PROGRAMMING FOR MACRO-FINANCE

Unit 4: Financial Derivatives Pricing

Daniel Arrieta
darrieta@faculty.ie.edu

IE University

Academic year: 24-25

Outline

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Introduction
Modern quantitative finance is based on two cornerstone valuation
frameworks
i. Martingale pricing based on the Fundamental Theorem of Asset
Pricing (FTAP).
ii. Replication pricing based on Partial Differential Equation (PDE)
modeling.
Martingale pricing is a financial valuation approach based on the
relation between absence of arbitrage and martingales. It can be
applied to a variety of financial contracts, e.g. futures, equity options,
interest rate products, credit derivatives, etc.
In contrast to the PDE approach to pricing, martingale valuation is
done by means of conditional expectations which can be numerically
computed using Monte Carlo algorithms.
The martingale pricing paradigm foundations, as well as its relation
with the replication approach, will be presented next.
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Next

Introduction and motivation

bookmaker example
main conclusions

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Bookmaker example
A unitary bet on Team i succeeding, has the following payout function
(
Qi − 1, if ω = ωi ,
X (ω) =
−1, otherwise.
Where X (ω) is the random variable modeling the bet result.
For example, ωi denotes the event of team i winning a match or a
championship. The quantity Qi is said to be the quote on Team i.
The number of unitary bets and historical probabilities for the winner
on a football game are gathered in the next table.

Team # Bets Probability

A 20 1/3
B 80 2/3

Table 1: Bookmaker available information.

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Bookmaker example (ii)
How can a bookie estimate fair quotes and hedge her risks? Initially
she only has the #Bets and the historical probabilities information.
If ω = ωi , then a zero result must verify
Ni · (1 − Qi ) + (N − Ni ) · 1 = 0. (1)
where Ni stands for the number of bets placed on Team i and N is
the total number of bets. Therefore a zero result quote verifies
N
Qi = .
Ni
Dividing both members of (1) by N yields
qi · (1 − Qi ) + (1 − qi ) · 1 = 0.
Where qi can be defined as an implied probability which verifies
Ni 1
qi := = . (2)
N Qi
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Bookmaker example (iii)
In our example i ∈ {A, B} and N = NA + NB = 20 + 80.
The quotes QA = 5.00 and QB = 1.25 imply the probabilities given in
the next table.
ω Implied Historical
ωA 1/5 1/3
ωB 4/5 2/3

Table 2: Historical versus implied probabilities.

Notice that, because of equation (1) being verified, under these

quotes the bookie’s result is certain and equal to zero.
Using these implied probabilities the expected value of the bookie’s
result is also zero, thus is
EQ [−X ] = 0.
where Q denotes the probability measure associated to the qi ’s.
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Bookmaker example (iv)
There is other way of managing the bookie’s position. It is based on
the historical probabilities, and it is said to be the actuarial approach.
This approach relies on the law of large numbers and it is a passive
risk managing technique.
In this latter pricing framework the quotes are deducted from the
historical probabilities.
If we denote by pi the historical probability of the ωi event, the
actuarial quote Qi′ is obtained by using pi instead of qi in (2), i.e.,

1
Qi′ = .
pi
The expected value of the bookie’s result under the historical
probability measure, denoted by P, is also zero.
Nevertheless, unlike the implied probabilities pricing framework, the
bookie’s profit or loss, is not certain.
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Next

Introduction and motivation

bookmaker example
main conclusions

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Main ideas from the example
Above simplified modeling of a bookmaker job rises the following
topics:
i. Bet result expectation as pricing functional;
ii. The bet risk could be totally hedged by means of a replicating
portfolio;
iii. This total risk hedge implies also a bet valuation procedure;
iv. Are both pricing approaches equivalent?;
v. There are several probability measures involved.
The goal of the current lecture is answering above questions in the
context of financial valuation.
The replication approach in continuous time was firstly presented by
Black and Scholes (1973), and Merton (1973).
The usage of the expectation as pricing tool was originally proposed
by Harrison and Kreps (1979).
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Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Next

Introduction and motivation

Mathematical foundations
probability theory
financial definitions

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Measure theory basics
Definition (measurable space)
Let Ω be a set and let F be a σ-algebra of its subsets, the pair (Ω, F)
is said to be a measurable space.

Definition (measure)
A measure is a mapping µ : F → [0, ∞] verifying for each An ∈ F
∞
!
∞
[
µ An = sumn=1 µ (An ) .
n=1

The triple (Ω, F, µ) is called measure space.

Definition (absolutely continuous)
Let µ and ν be two measures defined on (Ω, F), µ is said to be
absolutely continuous with respect to ν if µ(A) = 0 ⇒ ν(A) = 0 for
each A ∈ F. This is denoted as µ ≫ ν.

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Measure theory and probability
Definition (equivalent measures)
If µ ≫ ν and ν ≫ µ, µ and ν are said to be equivalent measures,
i.e., µ and ν are equivalent iif they agree in the zero measure sets.

