MAT4730
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26/08-2010
Probabilistic Setting
Working in a probabilistic setting, we first define the probability space. Ω:
the sample space, F : the σ-algebra of events and P: the probability measure.
We will use the filtration F = {Ft |t ≥ 0}, which is an increasing ensemble
(collection/family) of σ-algebras Ft . We have the “no forgetting” process,
Fs ⊆ Ft for s ≤ t and Ft ⊂ F which is true ∀t.
The filtration represents the accumulation of information on the market,
which is called an “information set” in statistics.
We have a source of noise - the random element.


W1 (t)


W (t) =  ...  , t ≥ 0.
WD (t)
This is a D-dimensional Brownian motion (BM) on (Ω, F , P).
To W (t) we can associate a particular filtration: the natural filtration
generated by W :
F W = {FtW , t ≥ 0}
where
FtW = σ(Wd (s), s ≤ t, d = 1, . . . , D).
Let N be the ensemble of P-null sets of Ω (the impossible events). In most
cases we consider F given by Ft = FtW ∨ N .
Definitions for a financial market
We need a concept of a market model. A financial market model on (Ω, F , P)
is given by the following elements.
(1) (Ω, F , F).
(2) T < ∞ (or T ≤ ∞), which is the planning horizon/time horizon.
(3) The driving noise: W (t), 0 ≤ t ≤ T . This is a simulation of the
randomness we observe in reality.
(4) F : information available on the market. All agents in the market see
everything and remember everything.
1
(5) A money market account S0 (risk free account, like a bank account). The
evolution of S0 is given by an ODE
dS0 (t) = S0 (t)r(t)dt
S0 (0) > 0
(and we usually use S0 = 1 to simplify the calculations). S0 is continuous,
strictly positive and F-adapted (at time t we know S0 (t)). Here r(t) is the
instantaneous, risk free interest rate at time t. In the Black & Scholes model
(B&S) r(t) is just a constant.
The process r(t), 0 ≤ t ≤ T is F-adapted such that
Z
T
|r(s)ds| > ∞
P − a.s.
0
(This is just a mathematical necessity to obtain a well defined solution for
the ODE dS0 (t)). A priori, r(t) can admit negative values.
(6) N stocks with price per share given by Sn (t), 0 ≤ t ≤ T , for n = 1, . . . , N.
These processes are assumed to be continuous, strictly positive and F-adapted
(so at time t we know the stock price).
Typically we will consider their dynamics to be of the form SDE (Ito
equation)
P
dSn (t) = Sn (t) bn (t)dt + D
d=1 σnd (t)dWd (t)
(*)
Sn (0) > 0 (a given constant)
where bn (t)dt is the drift and σnd (t)dWd (t) is the volatility. In the B&S-model,
the drift and volatility are constants.
The solution for (*) is
Sn (t) = Sn (0) exp
D
nZ tX
σnd (s)dWd (s) +
0 d=1
Z
t
0
D
bn (s) −
o
1X
σnd (s) ds
2 d=1
where t ∈ [0, T ].
Discounted stock prices S
We define the discounted price as S n (t) =
Sn (0) exp
nZ
0
t
. . . dW (s) +
Z
0
t
Sn (t)
S0 (t)
=
D
o
1X
σnd (s) ds
bn (s) − r(s) −
2
d=1
The processes bn (t): the mean rate of return for stock n.
2
σnd (t): the volatility matrix (covariance matrix).
These processes are F-adapted and
N Z
X
n=1
T
|bn (s)|ds < ∞ a.s
0
D Z
N X
X
n=1 d=1
T
2
σnd
(s)ds < ∞ a.s
0
The matrix [σnd (t)] is of maximal rank ∀t ∈ [0, T ], which means if it is square
we can invert it.
In the B&S both bn (t) and σnd (t) are constants.
A short summary on stochastic integration
Classical Ito integral.
