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Cheat Sheet: Mathematical Foundations for Finance

Steven Battilana, Autumn Semester 2017

1 1 b b 2 b 2 Thm. (dominated convergence theorem for stochastic integrals) • R u(τ)w(τ)dτ ≤ R u2(τ)dτ R w2(τ)dτ a a a Suppose that X is a semimartingale with decomposition X = M +A 1 1 n • |a(u, v)| ≤ (a(u, v)) 2 (a(u, v)) 2 as above, and let G , n ∈ N, and G be predictable processes. If Def. (Taylor) n j j Gt (ω) → Gt(ω) for anyt ≥ 0, almost surely, • f(x ± h) = P∞ (±h) d f 1 Appendix: Some Basic Concepts and Results j=0 j! dxj and if there exists a process H that is integrable w.r.t. X such that (n) P∞ f (a) n |Gn| ≤ H for any n ∈ N, then • T f(x; a) = n=0 n! (x − a) Rem. (uniform convergence on compacts in probability 1) GnX → G X u.c.p., as n → ∞. 1.1 Very Basic Things A mode of convergence on the space of processes which occurs often · · 2 in the study of stochastic calculus, is that of uniform convergence on If, in addition to the assumptions above, X is in H and kHkX < ∞ then even Def. compacts in probability or ucp convergence for short. First, a sequence of (non-random) functions fn : R+ → R converges n Q0 kG· X − G·XkH2 → 0 as n → ∞. (i) An empty product equals 1, i.e. j=1 sj = 1. uniformly on compacts to a limit f if it converges uniformly on each P0 bounded interval [0, t]. That is, ”Def.” (Wiki: null set) (ii) An empty sum equals 0, i.e. j=1 sj = 0. In set theory, a null set N ⊂ R is a set that can be covered by a Def. (geometric series) sup |fn(s) − f(s)| → 0 s≤t countable union of intervals of arbitrarily small total length. The no- n n+1 n+1 tion of null set in set theory anticipates the development of Lebesgue P k q6=1 q −1 1−q as n → ∞. (i) sn = a0 q = a0 = a0 measure since a null set necessarily has measure zero. More generally, q−1 1−q If stochastic processes are used rather than deterministic functions, k=0 on a given measure space M = (X, Σ, µ) a null set is a set S ⊂ X then convergence in probability can be used to arrive at the following ∞ |q|<1 s.t. µ(S) = 0. (ii) s = P a qk = a0 definition. 0 q−1 Def. (power set) k=0 Def. (uniform convergence on compacts in probability) We denote by 2Ω the power set of Ω; this is the family of all subsets A sequence of jointly measurable stochastic processes Xn converges Def. (conditional probabilities) of Ω. to the limit X uniformly on compacts in probability if Q[C ∩ D] = Q[C]Q[D|C] Def. (σ-fiel or σ-algebra) " # A σ-field or σ-algebra on Ω is a family F of subsets of Ω which con- Def. (compact) n P sup |Xs − Xs| > K → 0 tains Ω and which is closed under taking complements and countable Compactness is a property that generalises the notion of a subset of s≤t unions, i.e. Euclidean space being closed (that is, containing all its limit points) c and bounded (that is, having all its points lie within some fixed di- as n → ∞, ∀t, K > 0. A ∈ F ⇒ A ∈ F N S stance of each other). Examples include a closed interval, a rectangle, Rem. (uniform convergence on compacts in probability 2) Ai ∈ F, i ∈ ⇒ i∈N Ai is in F ucp or a finite set of points (e.g. [0, 1]). The notation Xn −−→ X is sometimes used, and Xn is said to Rem. Def. (P -trivial) converge ucp to X. Note that this definition does not make sense for F is then also closed under countable intersections. F0 is P -trivial iff P [A] ∈ {0, 1}, ∀A ∈ F0. arbitrary stochastic processes, as the supremum is over the uncoun- Def. (finite σ-field) Useful rules table index set [0, t] and need not be measurable. However, for right A σ-field is called finite if it contains only finitely many sets. 1 2 Z µ+ 2 σ 2 or left continuous processes, the supremum can be restricted to the Def. (measurable space) E[e ] = e , for Z ∼ N (µ, σ ). countable set of rational times, which will be measurable. A pair (Ω, F) with Ω 6= ∅ and F a σ-algebra on Ω is called a mea- Let W be a BM, then hW it = t, hence dhW is = ds. Def. surable space. Def. (Hilbert space) 2 2 Rem. M ([0, a]) := {M ∈ M ([0, a]) | t 7→ Mt(ω) RCLL for any ω ∈ Ω} A Hilbert space is a vector space H with an inner product hf, gisuch d X is measurable (or more precisely Borel-measurable) if for every that the norm defined by Def. B ∈ B(R), we have {X ∈ B} ∈ F. 2 Def. (indicator function) |f| = phf, gi (i) We denote by H the vector space of all semimartingales va- 2 For any subset A of Ω, the indicator function IA is the function de- nishing at 0 of the form X = M + A with M ∈ Md(0, ∞) ( turns H into a complete metric space. If the metric defined by the and A ∈ FV (finite variation) predictable with total variation 1, ω ∈ A fined by IA(ω) := norm is not complete, then H is instead known as an inner product (1) R ∞ 2 0, ω∈ / A V∞ (A) = 0 |dAs| ∈ L (P ). space. Def. Def. (cauchy-schwarz) (ii) R 0 2 Let Ω 6= ∅ and a function X :Ω → (or more generally to Ω ). • |hu, wi | ≤ hu, ui · hw, wi Then σ(X) is the smallest σ-field, say, on Ω s.t. X is measurable 1 1 " Z ∞ 2# PN PN 2 2 PN 2 2 2 2 (1) 2 with respect to G and B(R) (or G and F 0, respectively). We call • j=1 uj wj ≤ j=1 uj j=1 wj kXkH2 = kMkM2 +kV∞ (A)kL2 = E [M]∞ + |dA| 0 σ(X) the σ-field generated by X.

Steven Battilana p 1 / 23 Mathematical Foundations for Finance Rem. Intuitively, one can compute the conditional expectation We also consider a σ-field generated by a whole familiy of mappings; E[F (U, V )|G] by ’fixing the known value U and taking the ex- this is then the smallest σ-field that makes all the mappings in that pectation over the independent quantity V . Thm. 2.3 family measurable. Suppose (Un)n∈N is a sequence of random variables. Def. (probability measure, probability space) 1.2 Conditional expectations: A survival kit (i) If U ≥ XP -a.s. for all n and some integrable random variable If (Ω, F) is a masurable space, a probability measure on F is a map- n X, then ping P : F → [0, 1] s.t. P [Ω] = 1 and P is σ-additive, i.e. Rem.   E[lim inf Un|G] ≤ lim inf E[Un|G],P -a.s. Let (Ω, F,P ) be a probability space and U a real-valued random n→∞ n→∞ [ X N P  Ai = P [Ai],Ai ∈ F, i ∈ ,Ai ∩ Aj = ∅, i 6= j. variable, i.e. an F-measurable mapping U :Ω → R. Let G ⊆ F be (ii) If (Un) converges to some random variable UP -a.s. and if N N i∈ i∈ a fixed sub-σ-field of F; the intuitive interpretation is that G gives |Un| ≤ XP -a.s. for all n and some integrable random variable us some partial information. The goal is then to find a prediction for The triple (Ω, F,P ) is then called a probability space. X, then U on the basis of the information conveyed by G, or a best estimate Def. (P -almost surely) E[ lim U |G] = E[U|G] = lim E[U |G],P -a.s. for U that uses only information from G. n n A statement holds P -almost surely or P -a.s. if the set n→∞ n→∞ Def. (conditional expectation of U given G) A := {ω | the statement does not hold} A conditional expectation of U given G is a real-valued random variable Y with the following two properties: is a P -nullset, i.e. has P [A] = 0. We sometimes use instead the for- mulation that a statement holds for P -almost all ω. (i) Y is G-measurable Notation (ii) E[UIA] = E[YIA], ∀A ∈ G 1.3 Stochastic processes and functions E.g., X ≥ YP -a.s. means that P [X < Y ] = 0, or, equivalently Y is then called a version of the conditional expectation and is de- P [X ≥ Y ] = 1. Notation: noted by Y = E[U|G]. Def. (stochastic process) P [X ≥ Y ] := P [{X ≥ Y }] := P [{ω ∈ Ω | X(ω) ≥ Y (ω)}]. Thm. 2.1 A (real-valued) stochastic process with index set T is a family of Let U be an integrable random variable, i.e. U ∈ L1(P ). Then: random variables Xt, t ∈ T , which are all defined on the same pro- Def. (random variable) bability space (Ω, F,P ). We often write X = (Xt)t∈T . Let (Ω, F,P ) be a probability space and X :Ω → R a measurable (i) There exists a conditional expectation E[U|G], and E[U|G] is Def. (increment of stochastic process) function. We also say that X is a (real-valued) random variable. If again integrable. 0 For any stochastic process X = (Xk)k=0,1,...,T , we denote the in- Y is another random variable, we call X and Y equivalent if X = Y (ii) E[U|G] is unique up to P -nullsets: If Y,Y are random variables crement from k − 1 to k of X by P -a.s. satisfying the right above definition, then Y 0 = YP -a.s. Def. (p-integrable) ∆Xk := Xk − Xk−1. 0 0 Lem. (properties and computation rules) We denote by L or L (F) the family of all equivalence classes Next, we list properties of and computation rules for conditional ex- Rem. of random variables on (Ω, F,P ). For 0 < p < ∞, we denote by 0 A stochastic process can be viewed as a function depending on two p pectations. Let U, U be integrable random variables s.t. E[U|G] and L (P ) the family of all equivalence classes of random variables X 0 parameters, namely ω ∈ Ω and t ∈ T . p E[U |G] exist. We denote by bG the set of all bounded G-measurable which are p-integrable in the sense that E[|X| ] < ∞, and we write Def. (trajectory) p p ∞ random variables. Then we have: then X ∈ L (P ) or X ∈ L for short. Finally, L is the family If we fix t ∈ T , then ω 7→ Xt(ω) is simply a random variable. If of all equivalence classes of random variables that are bounded by a (i) E[UZ] = E[E[U|G]Z], ∀Z ∈ b 0 0 we fix instead ω ∈ Ω, then t 7→ Xω(t) can be viewed as a function constant c (where the constant can depend on the random variable). (ii) Linearity: E[aU1bU |G] = aE[U|G] + bE[U |G],P -a.s. ∀a, b, ∈ R R 0 0 T → , and we often call this the path or the trajectory of the Def. (Atom) (iii) Monotonicity: If U ≥ U P -a.s., then E[U|G] ≥ E[U |G] P -a.s. process corresponding to ω. If (Ω, F,P ) is a probability space, then an atom of F is a set A ∈ F (iv) Projectivity: E[U|H] = E[E[U|G]|H],P -a.s. ∀σ-field H ⊆ G Def. (continuous) with the properties that P [A] > 0 and that if B ⊆ A is also in F, (v) E[U|G] = U, P -a.s. if U ∈ G-measurable A stochastic process is continuous if all or P -almost all its trajecto- then either P [B] = 0 or P [B] = P [A]. Intuitively, atoms are the (vi) E[E[U|G]] = E[U] ries are continuous functions. (v) (viii) ’smallest indivisible sets’ in a σ-field. Atoms are pairwise disjoint up (vii) E[ZU|G] = E[ZE[U|G]|G] = ZE[U|G],P -a.s. ∀Z ∈ bG Def. (RCLL) to P -nullsets. (viii) E[U|G] = E[U],P -a.s. for U independent of G A stochastic process is RCLL if all of P -almost all its trajectories are Def. (atomless) right-continuous (RC) functions admitting left limits (LL). The space (Ω, F,P ) is called atomless if F contains no atoms; this Rem. Def. (Wiki: signed measure) can only happen if F is infinite. Finite σ-fields can be very conve- (i) Instead of integrability of U, one could also assume that U ≥ 0; Given a measurable space (X, Σ), i.e. a set X with a σ-algebra on niently described via their atoms because every set in F is then an then analogous statements are true. it, an signed measure is a function µ :Σ → R, s.t. µ(∅) = 0 and µ union of atoms. (ii) More generally, (i) and (vii) hold as soon as U and ZU are both is σ-additive, i.e. it satisfies Fatou’s Lemma integrable or both nonnegative. S∞ P∞ µ An = µ(An) If {fn} is a sequence of nonnegative measurable functions, then (iii) If U is Rd-valued, one simply does everything component by com- n=1 n=1 R R where the series on the right must converge absolutely, for any se- lim infn→∞ fndµ ≤ lim infn→∞ fndµ. ponent to obtain analogous results. quence A1,A2, ..., An, ... of disjoint sets in Σ. An example of an sequence of functions for which the inequality Lem. 2.2 Def. (Wiki: total variation in measure theory) becomes strict is given by Let U, V be random variables s.t. U is G-measurable and V is inde- Consider a signed measure µ on a measurable space (X, Σ), then it is R2 ( pendent of G. For every measurable function F ≥ 0 on , then possible to define two set functions W (µ, ·) and W (µ, ·), respectively 0, x ∈ [−n, n] fn(x) = E[F (U, V )|G] = E[F (u, V )]| =: f(U). 1, otherwise u=U