Definition (measurable function)

A mapping X : Ω → R is a measurable function if ∀ B ∈ B
X −1 (B) := {ω ∈ Ω : X (ω) ∈ B} ∈ F, where B is the Borel σ-algebra.

Definition (probability measure, probability space)

A measure P defined on a measurable space (Ω, F) verifying
P : F → [0, 1], and P(Ω) = 1 is said to be a probability measure,
and (Ω, F, P) is called a probability space.

Definition (random variable)

A measurable function S : Ω → R on a probability space (Ω, F, P) is
said to be a random variable.

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Filtrations, stochastic processes, and martingales
Definition (filtration)
Given a probability space (Ω, F, P) a (discrete) filtration is an
increasing sequence of sub-σ-algebras, {Ft }t≥0 of F, thus is
F0 ⊂ F1 ⊂ · · · ⊂ Ft ⊂ · · · F.
The tuple (Ω, F, {Ft }t≥0 , P) is called a filtered probability space.
Definition (adapted, predictable, stochastic process)
A sequence of random variables {St = St (ω)}t≥0 is a stochastic
process. St is adapted if it is Ft -measurable for all t, and it is said
to be predictable if St is Ft−1 -measurable.

Definition (martingale)
An adapted stochastic process St is a martingale regarding to Ft , if
it is integrable, and EP [ST |Ft ] = St , for each t ≤ T , almost surely.
When a property is fulfilled for each non zero measure set, it is said
that it holds almost surely (a.s.).
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Next

Introduction and motivation

Mathematical foundations
probability theory
financial definitions

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Portfolios and trading strategies
(m)
A financial market is composed by M + 1 financial assets. St
denotes the m-th security price process, for time t ≥ 0.
Definition (portfolio value, trading strategy)
A
 portfolio is an  adapted (M + 1)-dimensional stochastic process,
(0) (M)
St , . . . , St . The portfolio value process Vt is given by
M
(0) (0) X (m) (m)
Vt := ht St + ht St .
m=1
(m) (m)
Where each ht is a predictable stochastic process. The process ht
(m)
expresses the number of units of St held in the portfolio at time t
and is said to be a trading strategy.
Definition (numeraire)
A numeraire is any positive process used as reference for the portfolio
(0)
value. Without loss of generality, the process St is usually
considered the numeraire, and it is denoted by Nt .
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Self-financed portfolio and arbitrage opportunity
Definition (self-financed portfolio)
A portfolio is said to be self-financed if its value process Vt verifies
M  
X (m) (m) (m)
Vt − Vt−1 = ht St − St−1 , for each t > 0.
m=0

Definition (arbitrage opportunity)

An arbitrage opportunity is a self-financed portfolio which verifies

V0 < 0, VT ≥ 0, (a)
or
V0 = 0, VT ≥ 0, P (VT > 0) > 0. (b)

Type a arbitrage is an immediate profit without future cost at T .

Whereas type b arbitrage requires no initial investment, has some
possibility of a positive portfolio value at expiration T , and has no
probability of terminal portfolio value being negative.
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Derivative securities and complete markets
The basic market comprises the aforementioned M + 1 securities,
however some additional securities are usually included too.
Definition (derivative security payoff)
The payoff of a derivative security in a financial market on a filtered
probability space (Ω, Ft , P), is any random variable H measurable
with respect to FT , i.e., generated by the basic market securities up
to time T .
The market including additional derivatives securities is said to be an
extended market.
Definition (replicable derivative security, complete market)
A derivative security H is said to be replicable if there exists a
self-financing portfolio V such that HT = VT .
A financial market is said to be complete if every derivative security
can be replicated.

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Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

fundamental theorems
replication approach

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
First fundamental theorem and martingale pricing
Modern financial pricing is based on the absence of arbitrage notion,
this is summarized in the following result.
Theorem 1 (first Fundamental Theorem of Asset Pricing (FTAP))
A basic market on a filtered probability space (Ω, Ft , P) is
arbitrage-free, if and only if a martingale probability measure Q
equivalent to P exists, and it satisfies
(m)
" (m) #
St ST
= EQ Ft , ∀ m ∈ {1, . . . , M}, t ≤ T . (3)
Nt NT

Therefore there is no arbitrage opportunity if exists a probability

measure Q for which the quotient between each security price process
S (m) and the numeraire process N is a martingale.
Usual numeraire securities are the bank account, and the T -maturity
zero coupon bond. Their associated martingale measures are called
risk-neutral measure, and T -forward measure respectively.
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Second fundamental theorem
Theorem 1 (first FTAP) tells nothing about market completeness or
the uniqueness of the martingale measure, a result linking both
concepts is given next.
Theorem 2 (second FTAP)
An arbitrage-free market is complete if and only if exists a unique
martingale measure Q equivalent to P.
The seminal reference addressing the first Fundamental Theorem of
Asset Pricing (FTAP) in discrete time for a finite securities market is
Harrison and Kreps (1979).
Later Harrison and Pliska (1981) and Harrison and Pliska (1983),
Taqqu and Willinger (1987), and Delbaen and Schachermayer (1994),
extended and generalized both fundamental theorems.
The proofs of both theorems, for discrete time finite securities market,
can be found in Capinski and Kopp (2012). For a general continuous
time approach see for example Elliott and Kopp (2004).
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Fundamental theorems interpretation
An intuitive understanding of previous theorems is that arbitrage-free
derivatives valuation is an interpolation exercise.
h i
X
EQ N

market
securities

derivative
securities

X
Figure 1: Martingale pricing in a complete market.
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Fundamental theorems and incomplete markets
Arbitrage-free pricing in an incomplete market can be regarded as a
measure choice problem as illustrated below.
h i
X
EQj N

