50
3. Finite Market Models
Recall that during last lecture we defined iteratively the minimum wealth
process U for the seller of an ACC Y , i.e.
8
<UT = YT
:U = max Y , (1 + r) 1 EQ [U |F ]
t 1
t 1
t
t 1
(3.14)
8t = 1, . . . , T,
and we found a trading strategy (˜
↵, ˜) replicating U , but which is not selffinancing, i.e.
8t = 1, . . . , T,
↵
˜ t St + ˜t Bt = Ut
↵
˜ t St 1 + ˜t Bt 1 = (1 + r) 1 EQ [Ut |Ft 1 ]
(3.15)
Then, for all t = 1, . . . , T , we define the excess wealth ˜t in U at time t 1 over
what is needed to finance the strategy (˜
↵, ˜) over the time period (t 1, t],
i.e.
˜t := Ut
1
(1 + r) 1 EQ [Ut |Ft 1 ].
(3.16)
Finally, we defined another trading strategy (↵⇤ , ⇤ ) that trades in the stock
and bond equally to (˜
↵, ˜) and, in addition, it invests the excess wealth at
each time in the bond, i.e.
↵t⇤ = ↵
˜t,
⇤
t
(3.17)
= ˜t +
t
X
n=1
˜n
Bn
,
1
8t = 1, . . . , T.
(3.18)
Now, we see the properties of this particular trading strategy defined in
(3.17)-(3.18) and of the minimum wealth process defined in (3.14).
Lemma 3.11. The trading strategy (U0 , ↵⇤ ,
⇤
) is a self-financing super-
hedging strategy for the ACC Y . Moreover, the random time
⌧0⇤ := min{t
0 : Ut = Yt }
is an F-stopping time and
Vt↵
⇤, ⇤
= Ut
8t : 0  t  ⌧0⇤ .
(3.19)
3.1 The (Multi-Period) Binomial Model
Proof. For all t = 1, . . . , T
⇤
↵t+1
St +
⇤
t+1 Bt
1,
=↵
˜ t+1 St + ˜t+1 Bt +
t+1
X
˜n
n=1
Bn
= (1 + r) 1 EQ [Ut+1 |Ft ] +
= Ut +
t
X
˜n
Bn
n=1
⇤
t Bt .
+
t
X
˜n
n=1
Bn
Bt + ˜t+1
by (3.15)
1
Bt
by (3.16)
t
X
˜n
n=1
=
Bt
1
1
=↵
˜ t St + ˜t Bt +
↵t⇤ St
51
Bn
Bt
by (3.15)
1
In order to conclude the proof of the self-financing property for (U0 , ↵⇤ ,
⇤
),
it remains the condition at time t = 0:
↵1⇤ S0 +
⇤
1
=↵
˜ 1 S0 + ˜1 + ˜1
EQ [U1 ]
=↵
˜ 1 S0 + ˜1 + U0
= U0 .
Here, we used the fact that F0 is the trivial -algebra to write EQ [U1 |F0 ] =
EQ [U1 ]. The value of the self-financing strategy (↵⇤ ,
⇤
) at any time t =
1, . . . , T is
⇤
Vt↵ ,
⇤
=
↵t⇤ St
+
⇤
t Bt
=↵
˜ t St + ˜t Bt +
t
X
˜n
n=1
because ˜t
0 for all t = 1, . . . , T . Thus (↵⇤ ,
for the ACC Y . Then,
⌧0⇤
⇤
Bn
Bt
Ut ,
(3.20)
1
) is a super-hedging strategy
2 T[0,T ] , because 0  ⌧0⇤  T by definition and
since UT = YT , and for all t = 1, . . . , T
t 1
{⌧0⇤ = t} = [ {Yn < Un } [ {Yt = Ut } 2 Ft .
n=0
Finally, for all t = 0, . . . , ⌧0⇤
↵⇤ , ⇤
Vt
1, Yt < Ut , hence ˜t+1 = 0. This implies
= Ut for all t = 0, . . . , ⌧0⇤ , by (3.20).
52
3. Finite Market Models
Before going on, let us recall a fundamental theorem (also present in the
Appendix) in Probability theory that is a very useful tool when dealing with
stopped processes, as in the case of American contingent claims.
Theorem 3.12 (Doob’s Optional Sampling). Let ⌧, ⌫ be two bounded Fstopping times on a probabiilty space (⌦, F, P), such that ⌧  ⌫  T P-almost
surely for some T 2 N. If X is a sub-martingale, then
X⌧  E[X⌫ |F⌧ ].
In particular, if X is a martingale, then
X⌧ = E[X⌫ |F⌧ ].
