6: MULTI-PERIOD MARKET MODELS
Marek Rutkowski
School of Mathematics and Statistics
University of Sydney
Semester 2, 2016
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Outline
We will examine the following issues:
1
Trading Strategies and Arbitrage-Free Models
2
Risk-Neutral Probability Measures and Martingales
3
Fundamental Theorem of Asset Pricing
4
Arbitrage Pricing of Attainable Claims
5
Risk-Neutral Valuation of Non-Attainable Claims
6
Completeness of Multi-Period Market Models
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PART 1
SELF-FINANCING TRADING STRATEGIES
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Primary Traded Assets
In a multi-period market model M = (B, S 1 , . . . , S n ), we need to
examine the concept of a dynamic trading strategy φ and the associated
wealth process V (φ).
We first define primary traded assets
Let r be the interest rate. The money market account is denoted
by Bt for t = 0, 1, . . . , T where
Bt = (1 + r)t .
There are n risky assets, called stocks, with price processes denoted
by Stj for t = 0, 1, . . . , T .
We are given a probability space (Ω, F, P) endowed with a filtration F
generated by price processes of stocks.
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Dynamic Trading Strategy and Wealth Process
A dynamic trading strategy in a multi-period
market model is defined as
a stochastic process φt = φ0t , φ1t , . . . , φnt for t = 0, 1, . . . , T where:
φ0t is the number of ‘shares’ of the money market account B held at
time t.
φjt is the number of shares of the jth stock held at time t.
Definition (Value Process)
The wealth process (also known
as the value process) of a trading
0
1
n
strategy φ = φ , φ , . . . , φ is the stochastic process V (φ) given by
Vt (φ) := φ0t Bt +
n
X
φjt Stj .
j=1
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Self-Financing Trading Strategy
Definition (Self-Financing Trading Strategy)
A trading strategy φ is said to be self-financing strategy if for every
t = 0, 1, . . . , T − 1,
φ0t Bt+1
+
n
X
j
φjt St+1
=
φ0t+1 Bt+1
+
n
X
j
φjt+1 St+1
.
(1)
j=1
j=1
The LHS of (1) represents the value of the portfolio at time t + 1
before its revision, whereas the RHS represents the value at time t + 1
after the portfolio was revised.
Condition (1) says that these two values must be equal and this
means that no cash was withdrawn or added.
For t = T − 1, both sides of (1) represent the wealth at time T , that
is, VT (φ). We do not revise the portfolio at time T .
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Gains Process
Lemma (6.1)
For any self-financing trading strategy φ, the wealth process can be
alternatively computed by, for t = 1, . . . , T ,
Vt (φ) =
φ0t−1 Bt
+
n
X
φjt−1 Stj .
j=1
Proof.
The statement is an immediate consequence of formula (1).
Definition (Gains Process)
The gains process G(φ) = (Gt (φ))0≤t≤T of a trading strategy φ is given
by
Gt (φ) := Vt (φ) − V0 (φ).
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Multi-Period Market Model
The concept of a multi-period market model is a natural extension of
the notion of a single-period market model with only two dates: t = 0
and t = T = 1.
Definition (Market Model)
A multi-period market model M = (B, S 1 , . . . , S n ) is given by the
following data:
1
A probability space (Ω, F, P) with a filtration F = (Ft )0≤t≤T .
2
The money market account B given by Bt = (1 + r)t .
3
4
A number of risky financial assets with prices S 1 , . . . , S n , which are
assumed to be F-adapted stochastic processes.
The class Φ of all self-financing trading strategies.
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Increment Processes
As in the single-period model, we define the discounted value process,
gains process and discounted gains process.
It will be convenient to use the increment processes for traded assets.
Definition (Increment Processes)
The increment process ∆S j corresponding to the jth stock is
defined by
j
j
∆St+1
:= St+1
− Stj for t = 0, . . . , T − 1.
The increment process ∆B of the money market account is given by
∆Bt+1 := Bt+1 − Bt = (1 + r)t r = Bt r for t = 0, . . . , T − 1.
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Discounted Processes
Definition (Discounted Processes)
The discounted stock prices are given by
Sj
Sbtj := t
Bt
j
j
and the increments of discounted prices are ∆Sbt+1
:= Sbt+1
− Sbtj .
