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Ross Recovery theorem and itsextension
Ho Man Tsui
Kellogg College
University of Oxford
A thesis submitted in partial fulfillment of the MSc in
Mathematical Finance
April 22, 2013
Acknowledgements
I am sincerely grateful to my supervisor, Dr. Johannes Ruf, for the support and
guidance he showed me throughout my dissertation writing, also for giving me the
chance to work on this interesting topic.
I would also like to express my gratitude to Dr. Kostas Kardaras and Dr. Umut
Cetin from LSE, Dr. Aleksandar Mijatovic from Imperial College, Dr. Samuel
Cohen and Pedro Vitoria from Oxford University for their helpful advices and com-
ments on my thesis.
Finally I wish to thank my parents in Hong Kong and my brother in mainland
China, who always love and support me.
Abstract
Stephen Ross recently suggested the “Recovery Theorem”, which pro-
vides a way to reconstruct the real-world probability measure given the
risk-neutral measure, under a discrete time and state space context. Pe-
ter Carr and Jiming Yu modified Ross’s model and deduced a similar
result, under a univariate continuous diffusion context. This disserta-
tion is dedicated to access the theoretical basis and possible extensions
to the frameworks of both Ross’s recovery model and its extension Carr
and Yu’s recovery model.
To put the model in a more robust theoretical ground, we clarify each
of the assumptions, seeking the sufficient assumption set to derive each
model. Properties of the two models are established, highlighting the
limitations, similarities and other properties of the two models. Based
on the evidences presented in this thesis, we postulate that the existence
of the stationary distribution is a necessary condition for the recovery
theorem to succeed. A table of comparison between the two recovery
models is included to summrize the result in this thesis.
Contents
1 Introduction 1
1.1 Notations and market set-up . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Indexing of the assumptions . . . . . . . . . . . . . . . . . . . . . . . 3
2 Ross’s model - basic framework 4
2.1 Assumptions and definitions . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Ross’s model - analysis and discussion 12
3.1 The utility function in Ross’s model . . . . . . . . . . . . . . . . . . . 12
3.2 Existence of stationary distribution . . . . . . . . . . . . . . . . . . . 17
3.3 Interest rate process in Ross’s model . . . . . . . . . . . . . . . . . . 18
4 Carr and Yu’s model - basic framework 22
4.1 Assumptions and definitions . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Carr and Yu’s model - analysis and discussion 35
5.1 Boundary conditions of the numeraire portfolio . . . . . . . . . . . . 36
5.2 Market completeness in Carr and Yu’s model . . . . . . . . . . . . . . 37
5.3 Existence of stationary distribution . . . . . . . . . . . . . . . . . . . 38
5.4 Recovery theorem on unbounded domain . . . . . . . . . . . . . . . . 41
5.4.1 Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . 42
5.4.2 CIR process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Conclusion and further research 47
Bibliography 51
Appendices 53
A Perron Frobenius theorem 54
i
B Numeraire portfolio 56
C Regular Sturm Liouville theorem 61
D Markov chain 63
ii
Chapter 1
Introduction
The basic objective of derivative pricing theory is to determine the fair price of a
given security in terms of more liquid securities whose price is determined by the
law of supply and demand [23]. In a market with no arbitrage opportunities, one
can determine the price of the security by using the “risk-neutral” measure (also
known as equivalent local martingale measure) due to fundamental theorem of asset
pricing [6]. The idea of the risk-neutral probability measure has been extensively
used in deriving pricing and mathematical finance in general. A gentle introduction
of the concept can be found in [9]. The risk-neutral measure is typically denoted by
the blackboard font letter “Q”.
On the other hand, risk and portfolio management aims at modelling the prob-
ability distribution of the market prices of all the securities at the given future in-
vestment horizon [23]. Based on the “real-world” probability distribution, one could
take investment decisions in order to improve the prospective profit-and-loss profile
of their positions considered as a portfolio. The real-world probability measure can
be thought of the subjective probability measure perceived by the agents in the
market. The real-world probability measure is typically denoted by the blackboard
font letter “P”.
The difference between the real-world measure and the risk-neutral measure can
be understood as the “risk-premium” of the market. Risk premium is the expected
rate of return demanded by investors over the risk-free interest rate [24]. Intu-
itively, under the risk-neutral measure, the expected return of all asset prices are
the risk-free rate, so the risky assets are priced as if the investors have a risk neutral
preference. On the contrary, investors are typically risk averse in the real world,
and hence demand a higher return over risk-free rate for investing in risky assets.
The difference between the real world expected return and the risk-free rate is the
risk-premium.
1
The risk premium is neither tradable nor directly observable in the market. There
are a number of approaches to estimate this quantity, such as historical risk premium,
dividend yield models, market survey for market participants and institutional peers
[18]. However, the risk premium obtained from these approaches are often unstable,
conflicting and hence unreliable.
Despite the practical difficulties, there is a number of useful applications in es-
tablishing a connection between the real-world probability measure and the risk-
neutral measure. For example, based on the derivative prices of an equity asset one
can access the market subjective forecast of the return distribution of the underly-
ing equity, to derive the optimal investment strategy or to compute the risk control
measures.
Stephen Ross recently suggested the “Recovery Theorem” [19], which provides
a new way to connect the two sets of probability measures. When the risk-neutral
measure is provided, the theorem provides a way to reconstruct the real-world prob-
ability measure and deduce the risk-premium. His model is based on a discrete time
and state space with a restriction on preference of the representation agent. The
result of this paper has shed light on alternative approaches to risk management,
market forecast as well as many other applications.
Following this, Peter Carr and Jiming Yu [4] modified the model and deduced a
similar result by using a different set of assumptions. In particular, they assumed
a continuous diffusion driver of the market without assuming the existence of a
representative agent. They utilized a concept called the “numeraire portfolio”, which
is a strictly positive self-financing portfolio such that all assets, when measured in
the unit of the numeraire portfolio, are local martingales under real-world measure.
By restricting the form and the dynamics of the numeraire portfolio, they successful
proved the recovery theorem in an univariate bounded diffusion context. However,
to what extent the result is valid on the unbounded domain is still an open question.
Motivated by the two models of recovery theory, this thesis aims to review and
analyse the models in detail to access the theoretical basis and possible extensions
to the models. Our strategy to achieve this goal is twofold. First, to put the
frameworks to more robust theoretical grounds, we review and clarify each of the
assumptions in mathematical statements, seeking the sufficient assumption sets to
derive the conclusions for Ross’s model in Chapter 2 and for Carr and Yu’s model
in Chapter 4. Second, we discuss their properties, highlighting the limitations,
similarities, and other properties for Ross’s model in Chapter 3 and for Carr and
Yu’s model in Chapter 5. Notably, we prove the the existence of the stationary
2
distribution for Ross’s model in Section 3.2 and provide heuristic argument for the
existence for Carr and Yu’s model in Section 5.3.
Based on evidences presented in the thesis, in Chapter 6 we postulate that the
existence of the stationary distribution is a necessary condition for the recovery
theorem to succeed. We propose area for further research, and conclude the thesis
by a table of comparison between the two models in Figure 6.1, summarizing the
results in the thesis.
1.1 Notations and market set-up
For the convenience of both authors and readers, the notations and the market set-
up common to both models are provided in this section.
R and R≥0 denote the set of real numbers and positive real numbers with zero
respectively. N and N+ denote the set of natural numbers and positive natural
numbers respectively.
The analysis is placed over a filtered probability space (Ω,F , (Ft),P), where
F = σ(⋃tFt).
In addition, for some fixed n ∈ N, there exists a vector of n + 1 securities
St = (S0t , S
2t , S
3t , . . . , S
nt ) ∈ Rn+1
≥0 , where
1. S0t is the risk-free security,
2. Sit for i = 1, 2, . . . , n are risky securities.
All other assumptions of models will be stated in the corresponding chapter.
1.2 Indexing of the assumptions
To clearly express the model, all the assumptions are indexed with a letter and a
number. The number of the index is in line with Carr and Yu’s paper [4], and that
in Ross’s model is constructed in parallel with that in Carr and Yu’s model. All
the assumptions in Carr and Yu’s model are started with letter “C” followed by a
number (for example, C3), and assumptions in Ross’s model are started with “R”
followed by a number (for example, R3).
3
Chapter 2
Ross’s model - basic framework
This chapter aims at presenting the basic framework of Ross’s model, based on his
paper [19]. While Ross did not clearly specify the assumption set of the model in
mathematical terms, but rather descriptively explained the framework and derived
the result, this chapter could be treated as a formalization of the model. The pre-
sentation in this chapter is also different from Ross’s, in particular in the treatment
of the existence of representation agent in the market and his utility function, as
we found that there are inconsistencies in the derivation, which will be discussed in
Section 3.1.
We attempt to keep the material self-contained within the scope of this thesis,
therefore the definitions and theorems that are necessary to understand the model
and its conclusion are all included. Our discussion begins by laying out the assump-
tions and definitions of the models in the first section, followed by the deviation
of the result in the second section. The main result is Ross’s recovery theorem
(Theorem 2.2.1).
The objective of Ross’s recovery model is to determine the real-world transition
probabilities from the state-price securities, which is obtained from the market prices
of derivatives written on the underlying state variables.
The derivation of Ross’s model is summarized as follow:
1. The assumptions of the model are:
(a) Finite state space,
(b) Discrete time-step,
(c) No-arbitrage market (R3),
(d) Time-homogeneity of the state-price function and the real-world transi-
tion probability function (R4),
(e) The state prices are known (R5),
4
(f) Transition independent pricing kernel (R6), and
(g) Non-negativeness and irreducibility of the state-price matrix (R7).
2. The pricing kernel can be written as matrix form (2.8), which can be refor-
mulated as a eigenvector / eigenvalue problem (2.11). The Perron Frobenius
theorem (Appendix A) is applied (Theorem 2.2.1) to solve for the real-world
transition probability.
All assumptions in this chapter start with letter ’R’ (for example, R3).
2.1 Assumptions and definitions
With reference to the market set-up in Section 1.1, the economy in Ross’s model is
discrete in time with finite state space, i.e.
1. t = t1, t2, . . . is a discrete time with a fixed time-step.
2. The state space as Π is finite, with the state variable Xt ∈ Π and |Π| = M ∈ N.
Additionally, there exists a stochastic interest rate rt ∈ R≥0, such that S0t satis-
fies:
S0t+1 = S0
t (1 + rt) for all t ∈ N (2.1)
with
S00 = 1. (2.2)
The first assumption of the model is stated as follow:
R3: “No arbitrage” condition applies in the economy.
Definition 2.1.1 (State-prices)
A state-price (also called Arrow-Debreu price) is the price of a contract that agrees
to pay one unit of a numeraire of the economy if a particular state occurs in the
next time step and pays zero in all other states.1
By the First Fundamental Theorem of Asset pricing [6], no arbitrage implies
the existence of state prices. Notice that this does not imply the contracts of the
state-price securities are tradable in the market, but simply the set of state-prices
exists and can be theoretically used for pricing purpose. Also, the set of state-prices
1This definition is referenced to [21].
5
may not be unique.2
The following assumption considers the state-price function, which is the price
of a state-price security at a given time, given an initial state and a next transition
state.
R4 (part i): The state-price function, p, satisfies the Markov property, is time-
homogeneous and only depends on the initial state and the next transition state, i.e.
p : Π× Π→ [0, 1].
Each state price is denoted as pi,j, where i represents the initial state and j
represents the next state, as well as the state-price matrix of the model as P , where
P =
p1,1 p1,2 · · · p1,M
p2,1 p2,2 · · · p2,M...
.... . .