Steven Battilana p 2 / 23 Mathematical Foundations for Finance P 2 called upper variation and lower variation, as follows [B]t := lim (BsΠ − BsΠ ) . The discounted asset price S = (Sk)k=0,1,2,...,T are given by |Π|→∞ Π i i−1 S := Sek W (µ, E) = sup{µ(A) | A ∈ Σ,A ⊂ E}, ∀E ∈ Σ k S0 Note that the quadratic variation is itself a process [Bt]t≥0. Π here ek W (µ, E) = inf{µ(A) | A ∈ Σ,A ⊂ E}, ∀E ∈ Σ refers to a a partition of the interval [0, t], which is supposed to get finer and finer. This limit taking process is the same one found when Trading strategy ϕ The following clearly holds: one defines the Riemann integral. A trading strategy is an d-valued stochastic process ϕ = (ϕ0, ϑ). W (µ, E) ≥ 0 ≥ W (µ, E), ∀E ∈ Σ R 0 0 ϕ = (ϕk)k=0,1,...,T denotes the riskless asset. Def. (Wiki: bounded variation (finite variation)) 0 ϕ is R-valued and adapted. (i) The total variation of a real-valued (or more generally complex- 2 Financial Markets in Discrete Time ϑ = (ϑ ) denotes the d risky assets. valued) function f, defined on an interval [a, b] ⊂ R is the quan- k k=0,1,...,T d tity ϑ is R -valued and predictable. 0 b PnP −1 Initial value: ϕ = (ϕ , ϑ0 ≡ 0) V (f) = sup |f(xi+1) − f(xi)|. a P ∈P i=0 2.1 Basic setting (since there is no trading before time 0, i.e. investors start out If f is differentiable and its derivative is Riemann-integrable, its without any shares) total variation is the vertical component of the arc-length of its Basic setting of the market graph, that is to say, A trading strategy describes a dynamically evolving portfolio in Probability space: (Ω, F, P) the d + 1 basic assets available for trade. b b R 0 Finite discrete time horizon: k = 0, 1,...,T Va (f) = |f (x)|dx. a Value process V Flow of information over time: filtration F = (Fk)k=0,1,...,T . (ii) A real-valued function f on the real line is said to be of bounded This is a family of σ-fields Fk ⊆ F which is increasing. V (ϕ) = (V (ϕ)) denotes the discounted value process variation (BV function) on a chosen interval [a, b] ⊂ R if its total k k=0,1,...,T d of the strategy ϕ and is given by variation is finite, i.e. An (R -valued) stochastic process in discrete time: X = d (Xk)k=0,1,...,T of ( -valued) RVs which are all defined on the b R d f ∈ BV ([a, b]) ⇔ Va (f) < ∞. same probability space. X V (ϕ) := ϕ0 S0 + ϑ · S = ϕ0 + ϑi Si , for k = 0, 1, ..., T This describes the random evolution over time of d quantities. k k k k k k k k It can be proved that a real function f is of bounded variation in | {z } | {z } i=1 bank account portfolio [a, b] if and only if it can be written as the difference f = f1 − f2 A stochastic process is called adapted (w.r.t. F) if each Xk is of two non-decreasing functions on [a, b]: this result is known as (1) Fk-measurable, i.e. observable at time k. the Jordan decomposition of a function and it is related to the where S0 = 1. Jordan decomposition of a measure. A stochastic process is called predictable (w.r.t. F) if each Xk is k even Fk−1-measurable. V is -valued and adapted. Def. (Wiki: variation) R 0 The variation (also called absolute variation) of the signed measure Initial value: V0(ϕ) = ϕ0 = C0(ϕ). µ is the set function Frictionless financial market

|µ|(E) = W (µ, E) + |W (µ, E)|, ∀E ∈ Σ no transaction cost Cost process C and its total variation is defined as the value of this measure on the short-selling allowed (Ck(ϑ))k=0,1,...,T denotes the discounted cost process associated whole space of definition, i.e. kµk = |µ|(X). investors are small, i.e. their trading does not affect stock prices to ϕ: Def. (planetmath.org: finite variation process) In the theory of stochastic processes, the term finite-variation pro- Def. Ck(ϕ) := Vk(ϕ) − Gk(ϕ) cess is used to refer to a process X whose paths are right-continuous t (i) r describes the (simple) interest rate for the period (k − 1, k]. and have finite total variation over every compact time interval, with k (ii) Y is the growth factor for the time period (k − 1, k]. By construction, C (ϕ) describes the cumulative (total) costs for probability one. k k (iii) S0 models a bank account. the strategy ϕ on the time interval [0, k]. Def. (finite variation) e (iv) Sei models a stock. 0 A stochastic process is of finite variation if all or P -almost all its The initial cost for ϕ at time 0 comes from putting ϕ0 into the trajectories are functions of finite variation. bank account Def. (locally) 0 A stochastic process has a property locally if there exists C0(ϕ) = ϕ0 = V0(ϕ). a sequence of stopping times (τn)n∈N increasing to ∞ P - 2.2 Basic processes a.s. s.t. when restricted to the stochastic interval 0, πn = {(ω, t) ∈ Ω × T | 0 ≤ t ≤ τn(ω)}, the process hasJ theK property under consideration. Def. (discounted reference asset price) S0 Lem. (Brownian Motion and quadratic variation) The discounted price of the reference asset is S0 := ek = 1 at all k S0 If a continuous martingale X has quadratic variation [X] = t, then ek t times. it is a Brownian Motion. Def. (discounted asset price) Def. (quadratic variation)

Steven Battilana p 3 / 23 Mathematical Foundations for Finance Incremental cost: It then holds for the corresponding value process: Admissibility

d V (ϕ) = V (ϕ) + G (ϑ) = ϕ0 + G (ϑ) For a ∈ R, a ≥ 0, a trading strategy ϕ is called a-admissible if its 0 0 0 X i i i k 0 k 0 k ∆Ck+1(ϕ) : =( ϕk+1 − ϕk) Sk + ϑk+1 − ϑk Sk value process V (ϕ) is uniformly bounded from below by −a, i.e. |{z} k =1 i=1 0 X tr | {z } = ϕ0 + ϑj ∆Sj V (ϕ) ≥ −a, -a.s., ∀k ≥ 0 | {z } portfolio k P bank account j=1 tr A trading strategy is called admissible if it is a-admissible for rewrite the above by adding and subtracting ϕk+1Sk+1: Remarks: some a ≥ 0. 0 0 tr ∆Ck+1(ϕ) = ϕk+1 − ϕk + (ϑk+1 − ϑk) Sk The notion of a strategy being self-financing is a kind of economic Remark: 0 0 tr tr budget constraint. = ϕ − ϕ + (ϑk+1 − ϑk) Sk ± ϕ Sk+1 k+1 k k+1 An admissible strategy can be interpreted as a strategy having 0 tr 0 tr tr The notion of self-financing is numeraire (discounting) irrelevant, some credit line which imposes a lower bound on the associated = ϕk+1 + ϑk+1Sk+1 − ϕk − ϑk Sk − ϑk+1∆Sk+1 i.e. it does not depend on the units in which the calculations are value process. So one may make debts, but only within clearly (1) tr done. defined limits. = Vk+1(ϕ) − Vk(ϕ) − ϑk+1∆Sk+1 tr = ∆Vk+1(ϕ) − ϑk+1∆Sk+1 A first introduction to stopping time Notation: a ∧ b := min(a, b) 2.3 Some important martingale results Gains process G Def. (stopped stochastic process) τ τ (G (ϑ)) denotes the discounted gains process associa- The stochastic process S = (Sk )k=0,1,...,T defined by k k=0,1,...,T Def. (martingale, martingale property) ted to ϑ: τ S (ω) := Sk∧τ (ω) := Sk∧τ(ω)(ω) Let (Ω, F,Q) be a probability space with a filtration F = k is called the process S stopped at τ. (Fk)k=0,1,...,T . A (real-valued) stochastic process X = X tr Gk(ϑ) := ϑj ∆Sj ,k = 0, 1, ..., T Rem. (Xk)k=0,1,...,T is called martingale (w.r.t. Q and F) if it is ad- 1 j=1 τ τ apted to F, is Q-integrable in the sense that Xk ∈ L (Q), ∀k, and S could fail to be a stochastic process because Sk = Sk∧τ could = ϑ · S fail to be a random variable, i.e. could fail to be measurable. But (in satisfies the martingale property discrete time) this is not a problem if we assume that τ is measura- EQ[X`|Fk] = Xk,Q-a.s. for k ≤ `. ble, which is mild and reasonable enough. G is -valued and adapted. R Def. (stopping time) Def. (supermartingale, submartingale) If we think of a continuous-time model where successive trading Note that ϑk = I{k≤τ} is Fk−1-measurable for each k iff {τ ≥ k} ∈ (i) If we have ” ≤ ” in the above definition (a tendency to go down), dates are infinitely close together, then the increment ∆S beco- Fk−1 for all k or, equivalently by passing to complements, X is called a supermartingale. mes a differential dS and the sum becomes a stochastic integral {τ ≤ j} ∈ Fj , ∀j. (ii) If we have ” ≥ ” in the above definition (a tendency to go up), k Z X By definition, this means that τ is a stopping time (w.r.t. F). X is called a submartingale. G(ϑ) = ϑidSi, k = 0, 1, ..., T Example (a doubling strategy, p.16) Def. (local martingale, localising sequence) i=1 We denote by the following the (random) time of the first stock price An adapted process X = (Xk)k=0,1,...,T null at 0 (i.e. X0 = 0) is rise: called a local martingale (w.r.t. Q and F) if there exists a sequence N τ := inf{k | Yk = 1 + u} ∧ T of stoppint times (τn)n∈N increasing to T s.t. ∀n ∈ , the stopped τn F Self-financing strategy process X = (Xk∧τn )k=0,1,...,T is a (Q, )-martingale. We then which is a stopping time because N 0 call (τn)n∈ a localising sequence. A trading strategy ϕ = (ϕ , ϑ) is called self-financing if its cost Thm. 3.1 {τ ≤ k} = {Y = 1 + u} ∪... ∪ {Y = 1 + u} process C(ϕ) is constant over time. 1 k Suppose X = (X ) is an Rd-valued martingale or local | {z } | {z } | {z } k k=0,1,...,T 0 ∈Fk,∀k ∈F1⊆Fk ∈Fk Rd Proposition 2.1. A self-financing strategy ϕ = (ϕ , ϑ) is uniquely martingale null at 0. For any -valued predictable process ϑ, the determined by its initial wealth V0 and its risky asset compo- = {max(Y1, ..., Yk) ≥ 1 + u} ∈ Fk, ∀k. stochastic integral process ϑ · X defined by nent ϑ. k With P tr In particular, any pair (V0, ϑ) specifies in a unique way a self- ϑ · Xk := ϑj ∆Xj , k = 0, 1, ..., T financing strategy. 1 k−1 j=1 ϑk := 2 I{k≤τ}. 0 0 Sk−1 If ϕ = (ϕ , ϑ) is self-financing, then (ϕk)k=1,...,T is automati- is then a (real-valued) local martingale null at 0. If X is a martingale cally predictable. it follows that ϑ is predictable because each ϑk is Fk−1-measurable. and ϑ is bounded, then ϑ · X is even a martingale. Note that this uses {k ≤ τ} = {τ < k}c = {τ ≤ k − 1}c. Rem. It then holds for the corresponding (incremental) cost process: In continuous time, the above theorem no longer holds. 0 0 0 ∆Ck+1(ϕ) =( ϕk+1 − ϕk ) Sk + (ϑk+1 − ϑk) · Sk = 0,P -a.s. Rem. 0 If we think of X = S as discounted asset prices, then ϑ · S = G(ϑ) C(ϕ) = C0(ϕ) = V0(ϕ) = ϕ0

Steven Battilana p 4 / 23 Mathematical Foundations for Finance is the discounted gains process. Finally, define Y1, ..., YT by Cor. 3.2 For any martingale X and any stopping time τ, the stopped process Yk(ω) := 1 + yxk (2) τ 2.5 Properties of the market X is again a martingale. In particular, EQ[Xk∧τ ] = EQ[X0], ∀k. so that Yk(ω) = 1 + yj iff xk = j. We take as filtration the one Interpretation 1 A martingale describes a fair game in the sense that one cannot generated by Se (or, equivalently, by Y ) s.t. Characterisation of financial markets via EMMs (equivalent mar- predict where it goes next. F = σ(Y , ..., Y ), k = 0, 1, ..., T. tingale measures) The description of a financial market model via k 1 k EMMs can be summarized as follows: (i) Cor. 3.2 says that one cannot change this fundamental character Def. (atom) by cleverly stopping the game. Existence of an EMM ⇐⇒ the market is arbitrage-free A set A ⊆ Ω is an atom of Fk iff there exists a sequence (x1, ..., xk) st i.e. Pe(S) 6= ∅ by the 1 FTAP (ii) Thm. 3.1 says that as long as one can only use information from of length k with elements xi ∈ {1, ..., m} s.t. A consists of all those ω ∈ Ω that start with the substring (x , ..., x ), i.e. the past, not even complicated clever betting will help. 1 k Uniqueness of the EMM ⇐⇒ the market is complete nd A = A := {ω = (x , ..., x ) ∈ {1, ..., m}T | x = i.e. #(Pe(S)) = 1 by the 2 FTAP Thm. 3.3 x1,...,xk 1 T i Suppose that X is an Rd-valued local Q-martingale null at 0 and ϑ xi for i = 1, ..., k}. is an Rd-valued predictable process. If the stochastic integral process Consequences ϑ · X is uniformly bounded below (i.e. ϑ · X ≥ −b Q-a.s., ∀k, b ≥ 0), then ϑ · X is a Q-martingale. (i) Each Fk is parametrised by substrings of length k and therefore 2.5.1 Arbitrage contains precisely mk atoms. (ii) When going from time k to time k + 1, each atom 1st Fundamental Theorem of Asset Pricing (FTAP) Thm. 2.1 (Dalang/Morton/Willinger) A = Ax1,...,xk from Fk splits into precisely m subsets A = A , ..., A = A that are atoms of 1 x1,...,xk,1 m x1,...,xk,m (i) Consider a financial market model in finite discrete time. Fk+1. 2.4 An example: The multinomial model (ii) Then S is arbitrage-free iff there exists an EMM for S, i.e. (iii) The atoms of Fk are pairwise disjoint and their union is Ω. Fi- nally, each set B ∈ F is a union of atoms of F ; so the family k k (NA) ⇐⇒ e(S) 6= ∅ Def. (multiplicative model) k P F of events observable up to time k consists of 2m sets. The multiplicative model with i.i.d. returns is given by k In other words: If there exists a probability measure Q ≈ P on Def. (one-step transition probabilities) F s.t. S is a Q-martingale, then S is arbitrage-free. Se0 For any atom A = A of F , we then look at its m suc- T k = 1 + r > 0, ∀k, x1,...,xk k 0 cessor atoms A = A , ..., A = A of F , Limitations: The most important of these assumptions are fric- Se 1 x1,...,xk,1 m x1,...,xk,m k+1 k−1 and we define the one-step transition probabilities for Q at the node tionless markets and small investors—and if one tries to relax 1 Sek corresponding to A by the conditional probabilities these to allow for more realism, the theory even in finite discrete = Yk, ∀k, 1 Q[A ] time becomes considerably more complicated and partly does not Sek−1 j Q[Aj |A] = Q[A] , for j = 1, ..., m. even exist yet. 0 1 1 where Se0 = 1, Se0 = S0 > 0 is a constant, and Y1, ..., YT are i.i.d. Because A is the disjoint union of A1, ..., Am, we have Cor. 2.2 m and take the finitely many values 1 + y1, ..., 1 + ym with respective P The multinomial model with parameters y1 < ... < ym and r is 0 ≤ Q[Aj |A] ≤ 1 for j = 1, .., m and Q[Aj |A] = 1. probabilities p1, ..., pm. We assume that all the probabilities pj are j=1 arbitrage-free iff y1 < r < ym. > 0 and that ym > ym−1 > ... > y1 > −1. This also ensures that Rem. Cor. 2.3 Se1 remains strictly positive. The decomposition of factorisation of Q in such a way that for every The binomial model with parameters u > d and r is arbitrage-free Rem. trajectory ω ∈ Ω, its probability Q[{ω}] is the product of the succes- iff d < r < u. In that case, the EMM Q∗ for Se1/Se0 is unique (on Intuition suggests that for a reasonable model, the sure factor 1 + r sive one-step transition probabilities along ω. FT ) and is given as in Cor. 1.4. should lie between the minimal and maximal values 1+y1 and 1+ym Rem. of the (uncertain) random factor. We can describe Q equivalently either via its global weights Q[{ω}] Def. (canonical model) or via its local transition behaviour. Arbitrage opportunity The simplest and in fact canonical model for this setup is a path Def. (independent growth rates) Def. (arbitrage opportunity, arbitrage-free) space. Let The (coordinate) variables Y1, ..., YT from (2) are independent under (i) An arbitrage opportunity is an admissible self-financing strategy ∧ Q iff for each k, the one-step transition probabilities are the same ϕ = (0, ϑ) with zero initial wealth, with Vt(ϕ) ≥ 0, P -a.s. and Ω = {1, ..., m}T for each node at time k, but they can still differ across date k. with P [VT (ϕ) > 0] > 0. = {ω = (x , ..., x ) | x ∈ {1, ..., m} for k = 1, ..., T } Def. 1 T k (ii) The financial market (Ω, F, ,P,S 0,S) or shortly S is called Y1, ..., YT are i.i.d. under Q iff at each node throughout the tree, the F be the set of all sequences of length T formed by element of one-step transition probabilities are the same. arbitrage-free if there exist no arbitrage opportunities. {1, ..., m}. Take F = 2Ω, the family of all subsets of Ω, and de- Rem. (iii) Sometimes one also says that S satisfies (NA). fine P by setting Probability measures with this particular structure can therefore be Def. T described by m − 1 parameters; recall that the m one-step transition Q probabilities at any given node must sum to 1, which eliminates one P [{ω}] = px1 px2 · ... · pxT = pxk . (i) (NA+) := forbids to produce something out of nothing with 0- k=1 degree of freedom. admissible self-financing strategies.