The discounted wealth process Vb (φ) of φ is given by
Vt (φ)
.
Vbt (φ) :=
Bt
The discounted gains process Ĝ(φ) of φ equals
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bt (φ) := Vbt (φ) − Vb0 (φ).
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Discounted Processes
Proposition (6.1)
An F-adapted trading strategy φ = (φt )0≤t≤T is self-financing if and only
if any of the two equivalent statements hold:
1
for every t = 1, . . . , T
t−1
X
Gt (φ) =
φ0u
∆Bu+1 +
j
.
φju ∆Su+1
u=0 j=1
u=0
2
t−1 X
n
X
for every t = 1, . . . , T
bt (φ) =
G
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t−1 X
n
X
u=0 j=1
j
φju ∆Sbu+1
.
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Properties of Discounted Gains and Wealth
Proof of Proposition 6.1.
The proof is elementary and thus it is left as an exercise.
b
It is important to note that the process G(φ)
given by condition 2)
0
does not depend on the component φ of a trading strategy φ ∈ Φ.
In view of Proposition 6.1, the discounted wealth process of any
φ ∈ Φ satisfies
Vbt (φ) = Vb0 (φ) +
t−1 X
n
X
u=0 j=1
Hence for every t = 0, . . . , T − 1
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bt+1 (φ) =
∆Vbt+1 (φ) = ∆G
j
φju ∆Sbu+1
.
n
X
j=1
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j
φjt ∆Sbt+1
.
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PART 2
RISK-NEUTRAL PROBABILITY MEASURES
AND MARTINGALES
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Arbitrage Opportunity
As usual, we work under the standing assumption that the sample space Ω
is finite (or at most countable).
Definition (Arbitrage Opportunity)
A trading strategy φ ∈ Φ is said to be an arbitrage opportunity if
1
V0 (φ) = 0,
2
VT (φ)(ω) ≥ 0 for all ω ∈ Ω,
3
VT (φ)(ω) > 0 for some ω ∈ Ω or, equivalently, EP (VT (φ)) > 0.
We say that a multi-period market model M is arbitrage-free if no
arbitrage opportunities exist in the class Φ of all self-financing trading
strategies.
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Arbitrage Conditions
Observe that in the arbitrage conditions one can use either the
b
discounted wealth process Vb or the discounted gains process G
(instead of the wealth V ).
It is also important to note that conditions 1)–3) hold under P
whenever they are satisfied under some probability measure Q
equivalent to P.
The next step is to introduce the concept of a risk-neutral probability
measure for a multi-period market model.
The risk-neutral probability measures (also known as the martingale
measures) are very closely related to the question of arbitrage-free
property and completeness of a multi-period market model.
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Risk-Neutral Probability Measure
Definition (Risk-Neutral Probability Measure)
A probability measure Q on (Ω, FT ) is called a risk-neutral probability
measure for a multi-period market model M = (B, S 1 , . . . , S n ) if
1
2
Q(ω) > 0 for all ω ∈ Ω,
j
EQ (∆Sbt+1
| Ft ) = 0 for all j = 1, . . . , n and t = 0, . . . , T − 1.
We denote by M the class of all risk-neutral probability measures for the
market model M.
Observe that condition 2) is equivalent to the following equality for every
t = 0, . . . , T − 1
j
| Ft = Sbtj .
EQ Sbt+1
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Martingales (MATH3975)
Martingales are stochastic processes representing fair games.
Definition (Martingale)
An F-adapted process X = (Xt )0≤t≤T on a finite probability space
(Ω, F, P) is called a martingale whenever for all s < t
EP (Xt | Fs ) = Xs .
To establish the martingale property, it suffices to check that for every
t = 0, 1, . . . , T − 1
EP (Xt+1 | Ft ) = Xt .
In particular, this means that the discounted stock price Sbj is a martingale
under any risk-neutral probability measure Q ∈ M.
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Discounted Wealth as a Martingale (MATH3975)
Proposition (6.2)
Let φ ∈ Φ be a trading strategy. Then the discounted wealth process
b
Vb (φ) and the discounted gains process G(φ)
are martingales under any
risk-neutral probability measure Q ∈ M.
