CHAPTER 5
American Contingent Claim
dddddd contingent claim ddddd contingent claim. ddddddddd
dd contingent claim. dddd American contingent claim dd, ddddddddd
stopping time.
5.1. Stopping time
Consider a probability space (Ω, F, P) with filtration F = (Ft )t=0,1,2,...,T .
Definition 5.1. A random variable τ : Ω −→ {0, 1, 2, ...., T } ∪ {+∞} is called a
stopping time with respect to F if
{τ = t} ∈ Ft
for all t.
Example 5.2. Consider Ω = [0, 8), T = 4, and the filtration F given as follows:
F = σ ([0, 1/2), [1/2, 1), ..., [15/2, 8)) ,
F0 = σ ([0, 8)) ,
F1 = σ ([0, 4), [4, 8)) ,
F2 = σ ([0, 2), [2, 4), [4, 6), [6, 8)) ,
F3 = σ ([0, 1), [1, 2), ..., [7, 8)) ,
F4 = σ ([0, 1/2), [1/2, 1), ..., [15/2, 8)) .
113
114
5. AMERICAN CONTINGENT CLAIM
(1) Consider a random variable τ1 given in Figure 5.1. Since
{τ1 = 0} = ∅ ∈ F0 ,
{τ1 = 1} = [4, 8) ∈ F1 ,
{τ1 = 2} = [0, 2) ∈ F2 ,
{τ1 = 3} = [2, 4) ∈ F3 ,
{τ1 = 4} = ∅ ∈ F4 ,
τ1 is a stopping time.
3
2
1
1
2
3
4
5
6
7
8
Figure 5.1. The random variable τ1
(2) Consider a random variable τ2 given in Figure 5.2. Since {τ2 = 1} = [2, 4) ∈ F1 ,
τ2 is not a stopping time.
Remark 5.3.
(1) A random variable τ : Ω −→ {0, 1, 2, ...., T } ∪ {+∞} is a stop-
ping time if and only if {τ ≤ t} ∈ Ft for all t = 0, 1, 2, ..., T .
(2) If τ and σ are stopping times, then
τ ∨ σ,
are stopping times.
τ ∧ σ,
and
τ +σ
5.1. STOPPING TIME
115
3
2
1
1
2
3
4
5
6
7
8
Figure 5.2. The random variable τ2
Definition 5.4. Let τ be a stopping time, then
Fτ := {A ∈ F : A ∩ {τ ≤ t} ∈ Ft for all t ≤ T }
is called the σ-algebra of events determined prior to the stopping time τ .
Remark 5.5. Fτ is a σ-algebra.
Fτ dddddddd? ddddddd {τ = k} dddddddddd Fτ dd,
dd Fτ ddddd σ-algebra, dddddddd? ddd, dddddddd Fτ dd
dd {τ = k} dddd Fk dddddddddd σ-algebra.
Example 5.6.
(1) Consider the stopping time τ1 given in Example 5.2 (1). Then
Fτ1 = σ([1, 2), [2, 3), [3, 7/2), [7/2, 4), [4, 8)).
(2) Suppose that Ω = {ω1 , ω2 , ..., ω10 } and (Xn )n=0,1,2,3 is given in Figure 5.3. Then
116
5. AMERICAN CONTINGENT CLAIM
16
12
12
8
20
2
12
10
6
10
13
8
11
6
13
14
9
24
4
Figure 5.3. The stochastic process (Xn ) and τ
F = σ({ω1 }, {ω2 }, {ω3 }, ..., {ω10 })
F0 = σ(X0 ) = {∅, Ω},
F1 = σ(X0 , X1 ) = σ({ω1 , ω2 , ..., ω5 }, {ω6 , ω7 , ..., ω10 })
F2 = σ(X0 , X1 , X2 ) = σ({ω1 , ω2 }, {ω3 }, {ω4 , ω5 }, {ω6 }, {ω7 , ω8 }, {ω9 , ω10 }),
F3 = σ(X0 , X1 , X2 , X3 ) = F.
Let
τ = inf{n ≥ 0 : Xn ≥ 13 or Xn ≤ 8}.