Steven Battilana p 5 / 23 Mathematical Foundations for Finance (ii) (NA0) := forbids the same for all (not necessarily admissible) self- u > r > d. In that case, Q is unique (on F) and characterised We already know from Cor. 2.3 that this model is arbitrage-free and 1 1 0 1 financing strategies. by the property that Y1, ..., YT are i.i.d. under Q with parameter has a unique EMM for S = Se /Se . Hence S is complete by Thm. 2.2, and so every H ∈ L0 (F ) is attainable, with a price process (iii) Then we clearly have the implications: Q[Y = 1 + u] = q∗ = r−d = 1 − Q[Y = 1 + d]. + T k u−d k given by 0 (NA ) ⇒ (NA) ⇒ (NA+) H Vk = EQ∗ [H|Fk] for k = 0, 1, ..., T , Note, for finite discrete time, the three concepts are all equivalent. where Q∗ is the unique EMM for S1. We also recall from Cor. 2.3 ∗ Prop. 1.1 2.5.2 Completeness that the Yj are under Q again i.i.d., but with For a financial market in finite discrete time, the following statements Q∗[Y = 1 + u] = q∗ := r−d ∈ (0, 1). are equivalent: 1 u−d Def. (Completeness of the market) (i) S is arbitrage-free. All the above quantities S1,H,V H are discounted with Se0, i.e. ex- A financial market model (in finite discrete time) is called com- ∧ 0 pressed in units of asset 0. The undiscounted quantities are the stock (ii) There exists no self-financing strategy ϕ = (0, ϑ) with zero initial plete if every payoff H ∈ L (FT ) is attainable. + price S1 = S1S0, the payoff H := HS0 and its price process V He wealth and satisfying VT (ϕ) ≥ 0, P -a.s. and P [VT (ϕ) > 0] > 0. e e e eT e 0 Otherwise it is called incomplete. He H 0 In other words, S satisfies (NA ). with Vek := Vk Sek for k = 0, 1, ..., T . Thm. 2.1 (Valuation and hedging in complete markets) Cor. 3.1 (iii) For every (not necessarily admissible) self-financing strategy ϕ Consider a financial market model in finite discrete time and sup- In te binomial model, te undiscounted arbitrage-free price process of with V0(ϕ) = 0, P -a.s. and VT (ϕ) ≥ 0 P -a.s., we have VT (ϕ) = 0 pose that F0 is trivial and S is arbitrage-free and complete. Then any undiscountede payoff He ∈ L+(FT ) is given by 0 P -a.s. 0 for every payoff H ∈ L+(F), there is a unique price process (iv) For the space H H " # " 0 # 0 V = (Vk )k=0,1,...,T which admits no arbitrage. It is given by H 0 He Sek Sek V = S E ∗ F = E ∗ H F = E ∗ [H|F ] 0 d ek ek Q 0 k Q e 0 k 0 Q e k G := {GT (ϑ) | ϑ is -valued and predictable} H Se Se Se R Vk = EQ[H|Fk] = Vk(V0, ϑ), for k = 0, 1, ..., T T T T of all final wealths that one can generate from zero initial wealth ∧ for any EMM Q for S and any replicating strategy ϕ = (V0, ϑ) for for k = 0, 1, ..., T. through self-financing trading, we have H. 0 0 G ∩ L+(FT ) = {0} 0 2nd Fundamental Theorem of Asset Pricing (FTAP) where L+(FT ) denotes the space of all nonnegative FT - 0 Rn 0 Rn Thm. 2.2 2.6 Pricing of contingent claims H measurable RVs, i.e. L+(FT ) = +,(L (FT ) = ). Rem. Consider a financial market model in finite discrete time and assume that S is arbitrage-free, F0 is trivial and FT = F. Def. (general European option) 0 0 (i) The two sets L+(FT ) and G can be separated by a hyperplane, A general European option of payoff or contingent claim is a random Then S is complete iff there is a unique EMM for S 0 and the normal vector defining that hyperplane then yields (after variable H ∈ L+(FT ). suitable normalisation) the (density of the) desired EMM. Interpretation (NA) + completeness ⇐⇒ #(Pe(S)) = 1, The interpretation is that H describes the net payoff (in units of (ii) The existence of an EMM follows from the existence of a separa- asset 0) that the owner of this instrument obtains at time T ; so ha- ting hyperplane between two sets. i.e. Pe(S) is a singleton. ving H ≥ 0 is natural and also avoids integrability issues. Since H P is F -measurable, the payoff can depend on the entire information (iii) The set e(S) is convex, it is either empty, or contains exactly Remark: T one element, or contains infinitely (uncountably) many elements. up to time T ; and ”European” means that the time for the payoff is If a financial market in discrete time is complete, then F is finite fixed at the end T . Interpretation: Absence of arbitrage is a natural economic/financial T (i.e. completeness is quite restrictive). Def. (European call option, net payoff) requirement for a reasonable model of a financial market, since there

cannot exist ”money pumps” (at least not for long). Completeness is only an assertion about FT -measurable quanti- (i) A European call option on asset i, with maturity T and strike K ties. gives its owner the right, but not the obligation, to buy at time T Def. (equivalent) one unit of asset i for the price K, irrespective of what the actual Two probability measures Q and P on (Ω, F) are equivalent (on F), asset price Si then is. written as Q ≈ P on F, it they ahve the same nullsets (in F), i.e. if T for each set A (in F) we have P [A] = 0 iff Q[A] = 0. (ii) In monetary terms, any rational person will make use of (exercise) Lem. 1.2 Example: The binomial model that right iff Si (ω) > K, then the net payoff is Si (ω)−K, more Recall: T T If there exists a probability measure Q ≈ P on FT s.t. S is a Q- formal martingale, then S is arbitrage-free. We recall that this model is described by parameters p ∈ (0, 1) and 0 k 1 1 Qk H(ω) = max(0,Si (ω) − K) = (Si (ω) − K)+. Cor. 1.3 u > r > d > −1; then we have Sek = (1+r) and Sek = S0 j=1 Yj T T 1 In the multinomial model with parameters y1 < ... < ym and r, the- with S0 > 0 and Y1, ..., YT i.i.d under P taking values 1 + u or 1 + d As a random variable, this is clearly nonnegative and FT - 1 0 F i re exists a probability measure Q ≈ P s.t. Se0 /Se is a Q-martingale with probability p or 1 − p, respectively. The filtration is generated measurable since S is adapted. 0 1 1 iff yi < r < ym. by Se = (Se , Se ) or equivalently by Se or by Y . Note that F0 is then 0 1 1 Example (payoffs) Cor. 1.4 trivial because Se0 = 1 and Se0 = S0 is a constant. We also take In the binomial model with parameters u > d and r, there exists F = FT . 1 0 a probability measure Q ≈ P s.t. Se0 /Se is a Q-martingale iff Rem.

Steven Battilana p 6 / 23 Mathematical Foundations for Finance (i) If we want to bet on a reasonably stable asset price evolution, we Remark: Note that not all of these relationships necessarily hold Define the bank account/riskless asset by: might be interested in a payoff of the form H = IB with for financial markets in infinite discrete time or continuous time. k ”2) ⇒ 3)” in general only holds if H is bounded. Y S˜0 S˜0 := (1 + r ), k = 1 + r , S˜0 = 1 i i k j 0 k 0 B = a ≤ min min Sk < max max Sk ≤ b S˜ i=1,...,d k=0,1,...,T i=1,...,d k=0,1,...,T j=1 k−1 Approach to valuing and hedging payoffs For a given payoff H in This option pays at time T on unit of money iff all stocks remain a financial market in finite discrete time (with F0 trivial): Remarks: between the levels a and b up to time T . ˜0 (i) Check if S is arbitrage-free by finding at least one EMM Q for S. – Sk is Fk−1-measurable (i.e. predictable). (ii) A payoff of the form (ii) Find all EMMs Q for S. – rk denotes the rate for (k − 1, k]. T ! Define the stock/risky asset by: 1 X i (iii) Compute EQ[H] for all EMMs Q for S and determine the supre- H = IAg S ,A ∈ FT , g ≥ 0 T k mum of EQ[H] over Q. k=1 k S˜1 ˜1 1 Y k ˜1 ˜1 (iv) If the supremum is finite and a maximum, i.e. attained in some Sk := S0 Yj , = Yk, S0 = const., S0 ∈ R ∗ ˜1 gives a payoff which depends on the average price (over time) Q ∈ Pe(S), then H is attainble and its price process can be j=1 Sk−1 of asset i, but which is only due in case that a certain event A H P computed as Vk = EQ[H|Fk], for any Q ∈ (S). occurs. If the supremum is not attained (or, equivalently for finite discrete Remarks: time, there is a pair of EMMs Q1,Q 2 with EQ [H] 6= EQ [H]), ˜1 Def. (Attainability) 1 2 – S is Fk-measurable (i.e. adapted). then H is not attainable. k 0 A payoff H ∈ L+(FT ) is called attainable if there exists an admis- – Yk denotes the growth factor for (k − 1, k]. ∧ sible self-financing strategy ϕ=(V0, ϑ) with VT (ϕ) = HP -a.s. Invariance of the risk-neutral pricing method under a change of – The rate of return Rk is given by Yk = 1 + Rk. The stratety ϕ is then said to replicate H and is called a replica- numéraire ting strategy for H. The risk-neutral pricing method is invariant under a change of Thm. 1.1 (Arbitrage-free valuation of attainable payoffs) numéraire,i.e. all assets can be priced under a risk-neutral me- 2.7.1 Cox-Ross-Rubinstein (CRR) binomial model thod independent of the chosen asset used for discounting. Consider a financial market in finite discrete time and suppose that S is arbitrage-free and complete and F0 is trivial. ∗∗ ˆ0 S˜0 Denote with Q the EMM for S := ˜1 . Assumptions 0 S Then for every payoff H ∈ L (FT ) has a unique price process S˜1 + Denote with Q∗ the EMM for S1 = . Bank account/riskless asset: V H = (V H ) which admits no arbitrage. S˜0 k k=0,1,...,T Suppose all the r ∈ are constant with value r > −1. ˜0 ˜1 k R H Then it holds for a financial market (S , S ) and an undiscounted This means that we have the same nonrandom interest rate over V is given by: ˜ 0 payoff H ∈ L+(FT ) that: each period. ˜0 H Then the bank account evolves as Sk for k = 0, 1,...,T . Vk = EQ [H|Fk] = Vk(V0, ϑ) " ˜ # " ˜ # 0 H 1 H S˜ E ∗ F = S˜ E ∗∗ F k Q 0 k k Q 1 k Stock/risky asset: S˜ S˜ for k = 0, 1,...,T , for any EMM Q for S and for any replicating T T Suppose that Y1,...YT ∈ R are independent and only take two ∧ strategy ϕ=(V0, ϑ) for H. values, 1 + u with probability p, and 1 + d with probability 1 − p (i.e. all Y are i.i.d.). Rem.: Because it involves no preferences, but only the assumpti- EMMs in submarkets k Then the stock prices at each step moves either up (by a factor on of absence of arbitrage, the valuation from this Thm. is often 0 1 k If a market (S ,S ,...,S ) is (NA), i.e. there exists an EMM 1 + u) or down (by a factor 1 + d), i.e. also called risk-neutral valuation, and an EMM Q for S is called Q, then this EMM Q is also an EMM for all submarkets. a risk-neutral measure k i k i j ( (e.g. for (S ,S ), k 6= i, for (S ,S ,S ), k 6= i 6= j etc.) Se1 1 + u, with probability p k = Y = Thm. 1.2 (Characterisation of attainable payoffs) j 0 j 1 k 1 + d, with probability 1 − p If there exists a EMM Q for a submarket (S ,S ) which is not Sek−1 also an EMM for another submarket (S0,Sk), j 6= k, then the Consider a financial market in finite discrete time and suppose that 0 whole market (S0,S1,...,Sk) is not (NA), i.e. it admits arbitra- S is arbitrage-free and F0 is trivial. For any payoff H ∈ L+(FT ), the following are equivalent: ge. ˜1 Martingale property The discounted stock price S is a - (i) H is attainable. S˜0 P martingale iff r = pu + (1 − p)d. ∗ (ii) sup EQ[H] < ∞ is attained in some Q ∈ Pe(S), i.e. Q∈Pe(S) the supremum is finite and a maximum. 2.7 Multiplicative model EMM In other words, we have sup E [H] = E ∗ [H] < ∞ for Q Q In the binomial model, there exists a probability measure ≈ Q∈Pe(S) Suppose that we start with the RVs r , . . . , r and Y ,...,Y . Q P ∗ 1 T 1 T S˜1 some Q ∈ Pe(S). s.t. is a -martingale iff u > r > d. S˜0 Q (iii) The mapping Pe(S) → R, Q 7→ EQ[H] is constant, i.e. H has the same and finite expectation under all EMMs Q for S.

