Download 2.2 The Formal Set-Up

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2. Single Period Market Models
2.2
The Formal Set-Up
Let us define formally the simple case of a single-period model consisting
only of a risk-free security, called the bond, and a risky security, called the
stock. We then illustrate the economic principles that will also be considered
later on when dealing with more complex models.
The model is specified by the following elements:
• Two trading dates, the initial date t = 0 and the final date T = 1.
• A finite sample space ⌦ = {!1 , !2 , . . . , !n } with n < 1. Each ele-
mentary event !i , i = 1, . . . , n, represents a ‘state of the world’, also
called scenario; the investors at time t = 0 have no information on the
outcome, while at time T = 1 they know exactly which scenario has
materialized.
• A probability measure P on (⌦, P(⌦)), where P(⌦) is the powerset of
the sample space, such that P(!i ) := P({!i }) > 04 for all i = 1, . . . , n.
• A bank account, called the ‘bond’, with initial value B0 = 1 and deter-
ministic final value B1 = 1 + r, where r is the risk-free market interest
rate, fixed at time t = 0.
• A risky security, called the ‘stock’, whose initial value S0 is known
(quoted on the market) and whose final value S1 is a random variable
on (⌦, P(⌦)).
Definition 2.1. In a single-period market model, a trading strategy is a
triple of real constants (V0 , ↵, ), where V0 denotes the value of the initial
investment, while ↵ and
denote the number of shares of the stock and
the units of the bond, respectively, to be held in the portfolio over the time
4
By choosing a non-degenerate probability measure we avoid ambiguities when dealing
with equalities of random variables: if X, Y are two random variables on (⌦, P(⌦), P),
usually X = Y means equality P-almost surely, while in our setting it means X(!) = Y (!)
for all ! 2 ⌦.
2.2 The Formal Set-Up
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interval (0, 1]. The value of the trading strategy, also called portfolio value,
at time T = 1 is
V1 = ↵S1 + B1 .
(2.2)
Note that the investor decides the trading strategy at the initial date,
then they apply it over the next time period and wait for the outcome at
the final date. We will restrict our consideration to the strategies involving
niether cash inflow nor outflow aside the initial investment, therefore the
following relation must be satisfied:
V0 = ↵S0 + B0 = ↵S0 + .
(2.3)
This means that the whole and only amount of money invested in the bond
and stock over the time interval (0, 1] is exactly given by the initial investment
of the strategy. Such trading strategies are called self-financing. Since selffinancing trading strategies are univocally specified by the stock and bond
investments ↵, , or equivalently by the initial investment V0 and either ↵
or
, henceforth we will shorter the notation and write just (↵, ) for a
self-financing trading strategy.
Definition 2.2. The gain of a trading strategy (V0 , ↵, ) over the time period
(0, 1] is a random variable on (⌦, P(⌦)), defined by
G := r + ↵(S1
S0 ).
(2.4)
By simple rearrangement of the equations (2.2),(2.4), we get the following
self-financing condition, equivalent to (2.3), that relates the final value to the
gain of a trading strategy:
V1 = V0 + G.
(2.5)
In order to compare prices at di↵erent times, it is convenient to normalize
the market values with respect to the bond. Thus, we define the discounted
stock price S ⇤ as
St⇤ :=
St
,
Bt
t = 0, 1,
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2. Single Period Market Models
the discounted value V ⇤ of a trading strategy (↵, ) as
Vt⇤ :=
+ ↵St⇤ =
Vt
,
Bt
t = 0, 1,
and the discounted gain G⇤ of a trading strategy (↵, ) as
G⇤ := ↵(S1⇤
S0⇤ ).
The self-financing condition for the discounted market is the same as for the
original market:
V1⇤ = V0 + G⇤ .
2.3
(2.6)
Economic Considerations and
the No-Arbitrage condition
As anticipated in Lecture 1, it is desirable that the mathematical model
of a financial market satisfies some reasonable economic principles. In the
following, we see three related criteria, the strongest of which will be taken
as assumption for the models we study.
Definition 2.3. A trading strategy with value V is said to be dominant if
there exists another trading strategy, with value Ṽ , such that V0 = Ṽ0 and
V1 > Ṽ1 .
Note that by equalities and inequalities of random variables, we mean that
they hold in every scenario, for instance V1 (!i ) > Ṽ1 (!i ) for all i = 1, . . . , n
in Definition 2.3. The existence of dominant strategies in the market has a
few equivalent conditions.
Lemma 2.4. There exists a dominant strategy if and only if there exists a
trading strategy, with value V , satisfying V0 = 0 and V1 > 0.