Let φ = φ(t), t ∈ [0, T ] be F-adapted and
Z T
2
E
φ (s)ds < ∞
0
(In Oksendal this class of functions is φ ∈ V[0,T ] .), then
I(t) =
Z
t
φ(s)dW (s),
t ∈ [0, T ]
0
is well defined and it is an F-martingale: it is F-adapted, ∀t: E[|(t)|] < ∞,
and we have the Martingale property:
E[I(t)|Fs ] = I(s) ∀s ≤ t
(doesn’t grow on average). If we have ≥ we have a sub-martingale, and ≤ is
a super-martingale.
Generalized Ito integral.
R
(Relaxing the E[ φ2 ds] < ∞ assumption).
RT
Let φ = φ(t), t ∈ [0, T ] be F-adapted and 0 φ2 < ∞ a.s (Oksendal
φ ∈ W[0,T ] ), then
Z t
I(t) =
φ(s)dW (s)
0
is well defined and an F-local martingale, so it is F-adapted, ∃{τn }∞
n=1 for a
stopping time τ : Ω 7→ [0, T ], which means there is a F-stopping time such
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that τn ր T as n → ∞ and I(min(t, τn )) = In (t) is an F-martingale. (Local
over stochastic intervalls).
A lower bounded local martingale is a super martingale. If φ ∈ W[0,T ] , then
E[I(0)] = I)0) = 0 but E[I(t)] 6= 0 in general, so the martingale property
fails.
In MAT4730 we will deal mostly with the generalized Ito integrals.
Market price of risk, Girsanov’s Theorem.
EMM (equivalent martingale measure) or Pricing measure. From now on we
assume D = N, so the number of stocks equal the number of Brownian
motions. Then the volatility matrix [σnd (t)](= σ(t)) is a square matrix and
non-degenerate ∀t.
- Definition (Market price of risk/Risk premium/Sharp ratio)
−1
θ(t) := σnd
[b(t) − r(t)I] t ∈ [0, T ]
where


b1 (t)


b(t) =  ...  ,
 
1
 .. 
r(t) = r(t)  . 
bn (t)
1
(The equation within the []’s is sometimes called the excess return).
are all constants. If b is high and σ is small it is
For the B&S model, b−r
σ
a “good” stock. We can cosnider this a measurement of how well the stock
performs.
Defintion EMM.
Z0 (t) := exp −
N Z
X
d=1
t
θd (s)dWd (s) +
0
N Z
X
d=1
t
θd2 (s)ds
0
.
This is at least a non-negative super martingale on (F, P). We assume Z0 is a
martingale, which is equivalent to the assumption E[Z0 (T )] = 1. A sufficient
condition for this is the Novikov condition
Z TX
N
1
2
E
θ (s)ds < ∞ a.s
2 0 d=1 d
So if, on average, the deterministic part of Z0 (t) doesn’t blow, we have a
semimartingale.
Since Z0 is a martingale, we can define a measure P0 on (Ω, F ):
dP0 = Z0 (T )dP.
4
Then we can use Girsanov’s transform (theorem) to generate
Z t
W0 (t) = W (t) +
θ(s)ds
0
(i.e changing the drift). Now the W0 is a brownian motion on (F, P0 ), and
(t)
are F-martingales with respect to P0 . This
the discounted price S n (t) = SSn0 (t)
can be verified with Ito’s formula.
Consumption and Portfolio processes
Gain and Wealth processes
Definition: A portfolio process can be expressed as
(1) η0 (t), . . . , ηN (t) = (η0 , η)
where ηi is the number of shares held of stock i.
(2) π0 (t), . . . , πN (t) = (π0 , π)
where πi is the amount of money invested in stock i. Between these we have
the relation πi (t) = ηi (t)Si (t), i.e (Value) = (number of stocks)×(price).
(3) p0 (t), . . . , pN (t)
where pi is the proportion or fraction of the total investment. If X(t) is the
i (t)
total wealth at time t, then pi (t) = πX(t)
.
Definition: The gain process G(t) associated with a given portfolio (η0 , η)
is
Z t
G(t) =
η(s)dS(s).
0
We integrate over the price and express the gain by η.