Steven Battilana p 7 / 23 Mathematical Foundations for Finance In that case, Q is unique (on FT ) and characterised by the pro- Binomial call pricing formula Then it holds for the Q-expectation (risk-neutral) of the two perty that Y1,...,YT are i.i.d. under Q with parameter payoffs that: S˜0 ˜ H˜ ˜1 ∗∗ ˜ k ∗ r − d V = S Q [Wk,T > x] − K Q [Wk,T > x] " A # " E # q∗ = [Y = 1 + u] = (% up) k k ˜0 C C Q k ST k ≤ k u − d EQ 0 EQ 0 ˜ S˜ S˜ u − r log K − (T − k) log(1 + d) k k ∗ S˜1 1 − q = 1 − Q[Yk = 1 + d] = (& down) k u − d x = 1+u log 1+d Interpretation: Since the volatility of an Asian style contingent claim is lower than the one of a European style contingent claim, Arbitrage and completeness The following statements are equiva- Remark: the Asian option bears lower risk and thus yields also lower profit. lent: This is the discrete analogue of the Black-Scholes formula. (i) u > r > d American options ˜1 (ii) ∃ a unique EMM ∗ for S (on F ) Q S˜0 T Consider an American option with maturity T and nonnegative adapted payoff process U = (U ) . (iii) The market S is (NA) and complete. 2.7.2 Multinomial model k k=0,...,T ¯ ¯ Then the arbitrage-free price process V = (Vk)k=0,...,T w.r.t. Q EMM can be expressed as a backward recursive scheme such as: Put-Call parity 1 ¯ IOT construct an EMM for S , it needs to hold that: VT = UT Assuming T = 1, it holds that: V¯ = max(U , [V¯ |F ]), for k = 0,...,T − 1 [S1] = S1 k k EQ k+1 k C(K) P (K) 1 K EQ 1 0 V0 − V0 = S0 − m 1 + r X ⇐⇒ EQ[Y1] = 1 + r ⇐⇒ qk(1 + yk) = 1 + r k=1 Pricing binomial contingent claims H with the further conditions: Assume time horizon T = 1, strike K > 0, and a payoff function m H(x, K): R+ → R+ of a European style contingent claim with X strike K. qi = 1, q1, . . . , qm ∈ (0, 1) k=1 + H may be a European call function C(x, K) = (x − K) or a European put function P (x, K) = (K − x)+. H(K) Arbitrage condition The following statements are equivalent: Then H can be replicated using a self-financing strategy ϕ = H(K) H(K) (i) y1 < r < ym (V0 , ϑ ) s.t. S˜1 ˜ 1 (ii) ∃ an EMM Q ≈ P s.t. ˜0 is a Q-martingale. H(K) H(H1 ,K) S V1(ϕ = , P − a.s. 1 + r (iii) The market S is (NA).

and ϕH(K) is given by Completeness The multinomial model is 1 1 H(K) r − d H(S (1 + u),K) u − r H(S (1 + d),K) V = 0 + 0 complete whenever m ≤ 2 0 u − d 1 + r u − d 1 + r (i.e. there are no nodes that allow for more than two possible H(1 + u, K/S1) − H(1 + d, K/S1) ϑH(K) = 0 0 stock price evolutions) 1 u − d incomplete whenever m > 2 (i.e. there is at least one node that allows for more than two Note that this can also be expressed via the martingale pricing possible stock price evolutions) approach:

" ˜1 # H(K) H(S1 ,K) Inequality of the payoﬀs of Asian and European call options V = E 0 Q 1 + r E ˜1 + Consider a European call option Ck = (Sk) and an Asian call where option with  + h ˜1 0 i r − d k Q S1 = S1 (1 + u) = q = A 1 X 1 u − d C =  S˜ − K k k j h ˜1 0 i u − r j=1 Q S1 = S1 (1 + d) = 1 − q = u − d

Steven Battilana p 8 / 23 Mathematical Foundations for Finance R Y dQ = R Y DdP. Martingale and BMs Ω Ω Def. (martingale, supermartingale, submartingale) F 3 Martingales Rem. As in descrete time, a martingale w.r.t. P and is a (real-valued) stochastic process M = (Mt) s.t. M is adapted to F, M is P - The point of these fomulae is that the tell us how to compute 1 integrable in the sense that each Mt is in L (P ), and the martingale Q-expectations in terms of P -expectations and vice versa. property holds: for s ≤ t, we have 3.1 Martingales D is a random variable (density) E[Mt|Fs] = Ms,P -a.s.

Z is a process (density process) Martingales If we have the inequality ” ≤ ” instead of ”= ”, then M is a super- Def. martingale; if we have ” ≥ ”, then M is a submartingale. Of course, Let (Ω, F,P ) be a probability space with a filtration F = F = (Ft) and M = (Mt) should have the same time index set. (i) D := dQ (Fk)k=0,1,...,T . dP Rem. dQ QT Because our filtration satisfies the usual conditions, a general result A (real-valued) stochastic process X = (Xk)k=0,1,...,T is called (ii) D := dP := Z0 j=1 Dj a martingale (w.r.t. P and ) if: from the theory of stochastic processes says that any martingale has F Def. a version with nice (RCLL, i.e. right-continuous with left limits) tra- Q (i) X is adapted to F. We introduce the P -martingale Z (sometimes denoted by Z or jectories. ZQ;P ) by (ii) X is P -integrable in the sense that X ∈ L1(P ) for each k, Def. (local martingale, localising sequence) k An adapted process X = (X ) null at 0 (i.e. with X = 0) is i.e. h dQ i t t≥0 0 Zk := EP [D|Fk] = EP dP |Fk , for k = 0, 1, ..., T . called a local martingale (w.r.t. P and F) if there exists a sequence of stopping times (τ ) N increasing to ∞ s.t. for each n ∈ N, the EP [|X|] < ∞ Because D > 0 P -a.s., the process Z = (Z ) is strict- n n∈ k k=0,1,...,T τn F stopped process X = (Xt∧τn )t≥0 is a (P, )-martingale. We then ly positive in the sense that Zk > 0 P -a.s. for each k, or also call (τn)n∈N a localising sequence. (iii) X satisfies the martingale property: P [Zk > 0, ∀k] = 1. Z is called density process (of Q, w.r.t. P ).

EP [Xl|Fk] = Xk P -a.s. for k ≤ l Q is an EMM for the stochastic process (Sk)k≥0 iff S is a martingale under Q and if Q is equivalent to P . Interpretation: This means that the best prediction for the later 3.2 Stopping times The probability measure Q is equivalent to P (Q ≈ P ) iff: value Xl given the information Fk is just the current value Xk. Hence the changes in a martingale cannot be predicted. τ τ (i) Q[A] > 0 ⇐⇒ P [A] > 0 Stopping time The stochastic process S = (Sk )k=0,1,...,T defi- In other words, a martingale describes a fair game in the sense ned by that one cannot predict where it goes next. (ii) Q[Ω] = 1 Sτ (ω) := S (ω) := S (ω) A stochastic process (Sk)k≥0 is a Q-martingale iff: k k∧τ k∧τ(ω) A supermartingale is defined the same but with the property (i) S is adapted to the considered filtration. E [X |F ] ≤ X k is called the process S stopped at τ. It clearly behaves like S up to P l k k time τ and remains constant after time τ. (ii) Sk is integrable: EQ[|Sk|] < ∞. Thm 2.1 (Stopping theorem) A submartingale is defined the same but with the property (iii) Martingale property: EQ[Sk+1|Fk] = Sk. Suppose that M = (M ) is a (P, )-martingale with RC tra- E [X |F ] ≥ X t t≥0 F P l k k jectories, and σ, τ are -stopping times with σ ≤ τ. Local martingale F

If either τ is bounded by some T ∈ (0, ∞) or M is uniformly An adapted process X = (Xk)k=0,1,...,T null at 0 (i.e. with 1 Equivalent Martingale Measure (EMM) integrable, then Mτ , Mσ are both in L (P ) and Def. (equivalent martingale measure (EMM)) X0 = 0) is called a local martingale (w.r.t. P and F) if there exists a sequence of stopping times (τn) increasing to T s.t. n∈N E[Mτ |Fσ] = Mσ P -a.s. (i) An equivalent martingale measure (EMM) for S is a probability τn for each n ∈ N, the stopped process X = (Xk∧τn )k=0,1,...,T measure Q equivalent to P on FT s.t. S is a Q-martingale. is a (P, F)-martingale. Rem. P P (ii) We denote by e(S) or e the set of all EMMs for S. We then call (τn)n∈ a localising sequence. N (i) For any RC martingale M and any stopping time τ, we ha- Thm. (Radon-Nikodým) For any martingale X and any stopping time τ, the stopped pro- ve E[Mτ∧t|Fs] = Mτ∧s for s ≤ t, i.e. the stopped process Let (Ω, F) and filtration F = (Fk=0,1,...,T in finite discrete time. cess Xτ is again a martingale. τ τ M = (Mt )t≥0 = (Mt∧t)t≥0 is again a martingale (because we One (Ω, F), we have two probability measures Q and P , and we τ τ In particular, EP [Xk∧τ ] = EP [X0], ∀k. have E[M |Fs] = M ). assume that Q ≈ P . Then the Radon-Nikodýmtheorem tells us that t s dQ (ii) If M is an RC martingale and τ is any stopping time, then we there exists a density D := dP ; this is a random variable D > 0 P -a.s. (because Q ≈ P ) s.t. Q[A] = EP [DIA] for all A ∈ F, or always have for any t ≥ 0 that E[Mτ∧t] = E[M0]. If either more generally τ is bounded or M is uniformly integrable, then we also obtain E[Mτ ] = E[M0]. EQ[Y ] = EP [Y D], ∀RV Y ≥ 0. One sometimes writes this as

Steven Battilana p 9 / 23 Mathematical Foundations for Finance Cases of stopping times Radon-Nykodin theorem: There exists a density respectively. (This means that Zk is the density of Q w.r.t. P on Fk.) Define the stopping time τa for a ∈ R, a > 0 as: dQ := D 1 dP (ii) If j ≤ k and Uk is Fk-measurable and either ≥ 0 or in L (Q), τa := inf{t ≥ 0|Wt > a} then we have the Bayes formula This is a RV D > 0, P -a.s. s.t. for all A ∈ F and for all RVs k 1 Then it holds that: EQ[Uk|Fj ] = EP [ZkUk|Fj ] Q-a.s. Y ≥ 0 it holds that: Zj – τ ≤ τ ,P -a.s. for a < a . a1 a2 1 2 (This tells us how conditional expectations under Q and P are Q[A] = EP [DIA] ⇐⇒ EQ[Y ] = EP [Y D]. – P [τa < ∞] = 1. related to each other.) Written in terms of Dk, the Bayes formula for j = k − 1 becomes – Wτa = a,P -a.s. This can also be written as h i Z Z EQ[Uk|Fk−1] = EP [DkUK |Fk−1]. – E Wτ Fτ 6= Wτ , -a.s. a2 a1 a1 P Y dQ = Y DdP This shows that the rations Dk play the role of ”one-step condi- i.e. the stopping theorem fails for τ = τ and σ = τ . Ω Ω a2 a1 tional densities” of Q with respect to P . This formula tells us how to compute Q-expectations in terms of – limn→∞ τn(ω) = ∞, P-a.s. F P -expectations and vice-versa. (iii) A process N = (Nk)k=0,1,...,T which is adapted to is a Q- martingale iff the product ZN is a P -martingale. (This tells us Define the stopping time ρa for a ∈ R, a > 0 as: how martingale properties under P and Q are related to each ρa := sup{t ≥ 0|Wt > a} Density process in discrete time other.)

Then it follows that ρa = ∞ with probability 1 under P . Assume the same setting as before. Proof Lem. 3.1

Def. (stopping time) Radon-Nykodin theorem: The density process Z of Q w.r.t. P , (i) F or also called the P -martingale Z, is defined as Again exactly like in discrete time, a stopping time w.r.t. is a map- (vi) ping τ :Ω → [0, ∞] s.t. {τ ≤ t} ∈ F for all t ≥ 0. EQ[Y ] = EP [Y D] = EP [EP [Y D|Fk]] t dQ Def. (events observable up to time σ) Zk : = EP [D|Fk] = EP Fk for k = 0, 1,...,T Y Fk-meas. We define for a stopping time σ, the σ-field of events observable up dP = EP [YEP [D|Fk]] = EP [YZk] to time σ as Then for every F -measurable RV Y ≥ 0 or Y ∈ L1(Q), it holds F := {A ∈ F | A ∩ {σ ≤ t} ∈ F , for all t ≥ 0}. k σ t that (ii) a) LHS:

(One must and can check that Fσ is a σ-field, and one has Fσ ⊆ Fτ if Q-martingale for σ ≤ τ.) EQ[Y |Fk] = EP [YZk|Fk] EQ[Nk|Fj ] = Nj ⇒ Nj Zj = EP [NkZk|Fj ] Def. ⇒ NZP -martingale and for every k ∈ {0, 1,...,T } and any A ∈ Fk, it holds that We also need to define Mτ , the value M at the stopping time τ, by Q[A] = E [Z I ] b) RHS: (Mτ )(ω) := Mτ(ω)(ω). P k A 1 if P -martingale 1 Note that this implicitly assumes that we have a random variable EP [N Z |Fj ] = Nj Zj Properties: Z k k Z M∞, because τ can take the value +∞. j j Def. (hitting times) – Zk is a RV and Zk > 0, P -a.s. = Nj , i.e. EQ[Nk|Fj ] = Nj One useful application of Prop. 2.2 is the computation of the Laplace – A process N = (Nk)k=0,1,...,T which is adapted in F is a Q- ⇒ N is Q-martingale transforms of certain hitting times. More precisely, let W = (Wt)t≥0 be a BM and define for a > 0, b > 0 the stopping times martingale iff the product ZN is a P -martingale. (This tells us how martingale properties under P and Q are This concludes the proof for (ii) τa := inf{t ≥ 0 | Wt > a}, related to each other.) Def.

σa,b := inf{t ≥ 0 | Wt > a + bt}. Bayes formula: Zk Dk := , for k = 1, ..., T. 1 Zk−1 If j ≤ k and Uk is Fk-measurable and either ≥ 0 or in L (Q), then The process D is adapted, strictly positive and satisfies by its defini- 1 tion EQ[Uk|Fj ] = EP [ZkUk|Fj ] Q-a.s. Zj EP [Dk|Fk−1] = 1, 3.3 Density processes/Girsanov’s theorem This tells us how conditional expectations under Q and P are because Z is a P -martingale. related to each other. Rem. Again because Z is a martingale and by Lem. 3.1, Density in discrete time Lem. 3.1 Lem. 3.1 1) Assume (Ω, F) and a filtration = (Fk)k=0,1,...,T in finite dis- EP [Z0] = EP [ZT ] = EP [ZT IΩ] = Q[Ω] = 1, F (i) For every k ∈ {0, 1, ..., T } and any A ∈ Fk or any Fk-measurable crete time. random variable Y ≥ 0 or Y ∈ L1(Q), we have and we can recover Z from Z and D via On (Ω, F), we have two probability measures Q and P , and we 0 Q[A] = E [Z I ] ⇔ E [Y ] = E [Z Y ], Qk assume Q ≈ P . P k A Q P k Zk = Z0 j=1 Dj , for k = 0, 1, ..., T.

