Download One-Period Models

CHAPTER 3
One-Period Models
Assumptions:
(1) ddd t = 0, 1.
dddddddd t = 0.
(2) sample space Ω = {ω1 , ω2 , ..., ωK } with P({ωi }) > 0 for all i = 1, 2, ..., K.
at time 0, F0 = {∅, Ω}.
at time 1, F1 = the collection of all possible subsets of Ω.
Ft dddddddddddddd t d information dd.
(3) Suppose that there are 1 bond and N stocks⎛in ⎞
the financial market.
⎜Bt ⎟
security price S̄t = (Bt , St1 , St2 , · · · , StN )T = ⎝ ⎠ for t = 0, 1,
St
where T means the transpose of a matrix, St = (St1 , St2 , · · · , StN )T
dddddddddd B0 and S0i are constant, i.e., S̄0 is a deterministic vector.
dddddddddddddd B1 d S1 . dd B1 is a constant, the price of
the ith stock S1i : Ω −→ R+ is a random variable for i = 1, ..., N .
ddddddddd (See Figure 3.1).
Note: dddddddd probability space (Ω, F, P) d finite probability space ddd
d. ddddddddddddddd probability space d, ddddddddddd
dddd
∗
dd.
59
60
3. ONE-PERIOD MODELS
_
S1(w1)
_
S1(w2)
_
S0
_
S1(wK)
at time 0
at time 1
Figure 3.1. price in the one-period model
3.1. Portfolio
Definition 3.1.∗ A portfolio is a vector h̄ = (h0 , · · · , hN )T ∈ RN +1 , where hi denotes
the number of shares of the ith asset.
Remark 3.2.∗ The value of the portfolio h̄ at time 0 is given by
V0 (h̄) = h̄ · S̄0 = h0 B0 + h1 S01 + · · · + hN S0N .
The value of the portfolio h̄ at time 1 is given by
V1 (h̄) = h̄ · S̄1 = h0 B1 + h1 S11 + ... + hN S1N .
The profit of the portfolio h̄ is given by
G(h̄) := V1 (h̄) − V0 (h̄) = h̄(S̄1 − S̄0 ) = h̄ ·
Example 3.3. Suppose that
⎛ ⎞
⎛
⎞
⎜1⎟
⎜1.02⎟
S̄0 = ⎝ ⎠ ,
S̄1 (ω1 ) = ⎝
⎠,
10
12
ΔS̄
.
ddddd
⎛
⎞
⎜1.02⎟
S̄1 (ω2 ) = ⎝
⎠.
9
3.2. DERIVATIVE SECURITIES
⎛
⎞
⎜−10⎟1
If the portfolio h̄ = ⎝
⎠ , then its value at time 0 is given by
1
61
⎞
⎛ ⎞⎛
⎜ 1 ⎟ ⎜−10⎟
V0 (h̄) = h̄ · S̄0 = ⎝ ⎠ ⎝
⎠ = 0.
1
10
Moreover, at time 1,
V1 (h̄)(ω1 ) = h̄ · S̄1 (ω1 ) = −10 × 1.02 + 12 = 1.8,
V1 (h̄)(ω2 ) = h̄ · S̄1 (ω2 ) = −10 × 1.02 + 9 = −1.2.
Thus, the value of the portfolio h̄ at time 1 is given by
⎞ ⎛
⎛
⎞
⎜ V1 (h̄)(ω1 ) ⎟ ⎜ 1.8 ⎟
V1 (h̄) = ⎝
⎠=⎝
⎠.
V1 (h̄)(ω2 )
−1.2
The profit of h̄ is given by
⎛
⎞
⎜ 1.8 ⎟
G(h̄) = V1 (h̄) − V0 (h̄) = ⎝
⎠.
−1.2
3.2. Derivative securities
ddddddd, ddd, dddd securities dd, dd option (ddd), derivative
securities (ddddd), or contingent claim (dddddd).
Example 3.4.∗ Forward contract (dddd)
ddddddddddddddddddddddddddddd. One agent agrees
1dddddd,
dd, dddddd, dd/dd
62
3. ONE-PERIOD MODELS
to sell to another agent an asset at time 1 for a price K which is specified at time 0.
payoff = S1i − K.
Forward contract ddd (future contract) ddddd. dddddddddddd, d
dddddd, dddddd. d forward contract dddddddd, ddddddd
dddddd.
Example 3.5.∗ Call option (dd)
The owner has the right, but not the obligation to buy the asset at time 1 for a fixed
price K called the strike price.
payoff =
(S1i
+
− K) =
⎧
⎪
⎪
⎨S1i − K,
if S1i > K,
⎪
⎪
⎩0,
if S1i ≤ K.
Example 3.6.∗ Put option (dd)
The owner has the right, but not the obligation to sell the asset at time 1 for a fixed price
K.
payoff = (K −
S1i )+
=
⎧
⎪
⎪
⎨K − S i ,
if S1i < K,
⎪
⎪
⎩0,
if S1i ≥ K.
1
Definition 3.7.∗
(1) A contingent claim (d d d d d d) is a random variable C on a probability
space (Ω, F, P) such that
0≤C<∞
P-a.s.
3.3. ABSENCE OF ARBITRAGE
63
(2) A contingent claim C is called a derivative of B, S 1 , ..., S N if it is measurable
with respect to σ(B, S 1 , ..., S N ), i.e.,
C = f (B, S 1 , ..., S N )
for a measurable function f on RN +1 .
Question: What is the price of a contingent claim?
3.3. Absence of arbitrage
Definition 3.8.∗ A portfolio h̄ ∈ RN +1 is called an arbitrage opportunity if
(i) V0 (h̄) = h̄ · S̄0 ≤ 0
(ii) V1 (h̄) = h̄ · S̄1 ≥ 0 P-a.s. and P(V1 (h̄) > 0) > 0.
(i) dddddddddddd 0 dddddddddd 0, d (ii) ddddddd
1 ddddddddddddd 0, dddd 0 dddddddddd 0. dddd, d
dddddddddddddd, dddddddd.
