Stochastic calculus Introduction I. Stochastic Finance. C. Azizieh VUB 1/91. C. Azizieh VUB Stochastic Finance

Size: px
Start display at page:

Download "Stochastic calculus Introduction I. Stochastic Finance. C. Azizieh VUB 1/91. C. Azizieh VUB Stochastic Finance"

Transcription

1 Stochastic Finance C. Azizieh VUB C. Azizieh VUB Stochastic Finance 1/91

2 Agenda of the course Stochastic calculus : introduction Black-Scholes model Interest rates models C. Azizieh VUB Stochastic Finance 2/91

3 Stochastic calculus : introduction C. Azizieh VUB 3/91

4 Agenda Stochastic calculus Introduction I Stochastic calculus : introduction Probability space, stochastic process in continuous time, filtration, martingales, convergence of random variables Itô stochastic integral : motivation Itô stochastic integral : construction Itô lemma Stochastic differential equations Girsanov theorem 4/91

5 Probability space Stochastic calculus Introduction I We consider the set of all possible states of the world Ω, with a sigma-algebra F. We speak about the notion of measurable space (Ω, F). In financial modelling, Ω represents the different possible evolution scenarios in the market : each scenario ω Ω typically represents a possible evolution in time of the prices of a set of financial instruments. An element of the sigma-algebra, A F, represents an event, hence typically a set of scenarios, to which a probability can be attributed : 5/91

6 Probability space Stochastic calculus Introduction I A probability measure is a mapping : P : F [0, 1] : A P(A) such that : P( ) = 0, P(Ω) = 1, If A i (i IN 0 ) are 2 by 2 disjoint, then : P ( i IN0 A i ) = P(A i ) i=1 (sigma-additivity) We will always work on a probability space (Ω, F, P). 6/91

7 Stochastic process and filtration The prices of financial assets, or more generally, of market variables (e.g. : interest rates, exchange rates, etc), will be modelled by stochastic processes in continuous time, i.e. the given of a random variable at each instant t : X : [0, + ) Ω R : (t, ω) X(t, ω) such that X(t,.) : Ω R is a random variable for each fixed t. Hence we have : For a fixed scenario ω, we have a trajectory For a fixed instant t, we have a random variable Now, if t represents time, in order to model concepts like the progressive arrival of information, causality and predictability, we introduce the concept of filtration. 7/91

8 Stochastic process and filtration A filtration is an increasing family of sub-σ-algebras on the probability space : for all 0 < s < t. F 0 = {Ω, } F s F t F Interpretation : in a dynamical framework, when time passes, information is progressively revealed to the observer quantities seen as stochastic at t=0 are not stochastic anymore at a future instant t > 0 if their value is revealed to the observer in the mean time, i.e. given the available information at t > 0. The notion of filtration allows to enrich the modelling framework in order to take into account the time dimension and information flow. 8/91

9 Filtration and process Notions linked by two definitions : Definition (Natural filtration or filtration generated by a stochastic process) Every stochastic process X generates a filtration called the natural filtration of the process, and defined by : F t = σ(x(s) s t) t (filtration corresponding to the history of the process) Definition (Adapted process) A process X is said to be adapted to a filtration (F t ) if X(t) is F t measurable t 9/91

10 Filtration and process Concretely, if (F t ) represents the available information (in a market) through time, a process (X t ) is adapted if its value at t is known as soon as we have reach instant t. In finance, we will model prices of financial assets by adapted processes. Definition { A filtration {F t } is said to satisfy the usual conditions if F0 contains all sets of zero probability t 0 F t = F s (right continuous). s>t In what follows, we will always assume that the filtration satisfies the usual conditions (...) 10/91

11 Continuous stochastic process A continuous stochastic process is such that t X t (ω) is continuous for all ω Ω 0 Ω with P [Ω 0 ] = 1. In other words : a process whose trajectories are almost all continuous X(t) Time t 11/91

12 Continuous time martingale The notion of martingale is central in market finance. Definition (continuous time martingale) A stochastic process (M t ) t 0 is a martingale w.r.t. a filtration {F t } t if for all t : 1 M t is F t -measurable (adapted process) 2 M t is integrable (hence IE [ M t ] < ) 3 IE [M t F s ] = M s for all s < t. Interpretation : accumulated gains of a player in an equilibrated game In particular, E[M t ] = M 0. 12/91

13 Martingales - Doob s results in continuous time Definition (Stopping time in a continuous time framework) A random variable τ with values in R + { } is a stopping time w.r.t. the filtration {F t } t if {τ t} F t for all t 0. Interpretation : we don t need information on the future to know at each instant if τ has already occurred. Example : the first instant at which a stock price reaches some fixed level Counter-example : the last instant before a future date at which a stock reaches a given level. Other examples? 13/91

