Stochastic Calculus Financial Derivatives and PDE's Simone Calogero March 18, 2019 Contents 1 Probability spaces 3 1.1 σ-algebras and information . 3 1.2 Probability measure . 6 1.3 Filtered probability spaces . 10 1.4 The “infinite-coin tosses" probability space . 11 2 Random variables and stochastic processes 14 2.1 Random variables . 14 2.2 Distribution and probability density functions . 18 2.3 Stochastic processes . 27 2.4 Stochastic processes in financial mathematics . 32 3 Expectation 39 3.1 Expectation and variance of random variables . 39 3.2 Quadratic variation of stochastic processes . 52 3.3 Conditional expectation . 56 3.4 Martingales . 61 3.5 Markov processes . 64 4 Stochastic calculus 67 4.1 Introduction . 67 4.2 The It^ointegral of step processes . 68 4.3 It^o'sintegral of general stochastic processes . 72 4.4 Diffusion processes . 77 1 4.5 Girsanov's theorem . 82 4.6 Diffusion processes in financial mathematics . 84 5 Stochastic differential equations and partial differential equations 88 5.1 Stochastic differential equations . 89 5.2 Kolmogorov equations . 92 5.3 The CIR process . 96 5.4 Finite different solutions of PDE's . 100 6 Applications to finance 110 6.1 Arbitrage-free markets . 110 6.2 The risk-neutral pricing formula . 112 6.3 Black-Scholes price of standard European derivatives . 117 6.4 The Asian option . 128 6.5 Local and Stochastic volatility models . 134 6.6 Interest rate contracts . 143 6.7 Forwards and Futures . 155 6.8 Multi-dimensional markets . 163 6.9 Introduction to American derivatives . 174 A Numerical projects 183 A.1 A project on the Asian option . 183 A.2 A project on the CEV model . 184 A.3 A project on interest rate models . 185 B Solutions to selected problems 187 2 Chapter 1 Probability spaces 1.1 σ-algebras and information We begin with some notation and terminology. The symbol Ω denotes a generic non-empty set; the power of Ω, denoted by 2Ω, is the set of all subsets of Ω. If the number of elements in the set Ω is M N, we say that Ω is finite. If Ω contains an infinite number of elements 2 and there exists a bijection Ω N, we say that Ω is countably infinite. If Ω is neither finite nor countably infinite, we say$ that it is uncountable. An example of uncountable set is the set R of real numbers. When Ω is finite we write Ω = !1;!2;:::;!M , or Ω = !k k=1;:::;M . f g f g If Ω is countably infinite we write Ω = !k k2N. Note that for a finite set Ω with M elements, the power set contains 2M elements. Forf instance,g if Ω = ; 1; $ , then f~ g 2Ω = ; ; 1 ; $ ; ; 1 ; ; $ ; 1; $ ; ; 1; $ = Ω ; f; f~g f g f g f~ g f~ g f g f~ g g which contains 23 = 8 elements. Here denotes the empty set, which by definition is a subset of all sets. ; Within the applications in probability theory, the elements ! Ω are called sample points and represent the possible outcomes of a given experiment (or2 trial), while the subsets of Ω correspond to events which may occur in the experiment. For instance, if the experiment consists in throwing a dice, then Ω = 1; 2; 3; 4; 5; 6 and A = 2; 4; 6 identifies the event f g f g that the result of the experiment is an even number. Now let Ω = ΩN , N ΩN = (γ1; : : : ; γN ); γk H; T = H; T ; f 2 f gg f g where H stands for \head" and T stands for \tail". Each element ! = (γ1; : : : ; γN ) ΩN is called a N-toss and represents a possible outcome for the experiment \tossing a coin2 N N N ΩN 2 consecutive times". Evidently, ΩN contains 2 elements and so 2 contains 2 elements. We show in Section 1.4 that Ω1|the sample space for the experiment \tossing a coin infinitely many times"|is uncountable. Ω A collection of events, e.g., A1;A2;::: 2 , is also called information. The power set f g ⊂ 3 of the sample space provides the total accessible information and represents the collection of all the events that can be resolved (i.e., whose occurrence can be inferred) by knowing the outcome of the experiment. For an uncountable sample space, the total accessible informa- tion is huge and it is typically replaced by a