Note that we define inf ∅ = ∞. Then
τ (ω3 ) = ∞,
τ (ω1 ) = 2,
τ (ω2 ) = 2,
τ (ω5 ) = 2,
τ (ω6 ) = τ (ω7 ) = τ (ω8 ) = τ (ω9 ) = τ (ω10 ) = 1.
τ (ω4 ) = 2,
5.1. STOPPING TIME
We see that τ is a stopping time, since
{τ = 0} = ∅ ∈ F0 ,
{τ = 1} = {ω6 , ω7 , ..., ω10 } ∈ F1 ,
{τ = 2} = {ω1 , ω2 , ω4 , ω5 } ∈ F2 ,
{τ = 3} = ∅ ∈ F3
and
Fτ = σ({ω1 , ω2 }, {ω3 }, {ω4 , ω5 }, {ω6 , ω7 , ..., ω10 }).
(3) Consider Ω = {ω1 , ω2 , ..., ω16 } and
F = σ({ω1 }, {ω2 }, {ω3 }, ..., {ω16 })
F0 = {∅, Ω},
F1 = σ({ω1 , ω2 , ..., ω8 }, {ω9 , ω10 , ..., ω16 })
F2 = σ({ω1 , ..., ω4 }, {ω5 , ..., ω8 }, {ω9 , ..., ω12 }, {ω13 , ..., ω16 }),
F3 = σ({ω1 , ω2 }, {ω3 , ω4 }, ..., {ω13 , ω14 }, {ω15 , ω16 }),
F4 = F = σ({ω1 }, {ω2 }, {ω3 }, ..., {ω16 }).
Suppose that τ : Ω −→ {0, 1, 2, 3, 4} ∪ {∞} satisfies
{τ = 0} = ∅ ∈ F0 ,
{τ = 1} = ∅ ∈ F1 ,
{τ = 2} = {ω1 , ω2 , ..., ω8 } ∈ F2 ,
{τ = 3} = {ω9 , ω10 , ω15 , ω16 } ∈ F3 ,
{τ = 4} = {ω11 , ω12 } ∈ F4 .
117
118
5. AMERICAN CONTINGENT CLAIM
Then τ is a stopping time and
Fτ = σ({ω1 , ..., ω4 }, {ω5 , ..., ω8 }, {ω9 , ω10 }, {ω11 }, {ω12 }, {ω13 }, {ω14 }, {ω15 , ω16 }).
Lemma 5.7. Let τ be a stopping time, then the random variable
Xτ (ω) := Xτ (ω) (ω)
is Fτ -measurable.
Proof. For k ≥ 0,
{Xτ ≤ a} ∩ {τ ≤ k} =
k
({Xτ ≤ a} ∩ {τ = n})
n=0
=
k
({Xn ≤ a} ∩ {τ = n}) ,
n=0
Since both of {Xn ≤ a} and {τ = n} belong to Fn , which is a σ-subalgebra of Fk for
n ≤ k, we have
{Xτ ≤ a} ∩ {τ ≤ k} ∈ Fk ,
for all k. This implies that {Xτ ≤ a} ∈ Fτ . Thus, Xτ is Fτ -measurable.
Definition 5.8. For a stochastic process X = (Xt )0≤t≤T and a stopping time τ , the
stopped process X τ = (Xtτ )0≤t≤T is defined by
⎧
⎪
⎨ Xτ ,
τ
Xt = Xτ ∧t :=
⎪
⎩ Xt ,
if τ ≤ t,
if τ > t.
Theorem 5.9. Let M be an adapted process with Mt ∈ L1 (Q) for each t. Then the
following statements are equivalent.
(1) M is a Q-martingale;
(2) For any stopping time τ , the stopped process M τ is a Q-martingale;
5.1. STOPPING TIME
119
(3) EQ [Mτ ∧t ] = M0 for any stopping time τ .
Proof. (1) =⇒ (2): Since
τ
− Mtτ = (Mt+1 − Mt )I{τ >t} ,
Mt+1
we have
τ
EQ [Mt+1
− Mtτ |Ft ] = EQ [(Mt+1 − Mt )I{τ >t} |Ft ]
= I{τ >t} EQ [Mt+1 − Mt |Ft ] = 0
due to {τ > t} ∈ Ft .