14 Martingales - Doob s results in continuous time An example of link between stopping time and stochastic process : Definition (Process stopped by a stopping time) Let (Y t ) t be an adapted process to the filtration {F t } t and let τ be a stopping time. The process stopped at instant τ, denoted by (Y τ t ) t or (Y t τ ) t is defined by Y τ t (ω) = Y t τ(ω) (ω) t. 14/91

15 Martingales - Doob s results in continuous time Example : Let Y be a stochastic process and let us define the stopping time : T = inf{t 0 Y (t) M} for M R (M = barrier). The process Y stopped at M is then : { Y (t) if t T Z (t) = Y (t T ) = M if t > T 15/91

16 Martingales - Doob s results in continuous time When we stop a (continuous) martingale with a stopping time, we still have a (continuous) martingale : Theorem (Doob s stopping time theorem) Let (M t ) t be a continuous martingale w.r.t. filtration {F t }. Let τ be a stopping time w.r.t. {F t }. Then the process X t = M t τ is a continuous martingale w.r.t. {F t }. Proof : see [Steele, section 4.4]. 16/91

17 Martingales - Doob s results in continuous time Theorem (Doob s maximal inequalities in continuous time) Let (M t ) t be a continuous non-negative sub-martingale and λ > 0.. Then for all p 1, we have : ( ) λ p P sup M t > λ IE [ M p ] T (1) {t:0 t T } and, if M T L p (dp) for some p > 1, then : sup M t p p p 1 M T p. (2) {t:0 t T } Proof : see [Steele, section 4.4]. 17/91

18 Convergence of random variables Let {X n n IN} be a sequence of random variables. What is lim n X n?? One can actually define different types of convergence : almost sure convergence quadratic convergence (or L 2 ) convergence in probability convergence in distribution convergence in L p norm... 18/91

19 Almost sure convergence X n p.s. X iff P[ω Ω X n (ω) X(ω)] = 1 Example : Strong law of large numbers : If (X n ) is a sequence of random variables i.i.d. with finite expectation IE[X 1 ] = µ, then : 1 n p.s. X i E[X 1 ] = µ n i=1 19/91

20 Quadratic convergence (or in L 2 ) X n X in quadratic mean (or in L 2 ) iff X n X 2 L = IE[(X 2 n X) 2 ] = (X n (ω) X(ω)) 2 dp(ω) 0 Ω 20/91

21 Convergence in probability P X n X iff ɛ > 0 : P[ω Ω : Xn (ω) X(ω) ɛ] 0 Link between these different types of convergence : Almost sure convergence convergence in probability Convergence in L 2 convergence in probability 21/91

22 Convergence in distribution X n D X iff f bounded continuous : E[f (Xn )] E[f (X)] In particular this is equivalent to the convergence of characteristic functions : Definition : characteristic function of a random variable X : z R : Φ X (z) := E[exp(izX)] = e izx(ω) dp(ω) = e izx dµ X (x) (characteristic function = inverse Fourier transform of the density of X if X has such a density (up to some normalization constant)...) Result : (X n ) converges in distribution iff we have convergence of characteristic functions : X n D X iff ΦXn (z) Φ X (z) for all z R Ω R 22/91

23 Convergence in distribution convergence in probability convergence in distribution Theorem (Central limit theorem) If (X n ) is a sequence of i.i.d. random variables with finite variance σ 2, then n i=1 X i ne[x 1 ] σ D N(0, 1) n 23/91

24 Brownian motion : Motivation Suppose that we would like to define a stochastic model for the evolution of a financial asset in continuous time. Discrete time Continuous time Deterministic model S n = S 0 (1 + i) n S(t) = S 0 e δt Stochastic model S n = S 0 n k=1 (1 + i k)?? 24/91

25 : Motivation Continuous deterministic model We consider a financial asset which evolves deterministically in time : constant rate of return : δ value of the asset at t : S(t) we assume a double linearity hypothesis : S(t) = δs(t) t the evolution of the price is supposed to be proportional to time and to the invested amount 25/91

26 : Motivation Continuous deterministic model If we take the limit for t 0, we obtain the differential equation : ds(t) = δs(t)dt whose solution (passing by (0, S(0))) is given by the exponential function : S(t) = S(0)e δt Now, if we assume that the rate of return is not constant anymore, i.e. if δ = δ(t), then the differential equation becomes : whose solution is given by : ds(t) = δ(t)s(t)dt t0 δ(s)ds S(t) = S(0)e 26/91

27 : Motivation Binomial model on several periods (discrete time model) : Deterministic model : Stochastic model : S(n) = S(0) n (1 + i k ) = S(0) k=1 S(n) = S(0) n k=1 e Y k n k=1 où Y 1,..., Y n are i.i.d. : { δ + σ p = 1/2 Y i = δ σ q = 1 p = 1/2 e δ k We can rewrite : Y i = δ + σx i where X i = { 1 1/2 1 1/2. 27/91

