MAT4730
Karatzas & Shreve: Methods of Mathematical Finance
1 - Basic Definitions
1.1 Stocks and Money Market
We work within the framework of a financial market with N + 1 financial
assets. There is 1 risk-free asset: the money market, typically a bank account
or bond, and N risky assets, typically stocks (but can represent any risky
security). The stocks evolve continuously with BM (Brownian Motion) as its
driving force. This entails that we do not have any “jumps” in the stocks,
like we would have with more general Lévy processes (like Brownian motion
combined with a Poisson process). For the most part we assume a finite time
horizon: everything takes place in t ∈ [0, T ] for T < ∞.
We work in the complete probability space (Ω, F , P ), where we have a Ddimensional BM,


W (1) (t)


..
Wt = 
, 0 ≤ t ≤ T
.
(D)
W (t)
where W (0) = 0 almost surely. We define the augmented filtration
F (t) := σ(F W (t) ∪ N ),
∀t ∈ [0, T ]
where we combine the filtration generated by Brownian motion with the P null sets of it. This is the information available to investors at time t: for
A ∈ Ft and ω ∈ Ω, investors know if ω ∈ A.
Money Market
We denote the money market with S0 (t) (the price) at time t with S0 (0) = 1.
S0 (t) is continuous, strictly positive and Ft -adapted with finite variation.
Can be decomposed into absolutely continuous, S0ac and singularly continuous
parts S0sc :
Z t sc
d ac
S (t)
dS0 (u)
dt 0
,
A(t) :=
.
r(t) :=
S0 (t)
S0 (u)
0
We have the dynamics for the money market:
dS0 (t) = S0 (t)[r(t)dt + dA(t)],
1
∀t ∈ [0, T ],
with solution (normal ODE?):
Z t
S0 (t) = exp
r(u)du + A(t) ,
∀t ∈ [0, T ].
0
When S0 (t) is absolutely continuous, A(t) = 0 for all t, and the money market
evolves like the value of a savings account with risk-free interest rate r(t) at
time t.
Stocks
We have N stocks, with prices at time t given by S1 (t), . . . , SN (t), and where
S1 (0), . . . , SN (0) - the starting values at time t = 0 - are positive constants.
The processes Sn (t) (for n ∈ 1, . . . N are continuous, strictly positive and
satsify:
"
#
D
X
(d)
dSn (t) = Sn (t) bn (t)dt + dA(t) +
σnd (t)dW (t)
d=1
for all times t ∈ [0, T and n = 1, . . . , N. The function b(t) is the drift, or the
mean rate of return process, and σ is an N × D matrix called the volatility
matrix. We include the A(t) process here, as defined earlier, to prevent there
being any arbitrage possibilities. The solution of this SDE, is (proved by
multidimensional Ito)
(Z D
#
)
Z t"
D
tX
X
1
2
Sn (t) = Sn (0) exp
σnd (s)dW (d) (s) +
bn (s) −
σnd
(s) ds + A(t)
2
0 d=1
0
d=1
for all t ∈ [0, T ] or n = 1, . . . , N. The singularly continuous process A(t)
does not enter into the discounted
stock prices: (We multiply both sides with
Rt
−1
S0 (t) so we get exp(− 0 r(s)ds − A(t)) multiplied with the right side).
# )
(Z D
Z t"
D
tX
1X 2
Sn (t)
(d)
σnd (s) ds
= Sn (0) exp
σnd (s)dW (s) +
bn (s) − r(s) −
S0 (t)
2
0
0
d=1
d=1
Dividens
In some applications stocks have an associated dividend rate process,
dividend is money earned on the stock, δn (t) which is the rate of payment per
dollar invested in the stock at time t. By including a divident rate process
in the dynamics for the stock price, we can define a yield process (yield per
share) with Yn (0) = Sn (0) and
#
"
D
X
σnd (t)dW (d) (t) + δn (t)dt
dYn (t) = Sn (t) bn (t)dt + dA(t) +
d=1
2
for n = 1, . . . , N. In integral form:
Yn (t) = Sn (t) +
Z
t
Sn (u)δn (u)du
0
for all t ∈ [0, T ], n = 1, . . . , N. We set Y0 (t) = S0 (t).