Steven Battilana p 10 / 23 Mathematical Foundations for Finance R 2 Rem. (ii) An adapted process Y = (Yt)0≤t≤T is a (local) Q-martingale iﬀ In particular, if L = νdW for some ν ∈ Lloc(W ), then To construct an equivalent martingale measure for a given process the product ZY is a (local) P -martingale. R R Wf = W − h νdW, W i = W − νsds so that the P-BM S, all we need are an F0-measurable random variable Z0 > 0 R Rem. W = Wf + νsds becomes under Q a BM with (instantaneous) P -a.s. with EP [Z0] = 1 and an adapted strictly positive process loc drift ν. D = (Dk)k=1,...,T satisfying EP [Dk|Fk−1] = 1 for all k, and in If Q ≈ P , we can use Lem. 2.1 for any T < ∞ and hence obtain addition E [D (S − S )|F ] = 0 for all k. If we have a closer look at W ∗ deﬁned in the Black-Scholes chap- P k k k−1 k−1 a statement for processes Y = (Y )t≥0 on [0, ∞). Def. (i.i.d. returns) ter, we see that W ∗ is a Brownian motion under the probability One consequence of part 2) of Lem. 2.1 is also that 1 is a Q - measure Q∗ given by k Z 1 1 Q 0 k martingale, on [0, ∞] if Q ≈ P on FT , or even on [0, ∞) if Sek = S0 Yj , Sek = (1 + r) , ∗ Z j=1 loc dQ 1 2 Q ≈ P . := E − λdW = exp −λWT − λ T , dP 2 where Y , ..., Y are > 0 and i.i.d. under P . Rem. (construction of T 1 T Rem. an EMM Q) loc loc Suppose that Q ≈ P with density process Z, then Q ≈ P implies whose density process w.r.t. P is Choose Dk like Yk independend of Fk−1. Then we must have that Z is a local martingale. Z Dk = gk(Yk) for some measurable function gk, and we have to ∗ 1 2 choose g in such a way that we get Zt = E − λdW = exp −λWt − λ T for 0 ≤ t ≤ T. k Thm. 2.2 (Girsanov) t 2 1 = EP [Dk|Fk−1] = EP [gk(Yk)] loc (i) Suppose that Q ≈ P with density process Z. If M is a local and P-martingale null at 0, then

1 + r = EP [DkYk|Fk−1] = EP [Ykgk(Yk)]. Z 1 Mf := M − d[Z,M] (Note that these calculations both exploit the P independence of Yk Z from Fk−1.) If this choice is possible, we can then choose all the is a local Q-martingale null at 0. gk ≡ g1, because the Yk are (assumed) i.i.d. under P . To ensure that Dk > 0, we can impose gk > 0. (ii) In particular, every P-semimartingale is also a Q-semimartingale (and vice-versa, by symmetry). Density process in continuous time Def. (product rule)

Suppose we have P and a ﬁltration F = (Ft)t≥0. Z Z • ZM = Z dM + M dZ + [Z,M] Fix T ∈ (0, ∞) and assume only that Q ≈ P on FT . − −

If we have this for every T < ∞, we call Q and P locally equiva- • d(ZM) = Z−dM + M−dZ + d[Z,M] loc lent and write Q ≈ P . Lem. For an infinite horizon, this is usually strictly weaker tahtn Q ≈ P . Let Z,M be of finite variation, then it holds hMi, hZi ⇒ hM,Zi Def. (density process) Let us for simplicity fix T ∈ (0, ∞) and suppose that Q ≈ P on FT . is also of finite variation. Denote by Rem. " # When [Z,M] is of finite variation, the following holds dQ| Z := E FT F for 0 ≤ t ≤ T t P t 1 X 1 Z 1 dP |FT , [Z,M] = ∆ ∆[Z,M] = ∆ d[Z,M]. Z 2 Z the density process of Q w.r.t. P on [0,T ], choosing an RCLL version of this P -martingale on [0,T ]. Def. (alternative stochastic Itôintegral definition) Rem. ∀V ”that are nice enough” Since Q ≈ P on FT , we have Z > 0 on [0,T ], and because Z is a hR vdW, V i = R vdhW, V i P -(super)martingale, this implies that also Z− > 0 on [0,T ] by the so-called minimum principle for supermartingales. Thm. 2.3 (Girsanov (continuous version)) Lem. 2.1 loc Suppose that Q ≈ P with continuous density process Z. Write Suppose that Q ≈ P on FT . Then Z = Z0E(L). If M is a local P-martingale null at 0, then (i) For s ≤ t ≤ T and every Ut which is Ft-measurable and either 1 ≥ 0 or in L (Q), we have the Bayes formula Mf := M − [L, M] = M − hL, Mi 1 Q EQ[Ut|Fs] = EP [ZtUt|Fs] Q-a.s. is a local -martingale null at 0. Zs P Q Moreover, if W is a -BM, then Wf is a -BM.

Steven Battilana p 11 / 23 Mathematical Foundations for Finance Def. (alternative definition of BM without any filtration) (i) Law of large numbers: There is also a definition of BM without any filtration F. This is Wt a (real-valued) stochastic process W = (Wt) which starts at 0, lim = 0,P -a.s. t≥0 t→∞ t 4 Stochastic Integration and Calculus satisfies (BM2) and instead of (BM1) the following property: i.e. BM grows asymptotically less than linearly (as t → ∞). 0 (BM1 ) For any n ∈ N and any times 0 = t0 < t1 < ... < t < ∞, the increments W − W , i = 1, ..., n, (ii) (Global) law of the iterated logarithm (LIL): n ti ti−1 p are independent (under P ) and we have (under P ) that With ψglob(t) := 2t log(log t)), it holds for t ≥ 0 that:

Wti − Wti−1 ∼ N (0, ti − ti−1), or N (0, (ti −ti−1)Im×m) Wt if W is Rm-valued. lim sup = +1,P -a.s. t→∞ ψglob(t) 4.1 Brownian motion 0 Instead of (BM1 ), one also says (in words) that W has independent Wt stationary increments with a (speciﬁc) normal distribution. lim inf = −1,P -a.s. Brownian motion t→∞ ψglob(t)

i.e. for P -a.a. ω, the function t 7→ Wt(ω) for t → ∞ oscillates Transformations of BM Preliminery precisely between t 7→ ±ψglob(t). Prop. 1.1 Throughout this chapter, we work on a probability space (Ω, F,P ). Suppose W = (W ) is a BM. Then: (iii) (Local) law of the iterated logarithm (LIL): In particular, Ω cannot be finite or countable. A filtration F = (F ) t t≥0 q t With ψ (h) := 2h log(log 1 ), it holds for t ≥ 0 that: in continuous time; this is like in discrete time a family of σ-fields (i) W 1 := −W is a BM. loc h F ⊆ F with F ⊆ F for s ≤ t. The time parameter runs either t s t (ii) Restarting at a fixed time T : Wt+h − Wt through t ∈ [0,T ] with fixed time horizon T ∈ (0, ∞) or through lim sup = +1,P -a.s. h&0 ψloc(h) t ∈ [0, ∞). In the later case, we define 2 Wt := WT +t − WT W − W lim inf t+h t = −1,P -a.s.   h&0 ψ (h) _ [ for t ≥ 0 is a BM for any T ∈ (0, ∞). loc F∞ := Ft := σ  Ft . i.e. for P -a.a. ω, to the right of t, the trajectory u 7→ t≥0 t≥0 (iii) Rescaling in space and time: Wu(ω) around the level Wt(ω) oscillates precisely between h 7→ 3 ±ψloc(h). For technical reasons, we should also assume that F satisfies the Wt := cW t 2 so-called usual conditions of being right-continuous (RC) and P - c Prop. 1.3 complete. for t ≥ 0 is a BM for any c ∈ R, c 6= 0. Suppose W = (Wt)t≥0 is a BM. Then for P -a.a. ω ∈ Ω, the function t 7→ W (ω) from [0, ∞) to is continuous, but nowhere differentia- Def. (Brownian motion) (iv) Time-reversal: t R ble. A Brownian motion w.r.t. P and a filtration = (Ft)t≥0 is a (real- F 4 valued) stochastic process W = (Wt)t≥0 which [(BM0)] is adapted Wt := WT −t − WT Def. (partition) to F, starts at 0 (i.e. W0 = 0 P -a.s.) and satisfies the following Call a partition of [0, ∞) any set Π ⊆ [0, ∞) of time points properties: for 0 ≤ t ≤ T is a BM on [0,T ] for any T ∈ (0, ∞). with 0 ∈ Π and Π ∩ [0,T ] finite for all T ∈ [0, ∞). This implies that Π is at most countable and can be ordered increasingly as (BM0) null at zero (v) Inversion of small and large times: Π = {0 = t < t < ... < t < ... < ∞}. W is adapted to and null at 0 (i.e. W ≡ 0, P -a.s.). 0 1 m F 0 ( Def. (mesh size, sum) 5 tW 1 for t > 0 (BM1) independent and stationary increments Wt := t The mesh size of Π is then defined as |Π| := sup{ti − ti−1 | ti−1, ti ∈ Π}, 0 for t = 0 For s ≤ t, the increment Wt − Ws is independent (under P ) i.e. the size of the biggest time-step in Π. For any partition Π of of F and satisfies under P : W − W ∼ N (0, t − s)). [0, ∞), s t s for t ≥ 0 is a BM. Π P 2 (BM2) continuous sample paths 6 2 R t QT := (Wti∧T − Wti−1∧T ) (vi) W := (Wt) − t = 2 WsdWs, t ≥ 0 W has continuous trajectories, i.e. for P -a.a. ω ∈ Ω, the t 0 ti∈Π function t → Wt(ω) on [0, ∞) is continuous. 7 1 2 (vii) Wt := exp αWt − 2 α t for t ≥ 0 and for any α ∈ R. is then the sum up to time T of the squared increments of BM along Π. Remarks: Note that we always use here the definition of BM without an exo- We expect, at least for |Π| very small so that time points are close Brownian motion in m is simply an adapted m-valued stocha- genous filtration. 2 R R together, that (Wti∧T − Wti−1∧T ) ≈ ti ∧ T − ti−1 ∧ T and hence stic process null at 0 and with the increment Wt − Ws having the QΠ ≈ P t ∧ T − t ∧ T 2 for |Π| small. normal distribution N (0, (t − s)Im×m), where Im×m denotes Laws on BM T i i−1 ti∈Π the identity matrix. The next result gives some information about how trajectories of BM Rm Rm Thm. 1.4 (quadratic variation) Skript version: Brownian motion in is simply an adapted - behave. Suppose W = (Wt)t≥0 is a BM. For any sequence (Πn)n∈N of par- valued stochastic process null at 0 with (BM2) and s.t. (BM1) Prop. 1.2 titions of [0, ∞) which is refining (i.e. Πn ⊆ Πn+1∀n) and satisfies holds with N (0, t − s) replaced by N (0, (t − s)Im×m), where Suppose W = (Wt)t≥0 is a BM. Then: lim |Πn| = 0, we then have Im×m denotes the m × m identity matrix. n→∞ h i P lim QΠn = t, for every t ≥ 0 = 1. n→∞ t

Steven Battilana p 12 / 23 Mathematical Foundations for Finance We express this by saying that along (Πn)n∈N, the BM W has (with In mathematical finance, both τa and σa,b come up in connection probability 1) quadratic variation t on [0, t] for every t ≥ 0, and we with a number of so-called exotic options. In particular, they are write hW it = t. important for barrier options wohse payoff depends on whether 4.3 Poisson processes (We sometimes also say that P -a.a. trajectories W•(ω) : [0, ∞) → R or not a (upper or lower) level has been reached by a given time. of BM have quadratic variation t on [0, t], for each t ≥ 0.) When computing prices of such options in the Black-Scholes mo- Def. (finite variation) del, one almost immediately encounters the Laplace transforms in Poisson processes A function g : [0, ∞) → R is of finite variation or has finite 1- Prop. 2.3. A Poisson process N = (Nt)t≥0 with parameter λ ∈ R, λ > 0 variation if for every T ∈ (0, ∞), and w.r.t. (P, F) is a real-valued stochastic process satisfying the P following properties: sup |g(ti ∧ T ) − g(ti−1 ∧ T )| < ∞, Π ti∈Π (PP0) null at zero where the supremum is taken over all partitions Π of [0, ∞). N is adapted to F and null at 0 (i.e. N0 ≡ 0, P-a.s.). Interpretation 4.2 Markovian properties (PP1) independent and stationary increments The interpretation is that the graph of g has finite length on any time For 0 ≤ s < t, the increment N −N is independent (un- interval. More precisely, if we define the arc length of (the graph of) t s Rem. der ) of F and follows (under ) the Poisson distributi- g on the interval [0,T ] as P s P We have already seen in part 2) of Prop. 1.1 that for any fixed time on with parameter λ(t − s), i.e. Nt − Ns ∼ Poi(λ(t − s)), P p 2 2 sup (ti ∧ T − ti−1 ∧ T ) + (g(ti ∧ T ) − g(ti−1 ∧ T )) , T ∈ (0, ∞), the process i.e Π t ∈Π i W − W , t ≥ 0, is again a BM k k t+T T λ (t − s) −λ(t−s) with the supremum again taken over all partitions Π of [0, ∞), then P[Nt = k] = e k! g has finite variation on [0,T] iff it has finite arc length on [0,T ]. if (Wt)t≥0 is a BM. Moreover, one can show that the independence Rem. of increments of BM implies that Any monotonic (increasing or decreasing) function is clearly of finite 0 (PP2) counting process Wt+T − WT , t ≥ 0, is independent of FT , variation, because the absolute values above disappear and we get a N is a counting process with jumps of size 1, i.e. for P-a.a. 0 telescoping sum. Moreover, one can show that any function of finite where FT = σ(Ws; s ≤ T ) is the σ-field generated by BM up to ω, the function t 7→ Nt(ω) is RCLL, piecewise constant variation can be written as the difference of two increasing functions time T . and N0-valued, and increases by jumps of size 1. (and vice versa). Intuition Important properties of Poisson processes: if X ∼ Poi(λ), then: Rem. Intuitively, this means that BM at any fixed time T simply forgets its past up to time T (with the only possible exception that it remem- Every continuous function f which has nonzero quadratic variation E[X] = λ, Var[X] = λ bers its current position W at time T ), and starts afresh. along a sequence (Πn) as above must have infinite variation, i.e. T unbounded oscillations. We also recall that a classical result due to Def. (Markov property) The quadratic variation of a Poisson process equals itself, i.e. Lebesgue says that any function of finite variation is almost every- One consequence of the above remark is the following. Suppose that where differentiable. So Prop. 1.3 implies that Brownian trajectories at some fixed time T , we are interested in the behaviour of W after [N]t = Nt must have infinite variation, and Thm 1.4 makes this even quantita- time T and try to predict this on the basis of the past of W up tive. to time T , where ”prediction” is done in the sense of a conditional Examples of Poisson processes: the following Poisson processes expectation. Then we may as well forget about hte past and look Prop. 2.2 are (P, F)-martingales: Suppose W = (W ) is a (P, F)-Brownian motion. Then the fol- ony at the current value WT at time T . A bit more precisely, we can t t≥0 – Compensated Poisson process: lowing process are all (P, F)-martingales: express this, for functions g ≥ 0 applied to the part of BM after time T , as (i) W itself. N˜t = Nt − λt, t ≥ 0 E[g(Wu; u ≥ T ) | σ(Ws; s ≤ T )] = E[g(Wu; u ≥ T ) | σ(WT )]. (ii) W 2 − t, t ≥ 0. t This is called the Markov property of BM. – Geometric Poisson process: αW − 1 α2t (iii) e t 2 , t ≥ 0, for any α ∈ R. Rem. St = exp (Nt log(1 + σ) − λσt) , t ≥ 0 Prop. 2.3 BM has even the strong Markov property. Let W be a BM an a > 0, b > 0. Then for any λ > 0, we have Generalised: If we denote almost as above by FW the filtration where σ ∈ R, σ > −1. √ (i) E[e−λτa ] = e−a 2λ generated by W (and made right-continuous, to be accurate), – Two cases of squared compensated Poisson processes: W √ and if τ is a stopping time w.r.t. F and s.t. τ < ∞ P -a.s., then −λσ −λσ −a(b+ b2+2λ) 2 2 (ii) E[e a,b ] = E[e a,b Iσ <∞] = e . W ˜ ˜ a,b Wt+τ − Wτ , t ≥ 0, is again a BM and independent of Fτ . Nt − Nt, Nt − λt, t ≥ 0 Rem. This includes the first remark in this subsection as special case ˜ and one can easily believe that it is even more useful than the It follows that [N]t = Nt. In the proof of the above Prop. 2.3 the following was being used: 1 2 above definition of Markov property. αWt− α t Mt := e 2 , t ≥ 0. −λU For a general random variable U ≥ 0, the function λ 7→ E[e ] for λ > 0 is called the Laplace transform of U.

