Ross Recovery theorem and its extension

Transcription

1 Ross Recovery theorem and its extension Ho Man Tsui Kellogg College University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance April 22, 2013

2 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 Mijatović from Imperial College, Dr. Samuel Cohen and Pedro Vitória from Oxford University for their helpful advices and comments 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.

3 Abstract Stephen Ross recently suggested the Recovery Theorem, which provides a way to reconstruct the real-world probability measure given the risk-neutral measure, under a discrete time and state space context. Peter Carr and Jiming Yu modified Ross s model and deduced a similar result, under a univariate continuous diffusion context. This dissertation 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.

4 Contents 1 Introduction Notations and market set-up Indexing of the assumptions Ross s model - basic framework Assumptions and definitions Derivation Ross s model - analysis and discussion The utility function in Ross s model Existence of stationary distribution Interest rate process in Ross s model Carr and Yu s model - basic framework Assumptions and definitions Derivation Carr and Yu s model - analysis and discussion Boundary conditions of the numeraire portfolio Market completeness in Carr and Yu s model Existence of stationary distribution Recovery theorem on unbounded domain Black-Scholes model CIR process Conclusion and further research 47 Bibliography 51 Appendices 53 A Perron Frobenius theorem 54 i

5 B Numeraire portfolio 56 C Regular Sturm Liouville theorem 61 D Markov chain 63 ii

6 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 probability distribution of the market prices of all the securities at the given future investment 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]. Intuitively, 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

7 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 establishing a connection between the real-world probability measure and the riskneutral 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 underlying 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 probability 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

8 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 setup 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, (F t ), P), where F = σ( t F t). In addition, for some fixed n N, there exists a vector of n + 1 securities S t = (S 0 t, S 2 t, S 3 t,..., S n t ) R n+1 0, where 1. S 0 t is the risk-free security, 2. S i t 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

9 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 presentation 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 assumptions 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 transition probability function (R4), (e) The state prices are known (R5), 4

10 (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 reformulated 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 = t 1, t 2,... is a discrete time with a fixed time-step. 2. The state space as Π is finite, with the state variable X t Π and Π = M N. Additionally, there exists a stochastic interest rate r t R 0, such that St 0 satisfies: St+1 0 = St 0 (1 + r t ) for all t N (2.1) with S 0 0 = 1. (2.2) The first assumption of the model is stated as follow: R3: No arbitrage condition applies in the economy. Definition (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 1 This definition is referenced to [21]. 5

11 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 timehomogeneous 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 p i,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 1,1 p 1,2 p 1,M p 2,1 p 2,2 p 2,M P = (2.3) p M,1 p M,2 p M,M 1. From this one can also define the risk-neutral probability matrix Q, where Q = p 1,1 / M j=1 p 1,j p 1,2 / M j=1 p 1,j p 1,M / M j=1 p 1,j j=1 p 2,j p 2,M / M j=1 p 2,j p 2,1 / M j=1 p 2,j p 2,2 / M p M,1 / M j=1 p M,j p M,2 / M j=1 p M,j p M,M / M j=1 p M,j. (2.4) Notice that Q is a probability transition matrix, as the sum of each of its row is We now consider the transition density function of the model, which is the probability 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 f i,j, where i represents the initial state and j represents the next state, as well as the transition probability matrix of 2 If the set of state-prices is unique or the contracts are all tradable, then the market is complete. 6

12 the model as F, where f 1,1 f 1,2 f 1,M f 2,1 f 2,2 f 2,M F = (2.5) f M,1 f M,2 f M,M 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 (Pricing kernel in Ross s model) The pricing kernel from state i to state j is defined as the following quotient: ϕ i,j = p i,j f i,j. The pricing kernel matrix is defined as ϕ 1,1 ϕ 1,2 ϕ 1,M ϕ 2,1 ϕ 2,2 ϕ 2,M Φ = (2.6) ϕ M,1 ϕ M,2 ϕ M,M Definition (Transition independent pricing kernel) A pricing kernel is transition independent if it can be written in the following form: where ϕ i,j = p i,j f i,j = δ h(i) h(j), (2.7) 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