subclass of events 2Ω, which is imposed to form a σ-algebra. F ⊂ Definition 1.1. A collection 2Ω of subsets of Ω is called a σ-algebra (or σ-field) on Ω if F ⊆ (i) ; ; 2 F (ii) A Ac := ! Ω: ! = A ; 2 F ) f 2 2 g 2 F S1 (iii) Ak , for all Ak k2 . k=1 2 F f g N ⊆ F If is another σ-algebra on Ω and , we say that is a sub-σ-algebra of . G G ⊂ F G F Exercise 1.1. Let be a σ-algebra. Show that Ω and that k2 Ak , for all F 2 F \ N 2 F countable families Ak k2 of events. f g N ⊂ F Exercise 1.2. Let Ω = 1; 2; 3; 4; 5; 6 be the sample space of a dice roll. Which of the following sets of events aref σ-algebras ong Ω? 1. ; 1 ; 2; 3; 4; 5; 6 ; Ω , f; f g f g g 2. ; 1 ; 2 ; 1; 2 ; 1; 3; 4; 5; 6 ; 2; 3; 4; 5; 6 ; 3; 4; 5; 6 ; Ω , f; f g f g f g f g f g f g g 3. ; 2 ; 1; 3; 4; 5; 6 ; Ω . f; f g f g g Exercise 1.3 (Sol. 1). Prove that the intersection of any number of σ-algebras (including uncountably many) is a σ-algebra. Show with a counterexample that the union of two σ- algebras is not necessarily a σ-algebra. Remark 1.1 (Notation). The letter A is used to denote a generic event in the σ-algebra. If we need to consider two such events, we denote them by A; B, while N generic events are denoted A1;:::;AN . Let us comment on Definition 1.1. The empty set represents the \nothing happens" event, c while A represents the \A does not occur" event. Given a finite number A1;:::;AN of events, their union is the event that at least one of the events A1;:::;AN occurs, while their intersection is the event that all events A1;:::;AN occur. The reason to include the countable union/intersection of events in our analysis is to make it possible to \take limits" without crossing the boundaries of the theory. Of course, unions and intersections of infinitely many sets only matter when Ω is not finite. The smallest σ-algebra on Ω is = ; Ω , which is called the trivial σ-algebra. There is no relevant information containedF inf; theg trivial σ-algebra. The largest possible σ-algebra 4 is = 2Ω, which contains the full amount of accessible information. When Ω is countable, itF is common to pick 2Ω as σ-algebra of events. However, as already mentioned, when Ω is uncountable this choice is unwise. A useful procedure to construct a σ-algebra of events when Ω is uncountable is the following. First we select a collection of events (i.e., subsets of Ω), which for some reason we regard as fundamental. Let denote this collection of events. Then we introduce the smallest σ-algebra containing , whichO is formally defined as follows. O Definition 1.2. Let 2Ω. The σ-algebra generated by is O ⊂ O \ Ω O = : 2 is a σ-algebra and ; F F F ⊂ O ⊆ F i.e., O is the smallest σ-algebra on Ω containing . F O Recall that the intersection of any number of σ-algebras is still a σ-algebra, see Exercise 1.3, d hence O is a well-defined σ-algebra. For example, let Ω = R and let be the collection of all openF balls: O d = Bx(R) R>0;x2 d ; where Bx(R) = y R : x y < R : O f g R f 2 j − j g The σ-algebra generated by is called Borel σ-algebra and denoted (Rd). The elements O B of (Rd) are called Borel sets. B Remark 1.2 (Notation). The Borel σ-algebra (R) plays an important role in these notes, B so we shall use a specific notation for its elements. A generic event in the σ-algebra (R) will be denoted U; if we need to consider two such events we denote them by U; V , whileB N generic Borel sets of R will be denoted U1;:::UN . Recall that for general σ-algebras we use the notation indicated in Remark 1.1. The σ-algebra generated by has a particular simple form when is a partition of Ω. O O Definition 1.3. Let I N. A collection = Ak k2I of non-empty subsets of Ω is called a partition of Ω if ⊆ O f g (i) the events Ak k2I are disjoint, i.e., Aj Ak = , for j = k; f g \ ; 6 S (ii) k2I Ak = Ω. If I is a finite set we call a finite partition of Ω. O For example any countable sample space Ω = !k k2 is partitioned by the atomic events f g N Ak = !k , where !k identifies the event that the result of the experiment is exactly !k. f g f g Exercise 1.4.