Proof. If a strategy as in the statement of Lemma 2.4 exists, it is also dominant, since it dominates the strategy that starts with zero money and does
not invest in anything. Conversely, if a strategy (↵, ) dominates a strategy
2.3 Economic Considerations and the NA condition
(˜
↵, ˜), then by defining (¯
↵, ¯) = (↵, )
25
(˜
↵, ˜), we get a strategy as in the
statement of Lemma 2.4.
Lemma 2.4 describes the situation where an investor starts with zero
money and is able to end up with a sure strictly positive gain. Obviously
this situation is neither realistic nor reasonable and must be avoided. An
analogous reasoning is valid for a strategy as in Lemma 2.5.
Lemma 2.5. There exists a dominant strategy if and only if there exists a
trading strategy, with value V , satisfying V0 < 0 and V1
0.
Proof. Prove it as an exercise.
If we interpret the final value of an investment strategy as a contingent
claim, the initial value of such investment would be interpreted as the price
of such claim. This implies that the existence of a dominating strategy in
the market also leads to illogical pricing, because two claims with a surely
di↵erent value would have the same price at time t = 0. Instead, the pricing
of contingent claims would be logically consistent if the price of any claim
was unique and a claim that pays more than another one in every state of the
world had a higher price at time t = 0. There is a mathematical formalization
of these properties.
Definition 2.6. A linear pricing measure is a non-negative vector ⇡ =
(⇡1 , . . . , ⇡n ) 2 Rn+ such that the value V of any self-financing trading strategy
satisfies
V0 =
n
X
⇡i V1⇤ (!i ).
i=1
If there is a linear pricing measure, then the inconsistency disappears. An
equivalent way to characterize linear pricing measures is shown in the following lemma and it also gives an explanation for the appellation “measure”.
Lemma 2.7. ⇡ is a linear pricing measure if and only if it is a probability
measure on (⌦, P(⌦)) and satisfies
S0⇤ =
n
X
i=1
⇡i S1⇤ (!i ).
(2.7)
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2. Single Period Market Models
Proof. If ⇡ is a linear pricing measure, by Definition 2.6 we have
+
↵S0⇤
=
n
X
⇡i ( + ↵S1⇤ (!i )) .
(2.8)
i=1
By taking ↵ = 0, we get
Pn
i=1
⇡i = 1, which makes ⇡ a probability mea-
sure; then by taking ↵ = 1 we get the equation (2.7). Conversely, if ⇡ is a
probability measure satisfying (2.7), then (2.8) holds by substitution.
Note that, if a linear pricing measure exists, by (2.7) and Definition 2.6,
both the initial stock price and the initial value of any strategy are equal to
the expectation under such probability measure of the final discounted stock
price and of the final discounted value of the strategy, respectively. This
reminds us of the risk-neutral probability measure seen in the introductory
examples of Lecture 2. However, the two objects do not coincide, but we will
see that the risk-neutral probability is just a special kind of linear pricing
measure.
Lemma 2.8 shows that models which exclude dominant strategies can be
economically reasonable, in that they allow for a logically consistent pricing
of claims.
Lemma 2.8. There exists a linear pricing measure if and only if there are
no dominant strategies.
The proof pivots on the linear programming duality theory and we do
not show it.
The assumption of non-existence of dominant strategies is stronger than
the assumption mentioned in Lecture 1 and formalized as follows:
Definition 2.9. The law of one price holds if there do not exist two trading
strategies, with value V and Ṽ respectively, such that V1 = Ṽ1 but V0 > Ṽ0 .
It is clear how this law is crucial to avoid ambiguity about the initial price
of any claim.
Lemma 2.10. If there are no dominant strategies, then the law of one price
holds.
2.3 Economic Considerations and the NA condition
27
Proof. If the law of one price does not hold, then by definition there exist
two trading strategies (¯
↵, ¯) and (˜
↵, ˜) such that V̄1 = Ṽ1 but V̄0 > Ṽ0 . This
implies that G̃ > Ḡ. By defining a new strategy ↵ = ↵
˜
↵
¯,
=
↵S0 , this
has value V0 = 0, V1 > 0. Thus a dominant strategy exists.
Remark 2.11. The converse of Lemma 2.10 is not true. Indeed, you can
find a dominant strategy even in a model that satisfies the law of one price.
Example 2.12 presents a situation of this kind.
Example 2.12. Consider a single-period market model specified by the following data: the sample space contains only two elementary events, ⌦ =
{!1 , !2 }, the interest rate is r = 1, the stock price at time 0 is S0 = 10
and takes the following values at time T = 1, depending of the ‘state of the
world’: S1 (!1 ) = 12, S1 (!2 ) = 8. In this model, the law of one price is
satisfied: for all X = (X1 , X2 ) 2 R2 , there exists a unique trading strategy
(↵, ) such that
V1 (!1 ) = ↵S1 (!1 ) + (1 + r) = X1 ,
V1 (!2 ) = ↵S1 (!2 ) + (1 + r) = X2 .