13 Remark 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 intertemporal 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 (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. a i,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 (Irreducible matrix) A square n n matrix (A) i,j = a i,j is said to be reducible if the indices 1, 2,..., n can be divided into two disjoint non-empty sets i 1, i 2,..., i u and j 1, j 2,..., j v, with u + v = n, such that a iα,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 As mentioned by Ross [19], since the risk-neutral measure is equivalent to the realworld 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 p i,j of P is zero if and only if f i,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

14 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 equivalent 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: h(1) h(2) 0 D = h(m) D 1, the inverse of the matrix D, is: h(1) 1 D h(2) = h(m) The transition independent pricing kernel equation (2.7) can now be written in matrix form, P = δd 1 F D. Reorganize the equation, F = 1 δ DP D 1. (2.8) Lemma (Identity for transition probability) Let e be a M 1 column vector of 1, i.e. e = ( ) T. The following identity holds: F e = e. (2.9) Proof. The intuition for this lemma is simple: the rows on F are essentially probability to different states given the current state, the sums if each row must be equal to one. 9

15 As defined in (2.5), (F ) i,j = f i,j is the probability of transiting from the initial state i to j. Summing up all the transition probabilities one will get: M f i,j = 1 for all i = 1, 2,..., M. j=1 To write this equality in matrix form, we have F e = e. Multiply both sides of (2.8) by e and reorganize, we have P D 1 e = δd 1 e. Let Then x = D 1 e. (2.10) P x = δx. (2.11) Hence the equation has been transformed to an eigenvalue and eigenvector problem. Before we proceed, notice that: 1. The entries of x are actually the diagonal elements of the matrix D 1, which 1 is. Therefore they must be positive. h(i) 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 satisfies these two conditions. Theorem (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

16 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 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: p i,j = ϕ i,j f i,j = δ h(i) h(j) f i,j. Given only the knowledge of p i,j, the recovery theorem uses the transition independence to separately determine ϕ i,j and f i,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. 3 This argument will be elaborated in detail in Section

17 Chapter 3 Ross s model - analysis and discussion 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

18 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 additive 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 transition 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 ]. To begin with, the definition of the inter-temporally additive separable utility function is introduced. Definition (Inter-temporally additive separable utility function) An inter-temporally additive separable utility function U : C R in discrete twoperiod economy is where U(c) = u(c 0 ) + δe[u(c 1 )], (3.1) C is the space of all feasible consumption processes, c = (c 0, c 1 ) 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: sup u(c 0,i ) + δ c C M u(c 1,j )f i,j s.t. c 0,i + j=1 M c 1,j p i,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. j=1 13

19 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 intertemporally 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 consumption takes place. The consumption at state j Π is denoted as c j. We will first see how the transition independent kernel is derived and then explain 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: M sup u(c 0,i ) + δ c C u(c 1,j )f i,j s.t. c 0,i + j=1 M c 1,j p i,j = w. j=1 Define the Lagrangian L as: ( M L u(c 0,i ) + δ u(c 1,j )f i,j + λ w c 0,i j=1 The first order condition for the optimal solution are: ) M c 1,j p i,j. j=1 u (c 0,i) λ = 0 for all i Π and δu (c 1,j)f i,j λp i,j = 0 for all i, j Π. Solving these equations give formula for the pricing kernel in terms of the optimal consumption process: p i,j = δ u (c 1,j) f i,j u (c 0,i ) for all i, j Π. (3.3) 1 In Ross s paper [19], Ross didn t derive the transition independent kernel in detail, but rather explained it descriptively. The derivation here largely follows Carr and Yu s paper [4], in which they tried to provide the missing derivation steps in Ross s model. 14

20 Using assumption R6 (part ii), the optimal consumption process is timehomogeneous. Hence we can drop the time index of c in the expression: p i,j = δ u (c j) f i,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 consumption 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, because 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 indifferent 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 consumption 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 problem 2. This idea can be illustrated by the following simple example. Suppose, for simplicity, there are only two states, α and β, with the following transition probabilities and state prices: f α,α = 0.6, p α,α = 0.5, f α,β = 0.4, p α,β = 0.4, f β,α = 0.7, p β,α = 0.6, f β,β = 0.3, p β,β = This inconsistency was suggested by Carr and Yu on [4, p. 13]. 15