Proof.
bt (φ) for every t = 0, . . . , T ,
Recall that Vbt (φ) = Vb0 (φ) + G
Since Vb0 (φ) (the initial endowment) is a constant, it suffices to show
b
that the process G(φ)
is a martingale under any Q ∈ M.
From Proposition 6.1, we obtain
bt+1 (φ) = G
bt (φ) +
G
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n
X
j=1
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j
φjt ∆Sbt+1
.
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Proof of Proposition 6.2 (MATH3975)
Proof of Proposition 6.2 (Continued).
Hence
bt+1 (φ)|Ft ) = G
bt (φ) +
EQ (G
bt (φ) +
= G
bt (φ)
= G
n
X
j=1
n
X
j EQ φjt ∆Sbt+1
Ft
j φjt EQ ∆Sbt+1
Ft
|
{z
}
j=1
=0
We used the fact that φjt is Ft -measurable and the “take out what is
known” property of the conditional expectation.
b
We conclude that G(φ)
is a martingale under any Q ∈ M.
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PART 3
FUNDAMENTAL THEOREM OF ASSET PRICING
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Fundamental Theorem of Asset Pricing
We will show that the Fundamental Theorem of Asset Pricing
(FTAP) can be extended to a multi-period market model.
Recall that the class of admissible trading strategies Φ in a
multi-period market model is assumed to be the full set of all
self-financing and F-adapted trading strategies.
It possible to show that in that case, the relationship between
the existence of a martingale measure Q and no arbitrage for the
model M is “if and only if”.
We will only prove here the following implication:
Existence of Q ∈ M ⇒ Model M is arbitrage-free
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Fundamental Theorem of Asset Pricing
Theorem (FTAP)
Consider a multi-period market model M = (B, S 1 , . . . , S n ).
The following statements hold:
1
if the class M of martingale measures for M is non-empty, then there
are no arbitrage opportunities in the class Φ of all self-financing
trading strategies and thus the model M is arbitrage-free,
2
if there are no arbitrage opportunities in the class Φ of all
self-financing trading strategies, then there exists a martingale
measure for M so that the class M is non-empty.
To sum up:
Class M is non-empty ⇔ Market model M is arbitrage-free
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Proof of the FTAP (⇒)
Proof of the FTAP (⇒).
Let us assume that a martingale measure Q for M exists. Our goal it to
show that the model M is arbitrage-free.
To this end, we argue by contradiction. Let us thus assume that there
exists an arbitrage opportunity φ ∈ Φ. Such a strategy would satisfy the
following conditions:
b0 (φ) = 0,
1 the initial endowment V
2
3
bT (φ) ≥ 0,
the discounted gains process G
bT (φ)(ω) > 0.
there exists at least one ω ∈ Ω such that G
On the one hand, from conditions 2. and 3. above, we deduce easily that
bT (φ) > 0.
EQ G
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Proof of the FTAP (⇒)
Proof of the FTAP (⇒).
On the other hand, using Proposition 6.1, we obtain
EQ
X
X
n T
n T
−1
−1
X
X
j
j
j
bT (φ) = EQ
G
EQ φju ∆Sbu+1
φu ∆Sbu+1 =
j=1 u=0
j=1 u=0
=
−1
n T
X
X
j=1 u=0
=
−1
n T
X
X
j
| Fu )
EQ EQ (φju ∆Sbu+1
j
| Fu ) = 0.
EQ φju EQ (∆Sbu+1
|
{z
}
j=1 u=0
=0
This clearly contradicts the inequality obtained in the first step. Hence the
market model M is arbitrage-free.
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Martingale Argument (MATH3975)
Let us now present a simple martingale argument:
We already know from Proposition 6.2 that the discounted gains
b
process G(φ)
is a martingale under Q.
Hence
bT (φ)) = EQ (G
bT (φ) | F0 ) = G
b0 (φ) = 0
EQ (G
where we used the properties of the conditional expectation.
This observation shortens the argument in the proof of the
implication (⇒) in the FTAP.
The proof of the implication (⇐) in the FTAP requires a good
familiarity with the theory of martingales, although its idea is the
same as in the single-period market case.