Example
Consider the market with stocks
S1 S2
t1 (1, 0)
t2 (0, 1)
and price
S1 S2
t1 (3, 2)
t2 (1, 4)
5
We get,
G(2) = (1)(3) + (0)(2) + (0)(1) + (4)(1) = 7
|
{z
} |
{z
}
Period 1
Period 2
Gains processes by amount of money
Z
t
η(s)dS(s) =
0
=
Z
Z
η0 (s)dS0 (s) +
0
t
η0 (s)S0 (s)r(s)ds +
0
=
Z
n=1
N Z
X
n=1
0
t
=
t
0
ηn dSn (s)
0
n=1
t
π0 (s)r(s)ds +
N Z t
X
π0 (s)r(s) + π(s) · b(s) ds +
|{z} |{z}
N ×1
πn (s)[. . .]
0
n=1
1×N
t
N
h
i
X
ηn (s)Sn (s) bn (s)ds +
σnd (s)dWd (s)
0
Z
N Z
X
t
Z
t
0
π(s) · σ(s) dW (s)
|{z} |{z} | {z }
1×N
N ×N
1×N
G with respects to W0 :
Z th
Z t
i
G(t) =
π0 (s)r(s) + π(s) · b(s) − π(s)σ(s)θ(s) ds +
π(s) · σ(s)dW0 (s)
0
0
Substituting θ(s) = σ −1 (s) b(s) − r(s)I :
Z th
Z t
i
n
o
(
(
(
G(t) =
π(s)σ(s)θ(s)
ds+ π(s)σ(s)dW0 (s)
π0 (s)r(s)+π(s)
(((
σ(s)θ(s)+r(s)I −(
0
=
Z
0
t
0
π0 (s) + π(s)I r(s)ds +
Z
t
π(s)σ(s)dW0(s)
0
Formalising consumption
Definition: A consumption rate is a non-negative, F-adapted process c(t)
RT
such that 0 c(s)ds < ∞ P − a.s. Since it is non-negative |c(s)| = c(s).
We assume that an agent has an initial endowment x ≥ 0 and a given
consumption rate c(t). Then his/her cumulative income process Γ(t) is
Z t
Γ(t) = x −
c(s)ds.
0
Definition: The value process associated to a portfolio (π0 , π) is
π
X(t) = X (t) = η0 (t)S0 (t) +
N
X
n=1
6
ηn (t)Sn (t) = π0 (t)0π(t)I
which is the value of the portfolio at time t.
Definition: A portfolio (π0 , π) is self financing if X(t) = X(0) + G(t) which
means the portfolio can financially sustain itself.
More generally: for a given consumption rate c, a portfolio is called Γfinancing if
Z t
x,c,π
X(t) = X
(t) = π0 (t) + π(t)I = Γ(t) + G(t) = x + G(t) −
c(s)ds
def. prop.
0
Result (Application of Ito’s formula)
Given initial endowment x ≥ 0, c(t) and π(t) it is possible to find π0 such
that the portfolio (π0 , π) is Γ-financing.
Proof
Assume that π0 exists. Then
X(t) = π0 (t) + π(t)I = x + G(t) −
Γ-fin.
=x+
Z
t
0
Then,
π0 (t) = x+
Z
t
0
π0 (s) − π(s)I r(s)ds +
Z
Z
t
c(s)ds
0
Z
t
π(s)σ(s)dW0 (s) −
0
t
c(s)ds
0
Z t
Z t
Z t
π0 (s)r(s)ds− c(s)ds− π(s)Ir(s)ds +
π(s)σ(s)dW0 (s) − π(t)I
0
0
{z
}
|0
This diffed. is dA(t)
which gives
dπ0 (t) = π(t)r(t)dt + dA(t) − c(t)dt
We need
dY (t) = Y (t)r(t)dt,
Z(t) =
Then,
Y (0) = 1
=⇒
1
= exp −
Y (t)
Y (t) = exp
Z
0
t
r(s)ds
Z
t
r(s)ds
0
d Z(t)π0 (t) =
Ito
(
(
((
(
(
(
(
(
(
Z(t)
r(t)Z(t)π
−(
0 (t)r(t)dt + dA(t) − c(t)dt
0 (t)dt + (
((
((π(
= Z(t)dA(t) − Z(t)c(t)dt =⇒
Z t
Z t
π0 (t) = Y (t) π0 (0) +
Z(s)dA(s) −
Z(s)c(s)ds = x − π(0)I
0
0
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02/09-2010
We are eventually going to formulate the single agent consumption/investment problem. In this form we can assume self-financing portfolio exist, but recall that they only do so in a complete market. A complete
market is a unrealistic, but theoretically interesting map.
