturity T . Suppose Bt is increasing and satisfies B0 = 1.
Then for all P∗ ∈ P,
+ call
C
1
S
K
T
= E∗
(ST − K)+ = E ∗
−
= E∗
BT
BT
BT
BT
+ +
K
K
∗
∗
XT −
≥ E XT −
= E
BT
BT
+
K
∗
=
S0 − E
≥ (S0 − K)+ ,
BT
π call
4.4. ARBITRAGE-FREE PRICES FOR EUROPEAN CONTINGENT CLAIMS
107
due to Jensen’s inequality, BT ≥ 1 and the fact that (Xt ) is a P∗ -martingale.
Thus, we have
π call ≥ (S0 − K)+ .
(4.4)
ddddddddd: dd option dddddddddddddddd
option dddddd.
call price
+
(X-K)
time value
K
S0
Figure 4.8
(2) Consider a European put option
C put = (K − ST )+ .
However, (4.4) fails in general in this case unless Bt = constant for all t.
d d d d d d d d d d d d d d d arbitrage-free prices. d d d d d d d d d
one-period model ddd. dddd, ddddddddd.
Theorem 4.29. Let H be a discounted claim.
(1) If H is attainable, then #(Π(H)) = 1.
108
4. MULTI-PERIOD MODEL
+
(K-X)
K
S0
Figure 4.9
(2) If H is not attainable, then either Π(H) = ∅ or
Π(H) = (πinf (H), πsup (H)) .
Similar to Chapter 3, we have the following definition and results.
Definition 4.30. A (arbitrage-free) market model is called complete if every contingent claim is attainable.
Theorem 4.31. An arbitrage-free market model is complete if and only if #(P) = 1.
Exercise
(1) Consider a market model given as in Figure 5.9.
(a) If the sample space Ω = {ωi : i = 1, 2, ..., 8}. Describe the filtration in this
market model.
(b) Show that this market model is arbitrage-free and complete. Moreover, find
the equivalent martingale measure in this model.
4.4. ARBITRAGE-FREE PRICES FOR EUROPEAN CONTINGENT CLAIMS
109
(1,26)
(1,14)
(1,2)
(1, 14)
(1, 12)
(1,10)
(1, 8)
(1,10)
(1, 16)
(1, 12)
(1,0)
(1, 12)
(1,9)
(1,6)
(1,4)
Figure 4.10. Market model in Exercise (1)
(c) Find the arbitrage-free price of a call option at time 0 with maturity T = 3
and strike price K = 11.
(d) Find the arbitrage-free price of a put option at time 0 with maturity T = 3
and strike price K = 11.
(e) Find the arbitrage-free price of a call option at time 1 with maturity T = 3
and strike price K = 11.
(2) (Consider a market model given as in Figure 4.11.
(a) If the sample space Ω = {ωi : i = 1, 2, ..., 9}. Describe the filtration in this
market model.
(b) Show that this market model is arbitrage-free . Moreover, find the collection
of all equivalent martingale measures in this market model.
110
4. MULTI-PERIOD MODEL
(1,20)
(1,16)
(1,12)
(1,8)
(1,12)
(1,10)
(1,10)
(1,8)
(1,6)
(1,12)
(1,8)
(1,4)
(1,0)
Figure 4.11. Market model in Exercise (2)
(c) Find the arbitrage-free price of a call option at time 0 with maturity T = 2
and strike price K = 7.
(d) Find the arbitrage-free price of a put option at time 0 with maturity T = 2
and strike price K = 7.
(e) Find the arbitrage-free price of a call option at time 1 with maturity T = 2
and strike price K = 7.
4.4. ARBITRAGE-FREE PRICES FOR EUROPEAN CONTINGENT CLAIMS
111
(3) Consider a two-period market model with one bond and one stock. The sample
space is given by Ω = {ωi : i = 1, 2, 3, 4}. Suppose the stock price is given by
S0 (ωi ) = 5,
⎧
⎪
⎨ 8,
S1 (ωi ) =
⎪
⎩ 4,
⎧
⎪
⎪
9,
⎪
⎪
⎨
S2 (ωi ) =
6,
⎪
⎪
⎪
⎪
⎩ 3,
for all i,
if i = 1, 2,
if i = 3, 4,
if i = 1,
if i = 2, 3,
if i = 4.
Moreover, suppose the interest rate of the bond is given by r, i.e.,
B0 = 1,
B1 = (1 + r),
B2 = (1 + r)2 .
(a) Find the range of r such that the model is arbitrage-free. Furthermore, find
the collection of all equivalent martingale measures.
(b) Suppose this market model is arbitrage-free. Is this model complete? Justify
your answer
(c) If r = 0, find the arbitrage-free price of a call option at time 0 with maturity
T = 2 and strike price K = 7.