Let Y ⇤ = {Yt⇤ }t=0...,T , U ⇤ = {Ut⇤ }t=0...,T denote the discounted payo↵ and
minimum seller’s wealth, respectively, precisely
Yt⇤ = (1 + r) t Yt ,
Ut⇤ = (1 + r) t Ut ,
8t = 0, . . . , T.
Note that UT⇤ = YT⇤ and
⇤
Ut⇤ = max Yt⇤ , EQ [Ut+1
|Ft ] ,
8t = 0, . . . , T
1.
(3.21)
The process U ⇤ is called the Snell envelope of Y ⇤ .
Lemma 3.13. The Snell envelope U ⇤ of Y ⇤ has the following properties.
(i) U ⇤ is the smallest (F, Q)-martingale that dominates Y ⇤ , i.e. U ⇤
Y ⇤.
(ii) For all t = 0, 1, . . . , T , the random time
⌧t⇤ := min{s
t : Us = Ys }
is an element of T[t,T ] .
(iii) For all t = 0, 1, . . . , T , the stopped process
(F, Q)-martingale.
⇣
U⌧⇤t⇤ ^s
(3.22)
⌘
s=t,t+1,...,T
, is a
3.1 The (Multi-Period) Binomial Model
53
(iv) For all t = 0, 1, . . . , T ,
Ut⇤ = max EQ [Y⌧⇤ |Ft ] = EQ [Y⌧⇤t⇤ |Ft ].
⌧ 2T[t,T ]
(3.23)
The financial interpretation of (3.23) is that the Snell envelope is the
expectation under the risk-neutral measure of the payo↵ relative to the best
exercise time.
Proof. (i). The proof is in the solution of the exercise sheet 7.
(ii). We have already proved it for t = 0, and the proof for t = 1, . . . , T is
analogous.
(iii). Let us prove it for t = 0, for t = 1, . . . , T being analogous. First, U⌧⇤0⇤ ^·
is adapted by Remark A.46 and integrable by finiteness of the probability
space. Then, for all t = 1, . . . , T
U⌧⇤0⇤ ^(t+1)
1,
⇤
U⌧⇤0⇤ ^t = 1{⌧0⇤ >t} (Ut+1
⇤
= 1{⌧0⇤ >t} Ut+1
Ut⇤ )
⇤
EQ [Ut+1
|Ft ] ,
by (3.14) and the definition of ⌧0⇤ . Thus, taking the conditional expectation
with respect to Ft we get
EQ [U⌧⇤0⇤ ^(t+1)
⇥ ⇤
U⌧⇤0⇤ ^t |Ft ] = 1{⌧0⇤ >t} EQ Ut+1
since {⌧0⇤ > t} 2 Ft , by Remark A.43.
⇤
⇤
EQ [Ut+1
|Ft ]|Ft = 0,
(iv). Let t 2 {0, 1 . . . , T } and ⌧ 2 T[t,T ] arbitrary. By (i) and Remark A.46,
(U⌧⇤^s )s=t,t+1,...,T , is a (F, Q)-super-martingale, hence
Ut⇤ = U⌧⇤^t
EQ [U⌧⇤^T |Ft ] = EQ [U⌧⇤ |Ft ]
by Theorem A.47. Thus Ut⇤
EQ [Y⌧⇤ |Ft ]
max⌧ 2T[t,T ] EQ [Y⌧⇤ |Ft ], because ⌧ was arbi-
trarily chosen. Then, we prove that there exists a stopping time achieving
the equality, in particular that ⌧t⇤ does so. By (iii), Theorem A.47 and the
definition of ⌧t⇤ , we have
Ut⇤ = U⌧⇤t⇤ ^t = EQ [U⌧⇤t⇤ ^T |Ft ] = EQ [U⌧⇤t⇤ |Ft ] = EQ [Y⌧⇤t⇤ |Ft ].
This ends the proof.
54
3. Finite Market Models
It remains to determine, if it exists, the arbitrage-free price for the ACC,
that is a price which does not allow for abritrage opportunities in the market
where trading in the stock and bond is allowed at all trading dates t =
0, 1, . . . , T and the ACC is available to buy/sell only at time t = 0. We
already have a candidate, that is the initial value U0 of the minimum superreplicating strategy for the seller.
Di↵erently from the case of European contingent claims, when defining
the trading strategies and arbitrage opportunities in the extended market,
here we have to distiguish the long and the short position in the ACC, because
the first has the possibility to exercise the claim at any time and the existence
of one exercise satisfying certain conditions is enough to give an arbitrage,
while the latter must satisfy the arbitrage conditions for all possible exercise
times chosen by the long counterparty. Precisely, we consider the following
definitions of arbitrage in the extended market.