(2) =⇒ (3): By definition,
M0 = M0τ = EQ [MTτ |F0 ] = EQ [MTτ ] = EQ [Mτ ∧T ].
(3) =⇒ (1): For A ∈ Ft , consider the stopping time
τ1 = t IA + T IAc .
Then
M0 = EQ [Mτ1 ∧T ] =
Ω
Mτ1 ∧T dQ =
A
Mt dQ +
Ac
MT dQ.
(5.1)
MT dQ.
(5.2)
Moreover, consider the stopping time τ2 ≡ T ,
M0 = EQ [Mτ2 ∧T ] =
Ω
Mτ2 ∧T dQ =
A
MT dQ +
Ac
Comparing (5.1) and (5.2), we have
A
Mt dQ =
This implies that Mt = E[MT |Ft ].
A
MT dQ,
for all A ∈ Ft .
120
5. AMERICAN CONTINGENT CLAIM
Corollary 5.10. Let U be an adapted process such that Ut ∈ L1 (Q) for each t. Then
the following conditions are equivalent.
(1) U is a Q-supermartingale.
(2) For any stopping time τ , the stopped process U τ is a Q-supermartingale.
Proof. If U = M − A is the Doob decomposition of U , then U τ = M τ − Aτ is the
Doob decomposition of U τ . By Theorem 5.9 we can get the desired results.
Exercise
(1) Prove that Fτ is a σ-algebra.
5.2. American claims
Definition 5.11. An American contingent claim is a security that can be exercised at
any time t between 0 and its expiration time T .
Definition 5.12. An exercise strategy for an American contingent claim C is an FT measurable random variable τ taking values in {0, 1, 2, ..., T }. The payoff obtained by
using τ is equal to
Cτ (ω) = Cτ (ω) (ω),
ω ∈ Ω.
ddddddd exercise strategy ddddddd stopping time. ddddddd
d τ dddd 0 d T ddd FT -measurable random variable dd.
Example 5.13.
(1) An American put option on the ith asset and with strike
price K > 0 pays the amount
Ctput := (K − Sti )+
5.3. ARBITRAGE-FREE PRICES
121
if it is exercised at time t.
(2) The payoff at time t of the corresponding American call option is given by
Ctcall := (Sti − K)+
Remark 5.14. The concept of American contingent claim can be regarded as a generalization of European contingent claims: If C E is a European contingent claim, then we
can define a corresponding American claim C A by
CtA =
⎧
⎪
⎨ 0,
if t < T,
⎪
⎩ CE,
if t = T.
ddd T dd exercise dd, ddddddd T ddd.
Definition 5.15. The process
Ht =
Ct
,
Bt
t = 0, 1, 2, ..., T
of discounted payoff of C will be called the discounted American claim associated with
C.
5.3. Arbitrage-free prices
Notation 5.16. For fixed exercise strategy τ , we denote the set of arbitrage-free prices
by
Π(Hτ ) = {E∗ [Hτ ] : P∗ ∈ P, E∗ [Hτ ] < ∞}.
dd Hτ ddddd European contingent claim. ddddddddddddddd
d.
122
5. AMERICAN CONTINGENT CLAIM
16
13
2/5
1/2
3/5
10
1/8
1/2
1/2
2/7
12 w 2
16 w 3
11
5/7
9 w4
14
2/3
15 w 5
1/3
7
7/8
S0
20 w 1
1/2
S1
6
S2
12 w 6
10 w 7
1/3
2/3
S3
w8
4
Figure 5.4. The price process and the equivalent martingale measure in
every period
Example 5.17. Consider a financial market with interest rate r = 0 and the stock
prices are given in Figure 5.4.
The equivalent martingale measure is given by
1
2
1
Q({ω3 }) =
2
1
Q({ω5 }) =
2
1
Q({ω7 }) =
2
Q({ω1 }) =
2
5
3
·
5
1
·
8
7
·
8
·
1
2
2
·
7
2
·
3
1
·
3
·
1
,
10
3
= ,
35
1
= ,
24
7
= ,
48
=
1
2
1
Q({ω4 }) =
2
1
Q({ω6 }) =
2
1
Q({ω8 }) =
2
Q({ω2 }) =
2
5
3
·
5
1
·
8
7
·
8
·
1
2
5
·
7
1
·
3
2
·
3
·
Let
τ = inf{t : St ≥ 14 or St ≤ 8} ∧ 3.