X (i) price
under Q2

X (i) price
under Q1

X
Figure 2: Martingale pricing in an incomplete market.
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Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

fundamental theorems
replication approach

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Feynman-Kac theorem and replication pricing
The replication approach for arbitrage-free pricing generalizes the well
known Black and Scholes (1973) valuation framework.
Theorem 3 (Feynman-Kac)
Let xt follow the stochastic differential equation (SDE)
dxt = µ (xt , t) dt + σ (xt , t) dWtQ ,
where WtQ is Brownian motion under the measure Q.
Let V (xt , t) be a differentiable function of xt and t, which verifies the
partial differential equation (PDE)
∂V ∂V 1 ∂2V
+ µ (xt , t) + σ (xt , t)2 − r (xt , t) V (xt , t) = 0,
∂t ∂x 2 ∂x 2
with boundary condition V (xT , T ), then the PDE has the solution
 RT 
Q − r (xs ,s)ds
V (xt , t) = E e t V (xT , T ) Ft . (4)

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Feynman Kac theorem and martingale pricing
A proof of Feynman-Kac theorem can be found for example in
M. Capinski et al. (2012).
The expectation in Feynman-Kac theorem is taken under the measure

Q, and its associated numeraire is Nt = exp − 0T r (xs , s) ds .
R

The Black and Scholes valuation framework is a particular case of the

Feynman-Kac theorem where µ (xt , t) = r xt , σ (xt , t) = σxt , and
V (xT , T ) is the payoff of an European vanilla option.
In this model Q measure is unique because market is complete and
the numeraire is a deterministic risk free bank account.
Thus, the fundamental theorems, and the Black and Scholes pricing
framework “coincide” if the numeraire is the deterministic risk free
bank account and market is complete.
Both valuation frameworks yield the same prices and hence the
modeling assumptions could be considered “equivalent”.

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Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Next
Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

model set up
law of one price and arbitrage
working examples

Black-Scholes-Merton framework

Annex: proofs
Market model
The market is comprised by an adapted (m +1)-dimensional
(0) (j)
stochastic process, denoted by St , . . . , St , which models the
securities price process for each t ∈ {0, 1}.
 
(0) (j)
The initial prices, S0 , . . . , S0 , are positive scalars known at the
initial time t = 0.
At t = 0 the list of all possible states of the world is known, but which
one occurs is revealed at t = 1, i.e., the end of the investment period.
Their values at t = 1 are random variables, defined with respect to a
sample space Ω = {ω1 , . . . , ωn } of n possible states of the world.
A probability measure P satisfying P(ω) > 0, for all ω ∈ Ω, is defined
on Ω.
n o
S will denote the price process St : t ∈ {0, 1} , where St is the row
 
(0) (j)
vector St = St , . . . , St .

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Payoff matrix and riskless security
The values of the asset price process at t = 1 can be listed in the
following n × (m + 1) matrix
 
(0) (1) (j)
S (ω ) S1 (ω1 ) . . . S1 (ω1 )
 1(0) 1
S (ω2 ) S (1) (ω2 ) . . . S (j) (ω2 )

 1 1 1
S1 (Ω) :=  .

.. .. .. ..

 . . . . 

(0) (1) (j)
S1 (ωn ) S1 (ωn ) . . . S1 (ωn )

Above matrix is called the payoff matrix.

Since the assets are limited liability securities, the entries in S1 (Ω) are
non-negative scalars.
(0)
St is a strictly positive riskless security or bank account.
(0) (0)
Without loss of generality, S0 = 1 and S1 = 1 + r , where r ≥ 0 is
the deterministic interest rate over one period.

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Discounted price process
(0)
The riskless security St will be used as the numeraire. It defines for
each m ∈ {0, 1, . . . , M} the discounted price process

(j)
(j) S
Sbt := t(0) , t ∈ {0, 1}.
St

The payoff matrix of the discounted price process at t = 1 is given by

 
(1) (j)
1 Sbt (ω1 ) . . . Sbt (ω1 )

1 (1) (j) 
Sbt (ω2 ) . . . Sbt (ω2 )
S
b1 (Ω) :=  .

. .. .. ..
 .. . . . 
 