Remark 3.9. If Ω = {ω1 , ..., ωK } and there is an arbitrage opportunity, then there
exists a portfolio h̄ ∈ RN +1 such that
(1) V0 (h̄) ≤ 0
(2) V1 (h̄)(ωi ) ≥ 0 for all i and V1 (h̄)(ωj ) > 0 for some j.
Example 3.10.
Let
(1) Suppose that Ω = {ω1 , ω2 } with P({ωi }) > 0 for i = 1, 2.
⎛ ⎞
⎜1⎟
S̄0 = ⎝ ⎠ ,
10
⎛
⎞
⎜1.1⎟
S̄1 (ω1 ) = ⎝ ⎠ ,
11
⎛
⎞
⎜1.1⎟
S̄1 (ω2 ) = ⎝ ⎠ .
12
64
⎛
3. ONE-PERIOD MODELS
⎞
⎜−10⎟
Then h̄ = ⎝
⎠ is an arbitrage opportunity, since
1
⎛ ⎞ ⎛
⎞
⎜ 1 ⎟ ⎜−10⎟
V0 (h̄) = h̄ · S̄0 = ⎝ ⎠ · ⎝
⎠=0
10
1
⎛ ⎞ ⎛
⎞
⎜1.1⎟ ⎜−10⎟
V1 (h̄)(ω1 ) = h̄ · S̄1 (ω1 ) = ⎝ ⎠ · ⎝
⎠=0
11
1
⎛ ⎞ ⎛
⎞
⎜1.1⎟ ⎜−10⎟
V1 (h̄)(ω2 ) = h̄ · S̄1 (ω2 ) = ⎝ ⎠ · ⎝
⎠ = 1 > 0.
12
1
(2) Suppose that Ω = {ω1 , ω2 } with P({ωi }) > 0 for i = 1, 2. Let
⎛ ⎞
⎛ ⎞
⎛ ⎞
⎜1⎟
⎜1.2⎟
⎜1.2⎟
S̄1 (ω1 ) = ⎝ ⎠ ,
S̄1 (ω2 ) = ⎝ ⎠ .
S̄0 = ⎝ ⎠ ,
10
11
13
Then there is no arbitrage opportunity in this model.
(3) Consider Ω = [0, 1], F = B1 , P = Lebesgue measure m. Let
⎞
⎛ ⎞
⎛
⎜1⎟
⎜ 1 ⎟
S̄1 = ⎝
S̄0 = ⎝ ⎠ ,
⎠,
10
10Z
⎛
⎞
⎜ 10 ⎟
where Z is uniformly distributed on [0, 1]. Then h̄ = ⎝ ⎠ is an arbitrage
−1
opportunity, since
⎛ ⎞ ⎛ ⎞
⎜ 10 ⎟ ⎜ 1 ⎟
V0 (h̄) = h̄ · S̄0 = ⎝ ⎠ · ⎝ ⎠ = 0
−1
10
⎛ ⎞ ⎛
⎞
⎜ 10 ⎟ ⎜ 1 ⎟
V1 (h̄) = h̄ · S̄1 = ⎝ ⎠ · ⎝
⎠ = 10 − 10Z ≥ 0,
−1
10Z
3.3. ABSENCE OF ARBITRAGE
65
and
P(V1 (h̄) > 0) = P(10 − 10Z > 0) = P(Z < 1) = 1.
Assumption: Suppose the interest rate of the bond = r > −1, i.e.,
B0 = B,
B1 = B(1 + r).
Lemma 3.11.∗ The following statements are equivalent
(1) The market model admits arbitrage opportunity
(2) There exists a vector h ∈ RN such that
hS1 ≥ (1 + r)h · S0
P − a.s.
and
P[h · S1 > (1 + r)h · S0 ] > 0.
dd lemma ddd Definition 3.8 dddddddd bond ddddddd, ddd
dddddddd interest rate r dd constant dddddddd.
Proof. (1) =⇒ (2): Let h̄ = (h0 , h)T be an arbitrage opportunity. Then
0 ≥ h̄ · S̄0 = h0 B + h · S0
. Thus,
h · S1 − (1 + r)h · S0 ≥ h · S1 + (1 + r)h0 B
= h0 B1 + h · S1 = h̄ · S̄1 .
Since h̄ is an arbitrage opportunity,
h · S1 − (1 + r)h · S0 ≥ 0
P − a.s.
66
3. ONE-PERIOD MODELS
and
P[h · S1 > (1 + r)h · S0 ] > 0.
⎛ ⎞
0
h · S0
⎜h ⎟
.
(2) =⇒ (1): Suppose h satisfies the statement (2). Let h̄ = ⎝ ⎠ with h0 = −
B
h
Then
V0 (h̄) = h̄ · S̄0 = h0 B + h · S0 = −h · S0 + h · S0 = 0,
V1 (h̄) = h̄ · S̄1 = h0 (1 + r)B + h · S1 ≥ −(1 + r)h · S0 + h · S1 .
By assumption this implies that
V1 (h̄) ≥ 0
P − a.s.
and
P(V1 (h̄) > 0) > 0.
Thus, h̄ is an arbitrage opportunity.
Definition 3.12.∗ If there exists no arbitrage opportunity in a financial market we say
that there is no arbitrage (arbitrage-free, no free lunch) in this financial market.
No arbitrage dd financial mathematics dddddddd. ddddd financial
model dddddd.
3.4. No arbitrage and price system
ddd, dddddddd no arbitrage ddddd, dddd finite state space d
dddd. dddddddddddddddd case dddddd, dddddddd
dddddddddddd.
3.4. NO ARBITRAGE AND PRICE SYSTEM
Definition 3.13.
(1) The (N
⎛
⎜ B1 (ω1 ) B1 (ω2 )
⎜
⎜ 1
⎜ S1 (ω1 ) S11 (ω2 )
D=⎜
⎜ .
..
⎜ ..
.
⎜
⎝
S1N (ω1 ) S1N (ω2 )
+ 1) × K matrix D, defined by
⎞
· · · B1 (ωK ) ⎟
⎟
⎟
1
· · · S1 (ωK ) ⎟
⎟ = (S̄1 (ω1 ), ..., S̄1 (ωK ))
⎟
..