21 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(c α ) + 0.9[0.6 log(c α ) log(c β )] s.t. c α + 0.5c α + 0.4c β = 10. (3.5) f = 0.6, p = 0.5 α α β f = 0.4, p = 0.4 β Figure 3.1: Optimization problem when the current state is α The optimal solution to problem (3.5) is: (c α) 1 = and (c β) 1 = On the other hand, if the current state is β, the optimization problem is: sup u(c β ) + 0.9[0.7 log(c α ) + 0.3u(c β )] s.t. c β + 0.6c α + 0.2c β = 10. (3.6) α f = 0.7, p = 0.6 α β f = 0.3, p = 0.2 β Figure 3.2: Optimization problem when the current state is β The optimal solution to (3.6) is: (c α) 2 = and (c β) 2 =

22 Obviously (c α) 1 (c α) 2 and (c β ) 1 (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 X n 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. Informally, 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 (Stationary distribution of a Markov chain) Let {X n } be a time-homogeneous Markov chain having state-space Π and the transition probability matrix P. If π is a probability distribution such that: P π = π, 17

23 then π is called the stationary distribution of {X n }. 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 (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 = min i j f i,j r max i f i,j = 1, So 1 is a eigenvalue of F, and the corresponding Perron vector, v, satisfies: F v = v. Hence v is the stationary distribution of the Markov chain in Ross s model. j 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

24 3. If the interest rate process is a constant, then the real-world probability measure 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 (Time-homogeneity of interest rate process) In Ross s model, the interest rate for each period, r t, only depends on the current state, but is independent of time, i.e. it is a time-homogeneous process. Moreover, r t = 1 M j=1 p i,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 p i,j, j Π, one will be guaranteed a pay-off of one no matter what state is realized. Hence in a market without arbitrage (R3), the sum M j=1 p i,j should be equal to the one-period discounted value of one, i.e. M p i,j = 1 (3.7) 1 + r t Rearrange: j=1 r t = 1 M j=1 p i,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 discount factor, δ, is bounded above by the largest interest rate factor and bounded below by the smallest interest rate factor. Theorem (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 max i Π r i δ min i Π r i 19

25 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. min i From (3.7) we know that M j=1 p i,j δ max i M j=1 p i,j = r i. Substitute this into the equation above we have max i Π r i δ M p i,j. j= min i Π r i. Lastly, the following is proved: Theorem (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 p i,j = r = k for all i Π, where k is the constant interest discount factor. This can be rewritten in matrix form, P e = 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 k P. 20

26 From (2.4), the definition of the risk-neutral probability matrix Q is Q = 1 M j=1 p i,j P = (1 + r)p = 1 k P = 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 result as Ross did on a bounded continuous state space, based on a different set of assumptions. 21

27 Chapter 4 Carr and Yu s model - basic framework 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 dp dq. 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

28 (b) No free lunch with vanishing risk applies in the market, and the numeraire portfolio L t is equal to S0 t M t, where M t = dq (C3), dp (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 diffusion 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 (Theorem 4.1.1). 5. The problem of solving the diffusion coefficient can be transformed to a problem of eigenvalue and eigenfunction (4.21). 6. The regular Sturm Liouville theorem (Appendix C) is then applied (Theorem 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 dp dq (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]. C3). All the assumptions in this chapter model start with letter C (for example, 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. Ft 23

29 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 r t R 0, such that S 0 t satisfies: S 0 t = e t 0 rsds t [0, T ], (4.1) or equivalently, with ds 0 t = r t S 0 t dt t [0, T ], (4.2) S 0 0 = 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 (S i /S 0 ) t is a local martingale for all i = 0, 1,..., n under Q. Furthermore we can define the Radon-Nikodym derivative, M t, as M t = dq dp. (4.4) Ft 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 (The numeraire portfolio) A numeraire portfolio L t is a strictly positive self-financing portfolio, such that (S i /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, L t, is equal to S0 t M t. 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: 1 This assumption will be discussed in detail in Section