(1) (j)
1 Sbt (ωn ) . . . Sbt (ωn )

This matrix will be denoted by S

b1 when no explicit dependence of the
states is necessary.
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Trading strategy and portfolio value
Let h(j) be the trading strategy, i.e., a predictable stochastic process
(j)
reflecting the number of units of St held in the portfolio at time
t ∈ {0, 1}.
Let Vt denote the portfolio value process for t ∈ {0, 1}, then
m
(0) X (j)
Vt = h(0) St + h(j) St .
j=1

Let G be the random variable portfolio gain generated by following

the trading strategy h. Then
m
X
G = h(0) r + h(j) ∆S (j) ,
j=1

where
(j) (j)
∆S (j) := S1 − S0 .
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Discounted portfolio and gain processes
(0)
Using the riskless security St as numeraire, the discounted portfolio
value process Vbt generated by following the trading strategy h, is
given by
m
bt = h(0) +
X (j)
V h(j) Sbt .
j=1

The corresponding discounted portfolio gain process, denoted by G,

b is
yielded by
m
X
G
b = h(j) ∆S (j) .
j=1

The portfolio is self-financing if its value process verifies V1 = V0 + G.

If h ∈ Rm+1 and its j th entry is h(j) for each j ∈ {0, . . . , m}, then the
current portfolio value and the discounted portfolio payoff are given
by S0h and Sb1h , respectively.

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Next
Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

model set up
law of one price and arbitrage
working examples

Black-Scholes-Merton framework

Annex: proofs
Asset span and market completeness
n o
The set S = S1h : h ∈ Rm+1 is called the asset span.
h i
(0) (j)
S is the column space of the payoff matrix S1 (Ω) · · · S1 (Ω) ,
(j)
where S1 (Ω) is the j th column of S1 (Ω), for each j ∈ {0, 1, . . . , m}.
If a new security lies inside S, then its payoff can be expressed as a
(0) (j)
linear combination of S1 (Ω), . . . , S1 (Ω), and therefore it is a
replicable security, also called redundant security.
As defined previously, a discrete market model is complete if every
derivative’s payoff vector lies inside the asset span.
This occurs if and only if the dimension of the asset span equals the
number of possible states.
Recall that, in complete markets, all securities (e.g., derivatives) can
be replicated, if they can be replicated, why are they needed at all?

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Pricing functional and law of one price
Given a discounted portfolio payoff X , if X ∈ S then X = S
b1h for
some portfolio weights h ∈ Rm+1 .
Therefore the current value of X is S
b0h , and it can be regarded as a
X ) on the payoff X .
pricing functional F (X
Proposition 4 (law of one price (LOP))
Given two portfolios h and h ′ , if S1h = S1h ′ then S0h = S0h ′ .
According to this proposition all portfolios with the same payoff must
have the same price.
If the law of one price fails, then it is possible to have two portfolios
or trading strategies h and h ′ such that S1h = S1h ′ but S0h ̸= S0h ′ .
Hence no arbitrage implies that the law of one price holds.
Proposition 5
A necessary and sufficient condition for the law of one price to hold is
that a portfolio with zero payoff must have zero price.
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Arrow security and state price
If the law of one price holds, then the pricing functional is
single-valued. Furthermore, it is a linear functional, this is

X + βY
F (αX Y ) = αF (X
X ) + βF (Y
Y ),

for any scalars α and β and payoffs X and Y .

Definition (Arrow security)
Let e k denote the k th coordinate vector in the vector space Rn , i.e.,
e k takes the value 1 in the k th entry and zero in all other entries. The
vector e k is called the Arrow security of the state k and it can be
considered as the discounted payoff vector of a security.

Definition (state price)

If the securities model is complete and the law of one price holds,
then the pricing functional F assigns a unique value πk = F (ee k ) to
each Arrow security. This value πk is said to be the k th state price.
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Arbitrage and martingale measure
The above definition of an arbitrage opportunity, can be restated in
the current market model as some portfolio verifying:
V
b0 < 0, b1 ≥ 0,
V
or  
V
b0 = 0, b1 ≥ 0,
V P V
b1 > 0 > 0,

where Vbt stands for the portfolio discounted value process for
t ∈ {0, 1}, and P denotes the actual probability measure.
A probability measure Q on Ω is a martingale probability
measure,associated to a given numeraire, if it satisfies:
i. Q(ω) > 0 for all ω ∈ Ω, and
ii. EQ [∆Sb (j) ] = 0, for all j ∈ {0, 1, . . . , m}.
Latter property is equivalent to
n
(j) X (j)
Sb0 = Sb1 (ωk )Q(ωk ).
k=1
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State prices and martingale measure
In markets with no arbitrage opportunities, the fair value of a portfolio
is yielded by means of a martingale probability measure.
Such probability measure is not necessarily unique as can be deduced
from the Theorem 1 (first FTAP), where nothing about the
uniqueness is stated.
It is necessary the result of the Theorem 2 (second FTAP) for
ensuring the uniqueness.
Proposition 6
(0)
Assume there is a security with a strictly positive price process St .
If there is a set of positive state prices, then a martingale probability
(0)
measure, Q, exists with St as the numeraire security.
Furthermore, there is a one-to-one correspondence between sets of
positive state prices and risk-neutral probability measures.
The proof of this proposition is detailed in the Annex.
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Next
Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

model set up
law of one price and arbitrage
working examples

Black-Scholes-Merton framework

Annex: proofs
Example 1: law of one price and arbitrage
Consider the market given by
h i
S0 = 1 1 ,

and " #
1 1
S1 = .
1 2
LOP holds if and only if zero payoff has zero price.
Zero payoff implies S1h = 0 , which in the current example means
" #" # " #
1 1 h(1) 0
= .
1 2 h(2) 0

It is easy to verify that the unique solution is h(1) = 0, and h(2) = 0.