..
⎟
.
.
⎟
⎠
· · · S1N (ωK )
is called the payoff matrix.
(2) The vector b := S̄0 = (B0 , S01 , · · · , S0N )T is called the price vector.
(3) (S̄0 , S̄1 ) ∼
= (b, D) ∈ RN +1 × M(N +1)×K (R)2 is called the market model.
Example 3.14. As in Example 3.3,
⎛ ⎞
⎜1⎟
b = ⎝ ⎠,
10
⎛
⎞
⎜1.02 1.02⎟
D=⎝
⎠
12
9
⎛ ⎞
⎜ c1 ⎟
⎜.⎟
.⎟
Notation 3.15. Let C = ⎜
⎜ . ⎟, for c1 , ..., cn ∈ R.
⎝ ⎠
cn
(1) C ≥ 0 if ci ≥ 0 for all i = 1, 2, ..., n.
(2) C > 0 if ci ≥ 0 for all i = 1, 2, ..., n and ck > 0 for at least one k.
(3) C 0 if ci > 0 for all i = 1, 2, ..., n.
Remark 3.16. Remark 3.9 can be written as
h̄ · b ≤ 0
and
DT h̄ > 0.
ddddddddddd, arbitrage opportunity ddddddd.
2M
(N +1)×K (R)
means the collection of all (N + 1) × K matrices with real-valued entries.
67
68
3. ONE-PERIOD MODELS
Remark 3.17. An alternative definition of arbitrage opportunity is
h̄ · b ≤ 0
and
DT h̄ > 0
h̄ · b < 0
and
DT h̄ ≥ 0.
or
(3.1)
ddddd Remark 3.16 ddddddd (3.1) dddd. (3.1) ddddddddd
dd strong arbitrage opportunity. dd, ddddddddddd. ddddd no
arbitrage ddd. In fact, ddddddd (3.1) dd, ddddd arbitrage opportunity
(in the sense of Remark 3.16) dd, ddddddddddddddd no arbitrage d
ddd.
Claim: If there exists a portfolio h̄ satisfying (3.1), there exists an arbitrage opportunity
h̄∗ in the sense of Remark 3.16.
Let h̄ = (h0 , h)T . Set
∗
h̄ =
h̄ · b
h −
,h
B0
0
T
,
then
⎛
⎞
⎞ ⎛
h̄ · b
⎜ h − B0 ⎟ ⎜ B0 ⎟
h̄∗ · b = ⎝
⎠
⎠·⎝
S0
h
0
= h0 B − h̄ · b + h · S0 = h̄ · b − h̄ · b = 0,
3.4. NO ARBITRAGE AND PRICE SYSTEM
69
and
⎛
⎞
S11 (ω1 )
DT h̄∗
S1N (ω1 )
···
⎜ B1 (ω1 )
⎜
⎜
⎜ B1 (ω2 ) S11 (ω2 ) · · · S1N (ω2 )
= ⎜
⎜
..
..
..
..
⎜
.
.
.
.
⎜
⎝
B1 (ωK ) S11 (ωK ) · · · S1N (ωK )
⎞
⎛
⎜ B1 (ω1 ) ⎟
⎟
⎜
h̄
·
b
..
⎟
0
⎜
= DT h̄ −
.
⎟
B0 ⎜
⎠
⎝
B1 (ωK )
⎟⎛
⎞
⎟
⎟ h0 − h̄ · b
⎟⎜
B0 ⎟
⎟⎝
⎠
⎟
⎟
h
⎟
⎠
due to (3.1). Hence, h̄∗ is an arbitrage opportunity in the sense of Remark 3.16.
Theorem 3.18 (Fundamental Theorem of Asset Pricing). In the market model (b, D),
the following statements are equivalent:
(1) (b, D) is arbitrage-free.
(2) There exists ϕ ∈ RK+1 such that ϕ 0 and
ϕ · L(h̄) = 0
for all h̄ ∈ RN +1 ,
where L : RN +1 −→ RK+1 is a linear transformation given by
⎞ ⎛
⎞
⎛
T
⎜−h̄ · b⎟ ⎜ −b ⎟
L(h̄) = ⎝
⎠=⎝
⎠ h̄.
DT
DT h̄
(3) There exists a vector ψ ∈ RK , ψ 0 such that
b = Dψ.
(3.2)
70
3. ONE-PERIOD MODELS
Proof. (1) =⇒ (2): Suppose that the market model (b, D) is arbitrage-free. Then
there is no h̄ ∈ RN +1 such that
⎛
⎞
⎜−h̄ · b⎟
L(h̄) = ⎝
⎠ > 0.
T
D h̄
⎧⎛
⎫
⎞
⎪
⎪
⎨ −h̄ · b
⎬
⎜
⎟
N +1
Hence, the set ⎝
is a proper subset of RK+1 .
⎠ : h̄ ∈ R
⎪
⎪
T
⎩ D h̄
⎭
By ”Separating Theorem”3, there exists a vector φ ∈ RK+1 with φ 0 such that
φ · L(h̄) = 0
for all h̄ ∈ RN +1 .
Figure 3.2
⎛ ⎞
⎜φ0 ⎟
(2) =⇒ (3): Let φ = ⎝ ⎠ ∈ RK+1 , φ0 ∈ R, φ1 ∈ RK with
φ1
φ
0
3separating
and
φ · L(h̄) = 0
for all h̄ ∈ RN +1 .
theorem ddddddddd, ddddddd disjoint convex sets dddddddd
ddd. ddddddddddddddddddddddd: dddddddddddddddd (d
ddddddddddddd, dddddddddddddddddd) ddd, ddddddddd
ddddddddddddddd (d Figure 3.2).
3.4. NO ARBITRAGE AND PRICE SYSTEM
71
Since φ 0, we have φ0 > 0 and φ1 0. Thus,
⎞
⎛ ⎞ ⎛
⎜φ0 ⎟ ⎜−h̄ · b⎟
T
0 = φ · L(h̄) = ⎝ ⎠ · ⎝
⎠ = −φ0 h̄ · b + φ1 · D h̄.