30 Theorem (Dynamic of the numeraire portfolio under P) L t has a dynamics of the following form under P: dl t L t = (r t + σ 2 t )dt + σ t dw P t, (4.5) where W P t is a Brownian motion under P, and σ t is an adapted process. Proof. (S i /L) t is a local martingale for all i = 0, 1,..., n. In particular, by Martingale Representation Theorem, for some adapted process σ t. d(s 0 /L) t (S 0 /L) t = σ t dw P t (4.6) Let A be an Itô process. From Itô s formula we have ( ) 1 d = 1 A A da A 3 (da)2. Set A = (S 0 /L) t, ( ) ( ) L d = L2 t S 0 S 0 t (St 0 ) d 2 L t ( ( + L3 t S 0 d (St 0 ) 3 L Divide both sides by L t /St 0 and substitute (4.6) implies: ( ) ( ) L L d = σt 2 dt + σ t dwt P. S 0 Again by Itô s formula, ( ) L d = 1 dl S 0 t St 0 t L t (St 0 ) 2 ds0 t. Substitute the dynamics for St 0 as in (4.2) ( ) L d = 1 dl S 0 t St 0 t L tr t dt St 0 ( ) ( ) L L d / = dl t r S 0 t S 0 t dt t L t / t S 0 σ 2 t dt + σ t dw P t dl t L t t = dl t L t r t dt = (r t + σ 2 t )dt + σ t dw P t. ) ) 2. t 25

31 An important consequence of Theorem is the following relationship: Corollary (The market price of risk) The market price of risk, Θ t, (also known as the risk premium ) of the model is equal to the diffusion coefficient of the numeraire portfolio, i.e. Θ t = σ t. Proof. The market price of risk is defined as Θ t = α t r t σ t, where α t is the drift coefficient of a tradable asset, σ t is the drift coefficient of a tradable asset, and r t is the interest rate process. From Theorem the market price of risk can be deduced as: Θ t = r t + σ 2 t r t σ t. Hence Θ t = σ t. Definition (Time-homogeneous Itô diffusion 2 ) An one-dimensional bounded time-homogeneous Itô diffusion X t is an adapted stochastic process satisfying a stochastic differential equation of the form: dx t = b(x t )dt + a(x t )dw t, where W t is an one-dimensional Brownian motion, and also b : R R, a : R R satisfy 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 (Infinitesimal generator of an Itô diffusion) The infinitesimal generator G of an one-dimensional Itô diffusion X t is defined by Gf(x) = lim t 0 E [f(x t ) X 0 = x] f(x) t for x R. 2 This definition is referenced to [17, p. 116] 26

32 The infinitesimal generator can be shown to be equivalent to 3 : G = t + a2 (x) 2 2 x + b(x) 2 x. The driver of the model is now introduced, from which all security prices are derived. C4: There exists an univariate bounded time-homogeneous Itô diffusion X t [u, l] such that S i t = S i (X t, t) for i = 0, 1,..., n. X t is called the driver in the model. By definition the driver X t satisfies the following stochastic differential equation: where W Q is a Brownian motion under Q. dx t = b(x t )dt + a(x t )dw Q t, (4.7) (4.1) can be written as; r t = t log S0 t. From C4, St 0 = S 0 (X t, t), so r t = r t (X t, t). In other words, the interest rate process r t is also driven by the driver X t. 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) 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, L t, depends only on the current value of the driver X t and time t, i.e. L t L(X t, t) (4.8) 3 A reference for this can be found on [17, p. 123]. 4 The condition a(x) 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

33 C6 (part ii): The interest rate process, r t, is time-homogeneous, i.e. r(x, t) r(x) (4.9) C6 (part iii): The diffusion coefficient of L t, σ t, 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 r t. As L t is a self-financing portfolio, it also has the drift coefficient of r t. Therefore L t can be written as: dl t L t = r(x t )dt + σ(x t )dw Q t. (4.11) In other words, the numeraire portfolio L t 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. below. C7 (part i): a(x), b(x), and r(x) are continuous functions. r(x) is also bounded Together with a(x) 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): S i (x, t) for i = 0, 1,..., n are twice continuously differentiable C7 (part iii): L t = L(X t, t) is a twice differentiable function. They will be applied in different parts of the derivation. Theorem (PDE for asset prices) Assets in the market satisfy the following equation: GS i (x, t) = r(x)s i (x, t) for i = 0, 1,..., n. (4.12) Proof. Under the risk-neutral probability Q, the following identity holds: [ ] St i S = E Q i T F St 0 ST 0 t. 28

