Published on: **Mar 4, 2016**

- 1. Technical report, IDE1141, August 25, 2011 Master’s Thesis in Financial Mathematics Magdalena Antczak Marta Leniec School of Information Science, Computer and Electrical Engineering Halmstad University Pricing and Hedging of Defaultable Models
- 2. Pricing and Hedging of Defaultable Models Magdalena Antczak Marta Leniec Halmstad University Project Report IDE1141 Master’s thesis in Financial Mathematics, 15 ECTS credits Supervisor: Prof. Lioudmila Vostrikova-Jacod Examiner: Prof. Ljudmila A. Bordag External referees: Prof. Mikhail Babich August 25, 2011 Department of Mathematics, Physics and Electrical engineering School of Information Science, Computer and Electrical Engineering Halmstad University
- 3. Preface This thesis has been prepared at the University of Angers under the supervi- sion of Professor Lioudmila Vostrikova-Jacod. We would like to thank her for help in understanding the defaultable framework and useful remarks. The conversations at the Faculty and seminars were priceless. We also want to express our sincere gratitude to Professor Ljudmila Bordag for organizing our Erasmus in France.
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- 5. Abstract Modelling defaultable contingent claims has attracted a lot of interest in recent years, motivated in particular by the Late-2000s Financial Crisis. In several papers various approaches on the subject have been made. This thesis tries to summarize these results and derive explicit formu- las for the prices of ﬁnancial derivatives with credit risk. It is divided into two main parts. The ﬁrst one is devoted to the well-known theory of modelling the default risk while the second one presents the results concerning pricing of the defaultable models that we obtained ourselves. iii
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- 7. Contents 1 Introduction 1 2 Stochastic background 7 2.1 The probability space and ﬁltrations . . . . . . . . . . . . . . 7 2.2 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 The Brownian ﬁltration . . . . . . . . . . . . . . . . . . . . . 11 2.4 Stopping times . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 The Martingale Theory . . . . . . . . . . . . . . . . . . . . . . 14 3 The default setting 15 3.1 Basic assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The default process . . . . . . . . . . . . . . . . . . . . . . . . 16 4 The intensity-based approach in ﬁltration H 19 4.1 The H-intensity of τ . . . . . . . . . . . . . . . . . . . . . . . 19 4.1.1 The intensity of default . . . . . . . . . . . . . . . . . . 19 4.1.2 The hazard function Γ . . . . . . . . . . . . . . . . . . 26 5 The Carthaginian enlargement of ﬁltrations 33 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 General projection tools . . . . . . . . . . . . . . . . . . . . . 34 5.3 Measurability properties in enlarged ﬁltrations . . . . . . . . . 35 5.4 The E-hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.5 The change of measure on Gτ . . . . . . . . . . . . . . . . . . 38 5.5.1 The survival process under measure P and P∗ . . . . . 42 6 The initial enlargement framework 43 6.1 Expectation tools . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.2 The martingales characterization . . . . . . . . . . . . . . . . 45 6.3 The E-hypothesis and the absence of arbitrage in the ﬁltration Gτ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 v
- 8. 7 The progressive enlargement framework 47 7.1 The intensity approach . . . . . . . . . . . . . . . . . . . . . . 47 7.1.1 Expectation tools . . . . . . . . . . . . . . . . . . . . . 47 7.1.2 The F-hazard process (Γt)t≥0 . . . . . . . . . . . . . . 50 7.1.3 The G-intensity of τ . . . . . . . . . . . . . . . . . . . 52 7.1.4 H-hypothesis and the absence of arbitrage in the ﬁl- tration G . . . . . . . . . . . . . . . . . . . . . . . . . 53 7.1.5 The value of information . . . . . . . . . . . . . . . . . 54 7.2 The density approach . . . . . . . . . . . . . . . . . . . . . . . 55 7.2.1 Projection tools . . . . . . . . . . . . . . . . . . . . . . 55 7.2.2 The H-hypothesis and special property of the condi- tional density process . . . . . . . . . . . . . . . . . . . 56 7.2.3 The martingales characterization . . . . . . . . . . . . 57 8 Pricing and hedging of Black-Scholes type models with de- fault 59 8.1 The model evaluation and the description of the task . . . . . 59 8.2 Methods of pricing in arbitrage-free and incomplete market . . 63 8.2.1 The arbitrage-free market . . . . . . . . . . . . . . . . 63 8.2.2 The incomplete market . . . . . . . . . . . . . . . . . . 63 8.2.3 The f-divergence minimization approach . . . . . . . . 63 8.2.4 The utility approach . . . . . . . . . . . . . . . . . . . 64 8.3 Martingale measures on Gτ . . . . . . . . . . . . . . . . . . . 67 8.4 The distribution of τ with respect to P . . . . . . . . . . . . . 70 8.5 European call option pricing . . . . . . . . . . . . . . . . . . . 74 8.5.1 Pricing in the Black-Scholes market with default . . . . 75 8.5.2 The case when W(1) and W(2) are uncorrelated . . . . 76 8.5.3 The case when W(1) and W(2) are correlated with the correlation coeﬃcient ρ . . . . . . . . . . . . . . . . . . 79 9 Conclusions 91 Notation 93 Notation 94 Bibliography 95 Appendix 96 vi
- 9. Chapter 1 Introduction In the world of ﬁnance, it is crucial to consider the models based on the fact that the companies may default. Hearing the word ’default’ one can imagine the biggest defaults in the history of economy like that of Lehman Brothers in 2008. However, the exact deﬁnition of a default explains it only as a failure to meet debt obligations such as loans or bonds. The debtor is in default when he is either unable or unwilling to pay the debt. One has to distinguish the default from a state of being unable to pay the debts precisely which is called insolvency. The company is insolvent when it is unable to pay debts as they fall due (cash ﬂow insolvency) or when the liabilities exceed the assets (balance sheet insolvency). It is worth mentioning that the insolvency can lead to a bankruptcy which is the process of legally deﬁning a ﬁnancial situation as insolvent. While modelling credit risk, one usually takes under consideration the company’s default in general, without looking into the causes and hence distinguishing between being unable or unwilling to pay the debts. In the world of mathematics, the default appears as default time which is a strictly positive random variable. One can deﬁne this random variable in many ways. However, the most common one is the ﬁrst time of crossing a barrier by a certain process, e.g. a stock price process of a company (see a Figure 9.1). Modelling of the default event can be done in two manners. The ﬁrst one is called structural approach. It assumes that default time τ is a stopping time in the assets ﬁltration F. The second one, called reduced-form approach, is based on the assumption that τ is a stopping time in a larger ﬁltration and may no longer be measurable with respect to the prices ﬁltration. In our thesis, we focus on the last approach. 1
- 10. 2 Chapter 1. Introduction Figure 1.1: An example of a defaultable company stock price process. We consider a non-defaultable world which consists of riskless and risky assets. A ﬁltration generated by the prices of those assets is denoted by F and called the reference ﬁltration. It represents the information available to the regular investor in a non-defaultable world. However, when we take under consideration a possibility of a default we have to introduce default time τ and create a defaultable framework which may consists of default- free and defaultable assets, e.g. stock of the company that may default. We have to study diﬀerent types of information ﬂows available to agents trading in a defaultable market. On the one hand, the regular investors add the information about default to F when it occurs, i.e. they work in a progressive enlargement setting. On the other hand, we shall examine also the insider, i.e. the agent who possesses information about default time from the beginning. The information accessible to this agent is represented by a ﬁltration F initially enlarged by a positive random variable τ. In our thesis, we explore the special theory which establishes methods of enlarging the reference ﬁltration by the additional information, namely Carthaginian Enlargement of Filtrations (see [2]). We distinguish two methods of modelling default time in a reduced-form approach, namely the intensity (see [1]) and the conditional density-based
- 11. Pricing and Hedging of Defaultable Models 3 approach (see [2] and [3]). They are used to establish the expectation and projection tools which are necessary for pricing an hedging of ﬁnancial deriva- tives. An intensity of default is simply a ratio of probability that default will appear in a inﬁnitely small time interval (under the condition that there was no default before) and the time step. However, to determine the condi- tional density of default, we need to assume that the conditional law of τ is equivalent to the law of τ. In the ﬁrst chapter, we study some basic results concerning probability spaces and ﬁltrations, as well as stochastic processes, in particular a Brownian motion. We introduce some facts concerning stopping times and martingales. In the second chapter, we introduce crucial assumptions related to the ﬁltered probability space involving default time and all the price processes. Then, we introduce the law of τ and we give a deﬁnition of a default process. We determine the form of a random variable measurable with respect to the σ-algebra generated by that process and give some properties of the corresponding ﬁltration. Third chapter is devoted to the intensity approach in the ﬁltration gen- erated by the default process. In this framework, we give tools to compute expectations with respect to the σ-algebra generated by this process. Then, we value under the physical measure defaultable zero-coupon bond at time t in the case of zero and non-zero spot rate for the agent whose information ﬂow is the ﬁltration mentioned above. Finally, we give formulas and prop- erties of the survival and hazard function and we represent once again the defaultable zero-coupon bond value using these functions. In the fourth chapter, we present ﬁrstly the theory of Carthaginian En- largement of Filtrations and hence, the methods to enlarge reference ﬁltra- tion with an additional information. Secondly, we represent random variables with respect to the corresponding σ-algebras. Then, we introduce the crucial assumption that states that the conditional law of default time τ is equiv- alent to the law of τ. In addition, we present the density hypothesis which allows to express the distribution of τ conditioned on the information from the reference ﬁltration in terms of the conditional density process and the law of τ. We show that under the additional assumption concerning the law of τ, namely the property of being non-atomic, default time avoids stopping times from the reference ﬁltration. The second important part of this chap- ter is devoted to introducing the so-called decoupling measure which makes
- 12. 4 Chapter 1. Introduction τ and the underlying risky assets independent. We consider some proper- ties of the new measure and establish the expectation tools using obtained independence. What is more, we establish the form of the survival process under the physical and decoupling measure. Finally, we prove that initially enlarged ﬁltration inherits right-continuity from the reference ﬁltration. Fifth chapter presents some results obtained in the initially enlarged ﬁl- tration, i.e. the expectation tools and the characterization of martingales from the enlarged ﬁltration in terms of martingales from the reference ﬁltra- tion. We ﬁnish the chapter with establishing the conditions for the absence of arbitrage in the enlarged ﬁltration. In the sixth chapter we examine the progressive enlargement framework. We begin with the intensity-based approach and assume that a price process follows the log-normal distribution and the reference ﬁltration is generated by a standard Brownian motion. Firstly, we establish some expectation tools. Secondly, we introduce a hazard process in terms of the results obtained from the expectation tools. Then, we introduce the intensity in the progressively enlarged ﬁltration. We continue the chapter by studying the hypothesis that martingales from the reference ﬁltration remain martingales in the enlarged ﬁltration, namely H-hypothesis which is strongly related to the absence of arbitrage. We ﬁnish the intensity-based approach part with demonstrating the value of the default information, i.e. the diﬀerence between the price of a defaultable contingent claim for an agent who possesses the information about the default when it occurs and the one who does not have this in- formation. In the second part of this chapter, we analyse the density-based approach. We begin with establishing the projection of random variables on the progressively enlarged ﬁltration and we obtain the Radon-Nikodým on this ﬁltration. We continue with examining the relation between the density hypothesis and the H-hypothesis and ﬁnish with the martingales character- ization. The seventh chapter consists of our own results. We calculate the price of the option written on a investment consisting of both, default-free and defaultable assets. We consider a default-free market consisting of one risk- less asset and one risky asset and a defaultable market created by adding one defaultable asset to the preceding model. We deﬁne a reference ﬁltra- tion as a ﬁltration generated by a price process of a default-free asset. We deﬁne default time τ as the ﬁrst time when defaultable asset’s price crosses a certain barrier from interval (0, 1) and we establish distribution of τ. We
- 13. Pricing and Hedging of Defaultable Models 5 consider two agents trading in a defaultable market, a regular investor who observes only a price process of a default-free asset and a special agent who has additional information concerning default time τ from the beginning, i.e. its distribution. We put an accent on the fact that the defaultable market is arbitrage-free and incomplete for the regular investor and hence, we ﬁnd it interesting to calculate the price of the option for such an investor. We ﬁnd a pricing measure using the connection between two well-known methods, the utility maximization and the f-divergence minimization.