28 : Motivation We hence have : S(n) = S(0) n k=1 e δ+σx k = S(0)e nδ+σ n k=1 X k We hence obtain the cumulative log-return until n : log(s(n)/s(0)) = n Y i = i=1 }{{} δn + σ determ. trend n i=1 X i }{{} random walk In particular, the moments of cumulative log-returns are given by : E[log(S(n)/S(0))] = δn var[log(s(n)/s(0))] = σ 2 n (variance proportional to time) 28/91

29 : Motivation We will now consider passing to a continuous time model. This will be done in two steps : 1 Modification of the (space and time) scale of the random walk : The time periods of 1 are replaced by t, and jumps of +1 and -1 are replaced by jumps of + x and x on each period of time. The random walk then becomes : W (n) = x. and in terms of moments : E[W (n)] = 0 m X i, with m = n/ t i=1 Var[W (n)] = m( x) 2 = n ( x)2 t 29/91

30 : Motivation 2 Passing to the limit x, t 0 We will now consider a sequence of x k, t k with x k, t k 0 in such a way that the limiting process is non-trivial. For this purpose, we will require : (and not x k t k 1!) ( x k ) 2 t k 1 30/91

31 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 31/91

32 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 32/91

33 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 33/91

34 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 34/91

35 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 35/91

36 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 36/91

37 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 37/91

38 : Motivation Construction of BM as limit of a random walk : illustration X k (t) Time t 38/91

39 : Motivation One can show that the sequence of processes (W (k) (t)) defined above converges in distribution to a process with continuous trajectories called Brownian motion : where W (t) : t/ t k W (k) (t) = x k is a continuous time process is called Standard Brownian motion, or Wiener process, or infinitesimal random walk i=1 X i W (t) 39/91

40 : Motivation This convergence is actually the dynamic equivalent of the central limit theorem in the static case : If (X n ) is a sequence of i.i.d. random variables of mean µ and finite variance σ 2, then we note S n = n i=1 X i, S n nµ σ n D N(0, 1) 40/91

41 : Motivation By construction, this process W (t) has the following properties : (i) E[W (k) (t)] = 0 E[W (t)] = 0 (ii) var[w (k) (t)] = t ( X k ) 2 t k var[w (t)] = t (iii) W (k) (t) = sum of i.i.d. rvs W (t) N(0, t) (Central Limit Theorem) (iv) W is a process with independent and stationary increments 41/91

42 : Definition All these lead to the following definition : Definition (Standard ) A stochastic process {B t : 0 t < T } in continuous time is standard Brownian motion on [0, T ) if 1 B 0 = 0. 2 The increments of B t are independent, i.e. for all finite set of instants : 0 t 1 < t 2 <... < t n < T, the random variables are independent. B t2 B t1, B t3 B t2,..., B tn B tn 1 3 for all 0 s < t T, the increments B t B s have a Gaussian distribution of mean 0 and variance t s. 4 B t (ω) is a continuous function of t for almost all ω. 42/91

43 : Illustration A typical sample path : X(t) Time t 43/91

44 : back to the stochastic continuous model The random walk W (k) (t) converges to the standard Brownian motion W (t). The price of the asset S(t) hence becomes at the limit : S (k) (t) = S(0)e tδ+σ x t/ tk k X k=1 k δt+σw (t) S(t) = S(0)e This is what will be called later geometric Brownian motion. 44/91

45 : Historical Perspective 1829 : Brown : movement of pollen particles in suspension 1900 : Bachelier models financial stock prices by a Brownian motion 1905 : Einstein models particles in suspension in a liquid or a gaz, subject to collisions 1923 : Wiener proposes a rigorous construction of Brownian motion (also called Wiener process ) 1944 : Ito contributes to define a stochastic integral w.r.t. a Brownian motion BM has become a central process in finance for modelling the uncertainty present in the markets 45/91

46 Brownian motion and the Markov property By using the independence of disjoint increments we directly obtain : i.e. Markov property. P[X t+s B X u, 0 u t] = P[X t+s B X t ]. We also get : P[X t+s B X t = x] = P[X t+s X t B x X t X 0 = x] = P[X t+s X t B x] = P[X t+s X t + x B] = P[Y B], where Y N (x, s). We also have : P[X t+s B X t = x] = B 1 2πs e (y x)2 /(2s) dy. 46/91

47 Non standard Brownian motion from level a Z (t) = a + B(t) where (B t ) is a standard BM Brownian motion from level a, with drift µ and volatility σ Z (t) = a + µt + σb t In particular, E[Z (t)] = a + µt var[z (t)] = σ 2 t 47/91

48 Brownian motion and Gaussian processes Definition ( Gaussian processes) {X t : 0 t < } is a Gaussian process iff every linear combination of values of X at different instants is normally distributed : n α k X tk k=1 has a normal distribution for all coefficients α k and instants t k. Remark : such a process is characterized by : a mean function t E[X t ] and an autocovariance function (s, t) Cov[X s, X t ]. Example of Gaussian process : Brownian motion. 48/91