However, we can define a dominant strategy by taking ↵ =
1 and
= 10,
whose value at time 0 is V0 = 0 and at time T = 1 is V1 (!1 ) = 8 > 0,
V1 (!2 ) = 12 > 0.
Once we agreed upon restricting our attention to market models that do
not allow for the existence of dominant trading strategies, a further restriction
seems necessary: would you allow for trading strategies that starts with a null
initial investment, do not lose any money and end up with a strictly positive
gain in at least one (but not all) scenario? If you care for the equilibrium of
the market model, certainly not. This situation corresponds to the concept
of arbitrage opportunity mentioned in Lecture 1 and formalized below.
Definition 2.13. An arbitrage opportunity is a (self-financing) trading strategy with value V such that:
(i) V0 = 0,
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2. Single Period Market Models
(ii) V1
0,
(iii) EP [V1 ] > 0, i.e. P(V1 > 0) > 0.
Lemma 2.14. If there exists a dominant strategy, then there exists an arbitrage opportunity.
The statement is trivially true, by definition.
Remark 2.15. The converse of Lemma 2.14 is not true. Indeed, you may find
an arbitrage opportunity in a model excluding dominant strategies. Example
2.16 presents a situation of this kind.
Example 2.16. Consider a single-period market model specified by the following data: the sample space contains only two elementary events, ⌦ =
{!1 , !2 }, the interest rate is r = 0, the stock price at time 0 is S0 = 10
and takes the following values at time T = 1, depending of the ‘state of the
world’: S1 (!1 ) = 12, S1 (!2 ) = 10. The strategy defined by ↵ = 1,
=
10
is an arbitrage opportunity, because: V0 = 0 and
V1 (!1 ) = ↵S1 (!1 )+ (1+r) = 12 10 = 2,
V1 (!2 ) = ↵S1 (!2 )+ (1+r) = 10 10 = 0.
Nevertheless, we can prove that there are no dominant strategies in the market. Indeed: ⇡ = (0, 1) is a linear pricing measure, by Lemma 2.7, then
Lemma 2.8 concludes the proof.
The set of conditions in Definition 2.13 can be equivalently rewritten in
terms of the gain of the trading strategy.
Lemma 2.17. (↵, ) is an arbitrage opportunity if and only if it satisfies (i)
and its gain G satisfies:
(a) G
0,
(b) EP [G] > 0.
Proof. Suppose that (↵, ) is an arbitrage opportunity. Then, by definition,
G = V1
V0
0 and EP [G] = EP [V1 ]
V0 = EP [V1 ] > 0. Conversely, suppose
2.3 Economic Considerations and the NA condition
that (i), (a) and (b) are satisfied by some trading strategy (↵, ). Then
V1 = V0 + G = G
0 and EP [V1 ] = EP [G] > 0.
We can resume the relation between the models satisfying the conditions
explained in this section as:
NA ⇢ No dominant strategies ⇢ law of one price,
where we denote by NA the no-arbitrage condition, i.e. the non-existence
of arbitrage opportunities. This means that market models can be divided
in four categories: there are no arbitrage opportunities, there are arbitrage
opportunities but no dominant strategies, there are dominant strategies but
the law of one price holds, and the law of one price does not hold. An example
of model that does not even satisfy the law of one price is given in Example
2.18.
Example 2.18. Consider the market model specified in Example 2.12 but
change the specification of the stock price at time T = 1 in the state !2
as S1 (!2 ) = S1 (!1 ) = 12. Then, for any given real constant X 2 R, there
exist infinitely many trading strategies whose final value is V1 = X. That is
because we have to solve one equation in two variables:
V1 = ↵S1 + (1 + r) = X.
We want to work with models of the first category, that is satisfying the
NA condition. This is of course a theoretical assumption, because in real
markets much more frictions are involved and the markets are far from being
ideal. Nevertheless, in very liquid markets any arbitrage possibility that
arises is quickly eliminated (by all the investors taking advantage of it) and
the market equilibrium is restored. So our assumption is reasonable. The
NA condition is not easy to be checked, but we will see in future chapters a
fundamental theorem that provides us with an equivalent condition involving
a special probability measure.
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2. Single Period Market Models
2.4
The Binomial Model on a Single Period
A special case of this simple one period model is given by the so called
Binomial or Cox-Ross-Rubinstein (CRR) model, from the names of the authors who introduced it in 1979. We will consider here the single-period case
of the model, which is in general a multi-period model. We will study in
detail the general setting of the binomial model in the next chapter.