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PART 4
ARBITRAGE PRICING OF ATTAINABLE CLAIMS
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Replicating Strategy
Note that a contingent claim of European style can only be exercised
at its maturity date T (as opposed to contingent claims of American
style).
A European contingent claim in a multi-period market model is
an FT -measurable random variable X on Ω to be interpreted as
the payoff at the terminal date T .
For brevity, European contingent claims will also be referred to as
contingent claims or simply claims.
Definition (Replicating Strategy)
A replicating strategy (or a hedging strategy) for a contingent claim X
is a trading strategy φ ∈ Φ such that VT (φ) = X, that is, the terminal
wealth of the trading strategy matches the claim’s payoff for all ω.
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Principle of No-Arbitrage
Definition (Principle of No-Arbitrage)
An F-adapted stochastic process (πt (X))0≤t≤T is a price process for the
contingent claim X that complies with the principle of no-arbitrage if
there is no F-adapted and self-financing arbitrage strategy in the extended
f = (B, S 1 , . . . , S n , S n+1 ) with an additional asset S n+1 given by
model M
n+1
= πt (X) for 0 ≤ t ≤ T − 1 and STn+1 = X.
St
The standard method to price a contingent claim is to employ the
replication principle, if it can be applied.
The price will now depend on time t and thus one has to specify a
whole price process π(X), rather than just an initial price, as in the
single-period market model.
Obviously, πT (X) = X for any claim X.
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Arbitrage Pricing of Attainable Claims
In the next result, we deal with an attainable claim, meaning that
we assume a priori that a replicating strategy for X exists.
Proposition (6.3)
Let X be a contingent claim in an arbitrage-free multi-period market
model M and let φ ∈ Φ be any replicating strategy for X. Then the only
price process of X that complies with the principle of no-arbitrage is the
wealth process V (φ).
The arbitrage price at time t of an attainable claim X is unique and
it is also denoted as πt (X).
Hence the equality πt (X) = Vt (φ) holds for any replicating strategy
φ ∈ Φ for X.
In particular, the price at time t = 0 is the initial endowment of any
replicating strategy for X, that is, π0 (X) = V0 (φ) for any strategy
φ ∈ Φ such that VT (φ) = X.
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Example: Replication of a Digital Call Option
Example (6.1)
We will now examine replication of a contingent claim in a two-period
market model.
Consider a two-period market model consisting of the savings account
and one risky stock.
t
The interest rate equals r = 19 so that Bt = (1 + r)t = 10
.
9
The price of the stock is represented in the following exhibit
in which the real-world probability P is also specified.
It is easy to check that this model is arbitrage-free.
Our goal is to price a particular contingent claim X using replication.
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Example: Replication of a Digital Call Option
Example (6.1 Continued)
1
0.8
0.3
S2 = 3
ω1
0.7
S2 = 1
ω2
0.6
S1 = 1.5
0.4
0.6
0.2
0
S0 = 1
0.5
−0.2
−0.4
S2 = 1.5
ω3
S2 = 0.5
ω4
0.4
S1 = 0.75
−0.6
0.5
−0.8
−1
Figure: Stock Price Dynamics
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Example: Replication of a Digital Call Option
Example (6.1 Continued)
Consider a digital call option with the payoff function
1 if S2 (ω) > 2
X(ω) = g (S2 (ω)) =
0
otherwise
X = X(ω1 ), X(ω2 ), X(ω3 ), X(ω4 ) = (1, 0, 0, 0).
By the definition of replication, we have V2 (φ) = X or, more
explicitly,
V2 (φ) = φ01 B2 + φ11 S2 = g(S2 ) = (1, 0, 0, 0).
Observe that φi1 (ω1 ) = φi1 (ω2 ) and φi1 (ω3 ) = φi1 (ω4 ) for i = 0, 1
since φ is an F-adapted process.