- 14. 6 Chapter 1. Introduction
- 15. Chapter 2 Stochastic background In the Theory of Financial Markets pricing is based either on the stochastic or partial diﬀerential equations approach. We will focus on the former one. It is important to remind the most important deﬁnitions from the Theory of Stochastic Processes which will be used throughout our thesis. 2.1 The probability space and ﬁltrations While considering the randomness, it is necessary to introduce a proba- bility space (Ω, F, P) which is a mathematical form essential for modelling the stock prices and default processes consisting of the states which occur with uncertainty. A non-empty sample space Ω is an universe of all possi- ble random events ω. In our case it is a space of all possible scenarios that can happen on the ﬁnancial market. For further calculations and reasoning it is crucial to use a certain type of collections of these events ω ∈ Ω. Let us denote P(Ω) the set of all subsets of Ω. From the Theory of Probabil- ity we know how to treat the collections which are closed under countable unions and joints. Consequently, we introduce the most important algebraic structure, σ-algebra over Ω, as following. Deﬁnition 2.1. Let Ω be a non-empty sample space. F ⊂ P(Ω) is called a σ-algebra over Ω, if i) ∅ ∈ F, ii) F ∈ F ⇒ FC ∈ F, iii) ∀i ∈ I, Fi ∈ F ⇒ i∈I Fi ∈ F, where I ⊂ N. N is a set of natural numbers. 7
- 16. 8 Chapter 2. Stochastic background From the De Morgan’s laws we can easily combine ii) and iii) from the previous deﬁnition and get that the countable joints remain in the σ-algebra. Remark 2.1. If F is a σ-algebra over Ω, then i) Ω ∈ F, ii) ∀i ∈ I, Fi ∈ F ⇒ i∈I Fi ∈ F. Through equipping the sample space with the σ-algebra F we get a pair (Ω, F) called a measurable space. On such a space we can deﬁne a probability measure and obtain the probability space. Deﬁnition 2.2. Let Ω be a non-empty sample space and F a σ-algebra over Ω. The pair (Ω, F) is called a measurable space. In the Mathematical Finance, for pricing ﬁnancial derivatives, one can use several probability measures calculated from the actual market movements. For instance, a martingale measure is based on the risk-neutrality approach. Accordingly, in pursuance of the previous notations and assumptions we can deﬁne a probability measure P on measurable space (Ω, F) deﬁned on the set of events from Ω. Deﬁnition 2.3. We call a function P : F → [0, 1] a probability measure on (Ω, F) if i) P(∅) = 0, ii) P(Ω) = 1, iii) ∀i ∈ I Fi ∈ F are disjoint, i.e. Fi ∩ Fj = ∅ if i = j then P( i∈I Fi) = i∈I P(Fi), where I ⊂ N. Broadly speaking, a probability space is a measurable space such that the measure of the whole space is equal to one. In accordance with the previous suppositions we can deﬁne it more formally. Deﬁnition 2.4. We call a triplet (Ω, F, P) a probability space where Ω = ∅, F is a σ-algebra over Ω and P is a probability measure on (Ω, F). In mathematics there are some sets which can be ignored. In the Theory of Probability we call them P-negligible sets. They can be omitted when calculating integrals of measurable functions.
- 17. Pricing and Hedging of Defaultable Models 9 Deﬁnition 2.5. A set A ∈ F is called a P-negligible set if P(A) = 0. In general, the probability space (Ω, F, P) does not have to contain all P-negligible sets. However, it can be completed by incorporating all subsets of P-negligible sets in a suitable manner. Deﬁnition 2.6. A triplet (Ω, F, P) is called a complete probability space if F contains all P-negligible sets. It is important in the Theory of Martingales to deﬁne the ﬁltration on a measurable space (Ω, F). In the mathematical ﬁnance we understand the ﬁltration as the information available up to and including each time t which is more and more precise as more data from the stock becomes accessible. Deﬁnition 2.7. F is a ﬁltration if F is a family of non-decreasing sub-σ- algebras (PFt)t≥0 such that ∀t ≥ 0 Ft ⊂ F and ∀0 ≤ s < t < ∞ Fs ⊂ Ft. Similarly as before, we deﬁne a ﬁltered probability space (Ω, F, F, P) also known as a stochastic basis or a probability space with a ﬁltration of its σ-algebra. Deﬁnition 2.8. We call the quadruple (Ω, F, F, P) a ﬁltered probability space, where Ω = ∅, F is a σ-algebra over Ω, F is a ﬁltration and P is a probability measure. For further considerations we introduce a complete ﬁltered probability space. Deﬁnition 2.9. (Ω, F, F, P) a complete ﬁltered probability space if F con- tains all P-negligible sets and ∀t ≥ 0 F contains all P-negligible sets. 2.2 Stochastic processes In the study of stochastic processes there is an important reason to include σ-ﬁelds and ﬁltrations because they are necessary to keep the track of the information. The relating to time feature of stochastic processes implies the ﬂow of time. It means that at every moment t ≥ 0 we can talk about the past, present and future as well as ask how much the observer of the process knows about them at present. We can compare this information with how much he knew in the past or will know in some certain time in the future.
- 18. 10 Chapter 2. Stochastic background In this chapter we give the deﬁnition of a stochastic process, a natural ﬁltration and we distinguish three types of measurability. Deﬁnition 2.10. A stochastic process X = (Xt)t≥0 is a family of (Rd , B(Rd ))- valued random variables Xt, where ∀t ≥ 0 Xt is deﬁned on the probability space (Ω, F, P). We will assume that d = 1 in further considerations. Given the stochastic process X the most intuitive and the simplest way to choose the ﬁltration is to take the one generated by the stochastic process itself. Deﬁnition 2.11. A natural ﬁltration FX of a process X = (Xt)t≥0 is a ﬁltration FX = (FX t )t≥0, where FX t = σ(Xs, s ≤ t) is the smallest σ-algebra with respect to which Xs is measurable for every s ∈ [0, t]. One can interpret set A ∈ FX t as follows. By the time t the observer knows if the set A has occurred or not. To avoid problems with the measurability in the Theory of Lebesgue Integration, the probability measures are deﬁned on σ-algebras and consid- ered random variables are assumed to be measurable with respect to these σ-algebras. X is a function of two variables (t, ω) and it is convenient to have the following deﬁnitions of the measurability. Deﬁnition 2.12. The stochastic process X = (Xt)t≥0 is called B(R+) ⊗ F- measurable if for every A ∈ B(R), the set {(t, ω)|t ∈ R+, ω ∈ Ω : Xt(ω) ∈ A} belongs to the product σ-algebra B(R+) ⊗ F. One can be more precise and say that the stochastic process is B(R+)⊗F- measurable if ∀t ≥ 0 the mapping (t, ω) → Xt(ω) : (R+ × Ω, B(R+) ⊗ F) → (R, B(R)) is measurable.
- 19. Pricing and Hedging of Defaultable Models 11 The concept of measurability presented in the previous deﬁnition is rather weak. Given the deﬁnition of the ﬁltration we can deﬁne a stronger and more interesting concept. Deﬁnition 2.13. A stochastic process X is F-adapted if ∀t ≥ 0 Xt is F- measurable. Certainly, every process X is adapted to its natural ﬁltration FX . Further- more, if FX consists of all P-negligible sets and a process Y is a modiﬁcation of X then Y is also F-adapted. We can extend the previous study with the deﬁnition of a progressive measurability as follows. Deﬁnition 2.14. We say that a process X is progressively measurable if for every A ∈ B(R) the set {(s, ω)|s ≤ t, ω ∈ Ω : Xs(ω) ∈ A} belongs to the product σ-algebra B([0, t]) ⊗ Ft. In other words, X is a progressively measurable stochastic process if ∀s ≥ 0 the mapping (s, ω) → Xs(ω) : ([0, t] × Ω, B([0, t]) ⊗ Ft) → (R, B(R+)) is B([0, t]) ⊗ Ft-measurable. Fr the further calculations it is necessary to introduce the following lemma. Lemma 2.1. Let Y be an integrable random variable deﬁned on a probability space (Ω, F, P). Let (Ai)i∈N be a sequence of disjoint sets such that i∈N Ai = Ω. Then EP(Y ) = i∈N EP(Y |Ai)P(Ai). (2.1) 2.3 The Brownian ﬁltration In this section we will remind the deﬁnition of a standard Brownian motion and make discussion about the Brownian ﬁltration. In describing the Brow- nian motion we put an accent on the fact that it is important to distinguish diﬀerent ﬁltrations. Deﬁnition 2.15. A standard, one-dimensional Brownian motion is a con- tinuous adapted process B = (Bt, Ft)t≥0 deﬁned on some probability space (Ω, F, P) with the properties that:
- 20. 12 Chapter 2. Stochastic background i) B0 = 0 a.s., ii) for each t ≥ s ≥ 0, the increment Bt − Bs is independent of Fs, iii) for each t ≥ s ≥ 0, the increment Bt − Bs is normally distributed with mean 0 and variance t − s. Consequently, the ﬁltration F = (Ft)t≥0 is a part of the deﬁnition of a Brownian motion. However, if it is not precise which ﬁltration we are dealing with but we know that B has stationary independent increments and that Bt − Bs is normally distributed with mean 0 and variance t − s, then B = (Bt, FB t )t≥0 is a Brownian motion. FB = (FB )t≥0 is Brownian motion’s natural ﬁltration. Moreover, it ∀t FB t ⊂ Ft and Bt − Bs is independent of Fs then (Bt, Ft)t≥0 is also a Brownian motion. We mentioned before how to construct the natural ﬁltration FB = (FB )t≥0. We will study the deﬁnition of an augmented ﬁltration. Firstly, we denote by FB a σ-algebra generated by a Brownian motion, i.e. FB = σ(Bs, s ∈ R+). We remind that FB t = σ(Bs, s ≤ t). We consider the following deﬁnition of a collection of P-negligible sets relative to a σ-algebra F. Deﬁnition 2.16. We say that N is a collection of P-negligible sets relative to a σ-algebra F if for any set A ∈ N there exists a set B ∈ N such that A ⊂ B and P(B) = 0. Let us denote by N a collection of P-negligible sets relative to FB t . We consider the following ﬁltration. Deﬁnition 2.17. In the previous notations we call ˜FB = ( ˜FB t )t≥0 an aug- mentation of FB where ∀t ˜FB t = σ(FB t ∪ N). From this deﬁnition we also get a σ-algebra ˜FB. We can easily consider the process B on the ﬁltration (Ω, ˜FB, P) and get that (Bt, ˜FB t )t≥0 is a Brownian motion.
- 21. Pricing and Hedging of Defaultable Models 13 We can deﬁne the usual conditions for a ﬁltration. Deﬁnition 2.18. We say that the ﬁltration F satisﬁes the usual conditions if it is complete and right-continuous. Lemma 2.2. The augmented ﬁltration ˜FB = ( ˜FB t )t≥0 satisﬁes the usual conditions. We will be only considering ﬁltrations which satisfy the usual conditions. 2.4 Stopping times In the Financial Mathematics it is essential to introduce the Stopping Times Theory. Let us consider an American option. The buyer of such a ﬁnancial deriva- tive can decide when to exercise it. The choice of such a moment, let us call it τ, depends on the information about the stock price process up to time t. Then the value of an American call at τ is (Sτ − K)+ . When the agent pricing the option knows which stopping time the buyer will follow the cost of such a ﬁnancial derivative at time 0 will be EP∗ (exp(−rτ)(Sτ − K)+ ), where P∗ is the equivalent martingale measure. However, if we do not know which stopping time exactly will the observer use, he has to take the supremum. Accordingly, the price of the contingent claim at time 0 will be sup τ EP∗ (exp(−rτ)(Sτ − K)+ ). It is crucial to consider the following deﬁnition of a random time. Deﬁnition 2.19. A random time T is a strictly positive P-a.s. random variable. It is essential to deﬁne an F-stopping time τ, which is an example of a random time. Deﬁnition 2.20. A random variable τ such that τ : (Ω, F) → (R+ B(R+ )) is called an F-stopping time if ∀t ≥ 0 {τ ≤ t} is F-measurable.
- 22. 14 Chapter 2. Stochastic background Deﬁnition 2.21. XT = XT∧t is a process stopped at a stopping time T if i) X is a stochastic process, ii) T is a stopping time. 2.5 The Martingale Theory In this section we present a fundamental characteristic which underlies many important results in Finance, namely a martingale property. Its mo- tivation lies in the notion of a fair game. Broadly speaking, the martingale property states that tomorrow’s price is expected to be today’s and thus it is its best prediction. The martingale condition is assumed to be essential for an eﬃcient market in which the information included in the past prices is fully reﬂected in the current prices. Furthermore, the Fundamental Theorem of Asset Prices states that if the market is arbitrage-free then discounted assets prices are martingales under a risk-neutral measure. Here, we give a formal deﬁnition of a martingale and more general processes such as a submartingale and a supermartingale. Deﬁnition 2.22. An adapted, integrable stochastic process M = (Mt)t≥0 on a ﬁltered probability space (Ω, F, F, P) is a i) martingale if EP(Mt|Fs) = Ms ∀s ≤ t, ii) submartingale if EP(Mt|Fs) ≥ Ms ∀s ≤ t, iii) supermartingale if EP(Mt|Fs) ≤ Ms ∀s ≤ t.