...pM,1 pM,2 · · · pM,M
. (2.3)
From this one can also define the risk-neutral probability matrix Q, where
Q =
p1,1/
∑Mj=1 p1,j p1,2/
∑Mj=1 p1,j · · · p1,M/
∑Mj=1 p1,j
p2,1/∑M
j=1 p2,j p2,2/∑M
j=1 p2,j · · · p2,M/∑M
j=1 p2,j
......
. . ....
pM,1/∑M
j=1 pM,j pM,2/∑M
j=1 pM,j · · · pM,M/∑M
j=1 pM,j
. (2.4)
Notice that Q is a probability transition matrix, as the sum of each of its row is
1.
We now consider the transition density function of the model, which is the prob-
ability of transition under P from an initial state to the next transition state.
R4 (part ii): The transition density function, f , satisfies the Markov property,
is time-homogeneous and only depends on the initial state and the next transition
state, i.e. f : Π× Π→ [0, 1].
The transition density function is denoted as fi,j, where i represents the initial
state and j represents the next state, as well as the transition probability matrix of
2If the set of state-prices is unique or the contracts are all tradable, then the market is complete.
6
the model as F , where
F =
f1,1 f1,2 · · · f1,M
f2,1 f2,2 · · · f2,M...
.... . .
...fM,1 fM,2 · · · fM,M
. (2.5)
With the definitions above it is clear that the goal of the model is to recover the
matrix F from the matrix P . he following assumption:
R5: The state-price matrix P is known ex ante.
Now the pricing kernel of the model is defined:
Definition 2.1.2 (Pricing kernel in Ross’s model)
The pricing kernel from state i to state j is defined as the following quotient:
ϕi,j =pi,jfi,j
.
The pricing kernel matrix is defined as
Φ =
ϕ1,1 ϕ1,2 · · · ϕ1,M
ϕ2,1 ϕ2,2 · · · ϕ2,M...
.... . .
...ϕM,1 ϕM,2 · · · ϕM,M
. (2.6)
Definition 2.1.3 (Transition independent pricing kernel)
A pricing kernel is transition independent if it can be written in the following form:
ϕi,j =pi,jfi,j
= δh(i)
h(j), (2.7)
where
h : Π→ R>0 is a positive function, and
δ is a positive constant, called a discount factor.
The following assumption is pivotal in deriving the recovery theorem.
R6: The pricing kernel is transition independent.
7
Remark 2.1.1
In his paper [19], Ross introduced the model in a different way. Instead of assuming
R6, he introduced the model by the existence of a representative agent with inter-
temporal additive separable utility. The transition independent pricing kernel was
then derived from the optimization of the utility. However, as we will explain in
Section 3.1, the derivation is not consistent and the transition independent pricing
kernel does not follow from the utility function.
Before stating the last assumption in Ross’s model, we need the following two
definitions:
Definition 2.1.4 (Non-negative matrix)
A square n× n matrix A is said to be non-negative if all the elements are equal to
or greater than zero, i.e.
ai,j ≥ 0
for all i = 1, 2, . . .m and j = 1, 2, . . . n.
A square matrix that is not reducible is said to be irreducible.
Definition 2.1.5 (Irreducible matrix)
A square n × n matrix (A)i,j = ai,j is said to be reducible if the indices 1, 2, . . . , n
can be divided into two disjoint non-empty sets i1, i2, . . . , iu and j1, j2, . . . , jv, with
u+ v = n, such that
aiα,jβ = 0
for α = 1, 2, . . . , u and β = 1, 2, . . . , v.
R7: The state-price matrix, P , is non-negative and irreducible.
This assumption will be used to apply the Perron Frobenius Theorem in the next
section (Theorem 2.2.1).
Remark 2.1.2
As mentioned by Ross [19], since the risk-neutral measure is equivalent to the real-
world measure, the irreducibility of the matrix P is equivalent to the irreducibility
of the matrix F . We can justify this statement by the following: an entry pi,j of
P is zero if and only if fi,j of F is zero (by the definition of equivalent probability
measures), so the indices of P can be separated into two disjoint sets if and only if
the indices of F can be separated. Therefore the irreducibility of P is equivalent to
8
the irreducibility of F . The fact that the non-negativeness of P is equivalent to the
non-negativeness of F can be easily deduced.
Using a similar argument the irreducibility and non-negativeness of P is equiv-
alent to the irreducibility and non-negativeness of Q, because Q is just a rescaled
P .
The assumptions of Ross’s model will be further discussed in Chapter 3. In the
next section, we will see how the recovery theorem of Ross’s model is arrived based
on the set of assumptions presented in this section.
2.2 Derivation
The diagonal matrix D is defined as:
D =
h(1) 0 · · · 0
0 h(2) · · · 0...
.... . .
...0 0 · · · h(M)
.
D−1, the inverse of the matrix D, is:
D−1 =
1
h(1)0 · · · 0
0 1h(2)
· · · 0...
.... . .
...0 0 · · · 1
h(M)
.
The transition independent pricing kernel equation (2.7) can now be written in
matrix form,
P = δD−1FD.
Reorganize the equation,
F =1
δDPD−1. (2.8)
Lemma 2.2.1 (Identity for transition probability)
Let e be a M×1 column vector of 1, i.e. e =(1 1 · · · 1
)T. The following identity
holds:
Fe = e. (2.9)
Proof. The intuition for this lemma is simple: the rows on F are essentially proba-
bility to different states given the current state, the sums if each row must be equal
to one.
9
As defined in (2.5), (F )i,j = fi,j is the probability of transiting from the initial
state i to j. Summing up all the transition probabilities one will get:
M∑j=1
fi,j = 1 for all i = 1, 2, . . . ,M .
To write this equality in matrix form, we have
Fe = e.
Multiply both sides of (2.8) by e and reorganize, we have
PD−1e = δD−1e.
Let
x = D−1e. (2.10)
Then
Px = δx. (2.11)
Hence the equation has been transformed to an eigenvalue and eigenvector prob-
lem.
Before we proceed, notice that:
1. The entries of x are actually the diagonal elements of the matrix D−1, which
is 1h(i)
. Therefore they must be positive.
2. δ is the subjective discount factor so it must be a non-negative number less
than or equal to one.
The following theorem guarantees a unique solution of this problem, which sat-
isfies these two conditions.
Theorem 2.2.1 (Ross recovery theorem)
There exists a unique positive solution of x (up to a positive scaling) and δ to the
problem (2.11). Moreover, we can recover the matrix D (up to a positive scaling)
and the matrix F .
Proof. This proof uses the Perron Frobenius theorem (Appendix A).
From R7, P is an irreducible matrix. Applying the Perron Frobenius theorem,
the only eigenvector of P whose entries are all positive is its Perron vector, which
is unique up to a positive scaling. Therefore it is the only possible solution to x.
10
In addition, δ must be equal to the corresponding eigenvalue Perron root ρ(P ) >
0. Notice that δ is unique while x is unique up to a positive scaling.
δ is bounded above by the maximum sum of row and the minimum sum of row
of P . Since P is a state-price matrix, each of its sum of row must be non-negative
(by R7, each entry of P is non-negative). Moreover, the maximum pay-off of each
state-price security is one, if all state-price securities are purchased a pay-off of one
will be guaranteed. This implies that the sum of each of its row is always less than
or equal to one,3 i.e.
0 ≤ δ ≤ 1.
Once x is found, one can imply D by (2.10) (up to a positive scaling), and then
recover the matrix F by (2.8). The recover matrix F is unique since the scaling
factor is cancelled on the R.H.S. of (2.8).
Remark 2.2.1
Ross [19] pointed out that the transition independent pricing kernel (R6) is crucial in
deriving the recovery theorem - it allows the pricing kernel to be separated from the
real-world probability. To illustrate this point, notice that (2.7) can be rearranged
as:
pi,j = ϕi,jfi,j = δh(i)
h(j)fi,j.
Given only the knowledge of pi,j, the recovery theorem uses the transition inde-
pendence to separately determine ϕi,j and fi,j. Without the transition independent
pricing kernel, the recovery of real-world measure P is not feasible.
The foregoing deviation provides a theoretical basis to uniquely determine the
matrix F from the matrix P . By recovering the matrix F , the main objective of the
model is achieved - to obtain the real-world probability measure from the risk-neutral
probability.
The analysis and discussion for this model is contained in the next chapter.
3This argument will be elaborated in detail in Section 3.3.
11
Chapter 3
Ross’s model - analysis anddiscussion
While the last chapter gave the mathematical foundation for Ross’s framework, the
present chapter focuses the analysis and discussions of Ross’s model. We attempt to
highlight the features and limitations of the model, and these results will be contrast
with those of Carr and Yu’s model in the later chapters.
This chapter is divided into the following sections:
Section 3.1: The utility function in Ross’s model
We begin the discussion by pointing out a major gap in Ross’s paper [19]: the
transition independent pricing kernel (Definition 2.1.3) does not logically follow
from inter-temporal additive separable utility as Ross suggested. We will review
how Ross introduced a representative agent and his utility function, and explain the
inconsistencies in the derivation.
Section 3.2: Existence of stationary distribution
The discussion progresses to discuss a common feature of the two recovery model.
We analyse Ross’s model from a perspective of the Markov chain theory. The Perron
Frobenius theorem is then used to show the existence of the stationary distribution
in Ross’s model.
Section 3.3: Interest rate process in Ross’s model
Lastly a few properties relating to the interest rate process in Ross’s model are
explored.
3.1 The utility function in Ross’s model
In Chapter 2, we have reviewed how the real-world probability P is recovered from
the risk-neutral probability Q in Ross’s framework. The derivation starts from the
12
transition independent pricing kernel (2.1.3) as in A6, the equation is rewritten in
a matrix form and the Perron Frobenius theorem is applied to solve for the unique
solution of the matrix F .
However, in Ross’s paper [19], he introduced the model in a different way: he
first assumed the existence of a representative agent with an inter-temporal addi-
tive separable utility function, the transition independent pricing kernel was then
deduced from optimizing the agent’s utility, and the matrix F was uniquely solved
afterwards. In other words, instead of directly making the transition independent
pricing kernel as one of the assumptions, Ross deduced it by assuming the existence
of a representative agent with a specific form of utility function.
In fact, we deviate from Ross’s approach for a reason - it is because the transi-
tion independent pricing kernel does not logically follow from inter-temporal additive
separable utility function as Ross has suggested. In this section, we illustrate why
this is the case. This section is based on Carr and Yu’s paper [4, P. 11-13].
To begin with, the definition of the inter-temporally additive separable utility
function is introduced.
Definition 3.1.1 (Inter-temporally additive separable utility function)
An inter-temporally additive separable utility function U : C → R in discrete two-
period economy is
U(c) = u(c0) + δE[u(c1)], (3.1)
where
C is the space of all feasible consumption processes,
c = (c0, c1) ∈ C, is the consumption at t = 0 and t = 1,
u : R+ → R is a strictly concave function, and
δ is a constant impatient factor, with 0 < δ < 1.
Specializing this definition to the setting of Ross’s model, an agent faces the
following optimization problem:
supc∈C
u(c0,i) + δ
M∑j=1
u(c1,j)fi,j s.t. c0,i +M∑j=1
c1,jpi,j = w, (3.2)
where the first index of c is the time index and the second index is the state
index; w is the initial wealth of the representative agent.
13
Assuming in this section, R6 in Chapter 2 is replaced with the following two
assumptions, R6’ (part i) and R6’ (part ii).
R6’ (part i): There exists a representative agent in the economy with inter-
temporally additive separable utility.
The existence of the representative agent assumption is valid when, for example,
the market is complete and the economy is in equilibrium state. A reference for this
could be found in [7].