Then
τ (ω1 ) = τ (ω2 ) = 2,
τ (ω3 ) = τ (ω4 ) = 3,
τ (ω5 ) = τ (ω6 ) = τ (ω7 ) = τ (ω8 ) = 1.
1
,
10
3
= ,
14
1
= ,
48
7
= .
24
=
5.3. ARBITRAGE-FREE PRICES
123
Consider Ht = (St − 12)+ . Then
Hτ (ω1 ) = Hτ (ω2 ) = H2 (ω2 ) = (16 − 12)+ = 4,
Hτ (ω3 ) = H3 (ω3 ) = (16 − 12)+ = 4,
Hτ (ω4 ) = H3 (ω4 ) = (9 − 12)+ = 0,
Hτ (ω5 ) = H3 (ω6 ) = H3 (ω7 ) = H3 (ω8 ) = 0.
Hence,
EQ [Hτ ] = 4 ·
1
3
8
1
+4·
+4·
= ,
10
10
35
7
i.e., Π(Hτ ) = {8/7}.
dddddddddd American contingent claim ddddd exercise strategy τ
ddd, dddddd τ dddd, American contingent claim ddd, i.e., ddddd
dd American contingent claim H d Arbitrage-free price.
Definition 5.18. A real number π is called an arbitrage-free price of a discounted
American claim H if the following two conditions are satisfied.
(i) The price π is not too high in the sense that there exists some stopping time
τ ∈ T := {τ : stopping time with t ≤ T }
and π ∈ Π(Hτ ) such that π ≤ π .
(ii) The price π is not too low in the sense that there exists no τ ∈ T such that
π < π for all π ∈ Π(Hτ ).
(i) ddddd Π(Hτ ) ddddd π ddd.
(ii) ddddddd. ddddddddddd exercise strategy τ ddd π ddd
dddddd τ d arbitrage-free prices.
124
5. AMERICAN CONTINGENT CLAIM
Notation 5.19. The set of all arbitrage-free prices of H is denoted by Π(H), and we
define
πinf (H) = inf Π(H)
and
πsup (H) = sup Π(H).
Remark 5.20. Any arbitrage-free price π for H must be of the form π = E∗ [Hτ ] for
some P∗ ∈ P and τ ∈ T . This implies that
sup inf
E∗ [Hτ ] ≤ π ≤ sup sup E∗ [Hτ ]
∗
τ ∈T P ∈P
τ ∈T P∗ ∈P
for all π ∈ Π(H). In particular, if the market model is complete, i.e., P = {P∗ }, then
sup E∗ [Hτ ]
τ ∈T
is the unique arbitrage-free price.
Theorem 5.21. If Ht ∈ L1 (P∗ ) for all t and for each P∗ ∈ P, the set of arbitrage-free
prices for H is an interval with endpoints
πinf (H) = inf
sup E∗ [Hτ ] = sup inf
E∗ [Hτ ]
∗
∗
P ∈P τ ∈T
τ ∈T P ∈P
and
πsup (H) = sup sup E∗ [Hτ ] = sup sup E∗ [Hτ ].
P∗ ∈P τ ∈T
τ ∈T P∗ ∈P
Moreover, Π(H) either consists of one single point or does not contain its upper endpoint
πsup (H).
dddddddd Π(H) dddddd
(1) ∅;
(2) {πsup (H)};
(3) (πinf (H), πsup (H)),
(4) [πinf (H), πsup (H)).
5.3. ARBITRAGE-FREE PRICES
125
ddddddddddddd, ddddddd inf d sup dddddd. dddd
dddddddddddd minimax theorem. dddddddd πinf (H) ddddd
ddd Π(H) dd, d Example 5.24 dddddd πinf (H) dd Π(H) dddddd
Π(H) ddddd.
Example 5.22. Consider a market model with r = 0, T = 2. The price is given in
Figure 5.5.