Obviously, the zero portfolio always has zero price therefore LOP
holds.
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Example 1: law of one price and arbitrage (ii)
Let’s check now if there is any arbitrage in this market.
A type b arbitrage opportunity has zero price, this is S0h = 0 .
In the current market this implies
" # " #
h i h(1) 0
1 1 = . (E1.1)
h(2) 0

Type b arbitrages must also have portfolio holdings verifying S1h ≥ 0 ,

with at least one strict inequality.
In this first example
" #" # " #
1 1 h(1) 0
≥ . (E1.2)
1 2 h(2) 0

So equations (E1.1) and (E1.2) summarize non-arbitrage conditions.

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Example 1: law of one price and arbitrage (iii)
It is easy to verify that the solution to systems (E1.1) and (E1.2) are
the portfolios with holdings in the set
(" # )
−1
x : x ∈ R+ \ {0} .
1

The financial interpretation is quite intuitive, for example if x = 1

then the portfolio holdings are h(1) = −1, and h(2) = 1, i.e., short the
first security and long the other security.
This portfolio has zero price and, at t = 1, has the following payoff
" #" # " #
1 1 −1 0
= .
1 2 1 1

Thus is, the portfolio costs nothing and pays zero in the first state
and one in the second state.

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Example 2: martingale probability measure
Consider now the market given by
h i
S0 = 1.0194 3.4045 2.4917 ,

and  
1.03 3 2
1.03 4 1
S1 =  .
 
1.03 2 4
1.03 5 2

Using the first security as numeraire, does exist a martingale

probability measure?, if so, is it unique?
If a martingale probability measure Q exists, then

S
b0 = π S
b1 ,

being π the vector of Q probabilities.

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Example 2: martingale probability measure (ii)
The problem of finding π can be restated as solving the system
Ax = b ,
b T , x = π T , and b = S
where A = S bT .
1 0
Recall from linear algebra that the equation Ax = b solved for the x ,
could have no solution, infinitely many solutions, or a unique solution.
Definition (nullspace)
The nullspace of a m × n matrix A , denoted by N(A
A), is the set
A) := {xx : Ax = 0 } ⊆ Rn .
N(A
A) is a vector space.
It is easy to verify that N(A
If more than one solution exists so that Ax 1 = Ax 2 = b with x 1 ̸= x 2
then y = x 1 − x 2 is an element of the nullspace of A .
In particular, any solution can be written as the sum of a particular
solution and an element of the nullspace.
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Example 2: martingale probability measure (iii)
How do we apply this for finding the martingale probabilities π ,
assuming such probabilities exist?
Firstly, we have to compute a particular solution x 0 to
bTx = S
S bT .
1 0

For example computing the least squares solution. This can be donde
in Pyhton using “linalg.lstsq” method which is available from both
numpy, and scipy libraries.
This yields the following particular solution
 
0.197173
0.155735
x0 =  .
 
0.336672
0.310419

Notice that this x 0 is a martingale probability measure as previously

defined, but is it unique?.
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Example 2: martingale probability measure (iv)
b T , this is just what the Python
Now we have to find a basis for S 1
linalg.null_space method from scipy library does.
T
Using this the basis of N(Sb 1 ) is given by
 
−0.73721
 0.589768 
z = .
 
 0.294884 
−0.147442

In this market example, the dimensionality of the nullspace is 1, and

b T ) which we have
so there is only one element in the basis of N(S 1
denoted by z .
Next step is computing each solution to
b T πT = S
S bT ,
1 0

by means of the previously computed basis.

35/54
Example 2: martingale probability measure (v)
bTπT = S
Finally, every possible solution to the system S b T can be
1 0
written in the following form
     
π1 0.197173 −0.73721
π  0.155735  0.589768 
 2 
 =  + α ,
  
π3  0.336672  0.294884 
π4 0.310419 −0.147442

where α ∈ R.
We just need to find α such that the solution is strictly positive.
If no such α exists, then there is no martingale probability measure in
the analyzed market.
It is easy to verify that if
α < 0.264062,
then π components are strictly positive, and therefore they constitute
a martingale probability measure.
36/54
Example 2: Python code
Latter example can be solved using the Pyhton script displayed below.
''' Example showing how to find the null space of a
matrix using numpy and scipy . linalg '''
import numpy as np
from scipy . linalg import null_space
S0 = np . array ([[1.0194 ,3.4045 ,2.4917]])
S1 = np . array ([[1.03 , 3 , 2] ,
[1.03 , 4 , 1] ,
[1.03 , 2 , 4] ,
[1.03 ,5 , 2]])
S0_disc = S0 / S0 [0 ,0]
S1_disc = S1 / np . tile ( S1 [: ,0] , (3 ,1) ) . transpose ()
x0 , res , rnk , s = np . linalg . lstsq ( S1_disc .T , S0_disc .T ,
rcond = None )
ns = null_space ( S1_disc . T )
alpha0 = - x0 / ns