- 23. Chapter 3 The default setting 3.1 Basic assumptions We consider a probability space (Ω, F, P) equipped with a ﬁltration F =(Ft)t≥0, where F fulﬁlls the usual conditions, i.e. it is right-continuous and complete, F0 is trivial σ-ﬁeld. Let us specify that σ-algebra Ft represents a t-time information available to the agent in the default-free market. We can deﬁne default time τ as a R+ -valued ﬁnite random variable on (Ω, F, P). Let us determine the distribution of τ as a càdlàg function F such that F(t) = P(τ ≤ t), where F(0) = 0 and lims→t F(s) = P(τ < t) = F(t−). F deﬁnes a measure η which is the distribution of τ on R+ , e.g. η([a, b]) = F(b) − F(a−), [a, b] ∈ B(R+) and η(du) = P(τ ∈ du). Assumption 3.1. Let us assume that η is absolutely continuous with respect to the Lebesgue measure λ. Then, τ admits a Radon-Nikodým density fτ such that fτ = dη dλ . Moreover, if F is diﬀerentiable, then fτ = F . 15
- 24. 16 Chapter 3. The default setting Remark 3.1. Let us now interpret P(τ ∈ du). This is a probability of τ being in a small interval which we can denote also as (u, u + du). We know that P(τ ∈ (u, u + du)) = F(u + du) − F(u). If F is continuously diﬀerentiable, then from the Taylor series we have that F(u + du) = F(u) + F (u)du which gives us F (u)du = P(τ ∈ (u, u + du)) = P(τ ∈ du). Then, since in this case the law η of τ has a density with respect to the Lebesgue measure, the equality above becomes P(τ ∈ du) = f(u)du. In addition, ∀A ∈ B(R) P(τ ∈ A) = A P(τ ∈ du) = A η(du) = A fτ (u)du. 3.2 The default process We deﬁne a default process indicating whether the default occured or not as N =(Nt)t≥0 where Nt = I{τ≤t} is c`ad and increasing. We denote H=(Ht)t≥0 as a natural ﬁltration generated by N, i.e. Ht = σ(Nu, u ≤ t) and we complete H with all P-negligible sets. The σ-algebra Ht represents the in- formation generated by the observations of τ on the time interval [0, t]. It is necessary to mention two main properties of the ﬁltration H. First of all, H is the smallest ﬁltration such that τ is H-stopping time. Moreover, σ(τ) = H∞. Let us now establish the form of an Ht-measurable random variable with the following proposition. Proposition 3.1. A random variable Ut is Ht-measurable if and only if it is of the form Ut(ω) = ˜uI{τ(ω)>t} + h(τ(ω))I{τ(ω)≤t}, where h is a Borel function on [0, t] and ˜u is constant.
- 25. Pricing and Hedging of Defaultable Models 17 Proof. We can base the proof on the fact that Ht-measurable random vari- ables are generated by random variables of the form U0 t (ω) = h(t ∧ τ(ω)), where h is a bounded Borel function on R+ . Now we can specify h(t ∧ τ(ω)) on before the default set and after the default set, i.e. h(t ∧ τ(ω)) = h(t ∧ τ(ω))I{τ(ω)>t} + h(t ∧ τ(ω))I{τ(ω)≤t} = = h(t)I{τ(ω)>t} + h(τ(ω))I{τ(ω)≤t}. For a ﬁxed t, h(t) is constant. We denote it as ˜u and we have Ut(ω) = ˜uI{τ>t} + h(τ(ω))I{τ(ω)≤t}. Since we use function h only on the set {τ ≤ t} we can characterize the function h as a Borel function on [0, t] without loss of generality.
- 26. 18 Chapter 3. The default setting
- 27. Chapter 4 The intensity-based approach in ﬁltration H The intensity-based approach has a lot in common with the Reliability Theory. Clearly, default time is precisely expressed by the likelihood of the default event conditional on the information ﬂow. These considerations help us to deliver the reduced form of a price for a defaultable contingent claim. Speciﬁcally, we assume that the agent pricing the contingent claim knows only time of default. The assumption of the agent’s lack of knowledge about the price process is crucial for the ﬁrst glance at the valuation. Let τ, as deﬁned before, be a positive random variable on the probabil- ity space (Ω, F, P). Firstly, we study the distribution function F(t) of τ which is absolutely continuous with respect to the Lebesgue measure. In this case we can easily compute the intensity function which is a non-negative deterministic function deﬁned as follows. 4.1 The H-intensity of τ In this section we give the deﬁnitions of H-intensity of default time τ and deliver the expectation tools which are essential for pricing defaultable claims. 4.1.1 The intensity of default Let us deﬁne more formally an intensity of default time. Deﬁnition 4.1. An intensity of default time is a ratio of the probability that default will appear in a inﬁnitely small time interval ∆s, condition on 19
- 28. 20 Chapter 4. The intensity-based approach in filtration H that there was no default before, and the time step ∆s, i.e. λs = lim ∆s→0 P(τ ∈ (s, s + ∆s)|τ > s) ∆s . Consequently, from the Reliability Theory, we can obtain the following form of the intensity. Proposition 4.1. λs = fτ (s) 1−F(s) is the intensity function for a default time τ. Proof. Let us assume that the distribution function of τ is absolutely con- tinuous with respect to the Lebesgue measure. From the deﬁnition we have that λs = lim ∆s→0 P(τ ∈ (s, s + ∆s)|τ > s) ∆s . Using the deﬁnition of the conditional probability we can write λs = lim ∆s→0 P({τ ∈ (s, s + ∆s)} ∩ {τ > s}) P(τ > s)∆s . We have that {ω : τ(ω) ∈ (s, s + ∆s)} ∩ {ω : τ(ω) > s} = {ω : τ(ω) ∈ (s, s + ∆s)} . Thus, we can write λs = lim ∆s→0 P({τ ∈ (s, s + ∆s)}) P(τ > s)∆s . From the deﬁnition of the distribution function F(t) of τ and the fact that F(t) is absolutely continuous it follows that λs = lim ∆s→0 F(s + ∆s) − F(s) P(τ > s)∆s = fτ (s) 1 − F(s) . Recall that we introduced the intensity function for default time τ. Since we deﬁned the default process N =(Nt)t≥0 with Nt = I{τ≤t} and the ﬁltra- tion H is generated by the default process we can formulate the following deﬁnition. Deﬁnition 4.2. An H-adapted non-negative process λ = (λt)t≥0 is called an H-intensity of τ if (I{τ≤t}− t 0 λuI{τ≥u}du)t≥0 is an (P, H)-martingale.
- 29. Pricing and Hedging of Defaultable Models 21 Here, we give a proposition which is essential in delivering the expectation tools for pricing defaultable claims. Proposition 4.2. Let ζ be an F-measurable random variable, then EP(ζ|Ht) = I{τ>t} EP(ζI{τ>t}) P(τ > t) + I{τ≤t}EP(ζ|H∞). Proof. Since ζ is an F-measurable random variable, we can represent EP(ζ|Ht) on two sets {τ ≤ t} and {τ > t} in the following way EP(ζ|Ht) = I{τ>t}EP(ζ|Ht) + I{τ≤t}EP(ζ|Ht). Firstly, let us study the ﬁrst term on the right-hand side of the last equation. Then using the properties of conditional probability with respect to σ-algebra Ht we have I{τ>t}EP(ζ|Ht) = I{τ>t}I{τ>t} EP(ζI{τ>t}) P(τ > t) + I{τ>t}I{τ≤t} EP(ζI{τ≤t}) P(τ ≤ t) . We see that the second term on the right-hand side vanishes and we obtain I{τ>t}EP(ζ|Ht) = I{τ>t} EP(ζI{τ>t}) P(τ > t) . Secondly, let us ponder the term I{τ≤t}EP(ζ|Ht). To prove that I{τ≤t}EP(ζ|H∞) = I{τ≤t}EP(ζ|Ht) we use the fact that H∞ = σ(Ns, s ∈ R+) and ∀A ∈ H∞ A ∩ {τ ≤ t} ∈ Ht. From the properties of the conditional expectation we have A EP(ζI{τ≤t}|H∞)dP = A ζI{τ≤t}dP,
- 30. 22 Chapter 4. The intensity-based approach in filtration H which is also equal to A∩{τ≤t} ζI{τ≤t}dP. Again, using the property of the conditional expectation we obtain A EP(ζI{τ≤t}|H∞)dP = A∩{τ≤t} EP(ζI{τ≤t}|Ht)dP, which can be written as A I{τ≤t}EP(ζ|Ht)dP. Finally, we obtain the result A EP(ζI{τ≤t}|H∞)dP = A EP(I{τ≤t}ζ|Ht)dP. We give a lemma concerning previously deﬁned intensity of τ. Lemma 4.1. A process λ = (λt)t≥0, where λt = fτ (t) 1 − F(t) is an H-intensity of τ. Proof. The process λt = fτ (t) 1 − F(t) is deterministic and non-negative. Thus it is H-adapted. Now, we will check that M =(Mt)t≥t with Mt = I{τ≤t} − t 0 λuI{τ≥u}du is a (P, H)-martingale. Let us assume that s < t. We will show that EP(Mt − Ms|Hs) = 0
- 31. Pricing and Hedging of Defaultable Models 23 Using the previous notations and the additive property of integrals we obtain EP(Mt − Ms|Hs) = EP(Nt − Ns|Hs) − EP t s λuI{τ≥u}du|Hs . We will show that EP(Nt − Ns|Hs) = EP t s λuI{τ≥u}du|Hs . Let us use the fact that EP(I{τ>t}|Hs) = P(τ > t|Hs). Then, we can rewrite the right-hand side of the last equality on two sets, {τ ≤ s} and {τ > s}, and use the deﬁnition of the conditional probability to obtain P(τ > t|Hs) = I{τ>s} P(τ > t, τ > s) P(τ > s) + I{τ≤s} P(τ > t, τ ≤ s) P(τ ≤ s) . We easily see that the second term on the right-hand side of the last equation vanishes. We have that {τ > t} ∩ {τ > s} = {τ > t} . Thus we have EP(I{τ>t}|Hs) = I{τ>s} 1 − F(t) 1 − F(s) . We have that EP(Nt − Ns|Hs) = I{τ>s} F(t) − F(s) 1 − F(s) . Let us denote J = t s λuI{τ≥u}du. Then we can write J = t∧τ s∧τ λudu. Knowing that λt = fτ (t) 1 − F(t)
- 32. 24 Chapter 4. The intensity-based approach in filtration H we get J = ln 1 − F(s ∧ τ) 1 − F(t ∧ τ) . As previously, we can study J on two sets, before and after the default and get J = I{τ>s} ln 1 − F(s ∧ τ) 1 − F(t ∧ τ) + I{τ≤s} ln 1 − F(s) 1 − F(s) . Consequently, we get J = I{τ>s} ln 1 − F(s ∧ τ) 1 − F(t ∧ τ) . Thus J = JI{τ>s}. Now, we use the Proposition 4.2 and calculate the conditional expectation of J. EP(J|Hs) = I{τ>s} EP(JI{τ>s}) P(τ > t) + I{τ≤s}EP(J|H∞). Due to the fact that J = JI{τ>s} we get EP(J|Hs) = I{τ>s} EP(J) P(τ > s) . Using the deﬁnition of J and λu we get EP(J|Hs) = I{τ>s} EP( t s λuI{τ≥u}du) P(τ > s) . We can take the expectation operator inside the integral and get I{τ>s} t s λuEP(I{τ≥u})du 1 − F(s) . From the fact that EP(I{τ≥u}) = P(τ ≥ u) we obtain I{τ>s} t s λuP(τ ≥ u)du 1 − F(s) .
- 33. Pricing and Hedging of Defaultable Models 25 Consequently, from the form of the function λu we get I{τ>s} t s fτ (u) 1 − F(s) du, which is equal to I{τ>s} F(t) − F(s) 1 − F(s) . Finally, EP(Nt − Ns|Hs) = EP t s λuI{τ≥u}du|Hs ⇒ I{τ≤t} − t 0 λuI{τ≥u}du t≥0 is a (P, H)-martingale. Using those results we can value a defaultable zero-coupon bond which pays 1 if the default has not appeared before maturity time T. Let us consider a case when default time τ is exponentially distributed with a deterministic intensity function λs. Proposition 4.3. Expected value of this contingent claim for an agent who knows only that the default is exponentially distributed, is EP(I{τ>T}|Ht) = I{τ>t} exp − T t λsds . Proof. We use the Proposition 4.2. Firstly, we realize that I{τ>T} is an HT - measurable random variable. We have EP(I{τ>T}|Ht) = I{τ>t} EP(I{τ>t}I{τ>T}) P(τ > t) Using the property that EP(IA) = P(A) and the fact that τ is exponentially distributed we obtain EP(I{τ>T}|Ht) = I{τ>t} exp − T t λsds .