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Example: Replication of a Digital Call Option
Example (6.1 Continued)
At time t = 1 we obtain two linear systems for φ1 = (φ01 , φ11 )
For ω ∈ {ω1 , ω2 }
10
9
10
9
2
2
φ01 + 3φ11 = 1
φ01 + φ11 = 0
For ω ∈ {ω3 , ω4 }
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10
9
10
9
2
2
φ01 + 1.5φ11 = 0
φ01 + 0.5φ11 = 0
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Example: Replication of a Digital Call Option
Example (6.1 Continued)
We obtain the replicating strategy at t = 1:
81 1
0 1
φ1 , φ1 = −
,
if ω ∈ {ω1 , ω2 }
200 2
φ01 , φ11 = (0, 0)
if ω ∈ {ω3 , ω4 }
If S1 = 1.5 then the price of the digital call at t = 1 equals
V1 (φ) = φ01 B1 + φ11 S1 = −
81 10 1 3
·
+ · = 0.3
200 9
2 2
If S1 = 0.75 then the price of the digital call at t = 1 equals 0.
The price of X at time 1 equals π1 (X) = 0.3 1{S1 =1.5} .
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Example: Replication of a Digital Call Option
Example (6.1 Continued)
We now compute the price of the digital call at t = 0. The replicating
strategy at time t = 0 satisfies φ00 B1 + φ10 S1 = π1 (X), that is,
10 0
φ + 1.5φ10 = 0.3
9 0
10 0
φ + 0.75φ10 = 0
9 0
Then φ00 , φ10 = (−0.27, 0.4). The price of the digital call at time
t = 0 (recall that S0 = B0 = 1) thus equals
V0 (φ) = φ00 B0 + φ10 S0 = −0.27 + 0.4 = 0.13
Hence the arbitrage price of X at time 0 equals π1 (X) = 0.13.
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Example: Replication of a Digital Call Option
Example (6.1 Continued)
Summary of pricing and hedging results for a digital call option.
Recall that the price at time t = 2 equals π2 (X) = X.
Replicating strategy φ satisfies V2 (φ) = X.
The arbitrage price process of X equals π(X) = V (φ).
t=0
t=0
t=1
t=1
{ω1 , ω2 }
φ0t , φ1t = (−0.27, 0.4)
π0 (X)
= 0.13
81 1
φ0t , φ1t = − 200
,2
π1 (X) = 0.3
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{ω3 , ω4 }
φ0t , φ1t = (−0.27, 0.4)
π0 (X) = 0.13
φ0t , φ1t = (0, 0)
π1 (X) = 0
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PART 5
RISK-NEUTRAL VALUATION OF NON-ATTAINABLE
CONTINGENT CLAIMS
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Attainability of Contingent Claims and Completeness
Definition (Attainable Contingent Claim)
A contingent claim X is called to be attainable if there exists a trading
strategy φ ∈ Φ, which replicates X, i.e., VT (φ) = X.
For attainable contingent claims, it is clear how to price them by
the initial investment needed for a replicating strategy.
As in single period market models, for some contingent claims a
hedging strategy may fail to exist.
Definition (Completeness)
A multi period market model is said to be complete if and only if all
contingent claims have replicating strategies. If a multi period market
model is not complete, it is said to be incomplete.
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Risk-Neutral Valuation Formula
Proposition (6.4)
Let X be a contingent claim (possibly non-attainable) and Q any
risk-neutral probability measure for the multi-period market model M.
Then the risk-neutral valuation formula
X πt (X) = Bt EQ
Ft
BT
defines a price process π(X) = (πt (X))0≤t≤T for X that complies with
the principle of no-arbitrage.
Proof.
The proof hinges the same arguments as in the single-period case and thus
it is left as an exercise. If X is attainable then can also observe that Vb (φ)
is a martingale under Q and apply the definition of a martingale.
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Example: Risk-Neutral Valuation
Example (6.2)
1
0.8
0.3
S2 = 3
ω1
0.7
S2 = 1
ω2
0.6
S1 = 1.5
0.4
0.6
0.2
0
S0 = 1
0.5
−0.2
−0.4
S2 = 1.5
ω3
S2 = 0.5
ω4
0.4
S1 = 0.75
−0.6
0.5
−0.8
−1
Figure: Stock Price Dynamics
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Example: Risk-Neutral Valuation
Example (6.2 Continued)
Consider again the market model M = (B, S) introduced in Example
6.1. Recall that the conditional probabilities describe the mouvements
under the real-world probability P.