As aforementioned, calculations are performed using the numpy

library, and the linalg.null_space method from scipy library.
37/54
Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Next
Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework
market model
pricing by replication
martingale pricing approach

Annex: proofs
BSM market model
The Black-Scholes-Merton (BSM) valuation framework considers an
economy modeled by a filtered probability space (Ω, F, {Ft }t≥0 , P)
where P denotes the real or objective probability measure.
A finite time horizon T is assumed, i.e., 0 < T < ∞.
The evolution of the relevant asset (e.g. stock index), under the
risk-neutral measure Q, is given by the following stochastic differential
equation (SDE)
dSt
= bdt + σdWtQ , (5)
St
where St denotes the asset price at date t, b is the constant
risk-neutral drift, σ is the constant volatility of the asset log-return,
and WtQ denotes a standard Brownian motion under Q.
The default assumption will be that b = r the short riskless rate.
This hypothesis is dropped, i.e., b ̸= r , for taking into account
dividend yields, or the basis between rates for FX products.
38/54
BSM market model (ii)
The risk-less bank account follows the differential equation
dBt
= rdt. (6)
Bt
This later modeling assumption implies that the bank account is
deterministic.
The time t value of a zero-coupon bond paying one unit of currency
at T is
Pt (T ) = e −r (T −t) , 0 ≤ t < T,
with PT ≡ 1, i.e., the definition of a zero-coupon bond.
It is well-known that the BSM model is complete, and therefore that
the P-equivalent martingale measure Q is unique.
The proofs of these last statements can be found in Björk (2004),
Theorem 8.3 for completeness, and Theorem 10.17 for uniqueness of
the risk-neutral measure Q.
39/54
Next
Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework
market model
pricing by replication
martingale pricing approach

Annex: proofs
BSM partial differential equation
Next step is computing the value V of a contingent claim, e.g., an
European call option, on the underlying S.
Following Wilmott (1998), it is assumed that the value depends on S
value and time t only, i.e., V (S, t).
Itô’s lemma gives dV over time

∂Vt 1 ∂ 2 Vt 2 ∂Vt
dVt = dS + 2 νt dt + dt.
∂St 2 ∂St ∂t

Equation (5) models dS with νt = σSt , hence

∂Vt   1 ∂2V ∂V
t 2 2
dVt = rSt dt + σSt dWtQ + 2 σ St dt + dt
∂St 2 ∂St ∂t
! (7)
∂Vt ∂Vt 1 ∂ 2 Vt ∂Vt
= σSt dWtQ + rSt + σ 2 St2 + dt.
∂St ∂St 2 ∂St2 ∂t

40/54
BSM partial differential equation (ii)
Defining the delta of the contingent claim by
∂Vt
∆t := ,
∂St
and a portfolio Π := V − ∆S, which time t value is
Πt = Vt − ∆t St .
Using (5) and (7), dΠt is equal to
!
∂Vt ∂Vt 1 ∂ 2 Vt ∂Vt
 
σSt − ∆t dWtQ + rSt + σ 2 St2 2 + − r ∆St dt,
∂St ∂St 2 ∂St ∂t

which using definition of ∆, simplifies to

!
1 2 2 ∂ 2 Vt ∂Vt
dΠt = σ St + dt. (8)
2 ∂St2 ∂t

As a result, the portfolio is (locally) risk-less.

41/54
BSM partial differential equation (iii)
To avoid arbitrage, a risk-less portfolio must yield the risk-less short
rate, which according to (6), implies that

dΠt = r Πt dt.
Equating this with (8) yields
!
∂Vt 1 2 2 ∂ 2 Vt ∂Vt
 
r Vt − dt = σ St 2 + dt,
∂St 2 ∂St ∂t

rearranging all terms in left hand side, yields the famous BSM partial
differential equation

∂Vt 1 ∂ 2 Vt ∂Vt
+ σ 2 St2 2 + − rV = 0. (9)
∂St 2 ∂St ∂t

It is worth noticing that This equation holds for every contingent

claim whose value V depends on S and t only.
42/54
European option pricing
Each specific contingent claim will establish certain boundary
conditions based on its payoff at maturity T .
These boundary conditions applied to the partial derivative equation
given by (9), allow different theoretical prices to be obtained.
In the case of European call options
V (0, t) = 0,
lim V (St , t) = St ,
St →∞
V (ST , T ) = max (ST − K , 0) ,
and for European put options
V (0, t) = 0,
lim V (St , t) = K ,
St →0
V (ST , T ) = max (K − ST , 0) ,
where K is the option strike price.
43/54
European option pricing (ii)
For European options, also called vanilla options, the solution to BSM
PDE (9) is given by
 