Remark
Let (π0 , π) be a Γ-financing portfolio. Then the corresponding discounted
x,c,π
wealth process X(t) = X
(t) satisfies,
Z t
Z t
π(s) · σi (si )dW0 (s)
X(t)
=x−
c(s)ds +
X(t) :=
S0 (t)
S0 (s)
0
0
where x is the initial endowment and the integral containing c(s) is the total
consumption from 0 to t.
Proof
Assume Γ-financing and use Ito’s formula.
1
1
X(t)
d
=d
X(t) +
dX(t) + 0
S0 (t)
S0 (t)
S0 (t)
(The double derivative becomes 0 because we have two BM). Using an
expression from last lecture, we get
=
1 ((( + π(t)σ(t)dW (t)
π(t)
·(
Ir(t)dt)
−
r(t)X(t)dt
− c(t) + (π0 r(t) + (
((
0
S0 (t)
Cancel because of the Γ-financing property.
=
1 − c(t)dt + π(t)σ(t) · dW0 (t)
S0 (t)
Definition. The process (which in a sense is doubly discounted)
H0 (t) :=
Z0 (t)
S0 (t)
RT
is called the state-price density process. We assume that E[ 0 H0 (s)ds +
H0 (T )] < ∞.
A sufficient condition for this assumption to hold is that S0 (t), t ∈ [0, T ]
is bounded from 0, i.e ∃ε > 0, S0 (t) ≥ ε∀t.
8
In fact:
E
hZ
T
i
H0 (s)ds =
0
Z
T
E
0
h Z (s) i 1
0
≤ Z0 (0) T
S0 (s)
ε | {z }
=1
(Using the inequality due to Z0 being a super martingale).
1
1
E H0 (t) ≤ Z0 (0) =
ε
ε
Why do we introduce H0 ?
Using this process, we can rewrite the conditions involving P0 (and W0 )
in terms of the original probability P.
Example
Z t
Z t
X(t)
= H0 (t)X(t) = x− H0 (u)c(u)du+ H0 (t) π(u)σ(u)−X(u)θ(u) dW (u)
Z0 (t)
S0 (t)
0
0
Proof
This proof is left as an exercise. (Just apply Ito’s formula).
Moving towards Merton’s Problem
Definition:
For the initial endowment x ≥ 0, x ∈ R, we say that the consumption/investment scheme (c, π) is admissible at x if the wealth
X x,c,π (t) ≥ 0,
∀t P − a.s.
then (cπ) ∈ A(x). (This is a weak, minimal condition).
If x < 0, A(x) = ∅ where A(x): the set of all admissible portfolios. So if the
initial endowment is negative, there are no possible admissible sets.
The Merton Problem: an optimization/maximization problem over A(x).
We repeat the previous identity:
Z t
Z t
H0 (t)X(t) +
H0 (u)c(u)du = x +
H0 (u) π(u)σ(u) − x(u)θ(u) dW (u)
0
0
which we shorten as x+I(t). What does admissability mean in the expression?
Let (c, π) ∈ A(x), x ≥ 0. Then X x,c,π (t) ≥ 0 and consequently the left hand
side L.S≥ 0. This implies that I(t) is a lower bounded (F, P)-local martingale,
thus a super martingale. Then
Z T
i
h
H0 (u)c(u) du
E H0 (T )X(T ) +
{z
}
{z }
|
0 |
a)
b)
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where a is the discounted final wealth and b is the discounted cumulative
consumption. In the Merton Problem we use a utility function with
consumption and final wealth.
= x + E I(t) ≤ x + E[I(0)] = x
where the inequality comes from the super-martingale property.
This relation is called the ”budget constraint” and is a necessary condition
for admissibility of a given scheme (c, π). (The final wealth and cumulative
consumption cannot exceed the initial endowment).
Following this comes an important result: this constraint is not just
necessary, it is also sufficient. Since we are within the frameworks of a
complete market, we get a replication theorem.