- 34. 26 Chapter 4. The intensity-based approach in filtration H 4.1.2 The hazard function Γ In this section we deﬁne a survival and hazard function which are frequently used further. We begin with the assumption necessary for those functions to be well deﬁned. Assumption 4.1. We assume that ∀ t ≥ 0 F(t) < 1. Deﬁnition 4.3. We say that G(t) = 1 − F(t) is a survival function of τ if F(t) ∀t ≥ 0 is a distribution function of τ. From the Assumption above we have that ∀t ≥ 0 G(t) : R → (0, 1] because ∀t ≥ 0 F(t) : R → [0, 1). In the default framework we have that the survival function for τ is given by the following formula G(t) = P(τ > t). From the fact that ∀t ≥ 0 G(t) > 0 we can take a natural logarithm of G(t) and deﬁne a hazard function for τ. Deﬁnition 4.4. We call a function Γ(t) = − ln(G(t)) a hazard function of τ, where G(t) is a survival function for τ ∀t ≥ 0. If F(u) is diﬀerentiable we can approximate it by dF(u) = F (u)du. With the analogical argumentation we get dΓ(u) = Γ (u)du. We can write the hazard function in a form as follows. Proposition 4.4. Γ(t) = t 0 dF(s) G(s) is a hazard function ∀t ≥ 0. Proof. We have Γ(t) = t 0 dF(s) G(s) = t 0 dF(s) 1 − F(s) . We can easily obtain the result after realizing that the nominator of the fraction inside the integral is a derivative of the denominator but without the minus sign. By the formula t 0 dV (s) V (s) = ln(V (t)) − ln(V (0)) and the fact that G(0) = 1 we end the proof.
- 35. Pricing and Hedging of Defaultable Models 27 From this form of the hazard function it is obvious that Γ(t) satisﬁes the following property. Proposition 4.5. The hazard function Γ(t) of τ is increasing. Proof. From the deﬁnition of an integral and the fact that if the integrand does not change but we integrate on a larger interval the integral will be greater. More formally, ∀s < t Γ(s) = s 0 dF(u) G(u) < t 0 dF(s) G(s) = Γ(t). In the case when F(t) is continuous and has a derivative F (t) = fτ (t) we can write the hazard function of τ as Γ(t) = t 0 fτ (s) G(s) ds. Consequently, the derivative of Γ(t) is Γ (t) = t 0 fτ (s) G(s) ds = fτ (t) G(t) . Deﬁnition 4.5. We will call the derivative of Γ an H-generalized intensity of τ if I{τ≤t} − Γ(t ∧ τ) t≥0 is a (P, H)-martingale. Let us introduce and prove the following proposition which is important for further calculations. Proposition 4.6. Let h(τ) be a Borel function (i.e. h(τ) is σ(τ)-measurable random variable). Then EP(h(τ)|Ht) = I{τ>t} EP(h(τ)I{τ>t}) P(τ > t) + I{τ≤t}h(τ). Proof. We mentioned before that σ(τ) = H∞. According to the Proposition 4.2 we have EP(h(τ)|Ht) = I{τ≤t}EP(h(τ)|H∞) + I{τ>t} EP(h(τ)I{τ>t}) P(τ > t) . From the fact that h(τ) is an H∞-measurable random variable we get EP(h(τ)|Ht) = I{τ>t} EP(h(τ)I{τ>t}) P(τ > t) + I{τ≤t}h(τ).
- 36. 28 Chapter 4. The intensity-based approach in filtration H Let us study a zero-coupon defaultable contingent claim that pays h(τ) if the default has not appeared before the maturity time T. We assume that the spot rate r(s) ≡ 0. It is natural to reckon such a payoﬀ because the agent pricing the claim knows that it is a defaultable one and he studies the payoﬀ as a Borel function of τ. Here, we do not assume that the distribution function F of τ is absolutely continuous but we assume it is continuous. Proposition 4.7. The expected value of this derivative in the case of the knowledge only about the default time distribution is EP(h(τ)I{τ>T}|Ht) = I{τ>t} exp(Γ(t)) ∞ T h(u)dF(u). Proof. From the Proposition 4.6 we induce EP(h(τ)I{τ>T}|Ht) = I{τ>t} EP(h(τ)I{τ>T}I{τ>t}) P(τ > t) + I{τ≤t}I{τ>T}h(τ). The second term of the right-hand side of the equation above vanishes as well as the indicator I{τ>t} in the second term. From the deﬁnition of expected value we obtain EP(h(τ)I{τ>T}|Ht) = I{τ>t} R h(u)I{u>T}dF(u) 1 − F(t) . Using the correlation between F and Γ we obtain EP(h(τ)I{τ>T}|Ht) = I{τ>t} ∞ T h(u) 1 − F(u) 1 − F(t) dΓ(u). Substituting the terms with F by the terms with Γ we get EP(h(τ)I{τ>T}|Ht) = I{τ>t} exp(Γ(t)) ∞ T h(u) exp(−Γ(u))dΓ(u). Finally, after coming back to the terms with F we obtain EP(X(τ)I{τ>T}|Ht) = I{τ>t} exp(Γ(t)) ∞ T h(u)dF(u)
- 37. Pricing and Hedging of Defaultable Models 29 Now, let us derive a value similar to that one in the Proposition 4.3 but without any assumption about the distribution of τ except this one that the distribution is continuous. We consider a defaultable zero-coupon ﬁnancial derivative which pays 1 if the default has not appeared before the maturity time T. We assume that the spot rate r(s) ≡ 0. Proposition 4.8. The expected value of the payoﬀ for an agent who observes default when it occurs is EP(I{τ>T}|Ht) = I{τ>t} exp(−[Γ(T) − Γ(t)]). Proof. From the Proposition 4.7 we have EP(I{τ>T}|Ht) = I{τ>t} exp(Γ(t)) ∞ T dF(u). From the deﬁnition of the improper integral we induce I{τ>t} exp(Γ(t)) ∞ T dF(u) = I{τ>t} exp(Γ(t)) lim v→∞ v T dF(u). Then, after calculating the integral, taking the limit and writing F in terms of Γ, we obtain the result EP(I{τ>T}|Ht) = I{τ>t} exp(−[Γ(T) − Γ(t)]). Let us assume that there exists a deterministic spot rate r(s). Then the present value (at time t) of a zero-coupon bond which pays 1 when the default has not appeared before maturity time T is exp − T t r(s)ds , where t ∈ [0, T]. Let us study a ﬁrm which issues a zero-coupon bond which pays 1 at the maturity time T when the default has not appeared before T. On this ﬁnancial market we have the following. Proposition 4.9. We assume that τ admits an H-intensity λs. Then, the expected value at time t of described contingent claim calculated by an agent who has the information Ht is EP(exp(− T t r(s)ds)I{τ>T}|Ht) = I{τ>t} exp(− T t (r(s) + λs)ds).
- 38. 30 Chapter 4. The intensity-based approach in filtration H Proof. We can take the deterministic part outside the integral and obtain after taking under consideration the Proposition 4.8 that the left-hand side is equal to exp − T t r(s)ds I{τ>t} exp − [Γ(T) − Γ(t)] . We can take exp − [Γ(T) − Γ(t)] inside the integral and obtain I{τ>t} exp − T t r(s)ds − [Γ(T) − Γ(t)] . From the fact that τ admits a H-intensity λs and Γ (s) = λs we get I{τ>t} exp − T t r(s) + Γ (s))ds . Consequently, EP exp − T t r(s)ds I{τ>T}|Ht = I{τ>t} exp − T t r(s) + λs ds . However, we should not treat the last result as an actual price for a de- faultable zero-coupon bond. This is because we are calculating it under the initial measure P. What is more, it is impossible to hedge this default. We can only use this value to see that the default might act as a change in the interest rate r(s). The expected value calculated at time t of a contingent claim H under the condition that the default has not appeared before time T is EP H exp − T t r(s)ds I{τ>T}|Ht . This was the case when H was dependent on τ. Proposition 4.10. If ζ is independent of default time τ then EP ζ exp − T t r(s)ds I{τ>T}|Ht = I{τ>t} exp − T t (r(s)+λs ds EP(ζ).
- 39. Pricing and Hedging of Defaultable Models 31 Proof. We can take the exponent outside the expected value and obtain EP ζ exp − T t r(s)ds I{τ>T}|Ht = exp − T t r(s)ds EP ζI{τ>T}|Ht . Then, using the fact that I{τ>T} is Ht-measurable we can also take the in- dicator function outside and from the independence ζ of τ, we obtain the independence ζ of Ht and get EP(ζ) exp − T t r(s)ds I{τ>t} exp − [Γ(T) − Γ(t)] . Finally, analogically to the proof of the Proposition 4.9, we obtain the result EP ζ exp − T t r(s)ds I{τ>T}|Ht = I{τ>t} exp − T t (r(s)+λs)ds EP(ζ).
- 40. 32 Chapter 4. The intensity-based approach in filtration H
- 41. Chapter 5 The Carthaginian enlargement of ﬁltrations 5.1 Introduction To add the information about the default to the ﬁltration generated by the price process, we have to enlarge it by a positive random variable which is default time τ. It can be done in two diﬀerent manners: initially, i.e. from the beginning with the corresponding information σ(τ) or progressively with σ(τ ∧ t). The procedure of enlargement lets us to obtain three nested ﬁltrations, hence it was called Carthaginian Enlargement of Filtrations. The adjective "Carthaginian" was ﬁrst introduced by Callegaro, Jeanblanc and Zargari (see [2]) and it refers to three levels of diﬀerent civilizations which can be found at the archaeological site of Carthage. The initially enlarged ﬁltration Gτ = (Gτ t )t≥0 is generated by σ-algebras of the form Gτ t = Ft ∨ σ(τ). More generally Gτ t = Ft ∨ ˜F, where ˜F is σ-algebra. The progressively enlarged ﬁltration G = (Gt)t≥0 is generated by σ-algebras of the form Gt = Ft ∨ Ht, where H is the natural ﬁltration of the default process N =(Nt)t≥0 with Nt = I{τ≤t}. More generally Gt = Ft ∨ ˜Ft, where ˜F = ( ˜Ft)t≥0 is the natural ﬁltration generated by additional process. Usually we consider the right-continuous version of G, namely ∀t ≥ 0 Gt = Gt+ = s>t Fs ∨ Hs. The three acquired ﬁltrations represent diﬀerent sources of information available to the investors. The Enlargement of Filtrations Theory plays very 33
- 42. 34 Chapter 5. The Carthaginian enlargement of filtrations important role in modelling additional gain due to such asymmetric informa- tion as well as information itself. In the previous chapter we introduced the intensity approach in ﬁltration H. Hereafter, some of the results for progressively enlarged ﬁltration are also obtained using this approach. Nonetheless, the intensity process allows for a knowledge of the default conditional distribution only before the default. Thus, we have to consider density approach which gives the full characteri- zation of the links between the default time and the ﬁltration generated by the price process before and after the default. 5.2 General projection tools Working in the initially enlarged ﬁltration is easier since the whole infor- mation concerning the default is possessed by the insider from the beginning. However, we would like to represent the obtained results in terms of the pro- gressively enlarged ﬁltration so that they are accessible to the regular investor as well. Thus, we have to establish some projection tools. Let us introduce a following proposition determining a method of projecting martingale adapted to some arbitrary ﬁltration on the smaller ﬁltration. Proposition 5.1. [2] Let K and ˜K be ﬁltrations such that K ⊂ ˜K and let ζ =(ζt)t≥0 be uniformly integrable (P, K)-martingale. Then, there exists an (P, ˜K)-martingale ˜ζ = (˜ζt)t≥0 such that EP(˜ζt|Kt) = ζt, t ≥ 0. Proof. From ζ being a uniformly integrable (P, K)-martingale it follows that P-a.s. ζt = EP(ζ∞|Kt). We deﬁne ˜ζt as EP(ζ∞| ˜Kt). Let us check that it is a (P, ˜K)-martingale. For any s ≤ t we have that EP(˜ζt| ˜Ks) = EP(EP(ζ∞| ˜Kt)| ˜Ks). Applying the tower property we obtain P-a.s. EP(EP(ζ∞| ˜Kt)| ˜Ks) = EP(EP(ζ∞| ˜Ks)| ˜Kt) = EP(ζ∞| ˜Ks) = ˜ζs
- 43. Pricing and Hedging of Defaultable Models 35 and hence the martingale property. Let us now prove that EP(˜ζt|Kt) = ζt. Indeed, from the uniform integrability and the tower property we obtain that ζt = EP(ζ∞|Kt) = EP(EP(ζ∞|Kt)| ˜Kt) = EP(EP(ζ∞| ˜Kt)|Kt) = EP(˜ζt|Kt). 5.3 Measurability properties in enlarged ﬁltra- tions Let us now introduce some important results on the characterization of the random variables measurable with respect to the ﬁltrations Gτ and G. We begin with the representation of a Gτ t -measurable random variable. Proposition 5.2. [2] A random variable Zt is Gτ t -measurable if and only if it is of the form Zt(ω) = zt(ω, τ(ω)), where ∀t ≥ 0 zt(·, τ(·)) is a Ft ⊗ B(R+ )-measurable random variable. For the proof see [2]. Let us now give the analogous results about the representation of a Gt- measurable random variable. Proposition 5.3. [2] A random variable Xt is Gt-measurable if and only if it is of the form Xt(ω) = ˜yt(ω)I{τ(ω)>t} + ˆzt(ω, τ(ω))I{τ(ω)≤t}, where ˜yt is an Ft-measurable random variable and (ˆzt(ω, u)ω∈Ω,u∈R)t≥u is a family of Ft ⊗ B(R+ )-measurable random variables. Proof. Gt-measurable random variables are generated by the random vari- ables of the form X0 t (ω) = yt(ω)h(t ∧ τ(ω)), where yt is an Ft-measurable random variable and h is a Borel function on R+ . Specifying X0 t (ω) on before and after the default set we obtain X0 t (ω) = yt(ω)h(t ∧ τ(ω))I{τ(ω)>t} + yt(ω)h(t ∧ τ(ω))I{τ(ω)≤t}, which is equal to yt(ω)h(t)I{τ(ω)>t} + yt(ω)h(τ(ω))I{τ(ω)≤t}.