16
1/2
14
1/3
1/2
12
10
10
1/2
2/3
8
1/2
6
Figure 5.5. The stock price
This is a complete market model with equivalent martingale measure
1
3
2
Q({ω3 }) = Q({ω4 }) =
3
Q({ω1 }) = Q({ω2 }) =
1
1
= ,
2
6
1
1
· = .
2
3
·
Suppose Ht = (St − 11)+ , then its payoff is given in Figure 5.6.
126
5. AMERICAN CONTINGENT CLAIM
1/2
5
3
1/3
1/2
0
1/2
1
0
2/3
0
1/2
0
Figure 5.6. The payoff of H
The collection T of all stopping times ≤ 2 is {τ1 , τ2 , τ3 , τ4 , τ5 }1, where
τ1 ≡ 0,
1dddd
τ2 ≡ 1,
τ3 ≡ 2,
τ4 (ω1 ) = τ4 (ω2 ) = 1,
τ4 (ω3 ) = τ4 (ω4 ) = 2,
τ5 (ω1 ) = τ5 (ω2 ) = 2,
τ5 (ω3 ) = τ5 (ω4 ) = 1.
stopping time dddddddddddd. dddddddddd: d t = 0 dd.
{τ = 0} ∈ F0 implies that {τ = 0} = ∅ or Ω.
• {τ = 0} = Ω: This means that τ ≡ 0, i.e., τ1 .
• {τ = 0} = ∅: ddddddd. {τ = 1} ∈ F1 . This means that {τ = 1} = ∅, Ω, {ω1 , ω2 }, or
{ω3 , ω4 }.
– {τ = 1} = Ω. This is the case of τ2 .
– {τ = 1} = ∅. This implies that {τ = 2} = Ω. The case of τ3 .
– {τ = 1} = {ω1 , ω2 }. We have {τ = 2} = {ω3 , ω4 }. The case of τ4 .
– {τ = 1} = {ω3 , ω4 }. {τ = 2} = {ω1 , ω2 }. The case of τ5 .
5.3. ARBITRAGE-FREE PRICES
16
q
14
(4-5q)/3
1/3
127
12
-1/3+2/3 q
10
6
15
3/10
2/3
8
7/10
5
Figure 5.7. The stock price
EQ [Hτ1 ] = 0
EQ [Hτ2 ] =
4
Q({ωi })Hτ2 (ωi ) =
i=1
1
6
1
EQ [Hτ3 ] =
6
1
EQ [Hτ4 ] =
6
1
EQ [Hτ5 ] =
6
=
4
Q({ωi })H1 (ωi )
i=1
1
6
1
·5+
6
1
·3+
6
1
·5+
6
·3+
1
3
1
·1+
3
1
·3+
3
1
·1+
3
·3+
1
3
1
·0+
3
1
·0+
3
1
·0+
3
·0+
· 0 = 1,
· 0 = 1,
· 0 = 1,
· 0 = 1,
The arbitrage-free price is given by sup EQ [Hτ ] = 1, i.e.,
τ ∈T
Π(H) = {1}.
Example 5.23. Consider a market model with r = 0, T = 2. The price is given in
Figure 5.7.
128
5. AMERICAN CONTINGENT CLAIM
5
q
3
(4-5q)/3
1/3
-1/3+2/3 q
1
0
0
4
3/10
2/3
0
7/10
0
Figure 5.8. The payoff of H
Then the equivalent martingale measures are of the form
Qq ({ω1 }) =
Qq ({ω2 }) =
Qq ({ω3 }) =
Qq ({ω4 }) =
Qq ({ω5 }) =
1
q,
3
1
(4 − 5q),
9
1 2
− + q,
9 9
1
,
5
7
.
15
for
1
4
<q< ,
2
5
Consider Ht = (St − 11)+ , then its payoff is given in Figure 5.8.
The collection T of all stopping times ≤ 2 is given by {τ1 , τ2 , τ3 , τ4 , τ5 }, where
τ1 ≡ 0,
τ2 ≡ 1,
τ3 ≡ 2,
τ4 (ω1 ) = τ4 (ω2 ) = τ4 (ω3 ) = 1,
τ4 (ω4 ) = τ4 (ω5 ) = 2,
τ5 (ω1 ) = τ5 (ω2 ) = τ4 (ω3 ) = 2,
τ5 (ω4 ) = τ5 (ω5 ) = 1.