1.2 Portfolio and Gains Processes
First we consider a discrete time. We define a partition of the time horizon:
0 = t0 < t1 < . . . < tM = T . For n = 1, . . . , N and m = 1, . . . , M we define
ηn (tm ) to be the number of stocks of stock n held in the interval [tm , tm+1 )
and t0 (tm ) the number of shares in the money market in the same interval.
They are F (tm )-measurable (future information unknown/insider trading not
permissible).
We define the discrete gains process: the starting time gain is 0 G(0) = 0,
and
N
X
ηn (tm ) Yn (tm+1 − Yn (tm )
G(tm+1 ) − G(tm ) =
n=0
for 0 ≤ m ≤ M − 1, is the amount gained from tm to tm+1 . The investors
holding at time tm is the quantity:
N
X
ηn (tm )Sn (tm ),
0 ≤ m ≤ M,
n=0
which is the sum of all asset-prices multiplied by their value. If the portfolio
is self-financed, we have the relation:
G(tm ) =
N
X
ηn (tm )Sn (tm ),
0 ≤ m ≤ M,
n=0
that is; there are no withdrawals and no infusion of funds into the portfolio
in [0, T ].
T
We extend this to the continuous case, for η(t) = η0 (t), . . . , ηn (t) as
a continuous adapted process with the corresponding gains process defined
with starting value G(0) = 0 and the SDE for the evolution:
dG(t) =
N
X
ηn (t)dYn (t).
n=0
3
This will be an axiomatic equation. If we introduce the vector πn (t) :=
T
ηn (t)Sn (t), with π(t) = π0 (t), . . . , πn (t) , and use the expression for Sn (t)
and dYn (t), we can rewrite the self-financing equation as:
dG(t) = π0 (t)+π T (t)1 r(t)dt+dA(t) +π T (t) b(t)+δ(t)−r(t)1 dt+π T (t)σ(t)dW (t)
where 1 = (1, . . . , 1) and 1 ∈ RN . Note: πn (t) is the dollar amound invested
in stock n at time t.
See definition
2.1 for finiteness restrictions for a portfolio process
π0 (t), π(t) . The gains process associated with the portfolio process is given
by:
Z t
Z t
Z t
T
T
G(t) :=
π0 (s)+π (s)1 r(s)ds+dA(s) + π (s) b(s)+δ(s)−r(s)1 ds+ π T (s)σ(s)dW (s)
0
0
0
where we have a self-financing portfolio process if
G(t) = π0 (t) + π T (t)1,
∀t ∈ [0, T ],
or in other words: the value of the portfolio at time t is equal to the gains
earned from investments up to that time.
To simplify the notation, we can define the N-dimensional excess yield
processes:
Z t
Z t
σ(u)dW (u), 0 ≤ t ≤ T.
R(t) :=
[b(u) + δ(u) − r(u)1]du +
0
0
so we can write:
Z t
Z t
T
G(t) =
π0 (s) + π (s)1 r(s)ds + dA(s) +
π T (s)dR(s).
0
0
If the portfolio π is self financing, then we get:
G(t)
dS0 (t) + π T (t)dR(t)
S0 (t)
dG(t) =
which is solved by:
G(t) = S0 (t)
Z
0
t
1
π T (u)dR(u);
S0 (u)
4
0 ≤ t ≤ T.
1.3 Income and Wealth Processes
Introducing ways for an investor to make money or have expenses with the
portfolio.
For a financial market M, we define the cumulative income process Γ(t) which
is a semimartingale. The process Γ(t) is the cumulative wealth received by an
investor on the time interval [0, t], where the initial wealth is Γ(0). Expenses
are represented with a decrease in the Γ(t) process.
For a financial market M where π is a portfolio and Γ a cumulative income
process, the wealth process is defined as:
X(t) := Γ(t) + G(t),
the sum of the expenses and gains. We say the portfolio π is Γ-financed if
X(t) = π0 (t) + π T (t)1,
the wealth equals the bank account and the dollar values in each stock. Using
the excess yield process, R(t), we can write the differential form as:
X(t)
dS0 (t) + π T (t)dR(t)
S0 (t)
= dΓ(t) + X(t)[r(t)dt + dA(t)] + π T (t)[b(t) + δ(t) − r(t)1]dt + π T (t)σ(t)dW (t)
dX(t) = dΓ(t) +
We also have the discounted wealth process:
Z
Z t
X(t)
dΓ(u)
1
= Γ(0) +
+
π T (u)dR(u).