- 44. 36 Chapter 5. The Carthaginian enlargement of filtrations We can replace yt(ω)h(t) with the Ft-measurable random variable ˜yt(ω). What is more, it is well known that the measurable function of two variables can be approximated by the sum of the products of one variable measurable functions, i.e. f(x, y) = lim N→∞ N i=1 hi(x)gi(y). where in this case x ∈ Ω and y ∈ R+ . The random variable yt(ω)h(τ(ω)) is measurable with respect to the σ-algebra Ft ⊗ B(R+ ) and the sum of random variables of such form is also measurable with respect to Ft ⊗B(R+ ). Then, by passing to the limit with N → ∞, we obtain that the random variable ˆzt(·, τ(·)) which is an approximation of functions as in (5.3) is also an Ft ⊗ B(R+ )-measurable random variable. Finally we have that Xt(ω) = ˜yt(ω)I{τ(ω)>t} + ˆzt(ω, τ(ω))I{τ(ω)≤t}. 5.4 The E-hypothesis Let us consider now the crucial assumption which will be in force through- out the rest of our thesis. It is called E-hypothesis. Hypothesis 5.1. (E-hypothesis) We suppose that ∀t ≥ 0, P-a.s. P(τ ∈ du|Ft) ∼ η(du), i.e. the F-conditional law of τ is equivalent to the law of τ. As a result, there exists a strictly positive Ft⊗B(R+ )-measurable function (t, ω, u) → qt(ω, u), such that for every u ≥ 0, (qt(u))t≥0 is (P, F)-martingale and P(τ > θ|Ft) = ∞ θ qt(u)η(du) ∀t ≥ 0, P − a.s. or equivalently EP(Zt|Ft) = EP(zt(τ)|Ft) = ∞ 0 zt(u)qt(u)η(du), for any Ft ⊗ B(R+ )-measurable random variable Zt = zt(τ). The family of the processes q(u) is called the (P, F)-conditional density of τ with respect to η. In particular, P(τ > θ) = P(τ > θ|F0) = ∞ θ q0(u)η(du) and q0(u) = 1, ∀u ≥ 0.
- 45. Pricing and Hedging of Defaultable Models 37 Remark 5.1. One can consider a particular case when ∀u ≥ 0 qt(u) = qu(u), ∀t ≥ u dP − a.s. It means that P(τ > s|Ft) = P(τ > s|Fs), 0 ≤ s ≤ t and new information does not change the conditional distribution of τ. In the structural approach introduced by Merton τ is an F-stopping time. In the reduced-form approach which we work with this property is no longer fulﬁlled. Let us now present a proposition which shows that under the special assumption concerning the measure η, τ avoids F-stopping times. Assumption 5.1. We assume that the law of τ is non-atomic. Proposition 5.4. [3] The Assumption 5.1 and the Hypothesis 5.1 are satis- ﬁed. Then, we have for every F-stopping time ξ bounded by T that P(τ = ξ) = 0. Proof. From the tower property we have P(τ = ξ) = EP(I{τ=ξ}) = EP(EP(I{τ=ξ})|Ft) = EP(EP(I{τ=ξ}|Ft)). Let us prove ﬁrstly that EP(I{τ=ξ}|Ft) = 0. Again, using the tower property EP(I{τ=ξ}|Ft) = EP(EP(I{τ=ξ}|Ft)|FT ) = EP(EP(I{τ=ξ}|FT )|Ft). Since τ admits the conditional density we can write that EP(EP(I{τ=ξ}|FT )|Ft) = EP ∞ 0 I{u=ξ}qt(u)η(du)|Ft . The integral ∞ 0 I{u=ξ}qt(u)η(du) is a Lebesgue integral with respect to the measure η for each ﬁxed ω. Since the measure η is non-atomic, η({ξ(ω)}) = 0, the mentioned integral is also equal to 0, as well as its conditional expectation. Thus, EP I{τ=ξ}|Ft = 0 and P(τ = ξ) = 0.
- 46. 38 Chapter 5. The Carthaginian enlargement of filtrations 5.5 The change of measure on Gτ Due to the fact that working with τ independent of the prices ﬁltration F is easier we have to introduce a decoupling measure which provides this property. Proposition 5.5. [2] Let us suppose that E-hypothesis holds. There exists a process L = (Lt)t≥0 with Lt = 1 qt(τ) and EP(Lt) = L0 = 1 which is a strictly positive (P, Gτ )-martingale and thus deﬁnes a probability measure P∗ - locally equivalent to P such that dP∗ |Gτ t = LtdP|Gτ t , i.e. ∀A ∈ Gτ t P∗ (A) = A LtdP. The martingale L is called the Radon-Nikodým density of P∗ with respect to P. The measure P∗ has the following properties i) Under P∗ , the random time τ is independent of Ft, ∀ t ≥ 0; ii) ∀ t ≥ 0 P∗ |Ft = P|Ft ; iii) P∗ |σ(τ) = P|σ(τ); iv) P∗ (τ ∈ du|Ft) = P∗ (τ ∈ du); v) (P∗ , F)-martingales remain (P∗ , Gτ )-martingales. For the proof of the proposition and the properties, see [2] and [4]. The following lemma presents the Bayes formula which plays a crucial role in the proof of the next proposition. Lemma 5.1. [4] We assume that E-hypothesis holds, the measures P and P∗ are equivalent on Gτ t and Yt - an Ft-measurable, P∗ -integrable random variable. Then, for any s < t EP∗ (Yt|Gτ s ) = EP(LtYt|Gτ s ) Ls , where L is a Radon-Nikodým density of P∗ with respect to P.
- 47. Pricing and Hedging of Defaultable Models 39 Proof. Let us denote ζ as EP(LtYt|Gτ s ) Ls . We will show that ζ is a Gτ s -conditional expectation of Yt under the measure P∗ . We have that EP∗ (Yt|Gτ s ) = ζ. Let us modify ﬁrstly this condition. If we multiply both sides by a Gτ s - measurable random variable ˜Ys, as a result we get EP∗ (˜YsYt|Gτ s ) = ˜Ysζ. We take the expectation with respect to P∗ , again on both sides, and apply the tower property on the left-hand side to obtain EP∗ (˜YsYt) = EP∗ (˜Ysζ). (5.1) We transformed (5.1) to the equality above. Therefore, to prove (5.1) we can show that (5.1) is fulﬁlled. Starting from the left-hand side and changing the measure, we obtain EP∗ (˜YsYt) = EP(Lt ˜YsYt), since ˜YsYt is Gτ t -measurable. Then, we condition on Gτ s and we use the tower property. Therefore, we have that EP(Lt ˜YsYt) = EP(EP(Lt ˜YsYt|Gτ s )). Since ˜Ys is Gτ s -measurable, we can take it outside the conditional expectation. ˜YsEP(LtYt|Gτ s ) is a Gτ s -measurable random variable so we can, again, change the measure to obtain EP(˜YsEP(LtYt|Gτ s )) = EP∗ (L−1 s ˜YsEP(LtYt|Gτ s )). Replacing L−1 s EP(LtYt|Gτ s ) with ζ, we get that EP∗ (˜YsYt) = EP∗ (˜Ysζ) and we proved (5.1) which is equivalent to (5.1) being satisﬁed. Let us now analyse the proposition which allows to transform a Gτ t -expected value to an Ft-expected value under the decoupling measure. Proposition 5.6. [2] Let Zt = zt(τ) be Gτ t -measurable. For s ≤ t, if zt(τ) is P∗ -integrable and if zt(u) is P (or P∗ )-integrable for any u ≥ 0 then, EP∗ (zt(τ)|Gτ s ) = EP∗ (zt(u)|Fs)|u=τ = EP(zt(u)|Fs)|u=τ P (or P∗ )-a.s. See [2] for the proof.
- 48. 40 Chapter 5. The Carthaginian enlargement of filtrations Finally, using the proposition above, we prove in the following proposition that the ﬁltration Gτ inherits the right-continuity from the ﬁltration F. Proposition 5.7. [4] Let us assume that the Hypothesis 5.1 is satisﬁed. Then, ∀t ∈ [0, T) Gτ t = Gτ t+ (5.2) Proof. To prove that (5.2) is satisﬁed we have to show that any Gτ t+-measurable random variable is Gτ t -measurable. At the beginning, let us ﬁx t ∈ [0, T) and δ ∈ (0, T − t) which preserves δ + t being in the interval (t, T). The proof will be done according to the following plan. i) Firstly, we prove that Gτ t+-conditional expectation of the random vari- able Z0 t+δ = yt+δh(τ) (where ∀t ≥ 0 yt is Ft-measurable and h is a bounded Borel function on R+ ) is the same as a Gτ t -conditional expec- tation. ii) Then, we extend the obtained result to any Gτ t+δ-measurable random variable Zt+δ. iii) Finally, we use ii) to show that any Gτ t+-measurable random variable is also Gτ t -measurable. i) Let us assume that we are working at the beginning under the decou- pling measure P∗ , i.e. τ is independent of the ﬁltration F. ∀ε ∈ (0, δ) we get EP∗ (Z0 t+δ|Gτ t+) = EP∗ (yt+δh(τ)|Gτ t+). Since Gτ t+ = ε>0 Gτ t+ε = ε>0 Fτ t+ε ∨ σ(τ) and h(τ) is σ(τ)-measurable we have EP∗ (yt+δh(τ)|Gτ t+) = h(τ)EP∗ (yt+δ|Gτ t+). Using the tower property and the fact that ∀ε > 0, ε>0 Gτ t+ε ⊂ Gτ t+ε we obtain that h(τ)EP∗ (yt+δ|Gτ t+) = h(τ)EP∗ (EP∗ (yt+δ|Gτ t+ε)|Gτ t+). From the Proposition 5.6 and the deﬁnition of Gτ t+ε as Ft+ε ∨ σ(τ) it follows that EP∗ (yt+δ|Gτ t+ε) = EP∗ (yt+δ|Ft+ε)|u=τ = (yt+δ)|u=τ = yt+δ = EP∗ (yt+δ|Ft+ε).
- 49. Pricing and Hedging of Defaultable Models 41 From the right-continuity of F we get that lim ε→0 EP∗ (yt+δ|Ft+ε) = EP∗ (yt+δ|Ft) and lim ε→0 [h(τ)EP∗ (EP∗ (yt+δ|Ft+ε)|Gτ t+)] = h(τ)EP∗ (EP∗ (yt+δ|Ft)|Gτ t+). Since ∀ t ≥ 0, Gτ t+ ⊃ Gτ t ⊃ Ft, we can omit the conditional expectation with respect to σ-algebra Gτ t+ and in the result we obtain that h(τ)EP∗ (yt+δ|Ft). Now from the independence of τ and F we can replace Ft by Gτ t and put h(τ) inside the conditional expectation what follows that h(τ)EP∗ (yt+δ|Ft) = EP∗ (h(τ)yt+δ|Gτ t ) = EP∗ (Z0 t+δ|Gτ t ). As a result, we obtained that EP∗ (Z0 t+δ|Gτ t+) = EP∗ (Z0 t+δ|Gτ t ). ii) Since the property is fulﬁlled for the random variables Z0 t+δ of the form yt+δh(τ), which are Ft+δ ⊗ B(R+ ), using the property of the mathe- matical expectation, we can state that (5.7) is satisﬁed for the sum of such variables. From the Proposition 5.2 which establishes form of Gτ t+δ-measurable random variable and by passing to the limit, we obtain that (5.7) is satisﬁed for any Gτ t+δ-measurable random variable Zt+δ. iii) Since Gτ t+δ ⊃ Gτ t+ε ⊃ ε>0 Gτ t+ε = Gτ t+ we can apply the result from ii) to any Gτ t+-measurable random variable Zt+, hence Zt+ = EP∗ (Zt+|Gτ t+) = EP∗ (Zt+|Gτ t ). Since P∗ ∼ P and G0 contains all P-negligible events, Zt+ is also Gt- measurable.
- 50. 42 Chapter 5. The Carthaginian enlargement of filtrations 5.5.1 The survival process under measure P and P∗ Let us ﬁnally introduce the conditional survival process R applying the density approach under the measure P and P∗ . More precisely, Rt := P(τ > t|Ft) = ∞ t qt(u)η(du), R∗ t := P∗ (τ > t|Ft) = ∞ t η(du). The form of Rt is straightforward while the form of R∗ t requires more detailed explanation. Due to the properties of the measure P∗ in relation with the measure P (see section 5.5) we have P∗ (τ > t|Ft) = P∗ (τ > t) and P∗ (τ > t) = P∗ (τ > t|F0) = P(τ > t|F0) = P(τ > t) = ∞ t η(du). As a result we obtained that P∗ (τ > t|Ft) = ∞ t η(du). Remark 5.2. Properties of the process R i) (R∗ )t≥0 is a deterministic, continuous and decreasing function; i) (Rt)t≥0 is an (P, F)-supermartingale (called Azéma supermartingale).