Steven Battilana p 13 / 23 Mathematical Foundations for Finance 0 as n → ∞ s.t.   The operation [·, ·] is bilinear. X 2 4.4 Stochastic integration P [M]t(ω) = lim Mti∧t(ω) − Mti−1∧t(ω) ∀t ≥ 0 n→∞ From the characterisation of [M] in Thm 1.1, it follows that the ti∈Πn operation [·, ·] is bilinear, and also that B = [M,N] is the un- Def. (recall discrete stochastic integral) = 1 ique adapted RCLL process B null at 0, of finite variation with ∧ The trading gains or losses from a self-financing strategy ϕ = (V0, ϑ) ∆B = ∆M∆N and s.t. MN − B is again a local martingale. are described by the stochastic integral We call [M] the optional quadratic variation or square bracket process of M. Def. (predictable compensator) R P tr P tr G(ϑ) = ϑ · S = ϑdS = ϑ ∆Sj = ϑ (Sj − Sj−1). J j 2 There exists a unique increasing predictable process hMi = j j If M satisfies sup |Ms| ∈ L for each t ≥ 0 (and hence is 2 0≤s≤t (hMit) null at 0 s.t. [M] − hMi, and therefore also M − hMi 1 t≥0 Rem. in particular a martingale), then [M] is integrable (i.e. [M]t ∈ L [= M 2 − [M] + [M] − hMi], is a local martingale. The process hMi 2 for every t ≥ 0) and M − [M] is a martingale. is called the shapr bracket (or sometimes the predictable variance) Our goal in this section is to construct a stochastic integral pro- process of M. cess H · M = R HdM when M is a (real-valued) local martingale Thm. 1.1 Note, this might be useful since [M] is not necessarily predictable. null at 0 and H is a (real-valued) predictable process with a sui- (i) For any local martingale M = (Mt)t≥0 null at 0, there exists a Rem. and Cor. table integrability property (relative to M). unique adapted increasing RCLL process [M] = ([M]t)t≥0 null 2 (i) Any adapted process which is continuous is automatically locally To relate notation-wise to previous chapters we will the following at 0 with ∆[M] = (∆M) [this is an important property] and 2 bounded and therefore also locally square-integrable. notions: having the property M − [M] is also a local martingale. 2 ∧ (ii) If M is continuous, then so is [M], because ∆[M] = (∆M) = 0. (i) H = ϑ (ii) This process can be obtained as the quadratic variation of M in This implies then also that [M] = hMi. In particular, for a Brow- the following sense: There exists a sequence (Πn)n∈N of partiti- (ii) M ∧= S ons of [0, ∞) with |Πn| → 0 as n → ∞ s.t. nian motion W , we have [W ]t = hW it = t for all t ≥ 0. Rem. script: (iii) If bot M and N are locally square-integrable (e.g. if they are Throughout this chapter, we work on a probability space (Ω, F,P ) continuous), we also get hM,Ni via polarisation, i.e. F " # with a filtration (Ft)t≥0 satisfying the usual conditions of right- 1 W P 2 hM,Ni := (hM + Ni − hM − Ni). continuity and P -completeness. If needed, we define F∞ := Ft. P [M]t(ω) = lim (Mti∧t(ω) − Mti−1∧t(ω)) , ∀t ≥ 0 = 1. 4 n→∞ t ∈Π t≥0 i n (iv) If M is Rd-valued, then [M] becomes a d × d-matrix-valued pro- We also fix a (real-valued) local martingale M = (M ) null at t t≥0 lecture: cess with entries [M]ik = [M i,M k]. To work with that, one 0 and having RCLL trajectories. Since we want to define stochastic   needs to establish more properties. The same applies to hMi, if integrals R HdM and these are always over half-open intervals of the X 2 it exists. form (a, b] with 0 ≤ a < b ≤ ∞, the value of M at 0 is irrelevant P {ω : ∃[M]t(ω) = lim (Mt ∧t(ω) − Mt ∧t(ω)) , ∀t ≥ 0} n→∞ i i−1 In general the following holds: [M,M] = [M]. and it is enough to look at processes H = (Ht) defined for t > 0. ti∈Πn Def. (jump) = 1. Def. (Set of all bounded elementary processes) (i) For any process Y = (Y ) with RCLL trajectories, we denote We denote by bE the set of all bounded elementary process of the t t≥0 We call [M] the optional quadratic variation or square bracket by form process of M. ∆Yt := Yt − Yt− := Yt − lim Ys P 2 n−1 s%t (iii) If M satisfies |Ms| ∈ L for some T > 0 (and hence is in X 0≤s≤T H = hiI(ti,ti+1] the jump of Y at time t > 0. particular a martingale on [0,T ]), then [M] is integrable on [0,T ] i=0 1 2 t s (i.e. [M]t ∈ L ) and M − [M] is a martingale on [0,T ]. N R 2 R 2 R 2 with n ∈ , 0 ≤ t0 < t1 < . . . < tn < ∞ and each hi a bounded (ii) ∆( Hr d[M]r)t := Hr d[M]r − lim Hr d[M]r 0 s%t 0 (real-valued) Fti -measurable RV. (Optional) covariation process Def. (Stochastic integral) Optional quadratic variation/square bracket process For two local martingales M,N null at 0, we define the (optional) For any stochastic process X = (Xt)t≥0, the stochastic integral covariation process [M,N] by polarisation, i.e. R HdX of H ∈ bE is defined as For any local martingale M = (Mt)t≥0 null at 0, there exists a unique adapted increasing RCLL process [M] = ([M]t) and 1 t≥0 [M,N] := ([M + N] − [M − N]) Z t n−1 having the property that M 2 − [M] is also a local martingale. 4 X HsdXs := H · Xt := hi Xti+1∧t − Xti∧t for t ≥ 0. 0 i=0 This process can be obtained as the quadratic variation of M in the following sense. An alternative definition of the covariation process [M,N] There exists a sequence (Π ) of partitions [0, ∞) with |π| → Rd n n∈N If X and H are both -valued, the integral is still real-valued, [M,N]t := lim n→∞ and we simply replace products by scalar products everywhere. X Lem 1.2 (Isometry property) Mti∧t(ω) − Mti−1∧t(ω) Nti∧t(ω) − Nti−1∧t(ω) , ti∈Πn 2 Suppose M is a square-integrable martingale (i.e. Mt ∈ L for ∀t ≥ 0 all t ≥ 0).

Steven Battilana p 14 / 23 Mathematical Foundations for Finance R For every H ∈ bE, the stochastic integral process H·M = HdM (iv) This is the proof for (ii): Def. is then also a square-integrable martingale, We introduce the space Z Z t Z s h i 2 2 2 R t R t 2 ∆ H d[M]r = H d[M]r − lim H d[M]r 2 2 2 • [H · Mt] = HsdXs = Hs d[X]s r r r L (M) := L (M,P ) := L (Ω, P,PM ) 0 0 t 0 s%t 0 n−1 = {all (equivalence classes of) predictable H = (Ht)t>0 s.t. and we have the isometry property X 2 1 ) = hi ([M]ti+1∧t − [M]ti∧t) Z ∞ " # 2 1 2 2 Z ∞ 2 i=0 kHk 2 := (E [H ]) 2 = E H d[M]s < ∞ . 2 L (M) M s • E (H · M∞) = E HsdMs n−1 0 0 X 2 − lim hi ([M]ti+1∧s − [M]ti∧s) 0  2 s%t (As usual, by taking equivalence classes, we identify H and H if n−1 ! i=0 ∧ X they agree PM -a.e. (almost everywhere= ” a.s.) on Ω.) = E  hi(Mt − Mt )  i+1 i ti≤s0 predictable if it is P- R 0 0 0 measurable when viewed as a mapping H : Ω → . (N,N )M2 = (N∞,N∞)L2 = E[N∞,N∞]. Note, as a consequence, every H ∈ bE is then predictable since it 0 Rem. is adapted and left-continuous. Cor. Because of the above remark the mapping H 7→ H · M from bE (i) The argument in the proof of Lem. 1.2 actually shows that the Def. to M2 is linear and an isometry because of Lem. 1.2 says that for process (H · M)2 − R H2d[M] is a martingale. 0 We define the (possibly infinite) measure PM := P ⊗ [M] on (Ω, P) H ∈ bE, by setting (ii) It is ”not very difficult” to argue that 1 2 2 R 2 2 kH · MkM2 =(E[(H · M∞) ]) ∆( H d[M] = (∆(H · M)) for H ∈ bE, Z ∞ 0 EM [Y ] := E Ys(ω)d[M]s(ω) , for Y ≥ 0 predictable; 1 2 Z ∞ 2 by exploiting that H is piecewise constant and ∆[M] = (∆M) . 0 2 = E Hs d[M]s (iii) In view of Thm. 1.1 and the uniqueness there, the combination the inner integral is defined ω-wise as a Lebesgue-Stieltjes integral 0 of these two properties can also be formulated as saying that because t 7→ [M]t(ω) is increasing, null at 0 and RCLL and so =kHkL2(M). can be viewed as the distribution function of a (possibly infinite) Z Z Rem. 2 ω-dependent measure on (0, ∞). [H · M] = HdM = H d[M], for H ∈ bE. By general principles, this mapping can therefore be uniquely extended to the closure of bE in L2(M). In other words, we can define

Steven Battilana p 15 / 23 Mathematical Foundations for Finance a stochastic integral process H · M for every H that can be appro- Def. (continuous local martingale, locally bounded) R Kd(R HdM) = R KHdM. ximated, w.r.t. the norm k · k 2 , by processes from bE, and the L (M) (i) M is a continuous local martingale null at 0, briefly written as Behaviour under stopping 2 resulting H · M is again a martingale in M0 and still satisfies the c M ∈ M0,loc. 2 isometry property described in the equation above. – Suppose that M is a local martingale, H ∈ Lloc(M) and τ is a This includes in particular the case of a Brownian motion W . τ Prop. 1.3 stopping time. Then M is a local martingale by the stopping 2 2 theorem, H is in L2 (M τ ), HI is in L2 (M), and we Suppose that M is in M0. Then: (ii) Then M is in M0,loc because it is even locally bounded: loc 0,τ loc For the stopping times have K K (i) bE is dense in L2(M), i.e. the closure of bE in L2(M) is L2(M). (H · M)τ = H · (M τ ) = (HI ) · M = (HI ) · (M τ ). 2 τn := inf{t ≥ 0 | |Mt| > n}% is∞ P -a.s., 0,τ 0,τ (ii) For every H ∈ L (M), the stochastic integral process H · M = K K K K R 2 τn – In words: A stopped stochastic integral is computed by either HdM is well defined, in M0 and satisfies the above equation. We have by continuity that |M | ≤ n for each n, because first stopping the integrator and then integrating, or setting Rem. ( |M | ≤ n, t ≤ τ the integrand equal to 0 after the stopping time and then in- Let M ∈ M2, we then have E[|M |2] < ∞ for every t ≥ 0 s.t. every τn t n 0 t |Mt | = |Mt∧τn | = tegrating, or combining the two. 2 |Mτ | = n, t > τn. M ∈ M0 is also a square-integrable martingale. n However, the converse is not true; Brownian motion W for exam- Quadratic variation and covariation 2 2 ple is a martingale and has E[Wt ] = t. So sup E[Wt ] = ∞ which 2 2 (iii) The set Lloc(M) of nice integrands for M can here be explicitly – Suppose that M,N are local martingales, H ∈ Lloc(M) and t≥0 2 2 described as K ∈ Lloc(N). Then means that BM is not in M0. Def. (locally square-integrable, stochastic interval) L2 = {all predictable processes H = (H ) s.t. Z Z loc t t>0 HdM, N = Hd[M,N] (i) We call a local martingale M null at 0 locally square-integrable Z t 2 H2dhMi < ∞,P -a.s. ∀t ≥ 0 . and write M ∈ M0,loc if there is a sequence of stopping times s s 0 τn 2 and τn % ∞ P -a.s. s.t. M ∈ M0 for each n. R Z Z Z (ii) We say for a predictable process H that H ∈ L2 (M) if the- Finally, the resulting stochastic integral H · M = HdM is then, loc also a continuous local martingale, and of course null at 0. HdM, KdN = HKd[M,N]. re exists a sequence of stopping times τn % ∞ P -a.s. s.t. 2 HI 0,τn ∈ L (M) for each n. Here we use the stochastic in- K K terval notation 0, τn := {(ω, t) | 0 < t ≤ τn(ω). Properties – The covariation process of two stochastic integrals is obtained K K Def. (another definition of stochastic integral) by integrating the product of the integrands w.r.t. the cova- (Local) Martingale properties riation process of the integrators. 2 2 For M ∈ M and H ∈ L (M), defining the stochastic inte- 2 R R R 2 0,loc loc – If M is a local martingale and H ∈ Lloc(M), then HdM is – In particular, [ HdM] = H d[M]. (We have seen this al- gral is straightforward, we simply set 2 2 R a local martingale in M0,loc. If H ∈ L (M), then HdM is ready for H ∈ bE in the remark after Lem. 1.2) τn even a martingale in M2. H · M := (HI 0,τn ) · M , on 0, τn 0 K K K K Jumps – If M is a local martingale and H is predictable and locally 2 which gives a definition on all of Ω since τn % ∞, s.t. 0, τn Suppose M is a local martingale and H ∈ Lloc(M). Then we R 2 increases to Ω. K K bounded (∗), then HdM is a local martingale. already know that H · M is in M0,loc and therefore RCLL. Its (∗): (which means that there are stopping times τn % ∞ jumps are given by The only piont we need to check is that this definition is consis- P - a.s. s.t. HI 0,τn is bounded by a constant cn, say, for tent, i.e. tht the definition on 0, τn+1 ⊇ 0, τn does not clash each n ∈ N) K K Z ∆ HdM = H ∆M , for t > 0, with the defintion on 0, τn . ThisK canK be doneK byK using the pro- t t 2 t perties of stachastic integrals.K K – If M is a martingale in M0 and H is predictable and bounded, R 2 then HdM is again a martingale in M0. Of course, H · M is then in M2 . where ∆Yt := Yt − Yt− again denotes the jump at time t of a 0,loc – Warning: If M is a martingale and H is predictable and boun- process Y with trajectories which are RCLL. ded, then R HdM need not be a martingale; this is in striking Rem. contrast to the situation in discrete time. If M is Rd-valued with components M i that are all in M2 , one 0,loc Linearity can also define the so-called vector stochastic integral H · M for If M is a local martingale and H,H0 are in L2 (M) and a, b ∈ R, Rd-valued predictable processes in a suitable space L2 (M); the re- loc loc then also aH + bH0 is in L2 (M) and sult is then a real-valued process. However, one warning is indicated: loc 2 i 0 0 0 Lloc(M) is not obtained by just asking that each component H (aH +bH )·M = (aH)·M +(bH )·M = a(H ·M)+b(H ·M). 2 i P i i should be in Lloc(M ) and then setting H · M = H · M . In 2 i Associativity If M is a local martingale and H ∈ Lloc(M), then fact, it can happen that H · M is well defined whereas the individual we already know that H · M is again a local martingale. Then a i i 2 2 H · M are not. So the intuition for the multidimensional case is predictable process K is in Lloc(H ·M) iff KH is in Lloc(M), and that then Z Z X X Z K · (H · M) = (KH) · M, HdM = HidM i 6= HidM i. i i i.e.