R6’ (part ii): The optimal consumption process is time-homogeneous.
In other words, the consumption only depends on the state in which the con-
sumption takes place. The consumption at state j ∈ Π is denoted as cj.
We will first see how the transition independent kernel is derived and then ex-
plain why we found the derivation inconsistent.1
Suppose the current state is i. Since the state space Π is finite, and from (3.2)
the representative agent faces the following problem:
supc∈C
u(c0,i) + δM∑j=1
u(c1,j)fi,j s.t. c0,i +M∑j=1
c1,jpi,j = w.
Define the Lagrangian L as:
L ≡ u(c0,i) + δ
M∑j=1
u(c1,j)fi,j + λ
(w − c0,i −
M∑j=1
c1,jpi,j
).
The first order condition for the optimal solution are:
u′(c∗0,i)− λ = 0 for all i ∈ Π
and
δu′(c∗1,j)fi,j − λpi,j = 0 for all i, j ∈ Π.
Solving these equations give formula for the pricing kernel in terms of the optimal
consumption process:
pi,jfi,j
= δu′(c∗1,j)
u′(c∗0,i)for all i, j ∈ Π. (3.3)
1In Ross’s paper [19], Ross didn’t derive the transition independent kernel in detail, but ratherexplained it descriptively. The derivation here largely follows Carr and Yu’s paper [4], in whichthey tried to provide the missing derivation steps in Ross’s model.
14
Using assumption R6’ (part ii), the optimal consumption process is time-
homogeneous. Hence we can drop the time index of c∗ in the expression:
pi,jfi,j
= δu′(c∗j)
u′(c∗i )for all i, j ∈ Π. (3.4)
Therefore the kernel is the transition independent as defined in (2.7).
However, the derivation above is inconsistent for two reasons:
First, while (3.2) is an one-period optimization problem, the time-homogeneous
optimal consumption assumption (R6’ (part ii)) is satisfied only when the con-
sumption horizon is infinite. Given a one-period consumption horizon, a rational
agent will consume differently at time zero and time one to maximize his utility, be-
cause two consumptions contribute differently to his total utility, i.e. consumption
is discounted by factor δ only at time one but not at time zero. The same argument
could be applied to argue that optimal consumption is not time-homogeneous for
any finite-period consumption horizon.
On the other hand, if the consumption horizon is infinite, the agent will be indif-
ferent to the current time, since he will be faced with the same set of infinite period
optimization problems regardless of time. This implies that the optimal consump-
tion process will not depend on time but only depend on the current state of the
system. In other words, the optimal consumption process will be time-homogeneous.
Second, the optimal solution to (3.2) should depend on the initial state, and with
different initial states the optimal solutions should be different. However, in (3.3),
the optimal solutions are considered the same regardless of the initial state of the
optimization problem2.
This idea can be illustrated by the following simple example. Suppose, for sim-
plicity, there are only two states, α and β, with the following transition probabilities
and state prices:
fα,α = 0.6, pα,α = 0.5,
fα,β = 0.4, pα,β = 0.4,
fβ,α = 0.7, pβ,α = 0.6,
fβ,β = 0.3, pβ,β = 0.2.
2This inconsistency was suggested by Carr and Yu on [4, p. 13].
15
with w = 10, δ = 0.9, and function u(x) = log x.
If the current state is α, then the optimization problem is:
sup log(cα) + 0.9[0.6 log(cα) + 0.4 log(cβ)] s.t. cα + 0.5cα + 0.4cβ = 10. (3.5)
β
α α
β
f = 0.6, p = 0.5
f = 0.4, p = 0.4
Figure 3.1: Optimization problem when the current state is α
The optimal solution to problem (3.5) is:
(c∗α)1 = 5.40351 and (c∗β)1 = 4.73684.
On the other hand, if the current state is β, the optimization problem is:
supu(cβ) + 0.9[0.7 log(cα) + 0.3u(cβ)] s.t. cβ + 0.6cα + 0.2cβ = 10. (3.6)
β
α α
β
f = 0.7, p = 0.6
f = 0.3, p = 0.2
Figure 3.2: Optimization problem when the current state is β
The optimal solution to (3.6) is:
(c∗α)2 = 5.36184 and (c∗β)2 = 7.10526.
16
Obviously (c∗α)1 6= (c∗α)2 and (c∗β)1 6= (c∗β)2.
This example shows the optimal solutions to (3.2) are different with different
initial states. However, in the transition independent kernel expression (3.4), all
optimal consumptions are considered the same regardless of the initial states of the
optimization problems.
Based on the two reasons above we conclude that the transition independent
kernel (3.3) is not consistent with inter-temporally additive separable utility.
3.2 Existence of stationary distribution
In this section, we will analyse Ross’s model from the perspective of the Markov chain
theory to prove existence of the stationary distribution in Ross’s model. Similar to
the derivation of Ross’s recovery theorem, the proof utilizes the Perron Frobenius
theorem (Appendix A).
Intuitively, a finite state Markov chain is a system that undergoes transitions
from one state to another, between a finite number of possible states. It also satisfies
the Markov property (also known as “memoryless” property): the next state only
depends on the current state but not the sequence of events that precedes it [22]. A
formal definition of Markov chain and some of its useful properties could be found
in Appendix D.
It is easy to see that Ross’s model is a time-homogeneous finite state irreducible
Markov chain. The state variable is Xn ∈ Π, and with either
1. the transition density matrix F , as defined in (2.5), or
2. the risk-neutral probability matrix Q, as defined in (2.4)
as the transition probability matrix.
Stationary distributions play a important role in analysing Markov chains. In-
formally, a stationary distribution represents a steady state in the Markov chains
behaviour. The formal definition of the stationary distribution of a Markov chain is
given below:
Definition 3.2.1 (Stationary distribution of a Markov chain)
Let Xn be a time-homogeneous Markov chain having state-space Π and the tran-
sition probability matrix P . If π is a probability distribution such that:
Pπ = π,
17
then π is called the stationary distribution of Xn.
This definition basically means that if a chain reaches a stationary distribution,
then it maintains that distribution for all future time. The proof for the existence
of the stationary distribution in Ross’s model is given in the following theorem:
Theorem 3.2.1 (Existence of stationary distribution in Ross’s model)
Given the Markov chain formulation of Ross’s model, with
1. the transition density matrix F , or
2. the risk-neutral probability matrix Q,
as transition probability, there exists a unique stationary distribution in Ross’s
model.
Proof. This theorem is proved by the Perron Frobenius theorem (Appendix A).
We only prove for the transition density matrix F , but the same argument could
hold for Q.
For the transition probability matrix F , the sum of its row must be 1. By the
Perron Frobenius theorem, its spectral radius r must be
1 = mini
∑j
fi,j ≤r ≤ maxi
∑j
fi,j = 1,
So 1 is a eigenvalue of F , and the corresponding Perron vector, v, satisfies:
Fv = v.
Hence v is the stationary distribution of the Markov chain in Ross’s model.
3.3 Interest rate process in Ross’s model
In this section the important features of Ross’s model related to the interest rate
process are derived. This section is largely based on remarks and theorems in Ross’s
paper [19].
The following will be discussed:
1. The interest rate process is time-homogeneous.
2. The subjective discount rate δ is bounded above by the largest interest rate
factor and below by the lowest interest rate factor.
18
3. If the interest rate process is a constant, then the real-world probability mea-
sure will be the same as the risk-neutral probability measure.
First we show that the interest rate process is time-homogeneous. Note that
this feature is not an assumption but rather than an implication of the model, in
contrast with Carr and Yu’s model, where the interest rate process is assumed to
be time-homogeneous (C6 (part ii)).
Theorem 3.3.1 (Time-homogeneity of interest rate process)
In Ross’s model, the interest rate for each period, rt, only depends on the current
state, but is independent of time, i.e. it is a time-homogeneous process. Moreover,
rt =1∑M
j=1 pi,j− 1.
Proof. The conclusion is mainly followed by the R4 (part i).
Consider the sum of row i of the state-price matrix P . Given a state i as the
current state, if one purchases all state-price securities pi,j, j ∈ Π, one will be guar-
anteed a pay-off of one no matter what state is realized. Hence in a market without
arbitrage (R3), the sum∑M
j=1 pi,j should be equal to the one-period discounted
value of one, i.e.M∑j=1
pi,j =1
1 + rt(3.7)
Rearrange:
rt =1∑M
j=1 pi,j− 1.
Since the R.H.S. only depends on the current state, i, so the interest rate for
each period is a time-homogeneous process.
Second, based on the similar argument, one can prove that the subjective dis-
count factor, δ, is bounded above by the largest interest rate factor and bounded
below by the smallest interest rate factor.
Theorem 3.3.2 (Bounds for subjective discount rate)
In Ross’s model, the subjective discount rate δ is bounded above by the largest
interest discount factor and bounded below by the smallest interest discount factor,
i.e.1
1 + maxi∈Π ri≤ δ ≤ 1
1 + mini∈Π ri
19
Proof. First an argument from the Perron Frobenius theorem is used. The Perron
root δ is between minimum sum of row and maximum sum of row of P (inclusive),
i.e.
mini
M∑j=1
pi,j ≤ δ ≤ maxi
M∑j=1
pi,j.
From (3.7) we know thatM∑j=1
pi,j =1
1 + ri.
Substitute this into the equation above we have
1
1 + maxi∈Π ri≤ δ ≤ 1
1 + mini∈Π ri.
Lastly, the following is proved:
Theorem 3.3.3 (Implication for constant interest rate)
In Ross’s model, if the interest rate is also independent of the current state, i.e. the
interest rate is constant, then the real-world probability measure will be the same
as the risk-neutral measure, i.e.
P = Q.
Proof. If the interest rate is constant, from (3.7) each sum of rows of P is the same.
M∑j=1
pi,j =1
1 + r= k for all i ∈ Π,
where k is the constant interest discount factor.
This can be rewritten in matrix form,
Pe = ke,
where e be a M × 1 column vector of one.
From the Perron Frobenius theorem, the Perron vector of P is e and the Perron
root is one.
By (2.8),
F =1
kP .
20
From (2.4), the definition of the risk-neutral probability matrix Q is
Q =1∑M
j=1 pi,jP
= (1 + r)P
=1
kP
= F .
Therefore the real-world probability measure P is the same as the risk-neutral
measure Q.
In the next chapter, we will review how Carr and Yu arrive at a similar re-
sult as Ross did on a bounded continuous state space, based on a different set of
assumptions.
21
Chapter 4
Carr and Yu’s model - basicframework
As an extension to Ross’s model, Carr and Yu’s model aimed at establishing the
recovery theorem under a bounded diffusion context. They utilized the concept of
the numeraire portfolio, which is a strictly positive self-financing portfolio such that
all assets, when measured in the unit of the numeraire portfolio, are local martingales
under real-world measure P.
As Carr and Yu [4] pointed out, their model differs from the Ross’s model in two
ways. First, their model is based on a bounded diffusion context, compared with the
finite state Markov chain in Ross’s model. Second, they restrict a structure on the
dynamics of the numeraire portfolio to replace the restriction on the representative
agent’s preference as in Ross’s model.
Similar to Ross’s model in Chapter 2, in this chapter we present the framework of
Carr and Yu’s model, starting with its market setting and assumptions and followed
by the deviation of the recovery theorem. The main result is the recovery theorem
in Carr and Yu’s model (Theorem 4.2.3).
The steps in the derivation of Carr and Yu’s model are listed as follow:
1. The main objective is to recover the real-world measure P from the risk-neutral
measure Q, together with the Radon-Nikodym derivative dPdQ .
2. The main feature of Carr and Yu’s model is the role played by the numeraire
portfolio.