- 51. Chapter 6 The initial enlargement framework In this chapter we explore some propositions concerning the expectation tools and the martingales characterization in the initially enlarged ﬁltrations. We assume that the Hypothesis 5.1 is satisﬁed throughout the entire chapter and we show ﬁnally that it is a suﬃcient condition for the defaultable market to be arbitrage-free for the agent with initially enlarged ﬁltration as an information ﬂow. Let us introduce ﬁrstly an auxiliary lemma which will be used in the proofs below. Lemma 6.1. [2] Let Zt = zt(τ) be a Gτ t -measurable, P-integrable random variable and zt(τ) = 0 P − a.s. Then, for η-a.e. u ≥ 0, zt(u) = 0 P − a.s. Proof. Since zt(τ) is integrable, EP(|zt(τ)|) < ∞. On the other hand, zt(τ) = 0 P-a.s. Therefore, if we put the conditional expectation on both sides and apply the tower property thereafter, we will obtain that EP(zt(τ)) = EP(0) = 0 and 0 = EP(zt(τ)) = EP(EP(zt(τ))|Ft) = EP(EP(zt(τ)|Ft)). From the Hypothesis 5.1 we obtain that EP(EP(zt(τ)|Ft)) = EP ∞ 0 |zt(u)|qt(u)η(du) 43
- 52. 44 Chapter 6. The initial enlargement framework and from the previous results EP ∞ 0 |zt(u)|qt(u)η(du) = 0. Due to the fact that ∀t ≥ 0 zt(u) ≥ 0, ∀u ≥ 0 (qt(u))t≥0 is a strictly positive martingale P-a.s. and η is a positive measure, we get that ∞ 0 |zt(u)|qt(u)η(du) ≥ 0. Given that the expected value from this integral is equal 0, we conclude that ∞ 0 |zt(u)|qt(u)η(du) = 0 P − a.s. Again, from the fact that (qt(u))t≥0 is a strictly positive process P-a.s. and η is a positive measure, we obtain that for η − a.e. u ≥ 0 zt(u) = 0 − P-a.s. 6.1 Expectation tools In the following lemma we make precise how to express the Gτ s -conditional ex- pectation in terms of the Fs-conditional expectation under the same measure P. Lemma 6.2. [2] Let Zt = zt(τ) be Gτ t -measurable. For s ≤ t, if zt(τ) is P-integrable then, EP(zt(τ)|Gτ s ) = 1 qs(τ) EP(zt(u)qt(u)|Fs)|u=τ . Proof. Since P and P∗ are equivalent on Gτ s and L = ( 1 qt(τ) )t≥0 is a Radon- Nikodým density of P∗ with respect to P, we can apply the Bayes formula (see Lemma 5.1) to obtain EP(zt(τ)|Gτ s ) = EP∗ (L−1 t zt(τ)|Gτ s ) L−1 s . Using the explicit form for Lt, we get EP∗ (L−1 t zt(τ)|Gτ s ) L−1 s = EP∗ (qt(τ)zt(τ)|Gτ s ) qs(τ) . Eventually, from the Proposition 5.6, we have EP∗ (qt(τ)zt(τ)|Gτ s ) qs(τ) = EP∗ (qt(u)zt(u)|Fs)|u=τ qs(τ) .
- 53. Pricing and Hedging of Defaultable Models 45 Since P and P∗ coincide on Fs EP(zt(τ)|Gτ s ) = EP(qt(u)zt(u)|Fs)|u=τ qs(τ) . 6.2 The martingales characterization Our task now is to ﬁnd a characterization of (P, Gτ )-martingales in terms of (P, F)-martingales. Let us consider the following proposition. Proposition 6.1. [2] A process Z = z(τ) is a (P, Gτ )-martingale if and only if the process (zt(u)qt(u))t≥0 is a (P, F)-martingale, for η-a.e. u ≥ 0. Proof. Let us prove ﬁrstly the necessity condition by assuming that Z is a (P, Gτ )-martingale. As a result, we have zs(τ) = EP(zt(τ)|Gτ s ). Using the Lemma 6.2, we get that EP(zt(τ)|Gτ s ) = 1 qs(τ) EP(zt(u)qt(u)|Fs)|u=τ and hence, zs(τ)qs(τ) = EP(zt(u)qt(u)|Fs)|u=τ . zs(τ)qs(τ) − EP(zt(u)qt(u)|Fs)|u=τ is a Gτ s -measurable random variable and it is equal to 0. Therefore, we can use the Lemma 6.1 and write that η-a.s. for all u > 0 zs(u)qs(u) − EP(zt(u)qt(u)|Fs) = 0. Finally we have that η-a.s. for all u > 0 EP(zt(u)qt(u)|Fs) = zs(u)qs(u), which proves that (zt(u)qt(u))t≥0 is a (P, F)-martingale.
- 54. 46 Chapter 6. The initial enlargement framework To prove the suﬃciency part, let us assume that the process (zt(u)qt(u))t≥0 is a (P, F)-martingale, for η-a.e. u ≥ 0. We have to show that EP(Zt|Gτ s ) = Zs. If we apply the Lemma 6.2 for the left-hand side we obtain that EP(Zt|Gτ s ) = EP(zt(τ)|Gτ s ) = 1 qs(τ) EP(zt(u)qt(u)|Fs)|u=τ . From the martingale property stated above, we get 1 qs(τ) EP(zt(u)qt(u)|Fs)|u=τ = 1 qs(τ) (zs(u)qs(u))|u=τ = zs(τ). 6.3 The E-hypothesis and the absence of arbi- trage in the ﬁltration Gτ We shall remind in the beginning the general condition for the absence of arbitrage. It is a well-known fact that if there exists at least one martingale measure (a measure equivalent to the physical measure such that a stock price process is a martingale with respect to the given ﬁltration), i.e. the set of all martingales measures is not empty, then the market is arbitrage-free. Let us now consider a default-free and arbitrage-free market with assets remaining assets of the full ﬁltration Gτ . We set Q as one of the martingale measures equivalent to P on F and assume that the set of measures equivalent to P on Gτ is non-empty. We showed before that if E-hypothesis holds, then there exists a decoupling measure P∗ making τ independent of the reference ﬁltration and coinciding with Q on F (see section 5.5). As a result, P∗ preserves martingale property in the initially enlarged ﬁltration and a set of martingale measures equivalent to P on Gτ is non-empty. Therefore, E- hypothesis is a suitable condition to make the defaultable market arbitrage- free for the agent with the initially enlarged ﬁltration as an information ﬂow.
- 55. Chapter 7 The progressive enlargement framework 7.1 The intensity approach In this section we study the progressive enlargement of ﬁltration which we introduced before in the preceding chapter. Hereafter, we assume that the price process follows the log-normal distribution. Thus, the ﬁltration F is considered as a Brownian ﬁltration (i.e. F = (Ft)t≥0 and Ft = σ(Bs, s ≤ t), where Bs is a standard Brownian motion). It is not necessary to require the Hypothesis 5.1 to hold. We can easily describe Gt-measurable events on the set {τ > t}. Any Gt- measurable random variable Xt satisﬁes XtI{τ>t} = YtI{τ>t}, where Yt is an Ft-measurable random variable. 7.1.1 Expectation tools Proposition 7.1. Let ζ be an integrable random variable. Let T be a ﬁxed time horizon. Then, for any t < T, EP(ζ|Gt)I{τ>t} = I{τ>t} EP(ζI{τ>t}|Ft) EP(I{τ>t}|Ft) . Proof. Since ζ is an integrable random variable we can write the conditional expectation Xt = EP(ζ|Gt) 47
- 56. 48 Chapter 7. The progressive enlargement framework which is a Gt-measurable random variable. We make the assumption that Gt ⊂ G∗ t , where G∗ t = {A ∈ Gt : ∃B ∈ Ft A ∩ {τ > t} = B ∩ {τ > t}}. Thus, there exists an Ft-measurable random variable Yt such that XtI{τ>t} = YtI{τ>t}. Taking the conditional expectation with respect to Ft from both sides we obtain EP(XtI{τ>t}|Ft) = EP(YtI{τ>t}|Ft). Knowing that Yt is an Ft-measurable random variable we take it outside the conditional expectation and get EP(XtI{τ>T}|Ft) = YtEP(I{τ>t}|Ft). Thus, Yt = EP(XtI{τ>t}|Ft) EP(I{τ>t}|Ft) . We have EP(ζ|Gt)I{τ>t} = YtI{τ>t} = I{τ>t} EP(XtI{τ>t}|Ft) EP(I{τ>t}|Ft) . We get that Xt = EP(ζ|Gt) and I{τ>t} is Ft-measurable. Hence we have I{τ>t} EP(XtI{τ>t}|Ft) EP(I{τ>t}|Ft) = I{τ>t} I{τ>t}EP(EP(ζ|Gt)|Ft) EP(I{τ>t}|Ft) . From the fact that F ⊂ G we deduce I{τ>t} I{τ>t}EP(EP(ζ|Gt)|Ft) EP(I{τ>t}|Ft) = I{τ>t} I{τ>t}EP(ζ|Ft) EP(I{τ>t}|Ft) . Consequently, using again the fact that I{τ>t} is Ft-measurable, we obtain I{τ>t} I{τ>t}EP(ζ|Ft) EP(I{τ>t}|Ft) = I{τ>t} EP(ζI{τ>t}|Ft) EP(I{τ>t}|Ft) .
- 57. Pricing and Hedging of Defaultable Models 49 Corollary 7.1. Let X be a Gt-measurable integrable random variable. Let T be a ﬁxed time horizon. Then, for any t ∈ [0, T), EP(X|Gt)I{τ>t} = I{τ>t} EP(XI{τ>t}|Ft) EP(I{τ>t}|Ft) . Proof. When we put X = EP(X|Gt) we can use the proof of the Proposition 7.1 to obtain the result. Proposition 7.2. Let X be a GT -measurable random variable where T is a ﬁxed time horizon. Then EP(X|Gt)I{τ>t} = I{τ>t} EP(XI{τ>t}|Ft) EP(I{τ>t}|Ft) . Proof. We have that EP(X|Gt) is Gt-measurable. Hence, there exists an Ft- measurable random variable Yt such that EP(X|Gt)I{τ>t} = YtI{τ>t}. Taking the conditional expectation with respect to Ft from both sides we obtain EP(EP(X|Gt)I{τ>t}|Ft) = EP(YtI{τ>t}|Ft). Knowing that Yt is Ft-measurable and I{τ>t} is Ht-measurable, we can write I{τ>t}EP(EP(X|Gt)|Ft) = I{τ>t}EP(X|Ft) = EP(XI{τ>t}|Ft) = YtEP(I{τ>t}|Ft). We obtain Yt = EP(XI{τ>t}|Ft) EP(I{τ>t}|Ft) . Then EP(X|Gt)I{τ>t} = I{τ>t} EP(XI{τ>t}|Ft) EP(I{τ>t}|Ft) . Proposition 7.3. Let X be a GT -measurable integrable random variable and T a ﬁxed time horizon, then EP(XI{τ>T}|Gt) = I{τ>t} EP(XI{τ>T}|Ft) EP(I{τ>t}|Ft) .
- 58. 50 Chapter 7. The progressive enlargement framework Proof. We can write the expectation on two sets EP(XI{τ>T}|Gt) = I{τ>t}EP(XI{τ>T}|Gt) + I{τ≤t}EP(XI{τ>T}|Gt) Then we can take the indicators inside the expectations because they are Gt-measurable. We get EP(XI{τ>t}I{τ>T}|Gt) + EP(XI{τ≤t}I{τ>T}|Gt). The second term on the right-hand side vanishes and we get I{τ>t}EP(XI{τ>T}|Gt). Finally, from the Proposition 7.2 we have I{τ>t} EP(XI{τ>T}I{τ>t}|Ft) EP(I{τ>t}|Ft) = I{τ>t} EP(XI{τ>T}|Ft) EP(I{τ>t}|Ft) . 7.1.2 The F-hazard process (Γt)t≥0 In the previous chapter we introduced a hazard function in a framework of the ﬁltration H. We had a supposition that the agent pricing the defaultable contingent claims knows only the distribution function F(t) = P(τ ≤ t) of default time τ. Accordingly, the hazard function was purely deterministic. Nevertheless, in this chapter we assume that the agent also observes the price process. Thus, we add to our study this information and the hazard function is no more deterministic. Moreover, while calculating the probability of τ we take under consideration the ﬂow of information about the prices process F. We denote ˜F(t) = P(τ ≤ t|Ft) and make the following assumption. Assumption 7.1. We assume that ∀t ≥ 0 ˜F(t) < 1. Consequently, we deﬁne an F-hazard process as follows. Deﬁnition 7.1. We call a process Γ = (Γt)t≥0 an F-hazard process where Γt = − ln(1 − ˜F(t)). We can easily check that the process is a submartingale.