Definition 3.14. Let Y = (Yt )t=0,...,T be the payo↵ process of an ACC with
price C0 at time 0. An arbitrage opportunity for the seller is a self-financing
trading strategy (↵s ,
s
) in the stock and bond such that V0↵
s, s
= C0 and,
for all exercise times ⌧ 2 T[0,T ] ,
V⌧↵
s, s
Y⌧
0,
and EQ [V⌧↵
s, s
Y⌧ ] > 0.
(3.24)
An arbitrage opportunity for the buyer is a self-financing trading strategy
(↵b ,
b
) in the stock and bond such that V0↵
b, b
=
C0 and there exists an
exercise time ⌧ 2 T[0,T ] such that
V⌧↵
b, b
+ Y⌧
0,
and EQ [V⌧↵
b, b
+ Y⌧ ] > 0.
(3.25)
C0 is an arbitrage-free price for Y if it does not allow for any arbitrage
opportunity for the seller and any arbitrage opportunity for the buyer.
To better understand this definition, let us consider an investor who sells
the ACC at price C0 and invests in the stock and bond according to an
arbitrage strategy (↵s ,
s
) for the seller. At any time ⌧ his/her counterparty
decides to exercise the claim, he/she would pay o↵ Y⌧ and invest the resultant
3.1 The (Multi-Period) Binomial Model
wealth V⌧↵
s, s
55
Y⌧ at time ⌧ in the bond, fixed until maturity. The final value
at time T of the described investment is then (V⌧↵
s, s
Y⌧ )(1 + r)T
⌧
, which
is non-negative and strictly positive with positive probability. A symmetric
investment could be described for an arbitrage strategy for a buyer of the
ACC.
Before the main result, let us remark on a property of the discounted
value process.
Lemma 3.15. The discounted value process of a self-financing trading strategy (↵, ) in the stock and bond,
V ⇤ = (Vt⇤ )t=0,...,T ,
Vt⇤ =
Vt↵,
,
(1 + r)t
satisfies:
EQ [V⌧⇤ ] = V0⇤ .
8⌧ 2 T[0,T ]
Proof. The claim follows directly from Lemma 3.7 and Theorem A.47.
Theorem 3.16. The unique arbitrage-free price for the ACC Y is U0 .
Proof. (Uniqueness). We have to prove that if C0 6= U0 , then there are
abritrage opportunities either for the seller or for the buyer of the ACC.
Suppose first that C0 > U0 . Consider the strategy (↵s ,
stock and bond according to (↵⇤ ,
to C0
⇤
s
) that trades in the
) and invest an amount of money equal
U0 in the bond over the whole time period [0, T ]. Precisely: for all
t = 1, . . . , T , ↵ts = ↵t⇤ and
s
t
⇤
t
=
+ C0
U0 . The initial value of such
strategy, by self-financing, is
V0↵
s, s
= ↵1⇤ S0 +
⇤
1
+ C0
U0 = C0 ,
and, for all exercise times ⌧ 2 T[0,T ] , by Lemma 3.11 and the definition of U
in (3.14), we have
V⌧↵
s, s
Y⌧ = V⌧↵
⇤, ⇤
+ (C0
U⌧ + (C0
(C0
U0 )B⌧
U0 )B⌧
U0 )B⌧ ,
Y⌧
Y⌧
56
3. Finite Market Models
which is strictly positive. Thus, (↵s ,
s
) is an arbitrage opportunity for the
seller.
Then, suppose that C0 < U0 . Consider the strategy (↵b ,
the stock and bond according to ( ↵⇤ ,
equal to U0
⇤
b
) that trades in
) and invest an amount of money
C0 in the bond over the whole time period [0, T ]. Precisely:
for all t = 1, . . . , T , ↵tb =
↵t⇤ and
b
t
=
⇤
t
+ U0
C0 . The initial value of
such strategy, by self-financing, is
V0↵
b, b
↵1⇤ S0
=
⇤
1
+ U0
C0 =
V0↵
⇤, ⇤
+ U0
C0 =
C0 .
Now, choosing the exercise time ⌧0⇤ defined in (3.22), we get
s
V⌧↵0⇤ ,
s
Y⌧0⇤ =
=
⇤, ⇤
V⌧↵0⇤
+ (U0
U⌧0⇤ + (C0
(U0
U0 )B⌧0⇤
Y⌧0⇤
Y⌧0⇤
by (3.19)
C0 )B⌧0⇤ ,
which is strictly positive. Thus, (↵s ,
buyer.
C0 )B⌧0⇤
s
) is an arbitrage opportunity for the