Steven Battilana p 16 / 23 Mathematical Foundations for Finance finite variation and H is predictable and locally bounded, the pathwise defined integral H ·A can be chosen to be predictable 4.5 Extension to semimartingales again.) 4.6 Stochastic calculus – linearity: same formula as before. Def. (semimartingale, special semimartingale) – associativity: same formula as before. Good to know (i)A semimartingale is a stochastic process X = (Xt)t≥0 that can – behaviour under stopping: same formula as before. be decomposed as X = X0 + M + A, where M is a local martin- Def. (weak convergence of probability measure) – quadratic variation and covariation: same formula as before. gale null at 0 and A is an adapted process null at 0 and having Suppose µj is a sequence of measures on R. By the definition of RCLL trajectories of finite variation. – jumps: same formula as before. weak convergence of measures, µj weak converges to µ means that for any bounded continuous function f, there holds that (ii) A semimartingale X is called special if there is such a decompo- – If X is continuous, then so is H · X; this is clear from sition where A is in addition predictable. ∆(H · X) = H∆X = 0. Z n→∞ Z fµj −−−−→ fµ. Rem. (canonical decomposition, continuous semimartingale, optio- Thm. (sort of dominated convergence theorem, continuity proper- R R nal quadratic variation) ty) Thm. (Wiki: Lebesgue’s Dominated Convergence Theorem) (i) If X is a special semimartingale, the decomposition with A pre- (i) If Hn, n ∈ N , are predictable processes with Hn → 0 point- Let {fn} be a sequence of real-valued measurable functions on a mea- dictable is unique and called the canonical decomposition. The wise on Ω and |Hn| ≤ |H| for some locally bounded H, then sure space (S, Σ, µ). Suppose that the sequence converges pointwise uniqueness result uses that any local martingale which is predic- Hn · X → 0 uniformly on compacts in probability, which means to a function f and is dominated by some integrable function g in table and of finite variation must be constant. that the sense that (ii) If X is a continuous semimartingale, both M and A can be cho- n sup |H · Xs| → 0 in probability as n → ∞, ∀t ≥ 0. |fn(x)| ≤ g(x) sen continuous as well. Therefore X is special because A is then 0≤s≤t predictable, since A is adapted and continuous. ∀n in the index set of the sequence and x ∈ S. Then f is integrable (ii) This can also be viewed as a continuity property of the stocha- and (iii) If X is a semimartingale, then we define its optional quadratic stic integral operator H 7→ H · X, since (pointwise and locally n n Z variation or square bracket process [X] = ([X]t)t≥0 via bounded) convergence of (H ) implies convergence of (H · X), lim |fn − f|dµ = 0 in the sense of above formula. n→∞ S [X] := [M] + 2[M,A] + [A] := [M] + 2 P ∆M∆A + P(∆A)2. Rem. (further properties) which also implies One can show that this is well defined and does not depend on the chosen decomposition of X. Moreover, [X] can also be ob- (i) If X is a semimartingale and f is a C2-function, then f(X) is Z Z lim fndµ = fdµ. tained as a quadratic variation similarly as in Thm. 1.1. However, again a semimartingale. n→∞ X2 − [X] is no longer a local martingale, but only a semimartin- S S (ii) If X is a semimartingale w.r.t. P and R is a probability measure gale in general. equivalent to P , then X is a semimartingale w.r.t. R. Rem. Def. (stochastic integral for semimartingale) This will follow from Girsanov’s theorem, which even gives a de- The statement ”g is integrable” is meant in the sense of Lebesgue, If X is a semimartingale, we can define a stochastic integral H · X = composition of X under R. i.e R HdX at least for any process H which is predictable and locally Z (iii) If X is any adapted process with RC trajectories, we can always bounded. We simply set |g|dµ < ∞. define the (elementary) stochastic integral H · X for processes S H · X := H · M + H · A, H ∈ bE. If X is s.t. this mapping on bE also has the continuity n Lem. property from the above Thm. for any sequence (H ) N ∈ bE where H · M is as is the previous section and H · A is defined ω-wise n∈ One can use that any continuous local martingale of finite variation converging pointwise to 0 and with |Hn| ≤ 1 for all n, then X as a Lebesgue-Stieltjes integral. is constant. must in fact be a semimartingale. Throughout this chapter Rem. We work on a probability space (Ω, F,P ) with a filtration F = Properties Rem. The above result implies that if we start with any model where S is (Ft) satisfying the usual conditions of right-continuity and P - not a semimartingale, there will be arbitrage of some kind. completeness. For all local martingales, we then can and tacitly do The resulting stochastic integral then has all the properties from the previous section except those that rest in an essential way on Lem. choose a version with RCLL trajectories. For the time parameter t, the (local) martingale property. The family of semimartingales is invariant under a transformation we have either t ∈ [0,T ] with a fixed time horizon T ∈ (0, ∞) or by a C2-function, i.e. f(X) is a semimartingale whenever X is a t ≥ 0. In the latter case, we set The isometry property for example is of course lost. 2 semimartingale and f ∈ C . ! W S We still have, for H predictable and locally bounded: F∞ := Ft := σ Ft . t≥0 t≥0 – H · X is a semimartingale.

– If X is special with canonical decomposition X = X0 +M +A, then H · X is also special, with canonical decomposition H · X = H · M + H · A. (This uses the non-obvious fact that if A is predictable and of

Steven Battilana p 17 / 23 Mathematical Foundations for Finance Itôsformula I Suppose X = (Xt)t≥0 is a continuous real-valued semimartingale (i) If X has continuous trajectories: and f : R → R is in C2 (i.e. f is twice continuously differentiable). d Z t Def. (classical chain rule from analysis) X ∂f i f(Xt) =f(X0) + (Xs)dX 1 R R 1 ∂xi s If x ∈ C and f : → is in C , the composition f ◦ x : [0, ∞) → Then f(X) = (f(Xt))t≥0 is again a continuous (real-valued) i=1 0 R, t → f(x(t)) is again in C1 and semimartingale, and we explicitly have P -a.s.: d Z t 2 1 X ∂ f i j (i) its derivative is given by (i) Integral form: + (Xs)dhX ,X is 2 ∂xi∂xj i,j=1 0 d df dx Z t Z t (f ◦ x)(t) = (x(t)) (t), 0 1 00 f(Xt) = f(X0) + f (Xs)dXs + f (Xs)dhXis dt dx dt 2 or in more compact notation, using subscripts to denote partial 0 0 derivatives, (ii) in formal differential notation for all t ≥ 0. d d 0 X i 1 X i j d(f ◦ x)(t) = f (x(t))dx(t), (ii) Differential form: df(Xt) = f i (Xt)dX + f i j (Xt)dhX ,X is x t 2 x x i=1 i,j=1 0 1 00 (iii) in integral form df(Xt) = f (Xt)dXt + f (Xt)dhXit 2 Z t 0 0 1 00 (ii) if d = 1 (so that X is real-valued, but not necessarily continuous): f(x(t)) − f(x(0)) = f (x(s))dx(s). = f (Xt)dXt + f (Xt)dhMit 0 2 Z t Z t 0 1 00 In this last form, the chain rule can be extended to the case where for all t ≥ 0. f(Xt) =f(X0) + f (Xs− )dXs + f (Xs− )d[X]s 0 2 0 1 f ∈ C and x is continuous and of finite variation. We used that hXi = hMi. X + (f(Xs) − f(Xs− ) Def. (quadratic variation) Remarks: 0

Steven Battilana p 18 / 23 Mathematical Foundations for Finance R R 2 R R FW If f : → is in C , α, β ∈ and the semimartingale Itˆoprocess ψ has the additional property that ψdW is a (P, )- X = (Xt)t≥0 is given by Xt = αt + βNt, then: martingale on [0, ∞] (and is thus uniformly integrable). An Itˆoprocess is a stochastic process of the form Z t Z t Z t Rem. 0 X W f(Xt) = f(X0) + α f (Xs−)ds + (f(Xs) − f(Xs−)) Xt = X0 + µsds + σsdWs, ∀t ≥ 0 The assumptions on H say that H is integrable and F∞ -measurable. 0 0

Steven Battilana p 19 / 23 Mathematical Foundations for Finance General properties/results We obtain via Itˆo’sformula that S1 solves the SDE

Any continuous, adapted process H is also predictable and locally 1 1 dS = S ((µ − r)dt + σdWt). bounded. 5 Black-Scholes Formula t t It furthermore holds for any predictable, locally bounded process For later use, we observe that this gives 2 H that H ∈ Lloc(W ). 1 1 2 2 R R dhS it = (St ) σ dt Let f : → be an arbitrary continuous convex function. 5.1 Black-Scholes (BS) model Then the process (f(Wt))t≥0 is integrable and is a (P, F)- 1 for the quadratic variation of S , since hW it = t. submartingale. Rem. Rem. F The Black-Scholes model or Samuelson model is the continuous- As in discrete time, we should like to have an equivalent martingale Given a (P, )-martingale (Mt)t≥0 and a measurable function time analogue os the Cox-Ross-Rubinstein binomial model we have 1 g : R+ → R, the process measure for the discounted stock price process S . To get an idea seen at length in earlier chapters. how to find this, we rewrite (Mt + g(t))t≥0 Def. 1 1 Throughout this we will use the following setting. dSt =St ((µ − r)dt + σdWt). is a: A fixed time horizon T ∈ (0, ∞) and a probability space (Ω, F,P ) 1 1 µ − r 1 ∗ on which there is a Brownian motion W = (W ) . We take as ⇔ dS =S σ dWt + dt = S σdW , – (P, F)-supermartingale iff g is decreasing; t 0≤t≤T t t σ t t filtration F = (F0≤t≤T the one generated by W and augmented by – (P, F)-submartingale iff g is increasing. the P -nullsets form F 0 := σ(W ; s ≤ T ) s.t. F = FW satisfies the ∗ ∗ T s with W = (Wt )0≤t≤T defined by A continuous local martingale of finite variation is identically con- usual conditions under P . Z t stant (and hence vanishes if it is null at 0). Def. (undiscounted financial market model) ∗ µ − r The financial market model has two basic traded assets: a bank ac- Wt := Wt + t = Wt + λds for 0 ≤ t ≤ T. R R 1 σ 0 For a function f : → in C , the stochastic integral count with constant continuously compounded interest rate r ∈ R, R · 0 0 f (Ws)dWs is a continuous local martingale. and a risky asset (usually called stock) habing two parameters µ ∈ R Def. (market price of risk or Sharpe ratio) Furthermore, for f ∈ C2 it holds that f(W ) is a continuous local and σ > 0. Undiscounted prices are given by The quantity R · 00 martingale iff 0 f (Ws)ds = 0. 0 rt µ−r Set = e λ := σ If a predictable process H = (Ht)t≥0 satisfies 1 1 1 2 is often called the instantaneous market price of risk or infinitesimal Set = S0 exp σWt + µ − σ t 1 2 2 Sharpe ratio of S . E Hs ds < ∞, ∀T ≥ 0 1 Rem. R with a constant S0 > 0. then HdWs is a square-integrable martingale. Cor. R R Applying Itô’sformula to the above equations yields mean portfolio return−risk-free rate = Sharpe ratio. If f : → is bounded and continuous, then the stochastic standard deviation of portfolio return integral R f(W )dW is a square-integrable martingale. 0 0 dSet = Set rdt, Rem. 2 ∗ If a process H = (Ht)t≥0 is predictable and the map s 7→ E[H ] 1 1 1 By looking at Grisanov’s theorem, we see that W is a Brownian s dSe = Se µdt + Se σdWt, is continuous, then the stochastic integral R HdW is a square- t t t motion under the probability measure Q∗ given by integrable martingale. which can be rewritten as dQ∗ Z 1 0 2 If f : R → R is polynomial, then the stochastic integral dSe := E − λdW = exp −λWT − λ T , t = rdt, dP 2 R f(W )dW is a square-integrable martingale. 0 T Set 1 whose density process w.r.t. P is dSet = µdt + σdWt, 1 Z 1 Set ∗ 2 Zt = E − λdW = exp −λWt − λ T for 0 ≤ t ≤ T. This means that the bank account has a relative price change t 2 (S0 − S0)/S0 of rdt over a short time period (t, t + dt]; so r et+dt et et Rem. is the growth rate of the bank account. In the same way, the relative By price change of the stock has a part µdt giving a growth at rate µ, and a second part σdW ”with mean 0 and variance σ2dt” that t 1 1 µ − r 1 ∗ dS = S σ dWt + dt = S σdW , causes random fluctuations. We call µ the drift (rate) and σ the t t σ t t (instantaneous) volatility of Se1. Def. (discounted financial market model) the stochastic integral process 0 0 0 0 We pass to quantities discounted with Se ; so we have § = Se /Se ≡ Z t 1 1 0 1 1 1 ∗ 1, and S = Se /Se is by the undiscounted financial market model St = S0 + SuσdWu given by 0 1 1 1 2 S = S exp σWt + µ−r − σ t . t 0 2