49 Brownian motion and Gaussian processes Let us compute the covariance function of a Brownian motion. for all s t : cov (B s, B t ) = IE [(B t B s + B s ) B s ] = IE[B t B s ]IE[B s ]+IE [ B 2 s ] = IE [ B 2 s ] = s = cov (B s, B t ) = min (s, t) 0 s, t <. 49/91

50 Brownian motion and Gaussian processes Lemma Let {X t : 0 t T } be a Gaussian process with IE [X t ] = 0 for all 0 t T and let cov (X s, X t ) = min (s, t) for all 0 s, t T, then the process {X t } has independent increments. Moreover, if the process has continuous trajectories and X 0 = 0, then {X t } t is a standard Brownian motion on [0, T ]. 50/91

51 Martingales and Brownian motion Theorem Let (B t ) be a standard Brownian motion. Then : 1 (B t ) t 2 ( B 2 t t ) t ( ( 3 exp αbt α 2 t/2 )) t are martingales w.r.t. à {F t } t (= the family of sub-sigma algebra σ (B s, s t) completed by adding the sets of zero probability). 51/91

52 Martingales and Brownian motion Proof 1) B t F t-measurable, IE [ ] Bt 2 = t <, IE [B t B s F s ] = 0 since IE [B t B s] = 0. 2) ( Bt 2 t ) F t-measurable, IE [ B 2 t t ] IE[B t 2 ] + t = 2t, IE [ ] [ ] Bt 2 t F s = IE (B t B s) 2 Bs 2 + 2B tb s t F s ] = IE [(B t s) 2 F s t Bs 2 + 2Bs 2 = t s t + Bs 2 = Bs 2 s. 3) IE [ exp ( αb t α 2 t/2 ) ] F [ ( s ) = IE exp α (B t B s) α2 (t s) 2 = IE [exp (αb t s)] exp ( = exp αb s α2 s 2 ( α2 2 (t s) ) exp ] ( F s exp ( ) αb s α2 s 2 ) αb s α2 s 2 ). 52/91

53 Martingales and Brownian motion Remark : We can actually show that the converse is also true : if (X t ) is a continuous process such that (X t ) and (Xt 2 t) are martingales and X(0) = 0, then (X t ) is a standard Brownian motion Brownian motion appears as the fundamental example of continuous martingale 53/91

54 Brownian motion and bounded variation Let us consider a sequence of subdivisions of interval [0,t] : 0 = t 0 < t 1 <... < t n = t with thikness tending to 0 (lim n max 0 i n t i+1 t i = 0). We know that [ n ] n E ( B(t i )) 2 = t i = t. i=1 If f is a deterministic function with bounded variation, lim n n i=1 f (t i) exists, and n ( f (t i )) 2 = 0. lim n i=1 This implies that trajectories of a Brownian motion cannot be of bounded variation. In particular, they cannot be differentiable (in the classical sense). Hence, these trajectories are continuous everywhere but nowhere differentiable. i=1 54/91

55 Brownian motion and quadratic variation Lemma (Quadratic variation of a BM) Let t j = j 2 n t (j = 0,..., 2 n ) be a partition of [0,t]. Then Z n = (B(t j ) B(t j 1 )) 2 t in L 2 2 n j=1 Proof (i) E[Z n ] = t (ii) IE[(Z n t) 2 ] = Var[Z n ] = 2 n j=1 Var( B(t j) 2 ). Now, if X N(0, σ 2 ), then Var(X 2 ) = 2σ 4 (exercise), which implies : 2 n ( ) 2 t Var(Z n ) = 2 2 n = 2 n+1 t n j=1 55/91

56 Let us consider a market composed of d assets of price at t : S t = (S 1 t,..., S d t ), modeled like a vector process A trading strategy can be modelled by a vector φ t describing the quantities invested in each of the assets at instant t : φ t = (φ 1 t,..., φ d t ) The value at t of the portfolio obtained by following this strategy is then given by : d V t(φ) = φ k t St k = φ t.s t k=1 We will denote by 0 = T 0, T 1,..., T n+1 = T the rebalancing instants : between 2 such instants, the composition of the portfolio is supposed unchanged. 56/91

57 We can then denote by φ i = (φ 1 i,..., φ d i ) the composition of the portfolio between T i and T i+1, and rewrite φ t as : n φ t = φ 0 I t=0 + φ i I ]Ti,T i+1 ] i=0 ( ) Generally, the rebalancing instants T i are random (depending e.g. from the levels reached by some assets in the market...). The strategy φ t appears as a Stochastic process. Definition (Simple predictable process) A process φ t (t [0, T ]) that can be represented by ( ) is called a simple predictable process if moreover the stochastic instants T i are stopping times, and if φ i are bounded random variables with φ i F Ti, (which means that the value φ i is revealed at instant T i ). 57/91