3. The assumptions of the model are:
(a) A continuous time and space model,
22
(b) “No free lunch with vanishing risk” applies in the market, and the nu-
meraire portfolio Lt is equal toS0t
Mt, where Mt = dQ
dP
∣∣Ft
(C3),
(c) A continuous, time-homogeneous, bounded, and univariate driver, from
which all asset prices can be determined (C4),
(d) The dynamics of the interest rate and the driver under the risk-neutral
measure Q are known (C5),
(e) The numeraire portfolio depends only on the current value of the driver
and time (C6 (part i)),
(f) Time-homogeneity of the interest rate process (C6 (part ii)) and diffu-
sion coefficient of the numeraire portfolio (C6 (part iii)),
(g) Continuity and differentiability assumptions (C7), and
(h) The boundary conditions for the numeraire portfolio are known (C8).
4. Based on this set of assumptions, if the diffusion coefficient of the numeraire
portfolio is determined, the market price of risk can be determined (Theo-
rem 4.1.1).
5. The problem of solving the diffusion coefficient can be transformed to a prob-
lem of eigenvalue and eigenfunction (4.21).
6. The regular Sturm Liouville theorem (Appendix C) is then applied (Theo-
rem 4.2.3) to uniquely determine the diffusion coefficient.
7. The Girsanov’s theorem and the change of numeraire theorem are applied to
determine the dynamics of the driver under the real-world measure P and the
Radon-Nikodym density dPdQ (Theorem 4.2.4).
In this chapter we utilize the sufficient set of assumptions to derive the result of
Carr and Yu’s model. On top of the existing assumptions, we state the additional
assumptions to the model (C7, C8) which are necessary but have been omitted in
Carr and Yu’s paper [4].
All the assumptions in this chapter model start with letter ’C’ (for example,
C3).
4.1 Assumptions and definitions
With reference to the market set-up in Chapter 1.1, the economy in Carr and Yu’
model is continuous for time and state space, i.e.
23
1. t is a continuous time index on a finite interval t ∈ [0, T ].
2. The state space is continuous.
Moreover, there exists a stochastic interest rate rt ∈ R≥0, such that S0t satisfies:
S0t = e
∫ t0 rsds t ∈ [0, T ], (4.1)
or equivalently,
dS0t = rtS
0t dt t ∈ [0, T ], (4.2)
with
S00 = 1. (4.3)
C3 (part i): “No free lunch with vanishing risk” (NFLVR) condition applies in
the economy.
By the Fundamental Theorem of Asset Pricing [6], there exists an equivalent
local martingale measure (ELMM) Q such that (Si/S0)t is a local martingale for all
i = 0, 1, . . . , n under Q.
Furthermore we can define the Radon-Nikodym derivative, Mt, as
Mt =dQdP
∣∣∣∣Ft
. (4.4)
The concept of the numeraire portfolio is introduced, which plays an important
role in deriving the result of the model. More about the numeraire portfolio could
be found in Appendix B.
Definition 4.1.1 (The numeraire portfolio)
A numeraire portfolio Lt is a strictly positive self-financing portfolio, such that
(Si/L)t is a local martingale under P-measure for all i = 0, 1, . . . , n.
NFLVR condition implies the existence of numeraire portfolio, which is discussed
Appendix B.
C3 (part ii): The numeraire portfolio, Lt, is equal toS0t
Mt.1
This assumption can be satisfied, for example, if the market is complete. The
proof of this statement could also be found in Appendix B.
The numeraire portfolio has the following property:
1This assumption will be discussed in detail in Section 5.2.
24
Theorem 4.1.1 (Dynamic of the numeraire portfolio under P)
Lt has a dynamics of the following form under P:
dLtLt
= (rt + σ2t )dt+ σtdW
Pt , (4.5)
where W Pt is a Brownian motion under P, and σt is an adapted process.
Proof. (Si/L)t is a local martingale for all i = 0, 1, . . . , n.
In particular, by Martingale Representation Theorem,
d(S0/L)t(S0/L)t
= −σtdW Pt (4.6)
for some adapted process σt.
Let A be an Ito process. From Ito’s formula we have
d
(1
A
)= − 1
A2dA+
1
A3(dA)2.
Set A = (S0/L)t,
d
(L
S0
)t
= − L2t
(S0t )
2d
(S0
L
)t
+L3t
(S0t )
3
(d
(S0
L
)t
)2
.
Divide both sides by Lt/S0t and substitute (4.6) implies:
d
(L
S0
)t
/
(L
S0
)t
= σ2t dt+ σtdW
Pt .
Again by Ito’s formula,
d
(L
S0
)t
=1
S0t
dLt −Lt
(S0t )
2dS0
t .
Substitute the dynamics for S0t as in (4.2)
d
(L
S0
)t
=1
S0t
dLt −LtrtS0t
dt
d
(L
S0
)t
/
(L
S0
)t
=dLtLt− rtdt
σ2t dt+ σtdW
Pt =
dLtLt− rtdt
dLtLt
= (rt + σ2t )dt+ σtdW
Pt .
25
An important consequence of Theorem 4.1.1 is the following relationship:
Corollary 4.1.1 (The market price of risk)
The “market price of risk”, Θt, (also known as the “risk premium”) of the model is
equal to the diffusion coefficient of the numeraire portfolio, i.e.
Θt = σt.
Proof. The market price of risk is defined as
Θt =αt − rtσt
,
where αt is the drift coefficient of a tradable asset, σt is the drift coefficient of a
tradable asset, and rt is the interest rate process.
From Theorem 4.1.1 the market price of risk can be deduced as:
Θt =rt + σ2
t − rtσt
.
Hence
Θt = σt.
Definition 4.1.2 (Time-homogeneous Ito diffusion2)
An one-dimensional bounded time-homogeneous Ito diffusion Xt is an adapted
stochastic process satisfying a stochastic differential equation of the form:
dXt = b(Xt)dt+ a(Xt)dWt,
where Wt is an one-dimensional Brownian motion, and also b : R → R, a : R → Rsatisfy the Lipschitz continuity condition, that is,
|b(x)− b(y)|+ |a(x)− a(y)| ≤ D|x− y| x, y ∈ R for some constant D.
Definition 4.1.3 (Infinitesimal generator of an Ito diffusion)
The infinitesimal generator G of an one-dimensional Ito diffusion Xt is defined by
Gf(x) = limt→0
E [f(Xt)|X0 = x]− f(x)
tfor x ∈ R.
2This definition is referenced to [17, p. 116]
26
The infinitesimal generator can be shown to be equivalent to3:
G =∂
∂t+a2(x)
2
∂2
∂x2+ b(x)
∂
∂x.
The “driver” of the model is now introduced, from which all security prices are
derived.
C4: There exists an univariate bounded time-homogeneous Ito diffusion Xt ∈[u, l] such that Sit = Si(Xt, t) for i = 0, 1, . . . , n. Xt is called the driver in the model.
By definition the driver Xt satisfies the following stochastic differential equation:
dXt = b(Xt)dt+ a(Xt)dWQt , (4.7)
where WQ is a Brownian motion under Q.
(4.1) can be written as;
rt =∂
∂tlogS0
t .
From C4, S0t = S0(Xt, t), so rt = rt(Xt, t). In other words, the interest rate
process rt is also driven by the driver Xt.
The dynamics of the driver under Q is calibrated using the market data. This
is stated in the following assumption:
C5: a(x), b(x), and r(x, t) are known ex ante. In addition, a(x) 6= 0 for all
x ∈ [u, l].4
The next assumption is pivotal in deriving the conclusion of the model. It is
divided into three parts, but they are all related to the structure of the numeraire
portfolio.
C6 (part i): The numeraire portfolio, Lt, depends only on the current value of
the driver Xt and time t, i.e.
Lt ≡ L(Xt, t) (4.8)
3A reference for this can be found on [17, p. 123].4The condition a(x) 6= 0 is different from the assumption a(x) ≥ 0 as in Carr and Yu’s paper
[4]. As we will see in the next section this weaker condition is sufficient to derive the result.
27
C6 (part ii): The interest rate process, rt, is time-homogeneous, i.e.
r(x, t) ≡ r(x) (4.9)
C6 (part iii): The diffusion coefficient of Lt, σt, is time-homogeneous, i.e.
σ(x, t) ≡ σ(x) (4.10)
Under the risk-neutral measure Q, all self-financing portfolios have the drift
coefficient of rt. As Lt is a self-financing portfolio, it also has the drift coefficient of
rt. Therefore Lt can be written as:
dLtLt
= r(Xt)dt+ σ(Xt)dWQt . (4.11)
In other words, the numeraire portfolio Lt is a time-homogeneous process under
measure Q.
The next assumption C7 is also divided into three parts. They were not included
in Carr and Yu’s paper, but they are important in deriving the recovery theorem.
C7 (part i): a(x), b(x), and r(x) are continuous functions. r(x) is also bounded
below.
Together with a(x) 6= 0 as in C5, one can deduce that either a(x) > 0 or a(x) < 0
for all x ∈ [u, l], i.e. a(x) never changes sign.
C7 (part ii): Si(x, t) for i = 0, 1, . . . , n are twice continuously differentiable
C7 (part iii): Lt = L(Xt, t) is a twice differentiable function.
They will be applied in different parts of the derivation.
Theorem 4.1.2 (PDE for asset prices)
Assets in the market satisfy the following equation:
GSi(x, t) = r(x)Si(x, t) for i = 0, 1, . . . , n. (4.12)
Proof. Under the risk-neutral probability Q, the following identity holds:
SitS0t
= EQ[SiTS0T
|Ft]
.
28
Rearrange this identity and substitute (4.1), we have:
Si(x, t) = EQ[exp
(−∫ T
t
r(Xs)ds
)Si(XT , T )|Ft
].
Since the price functions Si are twice continuously differentiable (C7 (part ii))
and r is continuous and bounded below (C7 (part i)), we can apply Feynman-Kac
formula5,
∂Si(x, t)
∂t+a2(x)
2
∂2Si(x, t)
∂x2+ b(x)
∂Si(x, t)
∂x− r(x)Si(x, t) = 0.
Rewrite it using the infinitesimal generator,
GSi(x, t) = r(x)Si(x, t).
Like C7, the following assumption was not included in Carr and Yu’s paper.
C8: The boundary conditions of L(x, t) and ∂∂xL(x, t) at x = l and x = u are
known ex-ante.6
The assumptions of Carr and Yu’s model will be further discussed in Chapter 5.
In the next section we will see how the recovery theorem of Carr and Yu’ model is
arrived.
4.2 Derivation
In this section the recovery theorem of Carr and Yu’s model is derived.
We first derive the diffusion coefficient of the numeraire portfolio σt, which is
equivalent to the market price of risk Θt by Corollary 4.1.1. The Girsanov theorem
is then applied to recover the dynamics of the process Xt under P-measure, together
with the dynamics of all the security prices Sit . As mentioned, an important fea-
ture of Carr and Yu’s model is the assumption on the form and dynamics of the
numeraire portfolio (C6).
5A reference for this formula is on [20, p. 145].6Although not included in the paper, this condition was suggested by Peter Carr in an email
to Johannes Ruf. This assumption is needed in applying the regular Sturm Liouville theorem, aswe will see in the next section. This assumption will also be discussed in detail in Section 5.1.
29
Substitute (4.9) and (4.10) into (4.5), one can derive an expression for dynamics
of Lt under P:dLtLt
= [r(Xt) + σ2(Xt)]dt+ σ(Xt)dWPt . (4.13)
An expression for the dynamics of Xt under P is also needed:
Theorem 4.2.1 (Dynamics of the driver under P)
The dynamics of the driver, Xt, under P is given by
dXt = [b(Xt) + σ(Xt)a(Xt)]dt+ a(Xt)dWPt , (4.14)
which is a time-homogeneous diffusion process.