V (St , τ ) = e −r τ ϕ St e bτ N (ϕd+ ) − KN (ϕd− ) , (10)
where τ := T − t, N(x ) stands for the standard Gaussian cumulative
distribution function, the binary variable ϕ has the values 1 and −1
for call and put options respectively, and
bτ
ln Se 1 √
d± := √K ± σ τ .
σ τ 2
The notation d± means that to obtain d+ the sign + is used in the
right member of the definition, and similarly for d− .
It is usual also call d1 to d+ , and d2 to d− , and the following
relationship between the constants holds
√
d− = d+ − σ τ ,
√
d2 = d1 − σ τ .
44/54
Next
Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework
market model
pricing by replication
martingale pricing approach

Annex: proofs
Martingale pricing
Theorem 1 (first FTAP) states that the time t value of a replicable
contingent claim V maturing at T is given by
Nt
 
Vt = EQ V T Ft , 0 ≤ t ≤ T,
NT
where Q denotes the martingale probability associated to the
numeraire N.
For the sake of simplicity EQ
t [X ] := E [ X | Ft ] in the following.
Q

Above valuation in the BSM model framework with numeraire N = B

the risk-less bank account yields
h i
−rT
V0 = EQ
0 e VT , (11)
for t = 0, i.e., the relevant case when considering valuation.
As the market is complete, Q is unique according to Theorem 2
(second FTAP), and, as the numeraire is the risk-less bank account, it
is called the risk-neutral probability measure.
45/54
Pricing by Monte Carlo
Estimating expectations or integrals by Monte Carlo simulation is
straightforward. Applying this technique for computing (11) is easy if
the underlying process under Q is simulated without difficulty.
This is the case for the BSM model where the underlying dynamics is
given by the SDE (5).
For example, the contingent claim could be an European option with
payoff VT (ST ) = max (ST − K , 0) ,
i.e., a call, or VT (ST ) = max (K − ST , 0) ,
for a put, where K > 0 is the fixed strike price of the option.
This approach also handles effortlessly exotic path dependent options
as Asian, barrier, or cliquet.
The main limitation arises from American features, nevertheless, this
can be addressed by the well known Least Squares Monte Carlo
technique as first presented by Longstaff and Schwarz (2001).
46/54
Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
propositions
theorems
Proof of Proposition 6
Suppose a set of positive state prices [π1 , . . . , πn ]T exists, then
n
(j) X (j)
S0 = πk S1 (ωn )
k=1

n

n (0) (j)
(12)
X (0) X πk S1 (ωn ) S1
=  πj S1 (ωj ) Pn (0) (0)
.
j=1 k=1 j=1 πj S1 (ωj ) S1

Noticing that
n
(0) X (j)
S0 = πj S1 (ωj ).
j=1

This allows defining

(0)
πk S1 (ωn )
qk := P (0)
. (13)
n
j=1 πj S1 (ωj )

47/54
Proof of Proposition 6 (ii)
Using definition (13) then

Q := (q1 , . . . , qn ),

is a probability measure.
Equation (12) then implies
(j) n (j)
S0 X S1 (ωn )
(0)
= qk (0)
S0 k=1 S1 (ωn )
" (j) #
Q S1
=E (0)
,
S1

and so Q is a martingale probability measure, as desired.

The one-to-one correspondence between sets of positive state prices
and martingale probabilities is clear from (13). ■
48/54
Next

Introduction and motivation

Mathematical foundations

Asset pricing theory

One period discrete market

Black-Scholes-Merton framework

Annex: proofs
propositions
theorems
First Fundamental Theorem Proof
Next will be given a proof of the Theorem 1 (first Fundamental
Theorem of Asset Pricing (FTAP)).
Firstly will be shown the reverse implication, this is, if a martingale
probability measure exists then there are no arbitrage opportunities.
Proof ⇐
Assume a martingale probability measure Q exists, thus is

S
b0 = π S
b1 ,
 
where π = Q(ω1 ), . . . , Q(ωn ) .

Let h = (h0 , . . . , hm )T ∈ Rm+1 be a portfolio or trading strategy, such

b1h ≥ 0 for each ω ∈ Ω, and with strict inequality in some states.
that S
Then S b 0h = π S
b1h must hold, and S b0h > 0 since all entries in π are
strictly positive, and entries in S
b1h are strictly positive or zero.

Therefore no arbitrage opportunities exist.

49/54
First Fundamental Theorem Proof (ii)
Proof ⇒
Let U ⊂ Rm+1 be the subspace defined by
(" # )
−Sb0
m+1
U= h :h ∈R ,
Sb1

where h is any trading strategy.

m+1
Let R+ ⊂ Rm+1 be the convex set defined by
  
 x0
 
 .. 

m+1 m+1
R+ :=  .  ∈ R : xi ≥ 0 for all i ∈ {0, 1, . . . , K } .
 
xK
 

The absence of arbitrage opportunities implies that U ∩ Rm+1

+ = {00}.
This last fact will be proved next by reductio ad absurdum.