5.3. ARBITRAGE-FREE PRICES
Then
Eq [Hτ1 ] = 0,
2
1
· 3 + · 0 = 1,
3
3
1
1 56 10
1
Eq [Hτ3 ] =
q · 5 + (4 − 5q) · 1 + · 4 +
q,
3
9
5 45
9
⎧
⎪
⎪ inf Eq [Hτ ] = 56 + 10 · 1 = 9 ,
⎨
3
q
45
9 2
5
10
4
32
56
⎪
⎪
+
· = ,
⎩ sup Eq [Hτ3 ] =
45
9 5
15
q
Eq [Hτ2 ] =
1
9
1
·3+ ·4= ,
3
5
5
1
4 10
1
Eq [Hτ5 ] =
q · 5 + (4 − 5q) · 1 = +
q,
3
9
9
9
⎧
4 10 1
⎪
⎪
· = 1,
⎨ inf Eq [Hτ5 ] = +
q
9
9 2
4
4 10 4
⎪
⎪ sup Eq [Hτ5 ] = +
· = .
⎩
9
9 5
3
q
Eq [Hτ4 ] =
Thus, we have
9
9 9
πinf (H) = sup inf
E [Hτ ] = sup 0, 1, , , 1 = ,
∗
5 5
5
τ ∈T P ∈P
32 9 4
32
πsup (H) = sup sup E∗ [Hτ ] = sup 0, 1, , ,
= .
15 5 3
15
τ ∈T P∗ ∈P
∗
Hence,
9 32
Π(H) = ,
5 15
ddddddddddd open, ddd closed.
.
129
130
5. AMERICAN CONTINGENT CLAIM
Example 5.24. dddddddddddd πinf ddddddddddd Π(H)
d. As in Example 5.22, let Ω0 = {ω1 , ω2 , ω3 , ω4 } with P0 ({ωi }) > 0 for i = 1, 2, 3, 4 and
F 0 = σ({ω1 }, {ω2 }, {ω3 }, {ω4 })
F00 = {φ, Ω}
F10 = σ({ω1 , ω2 }, {ω3 , ω4 })
F20 = F 0
This market model (S, F0 ) is complete on (Ω0 , F 0 , P0 ). Enlarge this model by adding two
external states ω + and ω − . Let
Ω = Ω0 × {ω + , ω − } = {(ω1 , ω + ), (ω1 , ω − ), · · · , (ω4 , ω + ), (ω4 , ω − )}
P({(ωi , ω + )}) = P({(ωi , ω − )}) =
1 0
P ({ωi })
2
for
i = 1, 2, 3, 4,
and the filtration is given by
F0 = {φ, Ω}
F1 = σ({(ω1 , ω + ), (ω2 , ω + )}, {(ω1 , ω − ), (ω2 , ω − )}, {(ω3 , ω + ), (ω4 , ω + )}, {(ω3 , ω − ), (ω4 , ω − )})
F2 = σ({(ωi , ω + )}, {(ωi , ω − )},
i = 1, 2, 3, 4.)
5.3. ARBITRAGE-FREE PRICES
131
Then this market model is incomplete.2 Moreover, the collection of all equivalent martingale measures P consists of all probability measure P∗q with
1
q
6
1
P∗q ({(ω3 , ω + )}) = P∗q ({(ω4 , ω + )}) = q
3
1
P∗q ({(ω1 , ω − )}) = P∗q ({(ω2 , ω − )}) = (1 − q)
6
1
P∗q ({(ω3 , ω − )}) = P∗q ({(ω4 , ω − )}) = (1 − q)
3
P∗q ({(ω1 , ω + )}) = P∗q ({(ω2 , ω + )}) =
for 0 < q < 1.
(1) Consider the discounted American contingent claim H
H0 ≡ 0
H1 ≡ 1
⎧
⎪
⎪
⎨2
H2 =
⎪
⎪
⎩0
if ω = (ωi , ω + ),
i = 1, 2, 3, 4.
if ω = (ωi , ω − ),
i = 1, 2, 3, 4.