DT h̄
φ1
Let ψ =
φ1
, this implies that
φ0
h̄ · b =
φ1
· DT h̄ = ψ · DT h̄ = h̄ · Dψ
φ0
for all h̄ ∈ RN +1 .
Hence, b = Dψ for some ψ ∈ RK with ψ 0.
(3) =⇒ (1): Since
h̄ · b = ψ · DT h̄
for all h̄ ∈ RN +1 .
If DT h̄ > 0, due to ψ 0, we have ψ · DT h̄ > 0. Hence, h̄ · b > 0. By Remark 3.16, we
see that the market model (b, D) is arbitrage-free.
ddddddddd. ddddd (b, D) d, ddddd market model ddd no
arbitrage dddddd (ddd assertion (3)). ddddddddddddd: dd
dd finite probability space. dddddd, dddddddd, dddddddd
section ddd.
Example 3.19. (One-period, two states model) Suppose that the sample space Ω =
{ω1 , ω2 }. Consider a market model with
⎞
⎛
⎛ ⎞
⎜B(1 + r) B(1 + r)⎟
⎜B ⎟
b = ⎝ ⎠, D = ⎝
⎠,
S1 (ω1 )
S0
S1 (ω2 )
with S1 (ω1 )⎛> S⎞
1 (ω2 ). Suppose that the market model (bD) is arbitrage-free. Then there
⎜ψ1 ⎟
exists ψ = ⎝ ⎠ 0 such that
ψ2
b = Dψ,
72
3. ONE-PERIOD MODELS
i.e.,
⎛ ⎞ ⎛
⎞⎛ ⎞
⎜ B ⎟ ⎜B(1 + r) B(1 + r)⎟ ⎜ψ1 ⎟
⎝ ⎠=⎝
⎠⎝ ⎠.
S0
S1 (ω1 )
S1 (ω2 )
ψ2
In other words, ψ satisfies
⎧
⎪
⎪
⎨B = B(1 + r)ψ1 + B(1 + r)ψ2 ,
⎪
⎪
⎩S0 = S1 (ω1 )ψ1 + S1 (ω2 )ψ2 ,
and the corresponding solution is given by
⎧
⎪
(1 + r)S0 − S1 (ω2 )
1
⎪
⎪
·
,
⎨ψ1 =
1+r
S1 (ω1 ) − S1 (ω2 )
⎪
S1 (ω1 ) − (1 + r)S0
1
⎪
⎪
·
.
⎩ψ2 =
1+r
S1 (ω1 ) − S1 (ω2 )
Thus, ψ 0 if and only if
⎧
⎪
⎪
⎨(1 + r)S0 > S1 (ω2 ),
⎪
⎪
⎩S1 (ω1 ) > (1 + r)S0 ,
i.e., the market model is arbitrage-free if and only if the stock price at time 0 and 1
satisfies
S1 (ω2 )
S1 (ω1 )
< S0 <
.
1+r
1+r
Exercise
(1) Consider the market model
⎛⎛
⎞ ⎛
⎞⎞
⎜⎜ 56 ⎟ ⎜ 60 59 57 ⎟⎟
⎜⎜
⎟ ⎜
⎟⎟
⎜
⎟ ⎜
⎟⎟
(b, D) = ⎜
⎜⎜ 8 ⎟ , ⎜ 11 7 10 ⎟⎟ .
⎝⎝
⎠ ⎝
⎠⎠
33
32 36 41
(a) Show that (b, D) is arbitrage-free and complete, and find the vector ψ such
that b = Dψ.
3.5. MARTINGALE MEASURE
73
(b) Find the interest rate of the riskless asset in this market model.
3.5. Martingale measure
Remark 3.20. By Theorem 3.18, we have
b = Dψ.
Thus,
⎛
⎞
⎞⎛
⎛
⎜ B0 ⎟ ⎜ B1 (ω1 ) B1 (ω2 )
⎜ ⎟ ⎜
⎜ 1⎟ ⎜ 1
⎜ S0 ⎟ ⎜ S1 (ω1 ) S11 (ω2 )
⎜ ⎟=⎜
⎜ . ⎟ ⎜ .
..
⎜ .. ⎟ ⎜ ..
.
⎜ ⎟ ⎜
⎝ ⎠ ⎝
S1N (ω1 ) S1N (ω2 )
S0N
⎞
···
B1 (ωK ) ⎟ ⎜ ψ1 ⎟
⎟⎜ ⎟
⎟⎜ ⎟
· · · S11 (ωK ) ⎟ ⎜ ψ2 ⎟
⎟⎜ ⎟.
⎟⎜ . ⎟
..
...
⎟ ⎜ .. ⎟
.
⎟⎜ ⎟
⎠⎝ ⎠
ψK
· · · S1N (ωK )
In the form of the system of equations
⎧
⎪
⎪
⎪
⎪
B0 = B1 (ω1 )ψ1 + · · · + B1 (ωK )ψK
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨S 1 = S 1 (ω )ψ + · · · + S 1 (ω )ψ
0
1
1
1
1
K
K
⎪
..
⎪
⎪
.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩S0N = S1N (ω1 )ψ1 + · · · + S1N (ωK )ψK
If the interest rate = constant r, e.g. B0 = B, B1 = B(1 + r), then
ψ1 + · · · + ψK =
1
.
1+r
Thus, (3.3) can be written as
(1 + r)S0i = S1i (ω1 )(1 + r)ψ1 + · · · + S1i (ωK )(1 + r)ψK
= S1i (ω1 )
for all 1 ≤ i ≤ N .
ψ1
ψK
+ · · · + S1i (ωK )
ψ1 + · · · + ψK
ψ1 + · · · + ψK
(3.3)
74
3. ONE-PERIOD MODELS
Remark 3.21. Define
Q({ωj }) =
ψj
ψ1 + · · · + ψK
for all 1 ≤ j ≤ K.