58 Example of simple predictable process (one of its component) : 58/91

59 Gain process for a simple trading strategy If φ t is simple (we speak about simple trading strategy), then the gain realized by the trader between T i and T i+1 is equal to the scalar product φ i.(s Ti+1 S Ti ) Hence the accumulated gain on [0, T ] of the trader beginning with an initial composition given by φ 0 can be written as : G T (φ) = φ 0.S 0 + n φ i.(s Ti+1 S Ti ) i=0 59/91

60 Definition (Stochastic integral of a simple predictable process) G T is called the stochastic integral of the simple predictable process φ w.r.t. the price process (S t ), and will be denoted by : T 0 φ(u).ds(u) In finance, the cumulative gains of a trader following a strategy lead naturally to a new concept of integral : the stochastic integral 60/91

61 In view of getting some results concerning option prices in given models we cannot limit ourselves to the case where φ is a simple strategy. Hence we need to give a sense to this stochastic integral / gain process, also in the case where the composition continuously changes with time. We will first give a sense to this integral in the case where S t is a Brownian motion of dimension 1, and in the case where φ t is a more general process (not necessarily simple) : t 0 φ(s)db s 61/91

62 A first naive idea would consist to define the stochastic integral as a limit obtained along each trajectory : ( ) t n [ ] φ(s)db s (ω) = lim φ ti 1 (ω) B ti (ω) B ti 1 (ω) max(t i t i 1 ) 0 0 The problem is that (almost all) trajectories of a Brownian motion have no bounded variation... and that this limit hence does not exist in general for a given trajectory! i=1 62/91

63 Itô integral - Construction The stochastic integral will be defined as a limit of Riemann sums, but not in the sense of a.s. convergence, but in quadratic convergence (L 2 ) Objective : give a mathematical sense to : I (X) (ω) = T 0 X (ω, t) db t. where X(t) is a general stochastic process. 63/91

64 Contrarily to the Riemann integral, the result can depend on the point y i [t i 1, t i ). In the Itô integral, we will fix y i = t i 1 ; The final objective is to develop an integral allowing to introduce and to study stochastic evolution equations in continuous time ( Śtochastic differential equations, describing the dynamics of market variables) 64/91

65 Notations B [0, T ] = the set of Borel sets on [0, T ] (Borel sigma-algebra, i.e. the smallest σ algebra containing all open sets in [0, T ].) {F t } t the standard Brownian filtration. for all fixed t : F t B = the smallest σ algebra containing the product sets A B où A F t and B B. We say that a stochastic process X (.,.) is measurable if X (.,.) is F T B-measurable. We say that a process X (.,.) is adapted if X (., t) F t, t (0, T ). 65/91

66 In summary, Itô stochastic integral will be defined by following the subsequent steps : Definition of Itô integral for a simple process Itô isometry : the stochastic integral preserves the L 2 norm (in other words, is continuous for that norm) Density of the set of simple processes within the set of adapted square-integrable stochastic processes (with the L 2 ) Extension by density of the Itô integral on the set of square-integrable adapted stochastic processes thanks to the Itô isometry 66/91

67 Construction - Step 1 - Integral of indicator functions If X = I (a,b] is an indicator function, with (a, b] [0, T ], then the integral of X w.r.t. a Brownian motion is defined as : I (X) (ω) = T 0 X(s)dB(s) = B b (ω) B a (ω). (3) 67/91

68 Construction - Step 2 - Integral of step functions If X is a step function (or staircase function), i.e. n 1 X (ω, t) = c i I ]ti,t i+1 ] où 0 = t 0 < t 1 <... < t n = deterministic subdivision, and c i are constant, then by definition (additivity of the integral) : i=0 T 0 n 1 T X(s)dB(s) = c i I ]ti,t i+1 ](s)db(s) = i=0 0 i c i (B(t i+1, ω) B(t i, ω)) 68/91

69 Construction - Step 3 - Integration of simple stochastic processes X is square-integrable simple process if it can be written as : n 1 X (ω, t) = a i (ω) 1 (ti <t t i+1 ) with a i F ti, IE [ ] ai 2 <, et 0 = t0 < t 1 <... < t n 1 < t n = T (we will denote by H0 2 the set of such processes). Then we define the stochastic integral of X H0 2 as : i=0 I (X) (ω) = T 0 n 1 X(s)dB(s) = a i (ω) (B(t i+1, ω) B(t i ; ω)). (4) i=0 69/91

70 Construction - Step 4 - Extension by density X is a square-integrable process on (0,T) if : X is an adapted process [ ] T IE 0 X 2 (ω, s)ds <. The set of square-integrable adapted processes will be denoted by H 2 = H 2 [0, T ] Mathematicians show that H 2 is a closed vector subspace of L 2 (dp dt). We will denote by H0 2 (0, T ) the set of simple square-integrable (cf. preceding slide). 70/91