If q ≥ 1/2, τ ≡ 2 is an optimal stopping time for P∗q . Then
E∗q [Hτ ] = 2q.
2For
example, the contingent claim H with
H=
for all i, is obviously not attainable.
⎧
⎪
⎨ 0,
on (ωi , ω + ),
⎪
⎩ 1,
on (ωi , ω − ),
132
5. AMERICAN CONTINGENT CLAIM
This implies that
⎧
⎪
⎪
inf E∗q [Hτ ] = 1 inf 2q = 1
⎪
⎨ 1 ≤q<1
≤q<1
2
2
⎪
∗
⎪
⎪
⎩ 1sup Eq [Hτ ] = 1sup 2q = 2
2
≤q<1
2
≤q<1
If q < 1/2, τ ≡ 1 is an optimal stopping time for P∗q and
E∗q [Hτ ] = E∗q [H1 ] = 1
Thus, πinf (H) = 1, πsup (H) = 2, this implies that Π(H) = [1, 2).3
(2) Consider the discounted American contingent claim H̄ with
H̄0 ≡ 0
H̄1 ≡ 0
⎧
⎪
⎪
⎨2
H̄2 =
⎪
⎪
⎩0
if ω = (ωi , ω + ),
i = 1, 2, 3, 4.
if ω = (ωi , ω − ),
i = 1, 2, 3, 4.
Then τ = 2 is an optimal stopping time for P∗q for q ∈ (0, 1). Moreover,
E∗q [H̄τ ] = E∗q [H̄2 ] = 2q.
Hence,
⎧
⎪
⎪
⎨ inf E∗q [H̄τ ] = inf 2q = 0,
0<q<1
0<q<1
⎪
∗
⎪
⎩ sup Eq [H̄τ ] = sup 2q = 2,
0<q<1
0<q<1
i.e., πinf (H) = 0, πsup (H) = 2, this implies that Π(H) = (0, 2).4
3“1”
dddddddd q ≤ 1/2 d, d optimal exercise strategy τ = 1.
4dd,
“0” dd Π(H̄) ddddd q ∈ (0, 1), q = 0 ddddd.
5.3. ARBITRAGE-FREE PRICES
133
Definition 5.25. A discounted American claim H is attainable if there exists a stopping time τ ∈ T and a self-financing strategy h̄ whose value process V (h̄) satisfies
Vt (h̄) ≥ Ht
for all t
and
Vτ (h̄) = Hτ , P − a.s.
The trading strategy h̄ is called a hedging strategy for H.
If H is attainable, the seller in on the safe side even if the buyer would have full
knowledge of future prices.
Remark 5.26. If the market model is complete, every American contingent claim is
attainable.
Theorem 5.27. For a discounted American claim H satisfying Ht ∈ L1 (P∗ ) for all t
and for all P∗ ∈ P, the following conditions are equivalent.
(1) H is attainable.
(2) H admits a unique arbitrage-free price π(H), i.e., Π(H) = {π(H)}.
(3) πsup (H) ∈ Π(H).
Exercise
(1) Consider a stochastic process X = (Xn )n=0,1,2,3 whose values are given in Figure
1.
(a) Suppose the sample space Ω = {ωi : i = 1, 2, ..., 24}. What is the filtration
(Fn )n=0,1,2,3 generated by the process X.
(b) Assume that
T1 = inf{n ≥ 0 : Xn ≥ 13 or Xn ≤ 7} ∧ 3.
Please write down T1 (ωi ) for all i. Is T1 a stopping time?
134
5. AMERICAN CONTINGENT CLAIM
16
12
13
11
8
10
14
8
5
12
6
8
5
20
16
4
15
11
6
20
12
24
12
12
9
2
21
11
6
4
17
12
7
20
15
9
2
Figure 5.9. The stochastic process X = (Xn ) in Exercise 1.
(c) What is FT1 ?
(d) Please write down the random variable XT1 and the stochastic process X T1 .
(e) Let
T2 = arg max{X0 , X1 , X2 , X3 }.
Please write down T2 (ωi ) for all i. Is T2 a stopping time?
(f) Find the set of all arbitrage-free prices of an American call option with
maturity 2 and strike price 12.