(3.4)
Then Q is a probability measure.
Definition 3.22. The probability measure Q defined by (3.4) is called a risk-neutral
probability measure.
Remark 3.23. In general, Q in (3.4) is not unique.
dd Theorem 3.18 dddddddd.
Theorem 3.24. In an arbitrage-free market model (S̄0 , S̄1 ) ∼
= (b, D), there is a riskneutral measure Q such that
S0i
= EQ
S1i
1+r
for all 1 ≤ i ≤ N,
(3.5)
where EQ means the expectation with respect to the probability measure Q.
ddddddddddd, dddddddddddd, dddddddddddd
ddddddd.
Note that (3.5) is equivalent to the equation b = Dψ.
Example 3.25. As in Example 3.19
⎞
⎛ ⎞
⎛
⎜B ⎟
⎜B(1 + r) B(1 + r)⎟
b = ⎝ ⎠,
D=⎝
⎠,
S1 (ω2 )
S0
S1 (ω1 )
with S1 (ω1 ) > S1 (ω2 ), then
ψ1 =
(1 + r)S0 − S1 (ω2 )
1
·
1+r
S1 (ω1 ) − S1 (ω2 )
ψ2 =
S1 (ω1 ) − (1 + r)S0
1
·
.
1+r
S1 (ω1 ) − S1 (ω2 )
3.5. MARTINGALE MEASURE
75
Thus, the risk-neutral probability measure Q is given by
Q({ω1 }) =
ψ1
S0 (1 + r) − S1 (ω2 )
=
ψ1 + ψ2
S1 (ω1 ) − S1 (ω2 )
Q({ω2 }) =
ψ2
S1 (ω1 ) − S0 (1 + r)
=
ψ1 + ψ2
S1 (ω1 ) − S1 (ω2 )
and
EQ [S1 ] = S1 (ω1 )Q({ω1 }) + S1 (ω2 )Q({ω2 }) = (1 + r)S0 .
Thus,
S1
1
EQ [S1 ] = EQ
.
S0 =
1+r
1+r
Remark 3.26.∗ Let
X0i = S0i ,
X1i =
S1i
1+r
discounted stock price.
Then, (3.5) implies that
X0i = EQ [X1i ] = EQ [X1i | F0 ],
i.e., (Xki , Fk )k=0,1 is a martingale for all i. ddddddd one-period model dddd
dd. ddddddddd multi-period model. d multi-period model ddddddd
d. Hence, the risk-neutral measure Q is called a martingale measure.
Remark 3.27.∗
(1) (Xki )k=0,1 is a martingale with respect to Q for all 1 ≤ i ≤ N . This implies
that (hi Xki )k=0,1 is a martingale with respect to Q for all 1 ≤ i ≤ N . Thus,
h̄ · X̄k k=0,1 is a martingale with respect to Q, where X̄k = (Bk , Xk1 , ..., XkN )T .
76
3. ONE-PERIOD MODELS
(2) The random variable
Yi =
S1i
− S0i = X1i − X0i
1+r
is called the discounted net gain. Thus,
EQ [Y i ] = 0,
for all 1 ≤ i ≤ N.
Definition 3.28.∗
(1) Two probability measures P and Q are called equivalent, denoted by P ∼ Q, if
P(A) = 0
⇐⇒
Q(A) = 0,
for all A ∈ F.
(2) An equivalent risk-neutral measure is also called an pricing measure or an equivalent
martingale measure (EMM).
Example 3.29.
(1) Let Ω = {1, 2, 3, 4} and F the collection of all subsets of Ω.
Set
P1 ({1}) = 1/2,
P1 ({2}) = 1/4,
P1 ({3}) = 1/6,
P1 ({4}) = 1/12,
P2 ({1}) = 1/5,
P2 ({2}) = 1/5,
P2 ({3}) = 1/5,
P2 ({4}) = 2/5,
P3 ({1}) = 1/4,
P3 ({2}) = 1/4,
P3 ({3}) = 0,
P3 ({4}) = 1/2.
Then P1 ∼ P2 , but P1 ∼ P3 and P2 ∼ P3 .
(2) Consider a probability space (Ω, F, P) and let X be a random variable satisfying
X ≥ 1/2, P-a.s. and E[X] = 1. Define
Q(A) =
for all A ∈ F, then
(a) Q is a probability measure;
A
X dP,
3.5. MARTINGALE MEASURE
77
(b) P ∼ Q.
Theorem 3.30 (Fundamental Theorem of Asset Pricing).∗ A market model is arbitragefree if and only if the set
P = {Q : Q is a risk-neutral measure with P ∼ Q} = ∅.
Proof. “⇐=” Suppose that there exists a risk-neutral measure Q ∈ P. Let h̄ ∈ RN +1
with
h̄ · S̄1 ≥ 0 P-a.s.
and
E[h̄ · S̄1 ] > 04.
Since Q is a martingale measure,
h̄ · S̄0 = h̄ · EQ
S̄1
1
EQ [h̄ · S̄1 ] > 0.
=
1+r
1+r
This implies that the market model is arbitrage-free.
“=⇒” dddddddddddd, dddddd Föllmer and Schied [12] P.7.
dddddd risk-neutral measure ddddd.
Example 3.31. Consider a financial market with one bond and one stock. Consider
the sample space Ω = {ω1 , ω2 , ..., ωK } (K ≥ 2).
⎛ ⎞
⎜1⎟
b = ⎝ ⎠,
S0
4d
⎛
⎜ 1 + r 1 + r ···
D=⎝
S1 (ω1 ) S1 (ω2 ) · · ·
h̄ · S̄1 ≥ 0 dddd, ddd P(h̄ · S̄1 > 0) > 0 dddd.