71 Construction - Step 4 - Extension by density We will now extend the definition for all square-integrable processes X H 2. The key of this extension is Itô isometry Lemma (Ito Isometry on H 2 0 ) X H0 2, we have In other words : I (X) L 2 (dp) = X L 2 (dp dt) T IE((I(X)) 2 ) = IE( X 2 (s)ds). 0 71/91

72 Construction - Step 4 - Extension by density Proof of Itô isometry We first calculate X 2 L 2 (dp dt). X is a simple process, hence can be written in the form : X (ω, t) = n 1 a i (ω)1 (ti <t t i+1 ) i=0 with a i F ti, IE [ a 2 i Let us take the square of X : such that : IE ] < and 0 = t0 < t 1 <... < t n = T X 2 (ω, t) = n 1 ai 2 (ω) 1 (ti <t t i+1 ) i=0 [ ] T 0 X 2 (ω, t) dt = n 1 IE [ ai 2 i=0 ] (ti+1 t i ). 72/91

73 Construction - Etape 4 - Extension by density Let us compute now I (X) 2 L 2 (dp) : IE [ I (X) 2] = IE n 1 = IE i=0 a i (ω) ( ) ) 2 B ti+1 B ti ( n 1 i=0 [ a 2 i ( Bti+1 B t i ) 2 ] as the double products have a zero expectation. Then, as B ti+1 B ti is independent of a i F ti, we have : IE [I (X) 2] = n 1 IE [ ] ai 2 (ti+1 t i ). i=0 73/91

74 Construction - Step 4 - Extension by density We will now extend by density the stochastic integral to the set of square integrable processes. We will use the following result : Lemma (density of H 2 0 in H2 ) For each process X H 2, there exists a sequence {X n } with X n H0 2 such that [ ] T X X n 2 L 2 (dp dt) = IE (X(s) X n (s)) 2 ds 0 for n. 0 74/91

75 Itô integral - Construction - Step 4- Extension by density Consequence of the Itô isometry and the density result For each simple process of the sequence, we can define its stochastic integral : I(X n ) = T 0 X n(s)db(s) The idea is to define I (X) for a general process X (not necessarily simple) as the limit of the sequence I (X n ) n in L 2 : I (X) def = lim n (I (X n )) where I (X) L 2 (dp) and the convergence is such that I (X) I (X n ) L 2 (dp) 0. 75/91

76 Construction - Step 4 - Extension by density Let us check that the stochastic integral is well defined 1, i.e. that (1) the limit of the sequence I(X n ) exists and (2) does not depend on the considered sequence X n tending to X : (1) X X n L 2 (dp dt) 0 implies that (I (X n)) converges in L 2 (dp) : Indeed, the convergence of the sequence (X n) in L 2 (dp dt) implies that this is a Cauchy sequence in L 2 (dp dt), which thanks to the Itô isometry, implies that (I (X n)) is also a Cauchy sequence L 2 (dp). As L 2 (dp) is a complete metric space (convergence of Cauchy sequences towards limits belonging to the space), the Cauchy sequence (I (X n)) converges to an element of L 2 (dp), that we denote by I(X). 1. The argument developed here is classical in functional analysis 76/91

77 Construction - Step 4 - Extension by density (2) Is I(X) well defined? i.e. for another choice of the sequence (X n) n converging to X : X X n L2 (dp dt) 0, does the new sequence I (X n) converges to the same limit in L 2 (dp) as the initial sequence I (X n )? The answer is yes, since X n X n L2 (dp dt) 0 thanks to the triangle inequality, and Itô isometry implies that : I (X n ) I (X n) L2 (dp) 0. 77/91

78 Itô integral - Itô isometry Now, we will show that when extended by density to the whole space H 2, Itô integral is still an isometry : Theorem (Itô isometry in H 2 (0, T )) For X H 2 [0, T ], we have that I (X) L2 (dp) = X L 2 (dp dt). In other words, ( T ) 2 [ T ] IE X(s)dB(s) = IE X 2 (s)ds /91

79 Itô integral - Itô isometry Proof First, we chose (X n ) n H0 2 such that X n X L 2 (dp dt) 0 for n. The triangle inequality for the L 2 (dp dt) norm implies : X n L 2 (dp dt) X L 2 (dp dt). Similarly, since (I (X n )) n L 2 (dp) I (X), I (X n ) L2 (dp) I (X) L 2 (dp). But we know that on H0 2, Itô isometry holds : I (X n ) L2 (dp) = X n L2 (dp dt) n, and the uniqueness of the limit completes the proof. 79/91

80 Itô integral as a stochastic process I : H 2 L 2 (dp) has been defined, and maps a stochastic process on a random variable (and not a process). Now, we would like to be able to consider the evolution in time of the gains of the trader, i.e. be able to consider t X(u)dB(u) as a stochastic process (in mathematical 0 terms, we need a mapping which maps a process to a process). For that purpose, we introduce a truncation function m t H 2 [0, T ] defined by : { 1 if s [0, t] m t (ω, s) = 0 else. 80/91