- 59. Pricing and Hedging of Defaultable Models 51 Proposition 7.4. P(τ ≤ t|Ft) is an F-submartingale. Proof. We have EP(P(τ ≤ t|Ft)|Fs) = EP(EP(I{τ≤t}|Ft)|Fs) = EP(I{τ≤t}|Fs) = P(τ ≤ t|Fs) and clearly P(τ ≤ t|Fs) ≥ P(τ ≤ s|Fs) Thus, ˜F(t) is an F-submartingale. Proposition 7.5. Let T be a ﬁxed time horizon and Y be an FT -measurable integrable random variable. Then EP(Y I{τ>t}|Gt) = I{τ>t}EP(Y exp(Γt − ΓT )|Ft). Proof. If Y is FT -measurable, then Y is GT -measurable because FT ⊂ FT ∨ HT = GT . According to the Proposition 7.3 we have EP(Y I{τ>T}|Gt) = I{τ>t} EP(Y I{τ>T}|Ft) EP(I{τ>t}|Ft) . From the deﬁnition of a hazard process we have EP(I{τ>t}|Ft) = P(τ > t|Ft) = exp(−Γt), which is Ft-measurable. Function f(x) = 1 x , where x ∈ R+ {0}, is a Borel function. Hence, if exp(−Γt) is Ft-measurable then 1 exp(−Γt) is Ft-measurable. Thus, we can take 1 exp(−Γt) inside the expectation in the nominator and obtain I{τ>t}EP(Y exp(Γt)I{τ>T}|Ft). From the tower property we can condition the expectation in the nomina- tor with a bigger σ-algebra FT . exp(Γt) is Ft-measurable so it is also FT - measurable because Ft ⊂ FT . We can take Y exp(Γt) outside this expected value because the function f(x, y) = xy is a Borel function so Y exp (Γt) is FT -measurable. We get I{τ>t}EP(Y exp(Γt)EP(I{τ>T}|FT )|Ft).
- 60. 52 Chapter 7. The progressive enlargement framework Then again from the deﬁnition of a hazard process we have EP(I{τ>T}|FT ) = P(τ > T|FT ) = exp(−ΓT ). We obtain the result I{τ>t}EP(Y exp(Γt) exp(−ΓT )|Ft). Corollary 7.2. Let T be a ﬁxed time horizon. Then EP(I{τ>T}|Gt) = I{τ>t}EP(exp(Γt − ΓT )|Ft). Proof. The proof is straightforward from the Proposition 7.5. 7.1.3 The G-intensity of τ In the enlarged ﬁltration we can also deﬁne a G-intensity of default time τ. From the Deﬁnition 4.2, ˜λs is a G-intensity of τ if i) ˜λs is a G-adapted non-negative predictable process, ii) (I{τ≤t} − t 0 ˜λuI{τ≥u}du)t≥0 is a (P, G)-martingale. Proposition 7.6. If ( ˜kt)t≥0 is a G-predictable bounded process, then there exists an F-predictable bounded process (kt)t≥0 such that ˜ktI{τ≥t} = ktI{τ≥t}. Proof. On the set I{τ≥t}, i.e. before the default appears, we do not have any information about the distribution of default time. We observe the default only when it occurs. From this it follows that on I{τ≥t} any G-predictable process is an F-predictable process (kt)t≥0, i.e. ˜ktI{τ≥t} = ktI{τ≥t}. Corollary 7.3. There exists an F-predictable process λ = (λt)t≥0, such that ˜λtI{τ≥t} = λtI{τ≥t} and (Nt − t 0 λ(u)I{τ≥u}du)t≥0 is a G-martingale.
- 61. Pricing and Hedging of Defaultable Models 53 Proof. The existence follows from the Proposition 7.6. In the future calculations we change the measure so that τ is independent of F. Thus, we consider the following proposition. Proposition 7.7. If τ is independent of F then the G-intensity of τ on the set I{τ>t} is λs = f(s) 1 − F(s) . Equivalently, ˜λsI{τ>t} = f(s) 1 − F(s) I{τ>t}. 7.1.4 H-hypothesis and the absence of arbitrage in the ﬁltration G The H-hypothesis Let us consider the H-hypothesis (or the immersion property) which is strongly related to the absence of arbitrage in the progressively enlarged ﬁltration. Hypothesis 7.1. (H-hypothesis) Every square-integrable F-martingale re- mains square-integrable G-martingale. It is also pivotal to give the conditions equivalent to the H-hypothesis. Proposition 7.8. [1] The following statements are equivalent: (H) Every F-square integrable martingale is a G-square integrable mar- tingale, (H1) ∀t ≥ 0, ∀Y ∈ F∞, ∀X ∈ Gt E(Y X|Ft) = E(Y |Ft)E(X|Ft), (H2) ∀t ≥ 0, ∀X ∈ Gt E(X|F∞) = E(X|Ft), (H3) ∀t ≥ 0, ∀Y ∈ F∞, E(Y |Gt) = E(Y |Ft), (H4) ∀s ≤ t, P(τ ≤ s|F∞) = P(τ ≤ s|Ft). For the proof see [1]. The absence of arbitrage
- 62. 54 Chapter 7. The progressive enlargement framework Let us consider a default-free market with the property of the absence of arbitrage and assume that assets of the reference ﬁltration F remain assets of the full ﬁltration G. Moreover, Q is a martingale measure equivalent to P on F. Consequently, if the immersion property holds under a risk- neutral measure Q, i.e. F-martingales are G-martingales, then the set of all martingale measures equivalent to P on G contains already the measure Q. Therefore, it is not empty and the market is arbitrage-free. 7.1.5 The value of information To price the contingent claims we have to use the martingale measure. Let us consider a complete market with the risk-free interest rate r(t) ≡ 0 and B0 = 0. Equivalently, every FT -measurable claim Y is hedgeable and the price of Y is EQ(Y |Ft), where Q is a martingale measure equivalent to P. We consider default time such that {τ > T} is GT -measurable. Let us denote by Cun t the price of the defaultable contingent claim Y calculated by an agent who knows only the price process (i.e. the agent does not know the distribution of default time τ) and by Cin t - the price of Y calculated by the agent who knows the price process as well as observes the default when it happens (i.e. the agent knows the distribution of default time τ). Let us study the following proposition. Proposition 7.9. The diﬀerence between the prices calculated by these two agents is Cin t − Cun t = EQ(Y I{τ>T}|Ft) 1 EQ(I{τ>T}|Ft) − 1 . Proof. The informed agent knows the price process and the default distribu- tion. It means that at time t he has the information Gt. We can write Cin t = EQ(Y I{τ>T}|Gt. Further, from the Proposition 7.3 we have Cin t = I{τ>t} EQ(Y I{τ>T}|Ft) EQ(I{τ>t}|Ft) . At time t the uninformed agent has only the information Ft. Hence, Cun t = EQ(Y I{τ>T}|Ft). Finally, Cin t − Cun t = EQ(Y I{τ>T}|Ft) 1 EQ(I{τ>T}|Ft) − 1 .
- 63. Pricing and Hedging of Defaultable Models 55 7.2 The density approach Let us now present alternative approach for the default modelling, namely the density approach. It was proved in [3] that the G-intensity (see 7.1.3) can be completely derived from the conditional density process q(u). However, given the G-intensity, we can only obtain the knowledge of qt(u) for u ≥ t. As a result, the intensity-based approach is not suitable for after the default case. In the following subsection we study the projections tools. The lemmas below allow us to express σ(τ)- and a Gτ t -measurable random variable in terms of a Gt-measurable random variable, i.e. project the results obtained in Gτ and the ﬁltration generated by σ(τ) on the ﬁltration G. 7.2.1 Projection tools We begin with a G-projection of a σ(τ)-measurable random variable. Lemma 7.1. [2] Let V = h(τ) be σ(τ)-measurable and P-integrable random variable. Then, for s ≤ t, EP(V |Gs) = EP(h(τ)|Gs) = ˜ysI{τ>s} + h(τ)I{τ≤s}, with ˜ys = 1 Rs ∞ s v(u)qt(u)η(du), where ˜ys is an Fs-measurable random variable and h - a Borel function on R+ . See [2] for the proof. In the lemma below we establish analogous results for a Gτ t -measurable random variable. Lemma 7.2. [2] Let Zt = zt(τ) be a Gτ t -measurable and P-integrable random variable. Then, for s ≤ t, EP(Zτ t |Gs) = EP(zt(τ)|Gs) = ˜ysI{τ>s} + ˆzs(τ)I{τ≤s}, with ˜ys = 1 Rs E ∞ s zt(u)qt(u)η(du)|Fs , ˆzs(u) = 1 qs(u) EP(zt(u)qt(u)|Fs).
- 64. 56 Chapter 7. The progressive enlargement framework For the proof see [2]. As an application, let us consider the following lemma which by projecting the martingale L deﬁned earlier in section 5.5 gives us a Radon-Nikodým density on G. Lemma 7.3. [3], [2] Let us assume that P∗ is equivalent to P on G. Then, there exists a process l = (lt)t≥0 such that dP|Gt = ltdP∗ |Gt , i.e. ∀t ≥ 0 lt deﬁnes the corresponding Radon-Nikodým density on Gt. More- over, lt = EP(Lt|Gt) = I{τ>t} Rt R∗ t + I{τ≤t} 1 qt(τ) , where L was deﬁned earlier as (Lt)t≥0 with Lt = 1 qt(τ) . For the proof, see [3]. 7.2.2 The H-hypothesis and special property of the con- ditional density process Let us now study the relation between the H-hypothesis introduced in the Proposition 7.8 and the conditional density process q(u) with the special property shown in the Remark 5.1, namely qt(u) = qu(u), ∀t ≥ u ≥ 0 dP − a.s. One can consider the following proposition. Proposition 7.10. [3] We recall the H-hypothesis, in the form of (H2), which can be stated as: for any ﬁxed t and any bounded Gt-measurable random variable Xt, EP(Xt|F∞) = EP(Xt|Ft) P − a.s. Then, the H-hypothesis is fulﬁlled if and only if qt(u) = qu(u), ∀t ≥ u ≥ 0 dP − a.s. One can ﬁnd the proof in [3].
- 65. Pricing and Hedging of Defaultable Models 57 Remark 7.1. In the subsection 7.1.4 we established that the H-hypothesis satisﬁed under a risk-neutral measure is a suitable condition for the absence of arbitrage. If we combine this result with the proposition above, we can state that to provide the arbitrage-free market, it is suﬃcient to introduce the Hypothesis 5.1 and assume that the new information does not change the conditional distribution of τ. 7.2.3 The martingales characterization In this subsection we give some results concerning the characterization of (P, G)-martingales in terms of (P, F)-martingales. Proposition 7.11. [2] A G-adapted process X =(Xt)t≥0, given by Xt := ˜xtI{τ>t} + ˆxt(τ)I{τ≤t}, is a (P, G)-martingale if and only if the following con- ditions are satisﬁed i) the process (ˆx(u)qt(u))t≥u is a (P, G)-martingale for η-a.e. u ≥ 0; ii) the process m =(mt)t≥0 with mt := EP(Xt|Ft) = ˜xtZt + t 0 ˆzt(u)qt(u)η(du), is a (P, F)-martingale. For the proof, see [2].
- 66. 58 Chapter 7. The progressive enlargement framework
- 67. Chapter 8 Pricing and hedging of Black-Scholes type models with default 8.1 The model evaluation and the description of the task We consider two companies which may be related to each other. The default event is triggered by the second company while the ﬁrst company is default- free with respect to that default. However, it does not necessarily mean that the ﬁrst company is default-free in general. We assume that the regular investor observes only the stock prices of the default-free company (1) but he wants to price a European call option written on the investment consisting of • a stock of the company (1), • a defaultable corporate bond issued by the company (2) (see Figure 8.1). One may interpret this situation in the following way. The issuer of the option knows that the companies may be correlated. Thus, he adds to the stock of the default-free company, a defaultable corporate bond issued by the defaultable company as an additional gain opportunity. Basic assumptions 59
- 68. 60 Chapter 8. Pricing and hedging of Black-Scholes type models with default Figure 8.1: The task model. We ﬁx T as a maturity time for the option and from now on we consider all the price processes and ﬁltrations up to moment T. Let (Ω, G, P) be a probability space on which we deﬁne two-dimensional standard Brownian motion W = (W (1) t , W (2) t )t∈[0,T]. We endow (Ω, G, P) with a ﬁltration F(1) = (F (1) t )0≤t≤T generated by W(1) , i.e. ∀ t ∈ [0, T] F (1) t = σ(W (1) s , 0 ≤ s ≤ t). The default-free market We consider a default-free Black-Scholes market (B, S(1) ) consisting of one riskless asset B = (Bt)t∈[0,T] and one risky asset S(1) = (S (1) t )t∈[0,T]. Their prices follow the random walk with the dynamics dBt = rBtdt, t ∈ [0, T], B0 = 1, dS (1) t = S (1) t (µ(1)dt + σ(1)dW (1) t ), t ∈ [0, T], (8.1) where r, µ(1), σ(1) are real constants, σ(1) > 0. For the simplicity we put r = 0. The defaultable market Furthermore, we can establish a defaultable market (B,S(1) ,S(2) ) by adding to the default-free market one defaultable asset S(2) = (S (2) t )t∈[0,T] which follows the random walk with the dynamics dS (2) t = S (2) t (µ(2)dt + σ(2)dW (2) t ), t ∈ [0, T], where µ(2), σ(2) are real constants, σ(2) > 0. The processes S(1) and S(2) represent the stock price processes of respectively company (1) and (2). Ad- ditionally, we assume that there is a defaultable bond traded in the market. The bond consists of a payment of one monetary unit at time T if and only if default has not occurred before time T, i.e. the payment is I{τ>T}.