S0 (t)
S
(u)
S
(u)
0
(0,t] 0
0
We do not have π0 (t) in this expression, but we can find it using X(t).
1.4 Arbitrage and Market Viability
In a financial market M we say that a given tame, self-financed portfolio
process π is an arbitrage opportunity if the associated gains process G(t)
satisfies G(T ) ≥ 0 a.s and G(T ) > 0 with positive probability. A market M
where there is no arbitrage opportunity is said to be viable.
If athe number of stocks in a market exceeds the BM: N > D, we can
combine stocks into mutual funds (linear combinations of the underlying
stock) sand obtain N = D.
5
1.5 Standard Financial Markets
The market type we will mostly be concerned with.
Definition 1.5.1
A financial market model M is said to be standard if
(i) It is viable.
(ii) The number of stocks is not greater than the dimension D of the
underlying Brownian motion.
(iii) The D-dimensional, progressively measurable market price of risk θ(t)
satisfies
Z T
kθ(t)k2 dt < ∞ a.s
0
(iv) The positive local martingale
Z t
Z
1 t
′
2
Z0 (t) := exp −
θ (s)dW (s) −
kθ(s)k ds
2 0
0
is in fact a martingale.
For a standard market, we define the standard martingale measure P0 on
F (T ) by
P0 (A) := E[Z0 (T )1A ], ∀A ∈ F (T ).
By Girsanov’s theorem, the process
W0 (t) := W (t) +
Z
t
θ(s)ds,
0
is a BM under P0 relative to the filtration
F (t). Using W0 (t), the excess yield
Rt
process can be rewritten as R(t) = 0 σ(u)dW0 (u).
The process given by:
Z0 (t)
,
H0 (t) :=
S0 (t)
is called the state price density process, and will play a key role in subsequent
chapters. Using H0 (t) we can rewrite conditions involving the martingale
measure P0 in terms of the original probability measure P .
1.6 Completeness of Financial Markets
If an investor/agent is going to make a payment B at time T , the agent can
set aside an amount x at time 0, invest it such that the investment grows to
B over the time horizon: hedging the risk inherent in the random payment.
6
Definition
let M be a standard financial market, and let B be an F (T )-measurable
random variable such that B/S0 (T ) is almost surely bounded from below
and
B
x := E0
< ∞.
S0 (T )
(i) We say that B is financeable (replicable), if there is a tame, x-financed
portfolio process π whose associated wealth process satisfies X(T ) = B, that
is
Z T
1
B
=x+
π T (u)σ(u)dW0(u) a.s.
S0 (T )
S
(u)
0
0
(ii) If every B with these conditions is financeable, we say the market is
complete. (Not completeness in the normal, converging to a limit sense!).
3 - Single Agent Consumption/Investment
3.1 - Introduction
3.2 - The Market
We are working in a complete, standard financial market M, where the
money market is governed by the equation
dS0 (t) = S0 (t)[r(t)dt + dA(t)]
and the stock prices satisfy
"
dSn (t) = Sn (t) bn (t)dt + dA(t) +
N
X
d=1
#
σnd (t)dW (d) (t) .
with σ(t) is an N × N invertible (nonsingular) for almost every t ∈ [0, T ].
We also repeat additional important processes.
θ(t) := σ −1 (t) b(t) + δ(t) − r(t)1
Z t
W0 (t) := W (t) +
θ(s)ds
0
Z t
Z
1 t
′
2
Z0 (t) := exp −
θ (s)dW (s) −
kθ(s)k ds
2 0
0
H0 (t) :=
7
Z0 (t)
.
S0 (t)
In the standard market, Z0 (t) is a martingale which permits the definition
of the standard martingale measure P0 . We will use Z0 (t) only as a local
martingale, so we can use this theory for the unconstrained case in chapter
6. We make the three assumptions:
Z T
Z T
E
H0 (t)dt < ∞, E[H0 (T )] < ∞ and E
H0 (t)dt + H0 (T ) < ∞.
0
0
3.3 - Consumption and Portfolio processes
8