⎞
1+r ⎟
⎠
S1 (ωK )
78
3. ONE-PERIOD MODELS
(S1 is not a constant). If this financial market is arbitrage-free, there exists ψ ∈ RK ,
ψ 0 such that
⎛ ⎞
⎛
⎜1⎟
⎜ 1 + r 1 + r ···
⎝ ⎠ = b = Dψ = ⎝
S0
S1 (ω1 ) S1 (ω2 ) · · ·
Thus,
⎧
⎪
⎪
⎨ψ1 + · · · + ψK =
⎞
⎛
⎞
⎜ ψ1 ⎟
1 + r ⎟⎜ . ⎟
. ⎟
⎠⎜
⎜ . ⎟
⎠
⎝
S1 (ωK )
ψK
1
,
1+r
⎪
⎪
⎩ψ1 S1 (ω1 ) + · · · + ψK S1 (ωK ) = S0 .
How many strictly positive solutions does this system of equations?
(1) K = 2: We know that S1 (ω1 ) = S1 (ω2 ). Without loss of generality, we assume
that S1 (ω1 ) > S1 (ω2 ). By Example 3.25, the equivalent martingale measure is
unique and is given by
Q({ω1 }) =
ψ1
S0 (1 + r) − S1 (ω2 )
,
=
ψ1 + ψ2
S1 (ω1 ) − S1 (ω2 )
Q({ω2 }) =
ψ2
S1 (ω1 ) − S0 (1 + r)
=
.
ψ1 + ψ2
S1 (ω1 ) − S1 (ω2 )
(2) K > 2: the equivalent martingale measure is no more unique. In fact, there are
infinite many equivalent martingale measures in this case.
Theorem 3.30 ddddd, ddddddddd. dd, probability measure Q dd
d, d infinite many assets ddddd.
Example 3.32. Theorem 3.30 is not true in a market model with infinite many assets,
e.g., let
Ω = {1, 2, 3, ...} = N
with
P({ω}) > 0 for all ω ∈ Ω.
3.5. MARTINGALE MEASURE
79
Consider
B0 = B1 = 1
( i.e., interest rate r = 0).
For i = 1, 2, 3, ..., let the stock price S0i = 1, and
S1i (ω) =
⎧
⎪
⎪
⎪
0
⎪
⎪
⎪
⎨
ω = i,
2
⎪
⎪
⎪
⎪
⎪
⎪
⎩1
ω = i + 1,
otherwise.
(1) Claim: This market model is arbitrage-free.
Suppose that h̄ = (h0 , h1 , ..., hN , ...)T is a portfolio such that
h̄ · S̄1 ≥ 0 for all ω ∈ Ω
and
h̄ · S̄0 ≤ 0.
For ω = 1,
0 ≤ h̄ · S̄1 (1) = h0 +
∞
hk = h̄ · S̄0 − h1 ≤ −h1 .
k=2
For ω = i > 1,
∞
0 ≤ h̄ · S̄1 (i) =
hk + 2hi−1 = h̄ · S̄0 + hi−1 − hi ≤ hi−1 − hi .
k=0,k=i,i+1
This implies that
0 ≥ h1 ≥ h2 ≥ ... ≥ hi−1 ≥ hi ≥ · · ·
Since
h̄ · S̄0 ≤ 0
and
h̄ · S̄1 ≥ 0,
we have hi = 0 for all i = 0, 1, 2, ... Thus, there is no arbitrage opportunity in
this market model.
80
3. ONE-PERIOD MODELS
(2) Claim: There is no equivalent martingale measure.
Suppose there is an equivalent martingale measure Q, we have
S̄0 = EQ [S̄1 ]
This implies that
∞
1 = S0i = EQ [S1i ] = 2Q({i + 1}) +
Q({k})
k=1,k=i,i+1
= 1 + Q({i + 1}) − Q({i}).
This leads to Q({i}) = Q({i + 1}) for all i = 1, 2, 3, ... This is obviously a
contradiction, since Q(Ω) cannot be 1.
Theorem 3.33 (Law of one price).∗ Suppose that the market model is arbitrage-free
and suppose that
h̄ · S̄1 = k̄ · S̄1
for two different portfolios h̄ and k̄. Then h̄ · S̄0 = k̄ · S̄0 .
dddddddddddddddd. d arbitrage-free dddd, ddddddd
dddddddddd, dddd ddddddddd.
Proof. Since h̄ · S̄1 = k̄ · S̄1 P-a.s., we have
(h̄ − k) · S̄1 = 0
P − a.s.
By the equivalence of P and Q, we have
(h̄ − k̄) · S̄1 = 0
Q − a.s.
Hence,
0 = EQ [(h̄ − k̄) · S1 ] = (h̄ − k̄) · EQ [S̄1 ] = (1 + r)(h̄ − k̄) · S̄0 ,
3.6. PRICING
81
i.e.,
h̄ · S̄0 = k̄ · S̄0 .
Remark 3.34.∗ If V ∈ {h̄ · S̄1 : h̄ ∈ RN +1 }, then we can define the price of V as
π(V ) = h̄ · S̄0
if V ∈ h̄ · S̄1
whenever the market model is arbitrage-free (By Theorem 3.33, this definition is welldefined). Moreover, by Theorem 3.24,
π(V ) = EQ
V
.
1+r
3.6. Pricing
ddddddddd financial mathematics dddddddddd: dddddd
dddd.
Consider a derivative C, the price of C at time 0 π(C) =? ddddddddddd
ddddddd idea: dddddddddddddddd asset, ddddd (S̄0 , S̄1 )
dddddddd. Consider
⎧
⎪
⎪
⎨S1N +1 = C
⎪
⎪
⎩S N +1 = π(C) = π C
0
(3.6)
ddd: ddddddddddd: No arbitrage!
Definition 3.35.∗ A real number π C ≥ 0 is called an arbitrage-free price of a contingent
claim C if the market model extended according to (3.6) is arbitrage-free. The set of all
arbitrage-free prices for C is denoted by Π(C).
82
3. ONE-PERIOD MODELS
Theorem 3.36. Suppose P = ∅. Then Π(C) = ∅, and
EQ
Π(C) =
C
: Q ∈ P with EQ [C] < ∞ .
1+r
Proof. By Theorem 3.24 and Theorem 3.30, π C is arbitrage-free price for C if and
only if there exists Q ∈ P for the market model extended via (3.6), i.e.,
S0i
Thus,
= EQ
S1i
1+r
for i = 1, 2, ..., N + 1.