81 Itô integral as a stochastic process For X H 2 [0, T ], the product m t X H 2 [0, T ] t [0, T ], hence I (m t X) = T 0 m t(u)x(u)db(u) is a well defined element of L 2 (dp) One can show that we can construct a continuous martingale M t such that t [0, T ], we have P (M t = I (m t X)) = 1. The process {M t, t [0, T ]} is then the Ito integral considered as a process. 81/91

82 Itô integral as a stochastic process Theorem (Itô integral as martingales) for all X H 2 [0, T ], there exists a process {M t, t [0, T ]} which is a continuous martingale w.r.t. the standard Brownian filtration (F t ) t such that the event : {ω : M t (ω) = I (m t X) (ω)} has a probability 1 for all t [0, T ]. Proof : [Steele, pg 83-84, thm 6.2] 82/91

83 Itô integral as a stochastic process The integral sign : notation for all X H 2 [0, T ] and if {M t : 0 t T } is a continuous martingale such that we write : P [M t = I (m t X)] = 1 for all 0 t T, M t (ω) = t 0 X (ω, s) db s 0 t T t 0 X (ω, s) db s is a notation for what is well defined in the left-hand side of the equation. 83/91

84 Itô integral as a stochastic process But the notation is well chosen since ( ) 2 [ t ] t X H 2 = IE X (ω, s) db s = IE X 2 (ω, s) ds 0 0 t [0, T ]. Moreover, we have the following result : Proposition for all 0 s t and for all b H 2, we have ( ) 2 [ t ] t IE b (ω, u) db u F s = IE b (ω, u) 2 du F s. (5) s s 84/91

85 Itô integral as a stochastic process Proof Inequality (5) is equivalent to A F s : [ ( ) ] t 2 [ IE 1 A s b (ω, u) db ] t u = IE 1 A s b2 (ω, u) du. This is true by Theorem 1.4 on slide 78 for the modified integrand : b (ω, u) = 0 u [0, s) (t, T ] 1 A b (ω, u) u [s, t] 85/91

86 Riemann s representation Theorem For all f : R R continuous, if we consider the partition of [0, T ] given by t i = i T n with 0 i n, we have : lim n i=1 n f ( ) ( ) T B ti 1 Bti B ti 1 = f (B s ) db s (6) where the limit is taken in the sense of the convergence in probability. 0 86/91

87 Stochastic integral : explicit calculation In some cases, this integral can be explicitly computed from the definition. Example : t 0 B s db s =? If we denote i B := B ti B ti 1 and i t := t i t i 1, we have : ( B ti 1 i B ) = ( Bti 1 B t i Bt 2 i 1) i i = 1 [ (B 2 2 ti B 2 ) ( ] 2 ti 1 Bti B ti 1) = 1 B 2 2( t B0 2 ) 1 2 i i ( i B ) 2 = 1 2 B2 t 1 2 ( i B ) 2, i 87/91

88 Stochastic integral : explicit calculation We have seen that ( i B ) 2 t i in L 2. We then get : t 0 B s db s = 1 2 B2 t 1 2 t. Remark : We just saw that ( i B) 2 behaves like i t i.e., formally : (db t ) 2 = dt 88/91

89 Riemann s representation Proposition (Gaussian integrals) Let f C[0, T ] (a deterministic function), then the process defined by X t = t 0 f (s)db s for all t [0, T ] is a Gaussian process of zero mean with indepent increments and with covariance function cov(x s, X t) = s t 0 f 2 (u)du. Moreover, if we consider the partition of [0, T ] given by t i = it for 0 i n n and ti [t i 1, t i ], then lim n n i=1 f (t i ) ( B ti B ti 1 ) = T 0 f (s)db s, where the limit is taken in the sense of the convergence in probability. 89/91

90 Interpretation trajectory by trajectory of the Ito integral Theorem (Interpretation trajectory by trajectory of the Ito integral on H 2 ) If f H 2 is bounded and if ν is a stopping time such that f (ω, s) = 0 for almost all ω {ω : s ν}, then for almost all ω {ω : t ν}. X t (ω) = t 0 f (ω, s) db s = 0 90/91

91 Interpretation trajectory by trajectory of the Ito integral Theorem (Persistence of the identity on H 2 ) If f and g H 2 and if ν is a stopping time such that f (ω, s) = g (ω, s) for almost all ω {ω : s ν}, then the integrals X t (ω) = t 0 f (ω, s) db s and Y t (ω) = t 0 g (ω, s) db s are equal for almost all ω {ω : t ν}. 91/91

Drunken Birds, Brownian Motion, and Other Random Fun

Drunken Birds, Brownian Motion, and Other Random Fun Drunken Birds, Brownian Motion, and Other Random Fun Michael Perlmutter Department of Mathematics Purdue University 1 M. Perlmutter(Purdue) Brownian Motion and Martingales Outline Review of Basic Probability