Proof. With reference to (4.7), the change of measure only changes the drift coeffi-
cient by the Girsanov’s Theorem. Therefore the diffusion term of Xt under P is still
a(Xt).
From Corollary 4.1.1, the market price of risk is σ(Xt). By the Girsanov’s theo-
rem, the drift coefficient of Xt under P is b(Xt) + σ(Xt)a(Xt), i.e.
dXt = [b(Xt) + σ(Xt)a(Xt)]dt+ a(Xt)dWPt .
Theorem 4.2.2 (Diffusion coefficient of the numeraire portfolio)
The diffusion coefficient function, σ(x), satisfies the following equation:
σ(x) = a(x)∂
∂xlogL(x, t). (4.15)
Proof. As Lt is twice differentiable (C7 (part iii)), Ito’s formula can be applied to
(4.8),
dLt =∂Lt∂t
dt+∂Lt∂x
dXt +1
2
∂2Lt∂x2
(dXt)2
dLtLt
=1
Lt
∂Lt∂t
dt+1
Lt
∂Lt∂x
dXt +1
2
1
Lt
∂2Lt∂x2
(dXt)2.
Substitute (4.14) and compare only the diffusion terms of dLL
with equation (4.13):
σ(x) =1
Lt
∂Lt∂x
a(x).
Rearrange the equation,
σ(x) = a(x)∂
∂xlogL(x, t).
30
The equality in (4.15) will be reorganized such that it can fit into the regular
Sturm Liouville theorem (Appendix C).
Since a(x) 6= 0 (by C5), for some function f(t),
logL(x, t) =
∫ x
l
σ(y)
a(y)dy + f(t). (4.16)
Note that σ(x) and a(x) are continuous functions (C7 (part i)), so σ(x)a(x)
is
integrable.
Take exp(·) on the both sides and let:
π(x) = e∫ xlσ(y)a(y)
dy
p(t) = ef(t).
Thus, a separable expression of L(x, t) is obtained:
L(x, t) = π(x)p(t). (4.17)
Substitute this into the generator equation (4.12):
π(x)p′(t) +a2(x)
2π′′(x)p(t) + b(x)π′(x)p(t) = r(x)p(t)π(x).
Dividing by π(x)p(t) implies:
p′(t)
p(t)+a2(x)
2
π′′(x)
π(x)+ b(x)
π′(x)
π(x)= r(x)
a2(x)
2
π′′(x)
π(x)+ b(x)
π′(x)
π(x)− r(x) = −p
′(t)
p(t).
The two sides can only be equal if they are each equal to a constant λ ∈ R:
p′(t)
p(t)= λ, (4.18)
a2(x)
2
π′′(x)
π(x)+ b(x)
π′(x)
π(x)− r(x) = −λ. (4.19)
The solution to (4.18) is the following:
p(t) = p(0)eλt (4.20)
and λ is still an unknown.
Reorganize the problem for (4.19):
a2(x)
2π′′(x) + b(x)π′(x)− r(x)π(x) = −λπ(x), (4.21)
which is an eigenvalue and eigenfunction problem.
31
Theorem 4.2.3 (Recovery theorem in Carr and Yu’s model)
The only valid solution to (4.21), in which the corresponding the numeraire portfolio
is positive, is:
π(x) = φ(x)
λ = ρ,
where
1. ρ is the smallest eigenvalue of the problem, and
2. φ(x) is eigenfunction correspond to ρ.
Proof. The problem can be re-written in self-adjoint form:
a2(x)
2c(x)
∂
∂x
(c(x)
∂π(x)
∂x
)− r(x)π(x) = −λπ(x),
where
c(x) = exp
(∫ x
l
2b(y)
a2(y)dy
).
Dividing the function a2(x)2c(x)
the problem becomes
∂
∂x
(c(x)
∂
∂x
)− q(x)π(x) = −λπ(x)w(x), (4.22)
where
q(x) =2c(x)r(x)
a2(x), and
w(x) =2c(x)
a2(x).
We know that:
1. c(x), w(x) are positive,
2. c(x), c′(x), q(x) and w(x) are continuous (because from C7 (part i), a(x),
b(x) and r(x) are all continuous functions), and
3. The boundary conditions of π(x) and π′(x) at x = l and x = u are known
(because from C8, the boundary conditions of L(x, t) and ∂∂xL(x, t) at x = l
and x = u are known, which can be used to deduced the bounary conditions
for π(x) and π′(x)).
32
The problem (4.22) is a regular Sturm Liouville problem (Appendix C).
One can numerically solve for all of the eigenfunctions and eigenvalues. Moreover,
the eigenvalues of the problem can be denoted as:
λ1 < λ2 < · · · < λn < · · · → ∞,
and its corresponding eigenfunctions to be:
π1(x), π2(x), π3(x), . . .
Let
φ(x) = π1(x), and
ρ = λ1.
From (4.17),
L(x, t) = π(x)p(t).
Since L(x, t) is positive, of all the eigenfunctions only φ(x) is positive and never
changes sign. So φ(x) is the only valid eigenfunction to the problem (4.22), and
hence the only valid eigenvalue is the corresponding eigenvalue ρ.
Remark 4.2.1
Notice that the form and dynamics of the numeraire portfolio (C6) allow Lt to be
separated as a function of the driver x and a function of time t in (4.17), deriving
the recovery theorem. Without C6, the recovery theorem of Carr and Yu’s model is
not possible. This feature will be contrasted with the transition independent pricing
kernel in Ross’s model in Chapter 6.
Substitute the result of Theorem 4.2.3 and (4.20) into (4.17),
L(x, t) = φ(x)ef(0)eρt. (4.23)
Substitute (4.23) into expression (4.15),
σ(x) = a(x)∂
∂xlog φ(x).
Once the function σ(x) is found, we can derive the dynamics of Xt under P by
Theorem 4.2.1:
dXt = [b(Xt) + σ(Xt)a(Xt)]dt+ a(Xt)dWPt ,
and dPdQ in the following theorem.
33
Theorem 4.2.4 (Expression for the Radon-Nikodym derivative)
The Radon-Nikodym derivative dPdQ is given by:
dPdQ
= e−∫ T0 rsds
φ(XT )
φ(X0)eρT .
Proof. dPdQ can be determined by the change of numeraire theorem [20], using the
numeraire portfolio as the numeraire, we have
dPdQ
=S0
0
S0T
L(XT , T )
L(X0, 0).
Substitute (4.23) and (4.1) into this expression,
dPdQ
= e−∫ T0 rsds
φ(XT )
φ(X0)eρT .
In other words, the dynamics of Xt under the real-world measure P and the
Radon-Nikodym density dPdQ have been recovered.
In the next chapter we will analyse and discuss the features and implications of
Carr and Yu’s model.
34
Chapter 5
Carr and Yu’s model - analysisand discussion
In the preceding chapters we have reviewed the basic frameworks of Ross’s model
and Carr and Yu’s model, and also discussed the limitations and features of Ross’s
model. In this chapter Carr and Yu’s model is analysed to highlight its limitations,
similarities with Ross’s model and also other features.
This chapter is divided into the following sections:
Section 5.1: Diffusion coefficient of the numeraire portfolio
The discussion begins by analysing C8: the boundary conditions for the nu-
meraire portfolio is known ex-ante. We attempt to argue that if there is no general
method of obtaining the boundary conditions of the numeraire portfolio, the imple-
mentation of the recovery theorem could be difficult in practice.
Section 5.2: Market completeness in Carr and Yu’s model
An implicit restriction in Carr and Yu’s model is the market completeness con-
dition. We argue that, although not explicit assumed in the model, the market
completeness condition is necessary for Carr and Yu’s model.
Section 5.3: Existence of stationary distribution
The discussion will progress to illustrate the similarities between Ross’s model
and Carr and Yu’s model. In particular, we focus on the existence of the stationary
distribution. In this section we provide heuristic argument for the existence of the
stationary distribution in Carr and Yu’s model.
Section 5.4: Recovery theorem on unbounded domain
In addition to bounded diffusion considered in Chapter 4, we expand the scope
to consider examples of the model under unbounded domain. Two examples of the
recovery theorem of Carr and Yu’s model on unbounded domain are investigated:
35
the Black-Scholes model and the CIR model. The stationary distribution existence
condition is also checked for each model.
5.1 Boundary conditions of the numeraire port-
folio
In this section we discuss an important but often overlooked assumption, C8. We
argue that this assumption could be difficult to be satisfied in practice.
An important element in the derivation of Carr and Yu’s recovery model is the
use of the regular Sturm Liouville theorem to numerically solve eigenfunction and
eigenvalue problem (4.22) for the unique solution. The solution is then used to
deduce the solution of the numeraire portfolio and recover the real-world measure
P. However, in order to apply the regular Sturm Liouville theorem, one must have
the knowledge of the boundary conditions for the problem.
The knowledge of the boundary conditions is assumed in C8, which states that
the values of the numeraire portfolio L(x, t) and its partial derivative ∂∂xL(x, t) at
boundary values x = l and x = u are known ex-ante. Without this assumption, one
could only imply that the unique solution for the problem (4.22) exists, but would
not be able to solve for its solution, and therefore the knowledge of P cannot be
recovered.
While assuming the knowledge of the information required to apply the regular
Sturm Liouville theorem, Carr and Yu provided no practical method to obtain the
boundary conditions for the numeraire portfolio. In a typical calibration process, one
uses the prices of the derivative instruments to determine the information about the
risk-neutral measure Q, in Carr and Yu’s framework this is equivalent to knowing the
functions a(x), b(x) and r(x). But this does not imply that the boundary conditions
for the numeraire portfolio can be determined. After all, the numeraire portfolio is
a theoretical tool to derive the result of the recovery theorem, and the composition
of the numeraire portfolio is not explicitly known ex ante. In general, its value at
the boundaries could not be easily determined except for trivial cases.
This leads us to conclude that, without a general method of obtaining the bound-
ary conditions for the numeraire portfolio, the implementation of the recovery the-
orem of Carr and Yu’s model would be difficult in practice.
36
5.2 Market completeness in Carr and Yu’s model
The discussion in this section focuses on the market completeness condition in Carr
and Yu’s model. While the condition is not explicitly assumed, we attempt to argue
that in general the market completeness condition is necessary for the model.
Recall that following assumptions were made in the model:
1. “No free lunch with vanishing risk” (NFLVR) (C3 (part i)),
2. The numeraire portfolio Lt is equal toS0t
Mt(C3 (part ii)),
3. The dynamics of Xt, including its coefficient a(x) and b(x), is known by cali-
bration (C5).
First consider the trivial case when n ≥ 1, i.e. There exists at least one risky
asset. According to the meta-theorem1, if there is only 1 randomness in the market
(originated from the driver Xt), the market is complete.
Now consider the case when n = 0, i.e. There is only a risk-free asset S0 in the
market. Since the randomness in the driver Xt cannot be hedged, the market is
incomplete. We will illustrate that under this scenario the recovery theorem may
fail.
Market incompleteness implies both Q and Mt = dQdP
∣∣Ft
not unique. Let the set
of possible Mt asM. Since there is only one asset S0 in the market, the only possible
self-financing portfolio is S0 itself. By Definition 4.1.1, the numeraire portfolio Lt is
a self-financing portfolio. Thus the only possible solution is Lt = S0t , corresponds to
S0t
S0t
= 1 = M1t ∈ M, i.e. P = Q1 ∈ Q. For Mt ∈ M : Mt 6= M1
t , the numeraire
portfolio Lt =S0t
Mtis not a self-financing portfolio.