Let Q be a risk-neutral probability measure, that is, Q ∈ M.
We denote qi = Q (ωi ) for i = 1, 2, 3, 4.
By the definition of the risk-neutral probability, we have
q1 + q2 + q3 + q4 = 1
1
EQ (S2 |F1 ) = S1
1+r
1
EQ (S1 |F0 ) = S0
1+r
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Example: Risk-Neutral Valuation
Example (6.2 Continued)
The conditional probabilities under Q at time t = 1 are
q1
q1 + q2
q2
Q (S2 = 1 |{ω1 , ω2 } ) =
q1 + q2
q3
Q (S2 = 1.5 |{ω3 , ω4 } ) =
q3 + q4
q4
Q (S2 = 0.5 |{ω3 , ω4 } ) =
q3 + q4
Q (S2 = 3 |{ω1 , ω2 } ) =
and
Q (S2 = 1.5 |{ω1 , ω2 } ) = 0,
Q (S2 = 3 |{ω3 , ω4 } ) = 0,
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Q (S2 = 0.5 |{ω1 , ω2 } ) = 0
Q (S2 = 1 |{ω3 , ω4 } ) = 0
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Example: Risk-Neutral Probability Measure
Example (6.2 Continued)
Also, the probability distribution of S1 reads
Q (S1 = 1.5) = q1 + q2
Q (S1 = 0.75) = q3 + q4
Hence we obtain the following linear system:
q1 + q2 + q3 + q4
3q1
q2
9
+
10 q1 + q2
q1 + q2
9 3 q3
1 q4
+
10 2 q3 + q4
2 q3 + q4
3
9 3
(q1 + q2 ) + (q3 + q4 )
10 2
4
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= 1
3
=
= S1 (ωi ) for i = 1, 2
2
3
=
= S1 (ωi ) for i = 3, 4
4
= 1 = S0
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Example: Risk-Neutral Valuation
Example (6.2 Continued)
Equivalently,
q1 + q2 + q3 + q4 = 1
2q1 − q2 = 0
2q3 − q4 = 0
3
3
3
3
10
q1 + q2 + q3 + q4 =
2
2
4
4
9
The unique solution reads
q1 =
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13
26
14
28
, q2 = , q3 =
, q4 = .
81
81
81
81
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Example: Risk-Neutral Valuation
Example (6.2 Continued)
The price of the digital call option considered in Example 6.1 can be
computed as follows.
For ω ∈ {ω1 , ω2 }, we obtain
π1 (X)(ω) =
1
EQ (X |{ω1 , ω2 } )
1+r
9
=
10
1·
13
81
13
81
+
26
81
+0·
14
81
14
81
+
28
81
!
= 0.3
For ω ∈ {ω3 , ω4 }, we have π1 (X)(ω) = 0 since the payoff is 0 at
t = 2 if the stock price at time t = 1 equals S1 = 0.75.
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Example: Risk-Neutral Valuation
Example (6.2 Continued)
The price of the digital call option at time 0 equals
1
1
EQ (X |F0 ) =
EQ (X)
2
(1 + r)
(1 + r)2
81
26
14
28
13
=
+0·
+0·
+0·
1·
100
81
81
81
81
= 0.13
π0 (X) =
These pricing results coincide with those obtained in Example 6.1,
where we computed directly the wealth process V (φ) of the
replicating strategy φ for X.
As was indicated earlier, a multi-period market model M can be
decomposed into several single-period market models.
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Example: Backward Induction
Example (6.3)
1
0.8
0.3
S2 = 3
ω1
0.7
S2 = 1
ω2
0.6
S1 = 1.5
0.4
0.6
0.2
0
S0 = 1
0.5
−0.2
−0.4
S2 = 1.5
ω3
S2 = 0.5
ω4
0.4
S1 = 0.75
−0.6
0.5
−0.8
−1
Figure: Stock Price Dynamics
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Example: Backward Induction
Example (6.3 Continued)
We consider the market model in Example 6.1 once again.
The two-period market model is composed of the following
single-period market models:
1
2
3
S1 = 1.5, S2 = 3 and S2 = 1.
S1 = 0.75, S2 = 1.5 and S2 = 0.5.