34 Rearrange this identity and substitute (4.1), we have: ( T ) ] S i (x, t) = E [exp Q r(x s )ds S i (X T, T ) F t. t Since the price functions S i are twice continuously differentiable (C7 (part ii)) and r is continuous and bounded below (C7 (part i)), we can apply Feynman-Kac formula 5, S i (x, t) t + a2 (x) 2 S i (x, t) + b(x) S i(x, t) r(x)s 2 x 2 i (x, t) = 0. x Rewrite it using the infinitesimal generator, GS i (x, t) = r(x)s i (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 L(x, t) at x = l and x = u are x 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 σ t, which is equivalent to the market price of risk Θ t by Corollary The Girsanov theorem is then applied to recover the dynamics of the process X t under P-measure, together with the dynamics of all the security prices St. i As mentioned, an important feature of Carr and Yu s model is the assumption on the form and dynamics of the numeraire portfolio (C6). 5 A reference for this formula is on [20, p. 145]. 6 Although not included in the paper, this condition was suggested by Peter Carr in an to Johannes Ruf. This assumption is needed in applying the regular Sturm Liouville theorem, as we will see in the next section. This assumption will also be discussed in detail in Section

35 Substitute (4.9) and (4.10) into (4.5), one can derive an expression for dynamics of L t under P: dl t = [r(x t ) + σ 2 (X t )]dt + σ(x t )dwt P. (4.13) L t An expression for the dynamics of X t under P is also needed: Theorem (Dynamics of the driver under P) The dynamics of the driver, X t, under P is given by dx t = [b(x t ) + σ(x t )a(x t )]dt + a(x t )dw P t, (4.14) which is a time-homogeneous diffusion process. Proof. With reference to (4.7), the change of measure only changes the drift coefficient by the Girsanov s Theorem. Therefore the diffusion term of X t under P is still a(x t ). From Corollary 4.1.1, the market price of risk is σ(x t ). By the Girsanov s theorem, the drift coefficient of X t under P is b(x t ) + σ(x t )a(x t ), i.e. dx t = [b(x t ) + σ(x t )a(x t )]dt + a(x t )dw P t. Theorem (Diffusion coefficient of the numeraire portfolio) The diffusion coefficient function, σ(x), satisfies the following equation: σ(x) = a(x) log L(x, t). (4.15) x Proof. As L t is twice differentiable (C7 (part iii)), Itô s formula can be applied to (4.8), dl t = L t t dt + L t x dx t dl t = 1 L t L t L t t dt + 1 L t L t 2 L t x 2 (dx t) 2 x dx t L t 2 L t x (dx t) 2. 2 Substitute (4.14) and compare only the diffusion terms of dl L with equation (4.13): σ(x) = 1 L t L t x a(x). Rearrange the equation, σ(x) = a(x) log L(x, t). x 30

36 The equality in (4.15) will be reorganized such that it can fit into the regular Sturm Liouville theorem (Appendix C). Since a(x) 0 (by C5), for some function f(t), log L(x, t) = x l σ(y) dy + f(t). (4.16) a(y) Note that σ(x) and a(x) are continuous functions (C7 (part i)), so σ(x) a(x) integrable. Take exp( ) on the both sides and let: x σ(y) π(x) = e l a(y) dy p(t) = e f(t). Thus, a separable expression of L(x, t) is obtained: Substitute this into the generator equation (4.12): L(x, t) = π(x)p(t). (4.17) π(x)p (t) + a2 (x) 2 π (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) π (x) 2 π(x) + (x) b(x)π π(x) = r(x) a 2 (x) π (x) 2 π(x) + (x) b(x)π π(x) r(x) = (t) p p(t). The two sides can only be equal if they are each equal to a constant λ R: p (t) p(t) The solution to (4.18) is the following: and λ is still an unknown. = λ, (4.18) a 2 (x) π (x) 2 π(x) + (x) b(x)π r(x) = λ. (4.19) π(x) p(t) = p(0)e λt (4.20) is Reorganize the problem for (4.19): a 2 (x) 2 π (x) + b(x)π (x) r(x)π(x) = λπ(x), (4.21) which is an eigenvalue and eigenfunction problem. 31