However, a(x) and b(x) calibrated as in (C5) could correspond to any risk-neutral
probability measure Q2, which corresponds to M2t = dQ2
dP
∣∣∣Ft
. There is no guarantee
that Q2 = Q1, or equivalently M2t = M1
t = 1. If Q2 6= Q1, Lt is no longer a self-
financing portfolio, and the rest of the analysis will fail.
One possible remedy for this issue is to enforce the following assumption in the
model:
1A reference to this theorem is [3, p. 118].
37
C3’ (part iii): The calibrated risk-neutral measure Q corresponds to the Mt
which makes Lt =S0t
Mta self-financing portfolio.
While it can resolve the issue highlighted above, this assumption is a rather
unusual and awkward. There is no obvious economics reason to guarantee that such
a condition would be satisfied. Unless further analysis provides the basis for such
an assumption, it should generally be avoided.
Therefore, we argue that the market completeness condition is necessary for the
model. Otherwise without this condition one have resort to unusual assumption
such as C3’ (part iii) to derive the recovery theorem.
To further expand this argument, one can consider a more general case when
Carr and Yu’s model is extended to a multivariate diffusion driver, i.e.
dX = b(Xt) +d∑j=1
aj(Xi)dWjt ,
where W jt are correlated Brownian motions for j = 1, . . . , d, and there are n
traded risky assets in the market.
If n < d, the market is incomplete by the meta-theorem. Again the risk-neutral
measure Q is not unique and one must consider the issue highlighted above.
5.3 Existence of stationary distribution
After reviewing the limitation of Carr and Yu’s model, we turn our attention to its
similarities with Ross’s model. In this section we attempt to argue for the existence
of the stationary distribution in Carr and Yu’s model. The argument in this section
is heuristic in nature, and technicalities of the proof have not been dealt with. We
refer interested readers to [13, Chapter 3] for the conditions that the existence of
the stationary distribution of Markov processes required. Attentive readers will also
notice the similarities between this argument and the recovery theorem of Carr and
Yu’s model in Section 2.22.
The intuition behind the existence condition is simple. Recalled that Carr and
Yu’s model is established under a bounded domain with a time-homogeneous driver
Xt. As Xt diffuses within the bounded domain [u, l], the process will either return to
2The argument in this section is based on a discussion with Pedro Vitoria.
38
its original position or converge to a point in the domain. In either case the system
will exhibit an limiting behaviour and the distribution of Xt becomes ”stationary”.
We begin by introducing the transition density function:
Definition 5.3.1 (Transition density for the driver Xt)
The transition density for a diffusion process, f(t, x;T, y), is the density of the
probability distribution function for the stochastic process in that for any set A ⊂ R,
P(XT ∈ A|Xt = x) =
∫A
f(t, x;T, y)dy.
We focus on the transition density of the driver Xt under P, which is time-
homogeneous as shown in (4.14).
The driver Xt in Carr and Yu’s model as defined in (4.1.2):
dXt = b(Xt)dt+ a(Xt)dWt,
the transition density, f(x, t; y, T ), of Xt satisfies the Kolmogorov forward equa-
tion (KFE)3.
∂
∂Tf(x, t; y, T ) +
∂
∂y(b(y)f(x, t; y, T ))− 1
2
∂2
∂y2(a2(y)f(x, t; y, T )) = 0, (5.1)
together with the boundary conditions:∫ ∞−∞
f(x, t; y, T )dy = 1 and f(x, t; y, T ) ≥ 0.
As we only focus on the future time T and the future value y of Xt in the
argument, f(x, t; y, T ) is written as fx,t(y, T ).
A stationary distribution is a transition distribution which is independent of fu-
ture time T , which will be found below.
We try to find a solution f to (5.1) of a separable form:
fx,t(y, T ) = βx,t(T )αx,t(y).
Substitute into (5.1),
αx,t(y)∂βx,t(T )
∂T= −βx,t(t)
∂(b(y)αx,t(y))
∂y+ βx,t(T )
1
2
∂2(a2(y)αx,t(y))
∂y2.
Rearrange we get:
β′x,t(T )
βx,t(T )=
1
αx,t(y)
(− ∂
∂y(b(y)αx,t(y)) +
1
2
∂2
∂y2(a2(y)αx,t(y))
).
3A reference for Kolmogorov forward equation could be found in [8, p. 26].
39
The two sides can only be equal if both are equal to a constant λ ∈ R,
For L.H.S.,β′x,t(T )
βx,t(T )= λ.
Solving this we have
βx,t(T ) = βx,t(0)eλT .
For R.H.S.,(− ∂
∂y(b(y)αx,t(y)) +
1
2
∂2
∂y2(a2(y)αx,t(y))
)= λαx,t(y).
Expand the expression:
a2(y)
2α′′x,t(y)+[2a(y)a′(y)−b(y)]α′x,t(y)+[a(y)a′′(y)+(a′(y))2−b′(y)]α(y) = λαx,t(y),
which can be written as self-adjoint form:
a2(y)
2c(y)
∂
∂y
(c(y)
∂αx,t(y)
∂y
)− d(y)αx,t(x) = −λαx,t(y),
where
c(y) = − exp
(∫ y
l
2[2a(u)a′(u)− b(u)]
a2(u)du
), and
d(y) = a(y)a′′(y)− (a′(y))2 − b′(y).
Dividing the function a2(y)2c(y)
the problem becomes:
∂
∂y
(c(y)
∂αx,t(y)
∂y
)− q(y)αx,t(y) = −λαx,t(y)w(y),
where
q(y) =2c(y)d(y)
a2(y), and
w(y) =2c(y)
a2(y).
Similar to the recovery theorem in Carr and Yu’s, the equation becomes a regular
Sturm Liouville problem (Appendix C).
The eigenvalues of the problem are denoted as:
λ1 < λ2 < · · · < λn < · · · → ∞,
and its corresponding eigenfunctions to be
π1(y), π2(y), π3(y), . . .
40
Let
φ(y) = π1(y), and
ρ = λ1.
Since probability density is positive, of all the eigenfunctions only φ(y) is positive
and never changes sign. Hence φ(y) is the only valid eigenfunction to the problem,
and ρ is also the only valid eigenvalue.
Therefore the transition probability f must equal to:
fx,t(y, T ) = CeρTφ(y)
for some constant C which is independent of y and T .
Together with the boundary condition,∫ ∞−∞
fx,t(y, T )dy =
∫ ∞−∞
CeρTφ(y)dy = 1 for all T .
This implies:
C =1
eρT∫∞−∞ φ(y)dy
.
Therefore,
fx,t(y, T ) =φ(y)∫∞
−∞ φ(y)dy.
Therefore a transition density function f which is independent of future time T
has been deduced, i.e. It is the stationary probability distribution of the driver Xt.
However exploratory, the argument in this section has shed light on an impor-
tant property of Carr and Yu’s model: the existence of the stationary distribution.
Notice that the stationary distribution also exists in Ross’s model, which has been
proved in Section 3.2. In the next section we will explore the Carr and Yu’s model
unbounded domains, which will provide further evidence that the existence is a
necessary condition for the succeeding of the recovery theorem.
5.4 Recovery theorem on unbounded domain
So far, all the discussions of Carr and Yu’s model are established under a bounded
domain. As mentioned in Chapter 1, the extend that the recovery theorem is suc-
ceed in unbounded domain remains an open question. To provide insight on this
question, in this section we investigate two examples of the Carr and Yu’s model
41
under unbounded domain: the Black-Scholes model and the CIR model. Part of
this section is based on [5].
We examine if the recovery theorem succeeds in these two examples and also if
the stationary distribution exists.
5.4.1 Black-Scholes model
The driver of the Black-Scholes model is the stock prices St, which follows a geomet-
ric Brownian motion SDE under the risk-neutral measure. For simplicity we assume
no dividend and zero interest rate.
dSt = σStdWQt with St ≥ 0, (5.2)
where σ is a constant and WQt is the Brownian motion under the risk-neutral
measure.
As geometric Brownian motion is not a bounded process, the standard analysis
in Carr and Yu’s model does not apply.
Substitute a2(x) = σ2x2, b(x) = r(x) = 0 into (4.21), we have the following:
σ2x2
2π′′(x) = −λπ(x) for all x ≥ 0, (5.3)
which is a eigenvalue and eigenfunction problem.
Theorem 5.4.1 (Solution set to eigenvalue and eigenfunction problem in
Black-Scholes model)
There are uncountably many eigenvalues λ and eigenfunctions π satisfy the problem
(5.3) with π(x) ≥ 0 for all x ≥ 0.
Proof. If π is assumed to be of the form π(x) = xm, then π′′(x) = m(m− 1)xm−2.
Substitute into (5.3),
σ2x2
2m(m− 1)xm−2 = −λxm
σ2m2 − σ2m+ 2λ = 0
m =σ2 ±
√σ4 − 8σ2λ
2σ2.
This problem has real solution m when
σ4 − 8σ2λ ≥ 0
λ ≤ 8
σ2,
which has infinite many solutions for λ.
42
For each λ, the corresponding π is:
π(x) = xm with m =σ2 ±
√σ4 − 8σ2λ
2σ2,
which is positive for all x ≥ 0.
We can see that the real-world measure P can not be uniquely recovered here -
there are infinite many positive π, together with (4.17) implies the solution for the
numeraire portfolio is not unique. From Corollary 4.1.1, the market price of risk is
not unique, therefore the real-world measure cannot be uniquely recovered.
Theorem 5.4.2 (No stationary distribution exists in Black-Scholes model)
The stationary distribution does not exist in the Black-Scholes model.
Proof. The stationary distribution satisfies the stationary Kolmogorov forward equa-
tion:
− 1
2
∂2
∂S2(σ2S2f(S)) = 0, (5.4)
together with the boundary conditions:
f(S) ≥ 0 for all S ≥ 0 and
∫ ∞0
f(S)dS = 1.
Expand and reorganize (5.4),
S2f ′′(S) + 4Sf ′(S) + 2f(S) = 0.
Solving this equation one will get
f(S) =C1
S2+C2
S
for some constants C1 and C2.
Consider the second boundary condition:∫ ∞0
f(S)dS =
∫ ∞0
(C1
S2+C2
S
)dS.
However, this integral does not converge. Therefore the stationary distribution
does not exist.
43
5.4.2 CIR process
The CIR process is defined as
drt = (µ− κrt)dt+ σ√rtdW
Qt with rt ≥ 0,
where µ, κ, and σ are positive constant and WQt is a Brownian motion under Q.
Substitute into (4.21) with a2(x) = σ2x, b(x) = µ − κx, r(x) = x, l = 0, and
u =∞:
σ2x
2π′′(x) + (µ− κx)π′(x)− xπ(x) = −λπ(x), x ∈ (0,∞), (5.5)
which again is an eigenvalue and eigenfunction problem. Notice that the eigen-
function solution to this problem must be positive for x ≥ 0.
Theorem 5.4.3 (Solution to eigenvalue and eigenfunction problem in CIR
model)
There exists a unique solution to the problem (5.5), in which the eigenfunction is
positive for x ≥ 0. All other eigenfunction to (5.5) switch sign at least once.4
Proof. According to [10], the spectrum of eigenvalues is discrete and known in closed
form, and can be organized as
λ1 < λ2 < · · · < λn < · · · → ∞.
The smallest eigenvalue is
λ1 =µ
σ2(γ − κ),
where
γ =√κ2 + 2σ2.
The corresponding eigenfunction:
π1(x) = exp
(−γ − κ
σ2x
),
is positive for x ≥ 0. All other eigenfunctions to (5.5) switch signs at least
once.