A replicating theorem
Let x ≥ 0 and c(t), t ∈ [0, T ] be a given consumption rate.
Let ξ be an FT -measurable, non-negative random variable such that
hZ T
i
E
H0 (u)c(u)du + H0 (T )ξ = x,
0
then there exists a portfolio (π0 , π) such that the scheme (c, π) is admissible:
(c, π) ∈ A(x) and X x,c,π (T ) = ξ. (The final wealth equals ξ).
Comment:
• For a given claim ξ, a replicating portfolio exists.
• The ”budget constraint” is in this sense sufficient for admissibility.
Before we prove this theorem we need another few results. The replicating
theorem will be used a few times during the course.
Martingale Representation Theorem
Let M(t), t ∈ [0, T ] be an F -martingale such that E[M 2 (t)] < ∞, ∀t L2 (P),
RT
then ∃φ ∈ V[0,T ] for some adapted φ where E 0 φ2 (s)ds[ < ∞. With this
φ we know that
Z
t
M(t) = M(0) +
φ(s)dW (s)
0
Proof : Consult Oksendal.
Local Martingale Representation Theorem
If M(t), t ∈ [0, T ] is an F-local martingale, then ∃φ ∈ W[0,T ] for phi adapted
RT
and 0 φ2 (s) < ∞ a.s.. We note that this is a weaker condition than the
previous theorem. For these assumptions, we have
Z
M(t) = M(0) + 0t φ(s)dW (s)
Proof : Non-trivial. Omitted.
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Proof of the replication theorem
We will prove it using the martingale rep. thm. and Ito’s formula.
Z t
I(t) :=
H( u)c(u)du
0
M(t) := E I(T ) + H0 (T )ξ Ft .
Using the martingale representation theorem:
Z t
∃φ : M(t) = M(0) +
φ(s)dW (s).
0
From the budget constraint we have M(0) = x. We define
Z
S0 (t)
S0 (t) T
Y (t) :=
M(t) − I(t)
E
H+ (u)c(u)du + H0 (T )ξ =
Z0 (t)
Z0 (t)
t
| {z
}
H0−1
where we split the integral in half using the conditional expectation. Next
we use Ito’s formula:
Y (t) 1 1 1 d
=
dM(t)−dI(t) + M(t)−I(t) d
+dM(t)d
=
S0 (t)
Z0 (t)
Z0 (t)
Z0 (t)
i
1 h
1 1 φ(t)dW (t)−H0 (t)c(t)dt +
M(t)−I(t) θ(t)dW (t)+θ2 (t)dt +
φ(t)θ(t)dt
Z0 (t)
Z0 (t)
Z0 (t)
We apply the result from Girsanov’s theorem: dW0 (t) = dW (t) + θ(t)dt:
c(t)
1
φ(t)dW0 (t) + M(t) − I(t) θ(t)dW0 (t) −
dt
=
Z0 (t)
S0 (t)
i
c(t)
1 h
φ(t) + M(t) − I(t) θ(t) dW0 (t) −
dt
(*)
=
Z0 (t)
S0 (t)
On the other hand, recall the characterization of the discounted wealth
process of a Γ-financing portfolio:
X(t)
c(t)
1
d
=−
dW0 (t).
dt +
S0 (t)
S0 (t)
S0 (t)π(t)σ(t)
We set this integral equal to (*) above.
To construct the replicating portfolio, we set (superset T means
transpose)
iT
1 h
1
φ(t) + ; (t) − I(t) θ(t)
π(t)T σ(t) =
S0 (t) | {z }
Z0 (t)
column vector
11
and we obtain:
π(t)T =
iT
1 h
φ(t) + M(t) − I(t) θ(t) σ −1 (t)
H0 (t)
(we work in a complete market, so the σ-matrix is a non-degenerate, square
matrix).
i
1 −1 T h
π(t) =
σ (t) φ(t) + M(t) − I(t) θ(t)
H0 (t)
and we have obtained an explicit expression for the portfolio. Since all the
terms on the right side are adapted, so is π(t). This portfolio plays the
required role.