50/54
First Fundamental Theorem Proof (iii)
Proof ⇒ (ii)
Assume that exists x ∈ U ∩ Rm+1
+ such that x ̸= 0 .
Since there is a trading strategy h associated with every x ∈ U, it
suffices to show that the h associated with x always represents an
arbitrage opportunity.
If S
b0h < 0 all the entries S
b1h must be all non-negative. Consequently,
h is a type a arbitrage.
If −S
b0h = 0 then the entries of S b1h must be all greater than or equal
to zero, with at least one strict inequality.
Therefore h is a type b arbitrage opportunity.
Next part of the proof requires the use of the Separating Hyperplane
Theorem, which states that if A and B are two non-empty disjoint
convex sets in a vector space V , then they can be separated by a
hyperplane.
51/54
First Fundamental Theorem Proof (iv)
Proof ⇒ (iii)
Since U ∩ Rm+1
+ = {00} by the Separating Hyperplane Theorem, there
exists a hyperplane that separates Rm+1
+ \ {00} and U.
Let f ∈ Rm+1 and α ∈ R be such that, for all x ∈ Rm+1
+ \ {00} and
y ∈ U, holds
f ,x > α > f ,y , (14)
where ·, · denotes the Euclidean inner product.
if y ∈ U, as U is a subspace, then any multiple of y also belongs to U.
Hence the condition given by inequalities (14) hold only if f , y = 0
for each y ∈ U.
Therefore
f , x > 0, ∀ x ∈ Rm+1
+ \ {00},
i.e., all entries in f are strictly positive.

52/54
First Fundamental Theorem Proof (v)
Proof ⇒ (iv)
On the other hand, f , y = 0 implies, for all h ∈ Rm+1 , that
n
X
−f0 S
b 0h + fk S h = 0.
b1 (ωn )h
k=1

Where S b1 (ωn ) is the k th row of S

b1 , and f denotes the k th entry of f ,
k
for all k ∈ {0, 1, . . . , m}.
Latter equation can be rewritten
n
X
S
b0 = qk S
b1 (ωn ), (15)
k=1

where
fk
qk := .
f0
53/54
First Fundamental Theorem Proof (vi)
Proof ⇒ (v)
Lastly, applying (15) to the numeraire security yields
n
X
1= qk
k=1

As f , x > 0 implied that all entries in f are strictly positive, then

qk > 0 for all k ∈ {1, . . . , n}. Hence they defined a probability
measure.
Using (15) for the mth security results
n
(m) X (m)
Sb0 = qk Sb1 (ωn ).
k=1

Therefore the qk ’s define a a martingale probability measure. ■

54/54
References I
Björk, Tomas (2004). Arbitrage Theory in Continuous Time.
Ed. by Oxford. 2nd ed. Oxford University Press.
Black, Fisher and Myron Scholes (1973). “The Pricing of
Options and Corporate Liabilities”. In: Journal of Political
Economy 81, pp. 637–654.
Capinski and Kopp (2012). Discrete Models of Financial
Markets. Ed. by Cambridge. Mastering Mathematical Finance.
Cambridge University Press.
Capinski, Marek, Ekkehard Kopp, and Janusz Traple (2012).
Stochastic Calculus for Finance. Ed. by Cambridge. Mastering
Mathematical Finance. Cambridge University Press.
Delbaen, Freddy and Walter Schachermayer (Sept. 1994). “A
general version of the fundamental theorem of asset pricing”. In:
Mathematische Annalen 300, pp. 463–520.
References II
Elliott and Kopp (2004). Mathematics of Financial Markets.
Ed. by Springer. Springer Finance.
Harrison, J.M. and D.M. Kreps (1979). “Martingales and
arbitrage in multi-period securities markets.”. In: Journal of
Economic Theory 20, pp. 381–408.
Harrison, J.M. and S.R. Pliska (Aug. 1981). “Martingales and
stochastic integrals in the theory of continuous trading”. In:
Stochastic Processes and their Applications 11.3, pp. 215–260.
— (Aug. 1983). “A stochastic calculus model of continuous
trading: complete markets.”. In: Stochastic Processes and their
Applications 15.3, pp. 313–316.
Longstaff, F. and E. Schwarz (2001). “Valuing American Option
by Simulation: A Simple Least-Squares Approach”. In: The Review
of Financial Studies 14, No. 1, pp. 113–147.
References III
Merton, Robert C. (1973). “Theory of Rational Option Pricing”.
In: Bell Journal of Economics and Management Science 4, no. 1,
pp. 141–183.
Taqqu, Murad S. and Walter Willinger (1987). “The Analysis of
Finite Security Markets Using Martingales.”. In: Advances in
Applied Probability 19.1.
Wilmott, Paul (1998). Derivatives: The theory and practice of
financial engineering. Wiley.