4This theorem and its proof are referenced to [5].
44
Once the unique solution for π(x) is found, the rest is simple - the numeraire
portfolio Lt is equal to
L(x, t) = π1(x)ef(0)eλ1t,
and the rest of the analysis proceeds the same as we did in Section 4.2.
The following theorem shows that the stationary distribution exists in the CIR
process.
Theorem 5.4.4 (Stationary distribution exists in CIR model)
The distribution:
f(r) = Cr2µ−σ2
σ2 e−2κσ2r for all r ∈ R≥0.
is the stationary distribution in the CIR model, where
C =
(2κ
σ2
) 2µ
σ2 1
Γ(
2µσ2
) ,
and Γ is the gamma function
Γ(z) =
∫ ∞0
tz−1e−tdt.
Proof. The stationary distribution satisfies the stationary Kolmogorov forward equa-
tion:∂
∂r((µ− κr)f(r))− 1
2
∂2
∂r2(σ2rf(r)) = 0, (5.6)
together with the boundary conditions:
f(r) ≥ 0 for all r ≥ 0 and
∫ ∞0
f(r)dr = 1.
We will check the conditions one by one. First,
f ′(r) =2
σ2r
(µ− 1
2σ2 − κr
)f(r)
f ′′(r) =2
σ2r
(−1
r
(µ− 1
2σ2 − κr
)− κ+
2
σ2r
(σ − 1
2σ2 − κr
)2)f(r).
Now substitute these into L.H.S. of (5.6), it becomes
− κf(r)+(µ− κr)f ′(r)− σ2f ′(r)− 1
2σ2rf ′′(r)
= 0.
Second, it is obvious that:
f(r) ≥ 0 for all r ≥ 0.
45
Lastly, ∫ ∞0
f(r)dr = C
∫ ∞0
r2µ−σ2
σ2 e−2κσ2rdr
= C
(σ2
2κ
) 2µ
σ2
Γ
(2µ
σ2
)= 1.
Following the discussion and analysis of the two models, in the next chapter we
will summarize the result and draw a comparison between the two recovery models
to conclude this thesis.
46
Chapter 6
Conclusion and further research
In this thesis we have reviewed and clarified the assumptions of the recovery theorem
in Ross’s model and that in Carr and Yu’s model. We have also discussed a number
of properties of the two models, highlighting the limitations, similarities and other
features of the two recovery models. We have also considered the examples of Carr
and Yu’s model under unbounded domain.
For limitations of the models, we have discussed that the transition indepen-
dent pricing kernel does not follow from inter-temporal additive separable utility for
Ross’s model in Section 3.1, obtaining the boundary conditions for the numeraire
portfolio could be difficult in practice for Carr and Yu’s model in Section 5.1, and
market completeness condition is necessary Carr and Yu’s model in Section 5.2.
We have proved the existence of the stationary distribution for Ross’s model in
Section 3.2 and provided an heuristic argument for Carr and Yu’s model Section 5.3.
It is easy to notice the similarities between the argument for the existence of the
stationary distribution and deriving the recovery theorem in Ross’s model (using
the Perron Frobenius theorem) as well as in Carr and Yu’s model (using the regular
Sturm Liouville theorem).
Additionally, in Section 5.4, we reviewed that the recovery theorem of Carr
and Yu’s model succeeds in the CIR model but fails in the Black-Scholes model.
Interestingly, we also found that the stationary distribution exists only in the CIR
model but not in the Black-Scholes model, which has been also shown in the same
section.
These evidences lead us to postulate that the existence of the stationary distri-
bution is a necessary condition for the recovery theorem to succeed1. In other words,
in order to recover the real-world probability measure P from the risk-neutral prob-
ability measure Q, one must ensure that the stationary distribution exists in the
model being considered. This postulation, if valid, will be helpful in looking for the
1This idea was originally suggested by Dr. Samuel Cohen in a meeting.
47
minimal assumption sets to the models. This postulation could also provide insight
into the extension of the models, such as market with multivariate drivers, drivers
under unbounded domain or incomplete market.
In their paper [4], Carr and Yu took a different track: based on the fact that reg-
ular Strum Liouville theorem ensures the discreteness and countability of the point
spectrum in bounded domain, they conjectured the countability of point spectrum
is a condition for the recovery theorem under an unbounded domain. They also
provided further evidence by showing that the point spectrum of the CIR model
was discrete and countable while that of the Black-Scholes model was continuous.
Based on these two conjectures, exploring the connections between the existence
of the stationary distribution and the countability of point spectrum could be a
direction of future research. For example, one can check the set of assumptions
required for each of the two conditions and the relationship between them. One
can also examine other unbounded or multivariate models to access to what extent
these two conditions are valid, and their relationships with succeeding of the recovery
theorem.
The results of our analysis also highlight the fact that the assumption sets in
the two recovery models are relatively restrictive, such as transition independence of
pricing kernel in Ross’s model and the time-homogeneous dynamics of the numeraire
portfolio in Carr and Yu’s model. In a real financial market, these assumptions may
not be justified or even be verified. Therefore, in addition to the extensions to
the recovery theorem, the research community can benefit from searching for a less
restrictive and more realistic assumption sets for the current recovery models.
On the other hand, our discussion has been confined to the theoretical aspect
of the recovery theory, but the practicability of the theory should be verified by
empirical evidences. In principle the recovery theory can be applied to a wide range
of financial markets, and to find out which product types and markets most suitable
for applying the theorem is a worthy research topic. In his paper [19], Ross used
the S & P 500 index to verify his model. This result could be treated as a starting
point for this line of research. One could also explore how to practically apply the
theory into areas like risk management and market prediction.
We conclude this thesis with the following table, which draws a comparison
between the two models to summarize the results in this thesis.
48
49
Table 6.1: Comparison between the two recovery models
Item Ross’s model Carr and Yu’s model Remark
Setting Discrete time and state space. Continuous time and state space.Formulation Finite state irreducible Markov chain
(Section 3.2).Univariate diffusion process Xt on abounded domain.
Aim To recover the real-world transitionprobability matrix F from thestate-price matrix P .
To recover the dynamics of the driverXt under P from Q, deducing dP
dQ .
Marketcondition
No Arbitrage (R3). NFLVR and the numeraire portfolio
Lt is equal toS0t
Mt,
where S0t is the risk-free asset and
Mt = dQdP
∣∣Ft
(C3).
We argue for the necessity of marketcompleteness in Carr and Yu’smodel (Section 5.2).
Time homo-geneity
State-price function p and real-worldtransition density function f of thestate variable are time-homogeneous(R4).
The driver Xt is time-homogeneousunder Q (C4) and P(Theorem 4.2.1).
Calibration The state-price matrix P is known(R5), and the interest rate processrt can be deduced (Theorem 3.3.1).
Dynamics of the driver Xt under Qand the interest rate process rt areknown (C5).
Mainassumption
Pricing kernel ϕ is transitionindependent:ϕi,j =
pi,jfi,j
= δ h(i)h(j)
(R6).
1. Lt ≡ L(Xt, t),2. rt is time-homogeneous,3. Diffusion coefficient of thenumeraire portfolio, σt, istime-homogeneous (C6).
Despite suggested by Ross, weexplain that the transitionindependent pricing kernel does notfollow from inter-temporal additiveseparable utility function(Section 3.1).
Otherassumptions
P is non-negative and irreducible(R7).
1. Differentiability and continuityassumptions (C7),2. The boundary conditions for thenumeraire portfolio are known (C8).
We argue that C8 could be difficultto be satisfied in practice(Section 5.1).
50
(Continued from previous page)
Item Ross’s model Carr and Yu’s model Remark
Derivation Given its separable form, thetransition independent pricing kernelequation is formulated as aneigenvalue and eigenvector problem,for which a unique solution exists(Theorem 2.2.1).
The market price of risk, Θt, is equalto σt (Corollary 4.1.1). Byseparation of variables, an identityfor σt is formulated as an eigenvalueand eigenfunction problem, forwhich a unique solution exists(Theorem 4.2.3).
Carr and Yu’s for unboundeddomain:1. the Black-Scholes model (recoverytheorem does not succeed),2. the CIR model (recovery theoremsucceeds)(Section 5.4).
Separationof variables
Pricing kernel can be separated fromreal-world probability:pi,j = ϕi,jfi,j = δ h(i)
h(j)fi,j
(Remark 2.2.1).
The numeraire portfolio can beseparated by a function of the driverand a function of time:L(x, t) = π(x)p(t) (Remark 4.2.1).
R6 and C6 guarantee the separationof variables in the respective models.
Maintheoremused
The Perron Frobenius theorem(Appendix A).
The regular Sturm Liouville theorem(Appendix C).
Stationarydistribution
Existence proved (Section 3.2). Heuristic argument for existence(Section 5.3).
Carr and Yu’s for unboundeddomain:1. the Black-Scholes model (notexists),2. the CIR model (exists)(Section 5.4).We postulate that the existence ofstationary distribution is a necessarycondition for the recovery theoremto succeed (Chapter 6).
Interest rateprocess
Time-homogeneous (Theorem 3.3.1). Time-homogeneous (C6 (part ii)).
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52
Appendices
53
Appendix A
Perron Frobenius theorem
The Perron Frobenius theorem encompasses a number of related results about pos-
itive and irreducible matrices. In this appendix we focus on the theorem on irre-
ducible matrices, with the results that are useful in this thesis.
Definition A.0.1 (Spectral radius of a square matrix)
Let A be an n× n square matrix with real elements and with the set of eigenvalues
Λ = λ : Av = λv for some non-zero vector v . The spectral radius ρ(A) of A is
defined as:
ρ(A) = maxλ∈Λ|λ|.
Theorem A.0.5 (Perron Frobenius theorem for irreducible matrices)
Let A be an n× n irreducible matrix with real elements with ρ(A) = r, then
1. r is positive and it is an eigenvalue of the matrix A,
2. The entries of eigenvector v associated with eigenvalue r are all positive,
3. Eigenvalue r is simple, i.e. v is unique up to a positive scaling,
4. All of the other eigenvectors are not strictly positive, i.e. There are zero and
negative elements in the vectors, and
5. r is between minimum sum of row and maximum sum of row of A (inclusive),
i.e.
mini
∑j
ai,j ≤ r ≤ maxi
∑j
ai,j. (A.1)
r is called the Perron root and v is called the Perron vector.
Proof. A reference and the detailed proof of statement 1 to 4 of this theorem can be
found on [16, p. 673: Perron Frobenius Theorem]. As for statement 5, the inequality
A.1 can be proved by:
54
1. applying Collatz-Wielandt formula [16, p. 666] by setting x as a vector of
(1, 1, . . . , 1) to get the L.H.S. mini∑
j ai,j ≤ r, and
2. applying Gerschgorin Circles [16, p. 498] to get the R.H.S. r ≤ maxi∑
j ai,j.
55
Appendix B
Numeraire portfolio
In this appendix we first introduce the numeraire portfolio and provide background
and references on the topic. After that a statement mentioned in the Section 4.1 is
proved: if the market is complete then numeraire portfolio, Lt, is equal toS0t
Mt, where
S0t is the risk-free asset and Mt is the Radon-Nikodym derivative dQ
dP
∣∣Ft
.
The notion of the numeraire portfolio was first introduced by Long [15] in 1990.
The discrete time version of the numeraire portfolio, as in Long’s paper, is defined
as the following:
Definition B.0.2 (Numeraire portfolio (discrete time))
A numeraire portfolio Nt in discrete time economy is defined as a self-financing
portfolio with always positive value such that, for each asset j,
P jt
Nt
= E
[P jt+1
Nt+1
|Ft
]for all 0 ≤ t ≤ T with probability one,
where P jt is the price process of the asset j, and E denotes the expectation under
the real-world measure. (The definition here is simplified with the assumption that
there is no dividend.)