Steven Battilana p 20 / 23 Mathematical Foundations for Finance ∗ ∗ ∗ 1 ∗ is then a continuous local Q -martingale like W ; it is even a Q - In summary, every H ∈ L+(FT ,Q ) is attainable in the sen- where martingale since we get se that it can be replicated by a dynamic strategy trading in the Z t ∗ µ − r stock and the bank account in such a way that the strategy is self- W := Wt + t = Wt + λds 1 1 1 2 ∗ t S =S exp σWt + µ−r − σ t financing and admissible, and its value process is a Q -martingale. σ 0 t 0 2 In that sense, we can say that the Black-Scholes model is com- 1 ⇔ S1 =S1 exp σW ∗ − σ2t plete. Market price of risk t 0 t 2 The market price of risk or infinitemsimal Sharpe ratio of S1 is ∗ By the same arguments as in discrete time, we then also obtain by Itô’sformula, and so we can use Proposition 2.2 under Q . 1 ∗ the arbitrage-free value at time t of any payoff H ∈ L+(FT ,Q ) defined as Rem. (unique equivalent martingale measure) as its conditional expectation In the script on page 118 and 119 we have shown that in the Black- ∗ µ − r H ∗ λ = Scholes model, there is a unique equivalent martingale measure, Vt = Vt = EQ∗ [H|F] σ ∗ which is given explicitly by Q . So we expect that the Black-Scholes under the unique equivalent martingale measure Q∗ for S1. model is not only ”arbitrage-free”, but also ”complete” in a suitable

sense. This is in perfect parallel to the results we have seen for the CRR Def. binomial model. 0 Rem. (i) Take any H ∈ L+(FT ) and view H as a random payoff (in discounted units) due at time T . Recall that F is generated by W 5.2 Markovian payoffs and PDEs ∗ ∗ (i) All the above computations and results are in discounted units. and that Wt = Wt + λt, 0 ≤ t ≤ T , is a Q -Brownian motion. ∗ (ii) Itô’srepresentation theorem gives the existence of a strategy, but Rem. (martingale approach, PDE approach) (ii) Because λ is deterministic, W and W generate the same filtra- does not tell us how it looks. tion, and so we can also apply Itô’srepresentation theorem with The presentation in the previous subsection is often called the ∗ ∗ 1 ∗ 1 1 Q and W instead of P and W . So if H is also in L (Q ), the (iii) The SDE dSt = St ((µ − r)dt + σdWt) for discounted prices is martingale approach to valuing options, for obvious reasons. ∗ ∗ Q -martingale V := EQ∗ [H|Ft], 0 ≤ t ≤ T , can be represented t 1 If one has more structure for the payoff H, an alternative method as dSt 1 = (µ − r)dt + σdWt involves the use of partial differential equations (PDEs) and is Z t St ∗ H ∗ thus called the PDE approach. Vt = EQ∗ [H] + ψs dWs for 0 ≤ t ≤ T, 0 and this is rather restrictive since µ, r, σ are all constant. An ob- Def. (used throughout this subsection) H 2 ∗ R H ∗ ∗ vious extension is to allow the coefficients µ, r, σ to be (suitably 1 with some unique ψ ∈ L (W ) s.t. ψ dW is a Q - Suppose that the (discounted) payoff is of the form H = h(ST ) for loc integrable) predictable processes, or possibly funcitonals of S or martingale. some measurable function h ≥ 0 on R+. We also suppose that H is Se. This brings up several issue which are enlisted in the script on in L1(Q∗); here, H = (Se1 − Ke)+/Se0 = (S1 − Kee −rT )+. Rem. (trading strategy, self-financing) T T T page 121. Rem. (value process) If we define for 0 ≤ t ≤ T ∗ We start with the value process. Since we have Vt = EQ∗ [H|Ft] = 1 H E ∗ [h(S )|F ], we look at H ψt BS model (undiscounted, historical measure ) Q T t ϑt := 1 , P St σ S1 = S1 exp(σW ∗ − 1 σ2t) t 0 2 H ∗ H 1 0 rt 1 1 1 2 ηt := Vt − ϑt St S˜ = e S˜ = S˜ exp σWt + µ − σ t t t 0 2 and write H 1 (which are both predictable because ψ is), then we can interpret ˜0 ˜1 S1 = S1 ST = S1 exp(σ(W ∗ − W ∗) − 1 σ2(T − t)). H H H dSt dSt T t 1 t T t 2 ϕ = (ϑ , η ) as a trading strategy whose discounted value pro- = rdt = µdt + σdW St ˜0 ˜1 t cess is given by St St 1 In the last term, the first factor St is obviously Ft-measurable. Mo- H H 1 H 0 ∗ ∗ ∗ F Vt(ϕ ) = ϑ S + η S = V for 0 ≤ t ≤ T , reover, W is a Q -Brownian motion w.r.t. , and so in the se- t t t t t ∗ ∗ ∗ BS model (discounted, historical measure P) cond factor, WT − Wt is under Q independent of Ft and has an and which is self-financing in the (usual) sense that N (0,T − t)-distribution. Z t Z t 0 1 1 1 2 Def. H ∗ ∗ H ∗ H H 1 St = 1 St = S0 exp σWt + µ − r − σ t Vt(ψ ) = Vt = V0 + ψu dWu = V0(ψ )+ ϑu dSu, 0 ≤ t ≤ T. 2 Therefore we get 0 0 1 ∗ 1 1 dSt Vt = EQ∗ [h(S )|Ft] = v(t, St ) Moreover, = (µ − r)dt + σdWt T S1 H ∗ t VT (ϕ ) = VT = H a.s. shows that the strategy ϕH replicates H, and BS model (discounted, risk-neutral measure Q) Z H 1 H H ∗ ϑ dS = V (ϕ ) − V0(ϕ ) = V − EQ∗ [H] ≥ −EQ∗ [H] 1 1 µ − r 1 ∗ dS = S σ dWt + dt = S σdW t t σ t t ∗ H (because V ≥ 0, since H ≥ 0) shows that ϑ is admissible (for Z t 1 1 1 1 ∗ 1 ∗ 1 2 S ) int the usual sense. St = S0 + SuσdWu = S0 exp σWt − σ t Rem. 0 2

Steven Battilana p 21 / 23 Mathematical Foundations for Finance with the function v(t, x) given by explicitly, we can at least obtain v(t, x) numerically by solving the Rem. (European call option) above PDE. In the special case of a European call option, the value process and ∗ ∗ 1 2 Def. (discounted PDE) the corresponding strategy can be computed explicitly, and this has v(t, x) = EQ∗ h x exp σ(WT − Wt ) − σ (T − t) 2 If the undiscounted payoﬀ is H = h(S1 ) and the undiscounted value found widespread use in industry. √ e e eT h σ T −tY − 1 σ2(T −t)i 1 Def. = EQ∗ h xe 2 at time t is ve(t, Set ), we have the relations Suppose the undiscounted strike price is Ke s.t. the undiscounted Z ∞ √ 1 rT 1 rT rT 1 σ T −ty− 1 σ2(T −t) 1 − 1 y2 eh(Se ) = eh(e Se ) = He = e H = e h(S ) payoﬀ is = h xe 2 √ e 2 dy, T T T −∞ 2π 1 + and He = (SeT − Ke) . where Y ∼ (0, 1) under Q∗. rt −rt v(t, x) = e v(t, xe ). Then H = H/e Se0 = (S1 − Kee −rT )+ =: (S1 − K)+. Rem. Rem. e e e T T T ∗ For the function v, we then obtain from We obtain form This already gives a fairly precise structural description of Vt as a e 1 function of (t and) S , instead of a general Ft-measurable random 2 1 t ∂v 1 2 2 ∂ v ∗ ∗ 2 variable. 0 = + σ x , on (0,T ) × (0, ∞) v(t, x) = EQ∗ h x exp σ(WT − Wt ) − σ (T − t) ∂t 2 ∂x2 2 Rem. (strategy) √ h σ T −tY − 1 σ2(T −t)i As explained in the script on page 123, the PDE = EQ∗ h xe 2 2 ∗ 1 1 1 1 1 2 1 2 ∂v ∂v 1 ∂ v Z ∞ √ dV = vx(t, S )dS + (vt(t, S ) + vxx(t, S )σ (S ) )dt e e 2 2 e σ T −ty− 1 σ2(T −t) 1 − 1 y2 t t t t 2 t t 0 = + rx + σ x − rv = h xe 2 √ e 2 dy, ∂t e∂x 2 e ∂x2 e and e e −∞ 2π with the boundary condition Z t that the discounted value of H at time t is H ∗ H H 1 Vt(ϕ ) = Vt = V0(ϕ ) + ϑu dSu v(T, ·) = eh(·). 0 e √ + ∗ σ T −tY − 1 σ2(T −t) Vt = EQ∗ xe 2 − K . yield 1 x=St v (t, S1)dS1 = dV ∗ = ϑH dS1 x t t t t t Because we have Y ∼ N (0, 1) under Q∗, an elementary computation s.t. we obtain the strategy explicitly as 5.3 Black-Scholes PDE yields for x > 0, a > 0 and b ≥ 0 that

∂v x 1 2 ! x 1 2 ! H 1 2 + log + a log − a ϑt = (t, St ), Black-Scholes PDE aY − 1 a b 2 b 2 ∂x EQ∗ xe 2 − b = xΦ −bΦ , 2 a a ∂v˜ ∂v˜ 1 2 2 ∂ v˜ i.e. as the spatial derivative of v, evalueated along the trajectories of 0 = + rx˜ + σ x˜ − rv,˜ v˜(T, ·) = h˜(·) ∂t ∂x˜ 2 ∂x˜2 S1. where Def. (discounted PDE) Z y ∗ 1 − 1 z2 In fact, the vanishing of the dt-term means that the function Φ(y) = Q [Y ≤ y] = √ e 2 dz 1 2 2 −∞ 2π vt(t, x) + 2 vxx(t, x)σ x must vanish along the trajectories of the space-time process (t, S1) . But each S1 is by 5.4 Black-Scholes formula for option pricing t 0support. so √ Plugging in the above formula x = S1, a = σ T − t, b = K and the support of the space-time process contains (0,T ) × (0, ∞), and The discounted arbitrage-free value at time t of any discounted t 1 ∗ ˜0 ˜1 then passing to undiscounted quantities therefore yields the famous so v(t, x) must satisfy the (linear, second-order) PDE payoﬀ H ∈ L+(FT , Q ), HT = H(ST , ST ), is given by Black-Scholes formula in the form 2 ∗ 1 ∂v 1 2 2 ∂ v V = [H|Ft] := ϑ(t, S ) He 1 1 −r(T −t) 0 = + σ x , on (0,T ) × (0, ∞). t EQ t Vet = ve(t, Set ) = Set Φ(d1) − Kee Φ(d2) ∂t 2 ∂x2

Moreover, the definition of v via Then the discounted payoff H can be hedged via the replicating ∗ 1 1 strategy (V0, ϑ) s.t. Vt = EQ∗ [h(ST )|Ft] = v(t, St ) Z T gives the boundary condition 1 ˜0 ˜1 V0 + ϑudSu = H(ST , ST ) 0 v(T, ·) = h(·) on (0, ∞), Using Itô’srepresentation theorem the replicating strategy can be 1 ∗ 1 expressed as because v(T,ST ) = VT = H = h(ST ) and the support of the distribution of S1 contains (0, ∞). T Z t Rem. ∗ 1 1 Vt = ϑ(t, St ) = V0 + ϑsdSs + cont. FV process So if we cannot compute the integral in 0 | {z } ”usually” vanishes Z ∞ √ σ T −ty− 1 σ2(T −t) 1 − 1 y2 h xe 2 √ e 2 dy 1 ∂ϑ 1 V0 = ϑ(0,S ), ϑt = (t, S ) −∞ 2π 0 ∂x t

Steven Battilana p 22 / 23 Mathematical Foundations for Finance with intentionally left blank

1 log Set + r ± 1 σ2 (T − t) Ke 2 6 Appendix d1,2 = √ . σ T − t Markov’s inequality For X any nonnegative integrable RV and Rem. a ∈ , a > 0: Note that the drift µ of the stock does not appear here; this is ana- R logous to the result that the probability p of an up move in the CRR E[X] P[X ≥ a] ≤ binomial model does not appear in the binomial option pricing for- a mula. Rem. (replicating strategy) Chebyshev’s inequality For X an integrable RV with ﬁnite expec- To compute the replicating strategy, we recall that the stock price ted value µ ∈ and ﬁnite non-zero variance σ2, σ ∈ and for any holdings at time t are given by R R real number k > 0: H ∂v 1 1 ϑt = (t, St ). P[|X − µ| ≥ kσ] ≤ ∂x k2 −rt rt Moreover, v(t, x) = e ve(t, xe ) s.t. Jensen’s inequality For X a RV and f a convex function, it holds ∂v ∂v ∂v (t, x) = e−rt e(t, xert)ert = e(t, xert). that Abbreviations ∂x ∂xe ∂xe f(E[X]) ≤ E[f(X)] Computing the above derivative explicitly gives a.a. almost all

 1  a.s. almost surely log Set + r + 1 σ2 (T − t) Sets/families ∂v Ke 2 BM Brownian motion ϑH = e(t, S1) = Φ(d ) = Φ  √  , 0 t et 1   L+: familiy of all nonnegative RVs ∂xe  σ T − t  CDF cumulative distribution function

Correlation and Independence iff if and only if which always lies between 0 and 1. IOT in order to Let X and Y be two RVs. PDE partial differential equation Black-Scholes formula for a European call option Then X,Y are uncorrelated iff PDF probability density function ˜ H˜ ˜1 ˜1 ˜ −r(T −t) E[X · Y ] = E[X] · E[Y ] Vt =v ˜(t, St ) = St Φ(d1) − Ke Φ(d2) RCLL right-continuous with left limits S˜1 Then X,Y are independent iff log t + r ± 1 σ2 (T − t) K˜ 2 RV random variable d1,2 = √ P[X = x, Y = y] = P[X = x] · P[Y = y] σ T − t SDE stochastic differential equation Z y Note that independence of X,Y implies that X,Y are uncorrela- ∗ 1 − 1 z2 Φ(y) = Q [Y ≤ y] = √ e 2 dz ted (but not vice-versa!). s.t. such that 2π −∞ w.r.t. with respect to Remarks: Independence of equations

Φ denotes the CDF of the standard normal distribution N (0, 1). If there is e.g. a system of equations such as Note that the drift µ of the stock does not appear here. This is  a x + b y = c analogous to the result that the probability p of an up move in  1 1 1 Disclaimer the CRR binomial model does not appear in the binomial option a2x + b2y = c2  pricing formula. a3x + b3y = c3 This summary is work in progress, i.e. neither completeness nor then this system admits a solution iﬀ correctness of the content are guaranteed by the author. Replicating strategy for a European call option   This summary may be extended or modiﬁed at the discretion of a1 b1 c1 det a b c = 0 the readers. H ∂ν˜ ˜1  2 2 2 νt = (t, St ) = Φ(d1) a3 b3 c3 ∂x˜ Source: Lecture Mathematical Foundations for Finance, autumn semester 2015/16, ETHZ (lecture notes, script and exercises). Copyright of the content is with the lecturers. Greeks The derivatives of the option price w.r.t. the various para- Fubini’s lemma meters, i.e. the sensitivities of the option price w.r.t. to the parame- Z T Z T The layout of this summary is mainly based on the summaries of 2 2 ters, are called Greeks. E Hs ds = E Hs ds several courses of the BSc ETH ME from Jonas LIECHTI. 0 0

Steven Battilana p 23 / 23 Mathematical Foundations for Finance

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