Assumption 2.4.1 (Single-period binomial model). The initial stock price
S0 is strictly positive and the stock price at the final date T = 1 is proportional
to the initial stock price, in the following way:
S1 = S0 ⇠,
where ⇠ is a Bernoulli-like random variable, taking only two values u, d, u >
d > 0, with probabilities
P(⇠ = u) = p,
P(⇠ = d) = 1
p.
The sample space is the path space, that is the set of possible trajectories for
the stock price:
⌦ = {(S0 , S0 u), (S0 , S0 d)}.
Definition 2.19. A European contingent claim (ECC) is represented by a
random variable X on (⌦, P(⌦)) and it is said to be attainable if there
exists a self-financing trading strategy with final value V1 = X. Such trading
strategy, if it exists, is called a replicating, or hedging, strategy for X.
Now, given a European claim X, consider the problem of hedging, that
is to find the hedging strategy for X, if it exists. Since the claim is a random
variable on the path space, it can take only two values, which we denote by
X u in the scenario (S0 , S0 u) and X d in the scenario (S0 , S0 d). By imposing
the replicating condition, we get:
(
↵S0 u + (1 + r) = X u
↵S0 d + (1 + r) = X d
2.4 The Binomial Model on a Single Period
Solving for ↵,
31
we find
Xu Xd
,
(u d)S0
uX d dX u
= (1 + r) 1
u d
Thus the initial investment needed to replicate the derivative is
↵ =
(2.9)
(2.10)
V0 = ↵S0 +
=
1
(1 + r)(u
= (1 + r)
1
d)
(1 + r
qX u + (1
d)X u + (u
(1 + r))X d
q)X d
where
1+r d
.
(2.11)
u d
Note that q 2 (0, 1) if and only if d < 1 + r < u. In this case, we can define
q=
a probability measure Q by
Q(⇠ = u) = q,
Q(⇠ = d) = 1
q,
(2.12)
and the initial value of the hedging strategy can be rewritten as
V0 = EQ [X ⇤ ] ,
(2.13)
where X ⇤ = (1 + r) 1 X is the discounted claim. Furthermore, the initial
stock price can also be rewritten as the expectation under Q of the discounted
final stock value: S0 = EQ [S1⇤ ]. Thus, Q is the unique risk-neutral probability
of the model. The appellation “risk-neutral” refers to the fact that, under
Q, a risk-neutral investor, i.e. an investor who has no preference between
the expected value of a payo↵ and the random payo↵ itself, is indi↵erent to
investing either on shares of the stock or on bonds, since they have the same
expected payo↵.
Note that the risk-neutral probability is only a theoretical object and
it is unrelated to the market (or real-world, or objective) probability P that
specifies the dynamics of the stock price in the model. We remark that the
only property shared by P and Q is that they both assign a strictly positive
value to every possible event.
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2. Single Period Market Models
Theorem 2.20. The single-period binomial model satisfies the NA condition
if and only if d < 1 + r < u.
Proof. We leave the proof of the implication ) of the statement for homework.
(() If d < 1+r < u, we have already proved that we can define a probability
measure Q by (2.11) and that under this measure the initial stock price S0
is equal to the expected value of the discounted stock price at time T = 1.
Suppose that an arbitrage opportunity (↵, ) exists, by Definition 2.13, it
must satisfy the conditions (i)-(iii). Now, (i) translates to
V0 = ↵S0 +
=0
and (ii) to
V1 = ↵S1 + (1 + r)
0.
By taking the expectation under Q, we get
EQ [V1 ] = ↵EQ [S1 ] + (1 + r)
= ↵(1 + r)S0 + (1 + r)
= (1 + r)V0 = 0.
Thus, condition (iii) cannot be satisfied: we found a contradiction.
Henceforth we assume that the model satisfies the NA condition:
Assumption 2.4.2. We consider a single-period binomial model with d <
1 + r < u.
We have also seen that for any given European claim X there exists a
hedging strategy, defined in (2.9)-(2.10). By the lemmas proved in Section
2.3, the law of one price holds, hence the time 0 value of the hedging strategy
determines the price of the derivative. At this point, another concern arises:
by introducing the derivative security X in the market, with price determined
by (2.13), are we introducing a possibility for arbitrage opportunities? We
would like that the price of the derivative is such that also the secondary
2.4 The Binomial Model on a Single Period
market composed of the bond, the stock, and the derivative itself, is free of
arbitrage opportunities. In this case, the price of the derivative is called the
arbitrage-free price.
Fortunately, this is true for the initial value of the hedging strategy:
Theorem 2.21 (Thm 2.2.1 in ?). V0 = EQ [X ⇤ ] is the unique arbitrage-free
price for the European contingent claim X.
We defer the proof to the multi period case.
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