Long showed that the numeraire portfolio exists in a discrete time economy if
and only if there are no profit opportunities ([15, Theorem 1]). In addition, he
showed that the expected excess return of the numeraire portfolio is its the variance
of its rate of return. ([15, Theorem 3])
In fact the numeraire portfolio can be also obtained from the growth optimal
portfolio (GOP) which maximizes the expected utility of a log utility function. A
reference of this is [14].
The continuous time version of the numeraire portfolio, as described in [1], is the
following:
56
Definition B.0.3 (The numeraire portfolio (continuous time))
A numeraire portfolio Nt is a strictly positive self-financing portfolio, such that PtNt
is a supermartingale for all strictly positive self-financing portfolio, Pt under the
real-world measure.1
Karatzas and Kardaras [11] proved that “No unbounded profit with bounded
risk” (NUPBR) is sufficient to ensure the existence of the numeraire portfolio. This
condition is weaker than the usual ”No free lunch with vanishing risk” (NFLVR) [6]
as usually seen in financial literatures.
However, the existence of the numeraire portfolio does not automatically imply
that it is equal toS0t
Mt, as shown in the example in Section 5.2. However, in a com-
plete market with ”No Free Lunch with Vanishing Risk” (NFLVR), this is always
the case, as it will be shown below.
Definition B.0.4 (Self-financing portfolio)
A portfolio (φ0t , φ
1t , . . . , φ
nt ) is self-financing if
dVt =n∑i=0
φitdSit .
where
Vt =n∑i=0
φitSit ,
Lemma B.0.1 (Self-financing portfolio identity)
A portfolio (φ0t , φ
1t . . . , φ
nt ) with value Vt =
∑ni=0 φ
itS
it is self-financing if and only if
dVt =n∑i=1
φitdSit ,
where Yt = Yt/S0t denotes the discounted value of trading strategy or asset price
process.
Proof. By the definition of self-financing portfolio,
dVt =n∑i=0
φitdSit .
1Although the supermartingale condition is used in the referenced paper, in this thesis we usethe stricter definition used in Carr and Yu’s paper: Pt
Ntis a (local) martingale.
57
By Ito’s formula, for all i = 1, 2, . . . , n
dSit =1
S0t
dSi,t −Sit
(S0t )
2dS0
t
dSitS0t
= dSit +Sit
(S0t )
2dS0
t .
Consider dVt,
dVt =1
S0t
dVt −Vt
(S0t )
2dS0
t
=1
S0t
n∑i=0
φitdSit −
Vt(S0
t )2dS0
t
= φ0t
dS0t
S0t
+n∑i=1
φit
(dSit +
Sit(S0
t )2dS0
t
)− Vt
(S0t )
2dS0
t
=n∑i=1
φitdSit +
n∑i=0
φitSit
(S0t )
2dS0
t −Vt
(S0t )
2dS0
t
=n∑i=1
φitdSit .
Theorem B.0.6 (Numeraire portfolio in complete market)
If the market is complete with NFLVR, then the numeraire portfolio exists and
Lt ≡S0t
Mt
. (B.1)
Proof. First, we show that Sit/Lt is a local martingale under P-measure for all i =
0, 1, . . . , n, Pt.
By the Fundamental Theorem of Asset Pricing [6], there exists a unique equiv-
alent local martingale measure (ELMM) Q such that (Si/S0)t is a local martingale
for all i = 0, 1, . . . , n under Q, i.e. for each i = 0, 1, . . . , n, there exists a sequence of
stopping times τm →∞ almost surely such that:
EQ[
(SiT )τm
(SiT )τm|Ft]
=(Sit)
τm
(Sit)τm, t ∈ [0, T ], i = 0, 1, . . . , n (B.2)
for all m. (The notation P τmt denotes the stopped process Pmin(t,τm).)
Consider the following expression:
EP[M τm
T
M τmt
(SiT )τm
(S0T )τm|Ft]
=1
M τmt
EP[M τm
T
(SiT )τm
(S0T )τm|Ft]
58
(Split the expression into τm > t and τm ≤ t)
= 1τm>t1
M τmt
EP[M τm
T
(SiT )τm
(S0T )τm|Ft]
+ 1τm≤t1
M τmt
EP[M τm
T
(SiT )τm
(S0T )τm|Ft]
(On the first term, since MT is a martingale, we can write it as an expection of MT condi-tioned on time t)
= 1τm>t1
M τmt
EP[EP [MT |Fmin(τm,T )
] (SiT )τm
(S0T )τm|Ft]
+ 1τm≤t1
M τmt
EP[M τm
T
(SiT )τm
(S0T )τm|Ft]
= 1τm>t1
M τmt
EP[EP[MT
(SiT )τm
(S0T )τm|Fmin(τm,T )
]|Ft]
+ 1τm≤t1
M τmt
EP[M τm
T
(SiT )τm
(S0T )τm|Ft]
(From Tower property of conditional expectation, the first term is expressed as a singleexpectation.)
= 1τm>t1
M τmt
EP[MT
(SiT )τm
(S0T )τm|Ft]
+ 1τm≤t1
M τmt
EP[M τm
T
(SiT )τm
(S0T )τm|Ft]
(Expectation changed to Q using Generalized Bayes’ Theorem [12])
= 1τm>tEQ[
(SiT )τm
(S0T )τm|Ft]
+ 1τm≤tEQ[
(SiT )τm
(S0T )τm|Ft]
(Combined the two terms into one)
= EQ[
(SiT )τm
(S0T )τm|Ft]
(Using (B.2))
=(SiT )τm
(S0T )τm
. (B.3)
Hence
EP[
(SiT )τm
LτmT|Ft]
= EP[M τm
t
M τmt
(SiT )τm
(S0T )τm/M τm
T
|Ft]
= M τmt EP
[M τm
T
M τmt
(SiT )τm
(S0T )τm|Ft]
(Using (B.3))
= M τmt
(SiT )τm
(S0T )τm
=(SiT )τm
Lτmt.
Second, we show that Lt is a self-financing portfolio.
Since the Radon-Nikodym derivative Mt = dQdP
∣∣Ft
is a F-martingale, 1Mt
= dPdQ
∣∣∣Ft
is a Q-martingale.
Now consider
d
(L
S0
)t
= d
(S0/M
S0
)t
= d
(1
M
)t
.
This implies (L/S0)t is a martingale.
59
Under Q, S0t is a martingale for all i. By Martingale representation Theorem,
d
(L
S0
)t
= dLt =n∑i=1
φi,tdS0t .
From this one can apply Lemma B.0.1 to show Lt is a self-financing portfolio.
60
Appendix C
Regular Sturm Liouville theorem
In Section 4.2, the result of the regular Sturm Liouville theorem has been used to
prove the existence and the uniqueness of solution for the eigenfunction and eigen-
value problem (4.22). This step is crucial to deduce the solution of the numeraire
portfolio and recover the real-world measure P. In this appendix we state the main
result of the regular Sturm Liouville theorem and provides reference for it.
Theorem C.0.7 (Regular Sturm Liouville theorem)
A Sturm-Liouville equation is a second-order homogenous linear differential equation
of the following form:
d
dx
(p(x)
dy(x)
dx
)− q(x)y(x) = −λρ(x)y(x),
where p(x) and ρ(x) are positive functions, and p(x), p′(x), q(x), ρ(x) are con-
tinuous functions over a finite interval [a, b]. Also, the boundary conditions are of
the form:
α1y(a) + α2y′(a) = 0 α2
1 + α22 > 0, and
β1y(b) + β2y′(b) = 0 β2
1 + β22 > 0.
Then
1. The set of eigenvalues λ = λ1, λ2, . . . are real and discrete, with
λ1 < λ2 < · · · < λn <→∞.
2. Corresponding to each eigenvalue λn, there is an eigenfunction yn(x) which
has exactly n− 1 zeros in (l, u), and is uniquely determined up to a constant factor.
In particular, y1(x) has no zeros in (a, b) and can be taken to be positive.
3. All of the eigenfunctions and eigenvalues can be numeraically solved.
61
Proof. A reference for this is [2, p. 320 Theorem 5], which contains the proof and
detailed explanation of the theorem.
62
Appendix D
Markov chain
In Section 3.2, Ross’ model is reformulated as a finite state time-homogeneous
Markov chain. In this section we provide the relevant definitions and theorems
for completeness.
Definition D.0.5 (Markov chain and its transition probability)
A finite state Markov chain is a sequence of random variables X1, X2, X3, . . . with
values from finite state space S, and satisfies the Markov property, i.e.
P(Xn+1 = x|X1 = x1, X2 = x2, . . . , Xn = xn) = P(Xn+1 = x|Xn = xn),
where P denotes a probability measure.
The transition probability function of a Markov chain, p : Π× Π× N+ → [0, 1],
is defined as the probability
pi,j(n) = P(Xn = j|Xn−1 = i),
where N+ denotes the set of positive integers.
Definition D.0.6 (Time-homogeneity of a Markov chain)
A finite state Markov chain is time-homogeneous if it satisfies the following condition:
P(Xn+1 = i|Xn = j) = P(Xn = i|Xn−1 = j) for all i, j ∈ Π and n ∈ N+.
In other words, a Markov chain is time-homogeneous if its transition probability
function does not depend on time.
If a finite state Markov chain is time-homogeneous, one can define its transition
probability matrix as:
P =
p1,1 p1,2 · · · p1,M
p2,1 p2,2 · · · p2,M...
.... . .
...pM,1 pM,2 · · · pM,M
,
63
where M is the number of states in S.
Definition D.0.7 (Irreducibility of a Markov chain)
The irreducibility of a Markov chain can be defined through the following:
1. A state j is said to be accessible from a state i (written as i→ j) if a system
started in state i has a non-zero probability of transitioning into state j at
some point.
2. A state i is said to communicate with state j if both i→ j and j → i.
3. A set of states C is a communicating class if every pair of states in C commu-
nicates with each other.
4. A Markov chain is said to be irreducible if its state space is a single commu-
nicating class.
As the name suggested, the irreducibility of a Markov chain and the irreducibility
of its transition probability matrix are equivalent, which is proved in the following
theorem.
Theorem D.0.8 (Irreducibility of Markov chain and transition probability
matrix)
If a finite state Markov chain is time-homogeneous, then the irreducibility of the
Markov chain is equivalent to the irreducibility of its transition probability matrix
(as defined in Definition 2.1.5).
Proof. It is easier to prove the opposite: the reducibility of the Markov chain is
equivalent to the reducibility of its transition probability matrix.
⇒ )
If the Markov chain is reducible, then there exists two sets of communicating
classes C1 and C2 such that states i1, i2, . . . , iu in C1 does not communicate with
states j1, j2, . . . , jv in C2.
Consider the transition probability matrix (A)i,j = ai,j of the Markov chain, we
have:
aiα,jβ = 0
for α = 1, 2, . . . , u and β = 1, 2, . . . , v.
Therefore the transition probability matrix is reducible.
⇐ )
64
If the transition probability matrix (A)i,j = ai,j is reducible, then the indices
1, 2, . . . , n can be divided into two disjoint non-empty sets i1, i2, . . . , iu and j1, j2, . . . , jv,
with u+ v = n, such that
aiα,jβ = 0
for α = 1, 2, . . . , u and β = 1, 2, . . . , v.
Obviously the states in C1 = i1, i2, . . . , iu and C2 = j1, j2, . . . , jv are not commu-
nicating, since the probability of transition between the two sets is zero. Therefore
the Markov chain is reducible.
65