C
: Q ∈ P with EQ [C] < ∞ .
Π(V ) ⊆ EQ
1+r
Conversely, if
C
π = EQ
C
1+r
for some Q ∈ P,
then Q is also an equivalent risk-neutral measure for the extended market model. This
implies that
Π(C) ⊇
EQ
C
: Q ∈ P with EQ [C] < ∞ .
1+r
Example 3.37. Consider a market model with
⎛ ⎞
⎜1⎟
b = ⎝ ⎠,
10
⎛
⎞
⎜1 1 1 ⎟
D=⎝
⎠.
9 11 12
Then there exists ψ ∈ R3 , ψ 0 such that
⎛ ⎞
⎛ ⎞
⎛
⎛
⎞ ψ
⎞
⎜ 1⎟
⎟ ⎜ ψ1 + ψ2 + ψ3 ⎟
⎜1⎟
⎜1 1 1 ⎟ ⎜
⎟
⎝ ⎠ = b = Dψ = ⎝
⎠⎜
⎠.
ψ
⎜ 2⎟ = ⎝
10
9 11 12 ⎝ ⎠
9ψ1 + 11ψ2 + 12ψ3
ψ3
3.6. PRICING
Thus,
83
⎧
⎪
⎪
⎨ψ1 + ψ2 + ψ3 = 1,
⎪
⎪
⎩9ψ1 + 11ψ2 + 12ψ3 = 10.
Suppose ψ1 = a ∈ (0, 1), then
⎧
⎪
⎪
⎨ψ2 + ψ3 = 1 − a,
⎪
⎪
⎩11ψ2 + 12ψ3 = 10 − 9a.
Thus,
⎧
⎪
⎪
⎨ψ2 = 2 − 3a
with 1/2 < a < 2/3.
⎪
⎪
⎩ψ3 = 2a − 1
Obviously, the risk-neutral measure is not unique. Hence,
P = {Q : Q(ω1 ) = a, Q(ω2 ) = 2 − 3a, Q(ω3 ) = 2a − 1, with 1/2 < a < 2/3}.
(i) Consider a contingent claim C with
C(ω1 ) = 6,
C(ω2 ) = 8,
C(ω3 ) = 9.
Then
π C = EQ [C] = C(ω1 )Q({ω1 }) + C(ω2 )Q({ω2 }) + C(ω3 )Q({ω3 }) = 7.
(ii) Consider a contingent claim C with
C(ω1 ) = 10,
C(ω2 ) = 8,
C(ω3 ) = 12.
Then
EQ [C] = 10a + 8(2 − 3a) + 12(2a − 1) = 4 + 10a.
Therefore,
Π(C) = {4 + 10a : 1/2 < a < 2/3} = (9, 32/3).
84
3. ONE-PERIOD MODELS
dddddd, dddddddddddddd π C , ddddddddddd. d
dd?
Definition 3.38.∗ A contingent claim C is called attainable (or replicable) if
C = h̄ · S̄1
P − a.s.
for some h̄ ∈ RN +1 . Such a portfolio strategy h̄ is then called a replicating portfolio for
C.
Corollary 3.39. Suppose the market model is arbitrage-free and C is a contingent
claim.
(1) C is attainable if and only if it admits a unique arbitrage-free price.
(2) If C is not attainable , there exists a < b such that Π(C) = (a, b).
Proof.
(1) By Theorem 3.33.
(2) Since P is convex, Π(C) is convex. Hence, Π(C) is an interval.
It remains to show that Π(C) is open. ddddddddddddddd.
Remark 3.40. In fact, if C is not attainable,
Π(C) = (πinf (C), πsup (C)) ,
where
πinf (C) = inf EQ
Q∈P
C
,
1+r
and
πsup (C) = sup EQ
Q∈P
C
.
1+r
3.7. COMPLETE MARKET MODEL
85
Example 3.41. A financial market with one bond B0 = B1 = 1 and one stock S0 =
π = 1, S1 = S. Suppose S is a Poisson distributed random variable with parameter 1
under P, i.e.,
P(S = k) =
e−1
k!
for k − 0, 1, 2, ...
Then P is a risk-neural measure with E[S1 ] = 1 = π and the market model is arbitragefree.
Consider the contingent claim C = (S1 − K)+ . For any Q ∈ P, Due to Jensen’s
inequality,
EQ [C] = EQ [(S − K)+ ] ≥ (EQ [S] − K)+
= (π − K)+ = (1 − K)+ .
(3.7)
EQ [C] ≤ EQ [S] = π = 1.
(3.8)
Conversely, since C ≤ S,
This implies,
(1 − K)+ ≤ πinf (C) ≤ π C ≤ πsup (C) ≤ 1.
(3.7)
(3.8)
In fact, we can prove that
πinf (C) = (1 − K)+ ,
and
πsup (C) ≤ 1.
3.7. Complete market model
Definition 3.42.∗ A (arbitrage-free) market model is called complete if every contingent claim is attainable. Otherwise, this market model is called incomplete.
86
3. ONE-PERIOD MODELS
d Corollary 3.39 ddddddd complete market model ddd contingent claims
ddddddd.
Example 3.43.
(1) Consider
⎛ ⎞
⎜1⎟
b = ⎝ ⎠,
10
⎛
⎞
⎜1 1 ⎟
D=⎝
⎠.
9 11
Then (b, D) is a complete market model, since
⎧
⎪
⎪
⎨h0 · 1 + h1 · 9 = C(ω1 )
⎪
⎪
⎩h0 · 1 + h1 · 11 = C(ω2 )
has a unique solution (h0 , h1 )
(2) Consider
⎛ ⎞
⎜1⎟
b = ⎝ ⎠,
10
⎛
⎞
⎜1 1 1 ⎟
D=⎝
⎠.
9 11 12
Then (b, D) is an incomplete market model, since a contingent claim C with
C(ω1 ) = 10, C(ω2 ) = 8, C(ω3 ) = 12 is not attainable.