X x,c,π (t) = Y (t),
X x,c,π (0) = x,
X x,c,π (T ) = Y (T )
and
X x,c,π (T ) = Y (T ) =
1 1 M(T ) − I(T ) =
I(T ) + H0 (T )ξ − I(T )
H0 (t)
H0 (t)
=
1 H0 (T )ξ = ξ
H0 (t)
Utility Functions
Utility and preferences are central concepts in ”classical” economic theory.
The utility function describes agents preferences on ”well-being” in relation
to risk exposure. In utility there is a tradeoff between potential gain and risk.
Definition
A utility function is a concave, non-decreasing, upper semi-continuous (which
means lim supU(y) ≤ U(x)) function U : R 7→ [−∞, ∞] (the extended real
y→x
line), such that:
(i) dom U = {x ∈ R|U(x) > −∞} satisfies dom U 6= ∅, dom U ⊂ [0, ∞).
(ii) The derivative: the Marginal Utility, U ′ (x) is continuous, positive and
strictly decreasing on (dom U)O (the interior, or dom U\∂dom U.
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07/10-2010
Incomplete Markets
The complete market is a mathematically pleasent structure at the cost of
being unrealistic. There are several ways of making a market incomplete,
like e.g assymetric information, where the dealers dont have access to the
same information, or a market with an assymetric distribution of noise. If
we exchange the continuous Brownian motion for a Lévy process, we get
incompleteness.
The kind of incompleteness we will deal with are constraints on the
portfolio (like for instance imposing a lower bound on the portfolio, so we
can’t borrow the initial endowment or we can’t short sell).
Chapter 5
Pricing claims in an incomplete market. As opposed to the complete market
where we can find a specific price, we can only find a non-arbitrage price
range. Within this range the seller and buyer must agree, where seller
typically wants the highest possible price and the buyer the lowest possible.
We will focus our attention on the sellers price: the upper hedging price.
We introduce K: which is a closed, convex set in RN , which we regard as the
constrained set we must stay within. We repeat some important concepts.
The proportion of wealth invested in the different assets is given by the vector
p:
p = (p0 , p) = p0 (t) , p1 (t), . . . , pN (t)
| {z } |
{z
}
Risk free
Risky assets
In the same ways we have vectors π and η, given by
π = (π0 , π),
η = (η0 , η).
where πi is the amount of money invested in asset i and ηi is the number of
units of asset i (and in this setting we allow fractional units, so ηi ∈ R and
not N as is usual).
We have the relationship between this given by
pi (t) =
ηi (t)Si (t)
πi (t)
=
.
X(t)
X(t)
It’s easier to talk about convex sets when working with p since they are
bounded by X.
14
Pricing of claims B
The claim B is an Ft -measurable random variable. We only consider
European claims: i.e claims that can only be exercised at the strike time
T (so there are no τ < T ).
We will use the consumption process C(t) for t ∈ [0, T ]; the cumulative
consumption and not the consumption rate we used in the earlier chapter.
The consumption process C(t) is a right-continuous, non-decreasing process,
it is F-adapted and we have the helpful convention C(T ) < ∞. (All processes
we work with are measurable and adapted and all inequalities are P − a.s.
Digression: Why do we discount?
We will work with the discounted wealth process
X
x,C,π
(t) =
X x,C,π (0)
=⇒
S0 (t)
(The wealth process X x,C,π corresponding to the triple (x, C, π) is determined
by:
Z
Z t
X x,C,π (t)
dC(u)
π(u) · σ(u) · dW0 (u) x,c,π
= M
(t)
+
=x+
S0 (t)
S0 (u)
(0,t] S0 (u)
0
where we have the normal definitions, dW0 is the translated BM with θ:
Z t
W0 (t) = W (t) +
θ(s)ds,
θ(s) = σ −1 (t) [b(t) − r(t)I]
0
and we have the local martingale,
Z t
Z
1 t 2
θ ds .
Z0 (t) = exp −
θs dWs −
2 0
0
Rt
R
Whenever we write integrals 0 we really mean (0,t] . We omit the origin since
there can be a point with a certain weight that we do not want to include in
the calculations.
We define the corresponding portfolio-proportion process by,
p(t) =
15