S0 = 1, S1 = 1.5 and S1 = 0.75.
Note that these models are elementary market models.
Hence the unique martingale measure can be computed using
the formula
p̃ =
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1+r−d
,
u−d
q̃ = 1 − p̃ =
6: Multi-Period Market Models
u − (1 + r)
.
u−d
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Example: Backward Induction
Example (6.3 Continued)
Recall that if the elementary market model is arbitrage-free then
it is also complete and all contingent claims can be priced using
the risk-neutral probability.
In the first model, the martingale measure is
1 + 19 −
1 + r − d1
=
u1 − d1
2 − 23
2
= 1 − p̃1 =
3
p̃1 =
q̃1
2
3
=
1
3
The price of the digital call option in the considered model equals
1
1
2
u
π1 (X) =
1· +0·
= 0.3
3
3
1 + 19
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Example: Backward Induction
Example (6.3 Continued)
In the second model, the martingale measure is
p̃2
q̃2
1 + 19 −
1 + r − d2
=
=
u2 − d2
2 − 23
2
= 1 − p̃2 =
3
2
3
=
1
3
The price of the digital call option in this model equals 0 since its
payoff is 0. Formally,
1
2
1
d
0· +0·
π1 (X) =
=0
3
3
1 + 91
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Example: Backward Induction
Example (6.3 Continued)
In the last model, the martingale measure is
p̃3
q̃3
1 + 19 −
1 + r − d3
=
= 3 3
u3 − d3
2 − 4
14
= 1 − p̃3 =
27
3
4
=
13
27
We consider the contingent claim π1 (X) with the payoff at time
t = 1 given by
u
π1 (X) = 0.3 if S1 = u3 S0 = 1.5
π1 (X) =
if S1 = d3 S0 = 0.75
π1d (X) = 0
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Example: Backward Induction
Example (6.3 Continued)
The price of X at time 0 equals
π0 (X) =
1
1
EQ (π1 (X)) =
1+r
1+
1
9
0.3 ·
13
14
+0·
27
27
= 0.13
Hence π0 (X) = 0.13, π1 (X)(ω) = 0.3 if ω ∈ {ω1 , ω2 } and
π1 (X)(ω) = 0 if ω ∈ {ω3 , ω4 }.
The unique martingale measure Q in the two-period market model
can be recomputed as follows:
13 1
13
· =
27 3
81
26
13 2
· =
Q (ω2 ) = p̃3 (1 − p̃1 ) =
27 3
81
14
14 1
· =
Q (ω3 ) = (1 − p̃3 ) p̃2 =
27 3
81
14 2
28
Q (ω4 ) = (1 − p̃3 ) (1 − p̃2 ) =
· =
27 3
81
Q (ω1 ) = p̃3 p̃1 =
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PART 6
COMPLETENESS OF MULTI-PERIOD MARKET MODELS
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Completeness
As a handy criterion for the market completeness, we have the theorem,
which extends the known result for the single-period case.
Theorem (6.1)
Assume that a multi-period market model M = (B, S 1 , . . . , S n ) is
arbitrage-free. Then M is complete if and only if there is only one
e is a singleton.
martingale measure, that is, M = {P}
In the context of a model decomposition, the following statements are
known to hold:
If all single-period models which compose a multi-period model are
arbitrage-free then the multi-period model is also arbitrage-free.
If they are also complete then the multi-period model is also complete.
The converse of the above statement is also correct.
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Summary of Pricing and Hedging Approaches
We examined three pricing and hedging approaches:
1
The method based on the idea of replication of a contingent claim. It can
only be applied to attainable contingent claim in a complete or incomplete
model and it yields the hedging strategy and arbitrage price process.
2
The method relying on the concept of a martingale measure, which can
be used in either a complete or an incomplete model. It furnishes the unique
arbitrage price process for any attainable claim and a possible arbitrage price
process for a non-attainable contingent claim. In the latter case, the price
process depends on the choice of a risk-neutral probability.
3
The backward induction approach in which a multi-period market model is
decomposed into a family of single-period models. Pricing is performed in a
recursive way starting from the date T − 1 and moving step-by-step towards
the initial date 0. Hedging strategy can also be computed provided that the
claim is attainable.
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