Theorem 3.44.∗ An arbitrage-free market model is complete if and only if there exists
exactly one risk-neutral probability measure.
Proof. “=⇒” For A ∈ F. IA is a contingent claim. Then the arbitrage-free price is
unique and
π
IA
= EQ
1
IA
=
Q(A).
1+r
1+r
Thus,
Q(A) = (1 + r) π IA
for all A ∈ F .
This mens that Q is unique, i.e., the risk-neutral measure is unique.
3.7. COMPLETE MARKET MODEL
87
“⇐=” Suppose P = {Q}. Then any contingent claim has a unique arbitrage-free price .
This implies that C is attainable due to Corollary 3.39.
Example 3.45. Assume that Ω = {ω1 , ω2 } and N = 1, this implies that there are one
bond (with interest rate r) and one stock in the market model.
Moreover, suppose that the bond price is given by
B0 = 1,
B1 = 1 + r,
the stock price is at time 1 is given by 0 ≤ a = S1 (ω2 ) < b = S1 (ω1 ) and p = P({ω1 }) =
P(S1 = b) ∈ (0, 1).
(1) This market model does not admit arbitrage opportunity if nd only if
S0
S1
∈
EQ
:Q∼P
1+r
a
b
pb + (1 − p)a
: p ∈ (0, 1) =
,
=
.
1+r
1+r 1+r
(2) For any given S0 ∈
a
b
,
, the risk-neutral measure P∗ must satisfy
1+r 1+r
S0 (1 + r) = E∗ [S1 ] = p∗ b + (1 + p∗ )a.
Hence, P∗ is unique and is given by
⎧
⎪
S (1 + r) − a
⎪
⎨P∗ ({ω1 }) = 0
b−a
⎪
b
−
S
⎪
0 (1 + r)
⎩P∗ ({ω2 }) =
.
b−a
This means that the market model is complete.
(3) An alternative method to show that the market model is complete: for any
contingent claim C, find h̄ = (h0 , h1 )T such that C = h̄ · S̄1 , i.e., find h0 , h1 such
88
3. ONE-PERIOD MODELS
that
⎧
⎪
⎪
⎨h0 (1 + r) + h1 S1 (ω1 ) = C(ω1 ),
⎪
⎪
⎩h0 (1 + r) + h1 S1 (ω2 ) = C(ω2 ).
Its solution is given by
⎧
⎪
C(ω2 )b − C(ω1 )a
⎪
⎪
⎨h0 =
,
(1 + r)(b − a)
⎪
C(ω1 ) − C(ω2 )
⎪
⎪
⎩h1 =
.
b−a
This implies that C is attainable.
(4) The arbitrage-free price π C is given by
π
C
∗
= E
=
h̄ · S̄1
= h̄ · S̄0
1+r
C(ω1 ) − C(ω2 )
C(ω2 )b − C(ω1 )a
·1+
· S0
(1 + r)(b − a)
b−a
C(ω1 ) S0 (1 + r) − a C(ω2 ) b − S0 (1 + r)
C
∗
+
=E
=
.
1+r
b−a
1+r
b−a
1+r
In particular, if C = (S1 − K)+ with strike price K ∈ (a, b), then
+
π (S1 −K) =
b−K
1 (b − K)a
S0 −
.
b−a
1+r b−a
Exercise
(1) Consider the market model
⎛⎛
⎞ ⎛
⎞⎞
⎜⎜ 1 ⎟ ⎜ 1.1 1.1 ⎟⎟
(b, D) = ⎝⎝ ⎠ , ⎝
⎠⎠ .
5
8
4
(a) Investigate if the market model is complete and arbitrage-free.
(b) Find the price of the call option with strike price K = 6.
3.7. COMPLETE MARKET MODEL
(2) Consider the market model
⎛⎛
⎞ ⎛
89
⎞⎞
⎜⎜ 1 ⎟ ⎜ 1.1 1.1 1.1 ⎟⎟
(b, D) = ⎝⎝ ⎠ , ⎝
⎠⎠ .
5
8
4
6
(a) Investigate if the market model is complete and arbitrage-free.
(b) Find the price of the call option with strike price K = 6.
(3) Consider the market model
⎛⎛
⎞ ⎛
⎞⎞
⎜⎜ 1 ⎟ ⎜ 1.1 1.1 1.1 ⎟⎟
⎟⎟
⎜⎜
⎟ ⎜
⎟
⎜
⎟ ⎜
(b, D) = ⎜
4
6 ⎟
⎟⎟ .
⎜⎜ 5 ⎟ , ⎜ 7
⎠⎠
⎝⎝
⎠ ⎝
12 9
9
10
(a) Show that (b, D) is complete, but not arbitrage-free.
(b) Find an arbitrage opportunity.
(4) Consider the market model
⎛⎛
⎜⎜ 1
⎜⎜
⎜
(b, D) = ⎜
⎜⎜ 5
⎝⎝
10
⎞ ⎛
⎞⎞
⎟ ⎜ 1.1 1.1 1.1 1.1
⎟ ⎜
⎟,⎜ 7
4
6
3
⎟ ⎜
⎠ ⎝
12 9
9 13
⎟⎟
⎟⎟
⎟⎟ .
⎟⎟
⎠⎠
(a) Show that (b, D) is arbitrage-free, but not complete.
(b) Find the collection of all possible equivalent martingale measures.
(c) Find an contingent claim, which is not replicated.
(d) Find the set of all replicated contingent claims.
(5) Consider the market model
⎛⎛
⎞ ⎛
⎜⎜ 1 ⎟ ⎜ 1.1 1.1 1.1
⎟ ⎜
⎜⎜
⎜ 5 ⎟,⎜ 3
(b, D) = ⎜
4
7
⎜⎜
⎟ ⎜
⎝⎝
⎠ ⎝
10
12 9 11
⎞⎞
⎟⎟
⎟⎟
⎟⎟ .
⎟⎟
⎠⎠
(a) Show that (b, D) is arbitrage-free and complete.
90
3. ONE-PERIOD MODELS
(b) Find the price of a call option and a put option with strike price K = 10.