- 69. Pricing and Hedging of Defaultable Models 61 The default time Finally, we deﬁne default time τ as the ﬁrst time when the stock price process S(2) hits a barrier a, i.e. τ = inf{t ∈ [0, T] : S (2) t ≤ a}, (8.2) where 0 < a < 1. Since we determined all the price processes up to the Table 8.1: Deﬁnition of default time τ {t ∈ [0, T] : S (2) t ≤ a} = ∅ {t ∈ [0, T] : S (2) t ≤ a} = ∅ ↓ ↓ τ = inf{t ∈ [0, T] : S (2) t ≤ a} τ = T Figure 8.2: Default time occurs before time T Figure 8.3: Default time occurs after time T maturity time T we have to take under consideration the fact that the default may not occur before time T. If this is the case, then the set in (8.2) is empty and inf of such a set is equal to ∞. To avoid this situation, we put τ equal to T instead. Nevertheless, if the barrier a was crossed at least once by the process S(2) in the time interval [0, T], then τ is min of all time moments for which it occurred. As a result, we have to consider a random variable of the form τ ∧ T. The insider Let us now present a special type of an investor trading in such a de- faultable market, we call this investor an insider of the company (2). The insider observes the prices of the stock (1) and has access to some additional
- 70. 62 Chapter 8. Pricing and hedging of Black-Scholes type models with default information concerning default time τ of the company (2). In our case this additional information consists of the distribution of τ. Moreover, the in- sider has it from the beginning, i.e. the ﬁltration F(1) is enlarged with the default time τ in an initial manner (see chapter 5). Consequently, we make precise that the information available to the insider at time t is represented by σ-algebra Gτ t = Ft ∨σ(τ). We assume that the regular investor who wants to price the option knows that there exists an insider in the market. The wealth process Let us deﬁne Xφ , where Xφ = (Xφ t )t∈[0,T], as a wealth process obtained by the regular investor using a self-ﬁnancing strategy φ, where φ = (φt)t∈[0,T] with φt = (φS t , φB t ) - an F(1) -predictable strategy. We remind that the self- ﬁnancing property means that no money is withdrawn or added to the portfo- lio. More precisely, (φS t )t∈[0,T] indicates the ﬁnancial position of the investor in the stocks of the company (1) and (φB t )t∈[0,T] describes the position in riskless bonds. Speciﬁcally, if we denote by π = (πt)t∈[0,T] ratio of wealth from shares of the company (1) and the whole wealth Xφ then 1 − π = (1 − πt)t∈[0,T] is the ratio of wealth from bonds and the whole wealth Xφ . We can write φS uSu = Xφ u πu and φB u Bu = Xφ u (1 − πu). Therefore, the wealth at time t is deﬁned as Xφ t = x + t 0 φS udS(1) u + t 0 φB u dBu, where x is the initial capital. Furthermore, by (8.1) we obtain Xφ t = x + t 0 µ(1)πu + r(1 − πu) Xudu + t 0 πuXudW(1) u .
- 71. Pricing and Hedging of Defaultable Models 63 Let us make precise in the end of this chapter that our goal is to price a European call option written on the investment consisting of a default-free and defaultable assets. However, we want to ﬁnd this price for the regular investor whose information ﬂow is only the ﬁltration generated by the stock price process of the default-free company. Finally, we assume that the regular investor knows that there exists an insider in the market. 8.2 Methods of pricing in arbitrage-free and in- complete market In this section we begin with explaining why the defaultable market is arbitrage-free and incomplete for the regular investor. Then, we continue with ﬁnding a pricing measure via minimizing f-divergence method which is strongly related to the utility approach. 8.2.1 The arbitrage-free market From the fact that default time τ and the reference ﬁltration are independent under the physical measure P, P admits the properties of the decoupling measure. Thus, it preserves the martingale properties in the initially enlarged ﬁltration and according to the section 6.3 the market is arbitrage-free. As a result, there exists at least one martingale measure. 8.2.2 The incomplete market The incompleteness of the market is caused by the inﬂuence on the stock prices by an informed investor. The additional information is considered as a strong initial information which we model by the initial enlargement. On the one hand, for an informed investor the inﬂuenced market is complete. On the other hand, for a common investor it is incomplete which means that there exists more than one martingale measure. Consequently, one of the most challenging tasks is to choose a martingale measure for pricing ﬁnancial derivatives. 8.2.3 The f-divergence minimization approach A common method of pricing derivatives in incomplete market is to base the prices on a martingale measure which minimizes certain distance, namely f- divergence which measures the diﬀerence between the probability measures P and Q.
- 72. 64 Chapter 8. Pricing and hedging of Black-Scholes type models with default Deﬁnition 8.1. If f is a convex function on [0, ∞), Q and P are the prob- ability measures such that Q << P and dQ dP is a Radon-Nikodým density of Q with respect to P, then we call f(Q|P) = EP f dQ dP an f-divergence of Q with respect to P. As a function f we can take for example f(x) = ( √ x − 1)2 , f(x) = 2(1 −√ x) (Hellinger distances) or f(x) = |x − 1| (total variation distance). The standard approach is to choose as a pricing measure f-divergence minimal equivalent martingale measure Q∗ such that EP f dQ∗ dP = inf Q∈M(P) EP f dQ dP , where M(P) is the set of all martingale measures equivalent to P. It is crucial in our case that the f-divergence minimization is closely related to the utility maximization via the Legendre transform. Let us introduce now the utility approach in more details. 8.2.4 The utility approach The utility approach is based on the fact that one has to estimate the value of some (defaultable) contingent claim seen from the perspective of an agent who optimizes his behavior relative to some utility function. The utility function measures the investor’s satisfaction. Therefore, in this section we have to introduce brieﬂy some known results concerning the maximizing expected utility theory. Let us begin with the following deﬁnition. Deﬁnition 8.2. We deﬁne the utility function u as a strictly increasing, strictly concave, twice continuously diﬀerentiable function on dom(u) = {x ∈ R, u(x) > −∞} which satisﬁes u (∞) = lim x→∞ u (x) = 0, u (x) = lim x→x u (x) = ∞, where x = inf{u ∈ dom(u)}.
- 73. Pricing and Hedging of Defaultable Models 65 It is important to comment the deﬁnition above. We require that the utility function is an increasing function of wealth because with the growth of wealth the usefulness which the investor has also grows. The concavity of the function stands for an investor who is risk-averse. The utility function’s slope gets ﬂatter as the wealth increases. It means that the ﬁrst unit of wealth yields more utility (satisfaction) than the second and subsequent units. The standard utility functions Let us consider three standard utility functions: i) u(x) = 1 − e−x , (8.3) ii) u(x) = lnx, (8.4) iii) u(x) = xp p , p ∈ (−∞, 0) ∪ (0, 1). (8.5) The maximization of the expected value of the power utility u(x) = xp p , p ∈ (−∞, 0) ∪ (0, 1) is equivalent to the maximization of the expected rate of return compounded 1 pT times per year: 1 pT E XT x p − 1 . The values p < 0 correspond to the discount rate. With the increase in p the investor’s risk tolerance also increases. The case of the logarithmic utility function u(x) = lnx we consider as a limiting case of a power utility function as p → 0. The application is in the maximization of the expected continuously compounded growth rate: 1 T E ln XT x . The exponential utility function u(x) = 1−e−x corresponds to the entropic measure.
- 74. 66 Chapter 8. Pricing and hedging of Black-Scholes type models with default Reformulating the problem Our task now is to formulate the problem in terms of the expected utility theory. We begin with reminding that the wealth at time t obtained using strategy φ is deﬁned as Xφ t = Xφ 0 + t 0 φS udS(1) u + t 0 φB u dBu, where Xφ 0 is the initial capital and φ = (φt)0≤t≤T with φt = (φS t , φB t ) is the self-ﬁnancing strategy (see section 8.1). Let us assume for the simplicity that the risk-free interest rate r = 0. It means that Xφ t = Xφ 0 + t 0 φS udS(1) u . According to [5], to avoid phenomena like doubling strategies (doubling the position), we make an assumption that during the trading the losses do not exceed a ﬁnite credit line, i.e. ∃b > 0 such that ∀t ∈ [0, T] t 0 φS udS(1) u ≥ −b. We say that such a strategy φ is admissible. The preferences of the in- vestor are represented by the utility functions described above. The resulting optimization problem is of the form sup φ∈A EP(u(Xφ T )) = sup φ∈A EP(u(Xφ 0 + T 0 φS udS(1) u )) = EP(u(Xφ 0 + T 0 φ∗S u dS(1) u )), where A is a set of admissible strategies. The Legendre transformation and the dual approach Let us now explain brieﬂy the relationship between the utility maximiza- tion and the f-divergence minimization. We start with the deﬁnition of the Legendre transformation. Deﬁnition 8.3. If u : R → R is twice continuously diﬀerentiable and ∀x ∈ R u (x) < 0 (u is concave), then we call ˆu(x) = u(I(y)) − yI(y) a Legendre transform of u, where I = (u )−1 .
- 75. Pricing and Hedging of Defaultable Models 67 The function ˆu which we obtain in this case is convex. However, u is also a Legendre transform of ˆu and u(x) = inf y∈R {ˆu(y) + xy} = ˆu(I(x)) + xI(x). This property is called duality. Here we give the form of the convex duals of the standard utility functions given by (8.3), (8.4) and (8.5). i) u(x) = 1 − e−x → ˆu(x) = 1 − x + xlnx (8.6) ii) u(x) = lnx → ˆu(x) = −lnx − 1 (8.7) iii) u(x) = xp p , p ∈ (−∞, 0) ∪ (0, 1) → ˆu(x) = − p − 1 p x p p−1 . (8.8) Via the Legendre transformation one can obtain the equivalent problem in the following form sup φ∈A EP(u(Xφ T )) = inf y>0 {Xφ 0 y + EP(ˆu(y dQ∗ T dPT ))}, (8.9) where Q∗ is ˆu-minimal equivalent martingale measure. As a result we can base the price of the option in our task on the ˆu-minimal equivalent martin- gale measure. The problem now is to ﬁnd Q∗ such that EP ˆu dQ∗ dP = inf Q∈M(P) EP ˆu dQ dP . 8.3 Martingale measures on Gτ Let us denote the set of martingale measures equivalent to P on F(1) as MF(1) (P) and the set of martingale measures equivalent to P on Gτ as MGτ (P). Our goal now is to choose one measure from the set MGτ (P) as a pricing measure. We remind ﬁrstly that Q equivalent to P is a martingale measure on Gτ when the discounted price process S (1) t t∈[0,T] is a (Q, Gτ )- martingale. We have to consider prices as (Q, Gτ )-martingales since the reg- ular investor knows that in the market there is also an insider who inﬂuences the prices. For the simplicity we assumed that the risk-free interest rate r is equal to 0 and then ∀t ∈ [0, T] Bt = 1.
- 76. 68 Chapter 8. Pricing and hedging of Black-Scholes type models with default The ordinary investor with the public information ﬂow F(1) does not have the arbitrage opportunities since the default-free market in our case is arbitrage- free. In addition, it is complete. This means that there exists a uniquely deﬁned martingale measure Q equivalent to P such that the discounted price process S (1) t t∈[0,T] is a (Q, F(1) )-martingale, i.e. MF(1) (P) = {Q}. Let us denote p = (pt)t∈[0,T] as a corresponding density process, i.e. dQ|F (1) t = ptdP|F (1) t , i.e. ∀A ∈ F (1) t Q(A) = A ptdP. (8.10) It is well known that for a Black-Scholes market (B, S(1) ) and the ﬁltration F(1) (see section 8.1 for the deﬁnition) the density process p is deﬁned as such that pt = exp − θ2 t 2 − θW (1) t , where θ = µ(1) − r σ(1) . (8.11) In the end of the previous chapter we established that to ﬁnd a pricing measure it is necessary to ﬁnd a density process such that EP ˆu dQ∗ dP = inf Q∈M(P) EP ˆu dQ dP , where dQ dP has simply the following form dQ|Gτ T = PT (τ)dP|Gτ T . However, let us remind that PT (τ) has to be a positive random variable with EP(PT (τ)) = 1. We start with bounding EP ˆu dQ dP from below. Firstly we need to con- dition EP ˆu PT (τ) on F (1) T to obtain EP EP ˆu PT (τ) |F (1) T which from the tower property is equal to EP EP ˆu PT (τ) |F (1) T . Using Jensen inequality we get that EP EP ˆu PT (τ)|F (1) T ≥ EP ˆu EP PT (τ)|F (1) T .