More information

S t d with probability (1 p), where

S t d with probability (1 p), where Stochastic Calculus Week 3 Topics: Towards Black-Scholes Stochastic Processes Brownian Motion Conditional Expectations Continuous-time Martingales Towards Black Scholes Suppose again that S t+δt equals

More information

Homework Assignments

Homework Assignments Homework Assignments Week 1 (p. 57) #4.1, 4., 4.3 Week (pp 58 6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15 19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9 31) #.,.6,.9 Week 4 (pp 36 37)

More information

AMH4 - ADVANCED OPTION PRICING. Contents

AMH4 - ADVANCED OPTION PRICING. Contents AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5

More information

Introduction to Probability Theory and Stochastic Processes for Finance Lecture Notes

Introduction to Probability Theory and Stochastic Processes for Finance Lecture Notes Introduction to Probability Theory and Stochastic Processes for Finance Lecture Notes Fabio Trojani Department of Economics, University of St. Gallen, Switzerland Correspondence address: Fabio Trojani,

More information

Stochastic Calculus, Application of Real Analysis in Finance

Stochastic Calculus, Application of Real Analysis in Finance , Application of Real Analysis in Finance Workshop for Young Mathematicians in Korea Seungkyu Lee Pohang University of Science and Technology August 4th, 2010 Contents 1 BINOMIAL ASSET PRICING MODEL Contents

More information

STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL

STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce

More information

1 The continuous time limit

1 The continuous time limit Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1

More information

An Introduction to Stochastic Calculus

An Introduction to Stochastic Calculus An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics and Statistics Washington State University Lisbon, May 218 Haijun Li An Introduction to Stochastic Calculus Lisbon,

More information

BROWNIAN MOTION Antonella Basso, Martina Nardon

BROWNIAN MOTION Antonella Basso, Martina Nardon BROWNIAN MOTION Antonella Basso, Martina Nardon basso@unive.it, mnardon@unive.it Department of Applied Mathematics University Ca Foscari Venice Brownian motion p. 1 Brownian motion Brownian motion plays

More information

Stochastic Dynamical Systems and SDE s. An Informal Introduction

Stochastic Dynamical Systems and SDE s. An Informal Introduction Stochastic Dynamical Systems and SDE s An Informal Introduction Olav Kallenberg Graduate Student Seminar, April 18, 2012 1 / 33 2 / 33 Simple recursion: Deterministic system, discrete time x n+1 = f (x

More information

Continuous Time Finance. Tomas Björk

Continuous Time Finance. Tomas Björk Continuous Time Finance Tomas Björk 1 II Stochastic Calculus Tomas Björk 2 Typical Setup Take as given the market price process, S(t), of some underlying asset. S(t) = price, at t, per unit of underlying

More information

1.1 Basic Financial Derivatives: Forward Contracts and Options

1.1 Basic Financial Derivatives: Forward Contracts and Options Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables

More information

Risk Neutral Valuation

Risk Neutral Valuation copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential

More information

Modeling via Stochastic Processes in Finance

Modeling via Stochastic Processes in Finance Modeling via Stochastic Processes in Finance Dimbinirina Ramarimbahoaka Department of Mathematics and Statistics University of Calgary AMAT 621 - Fall 2012 October 15, 2012 Question: What are appropriate

More information

An Introduction to Stochastic Calculus

An Introduction to Stochastic Calculus An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2-3 Haijun Li An Introduction to Stochastic Calculus Week 2-3 1 / 24 Outline

More information

BROWNIAN MOTION II. D.Majumdar

BROWNIAN MOTION II. D.Majumdar BROWNIAN MOTION II D.Majumdar DEFINITION Let (Ω, F, P) be a probability space. For each ω Ω, suppose there is a continuous function W(t) of t 0 that satisfies W(0) = 0 and that depends on ω. Then W(t),

More information

An Introduction to Stochastic Calculus

An Introduction to Stochastic Calculus An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 5 Haijun Li An Introduction to Stochastic Calculus Week 5 1 / 20 Outline 1 Martingales

More information

Martingales. by D. Cox December 2, 2009

Martingales. by D. Cox December 2, 2009 Martingales by D. Cox December 2, 2009 1 Stochastic Processes. Definition 1.1 Let T be an arbitrary index set. A stochastic process indexed by T is a family of random variables (X t : t T) defined on a

More information

MATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS

MATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.

More information

Stochastic Processes and Financial Mathematics (part two) Dr Nic Freeman

Stochastic Processes and Financial Mathematics (part two) Dr Nic Freeman Stochastic Processes and Financial Mathematics (part two) Dr Nic Freeman April 25, 218 Contents 9 The transition to continuous time 3 1 Brownian motion 5 1.1 The limit of random walks...............................

More information

RMSC 4005 Stochastic Calculus for Finance and Risk. 1 Exercises. (c) Let X = {X n } n=0 be a {F n }-supermartingale. Show that.

RMSC 4005 Stochastic Calculus for Finance and Risk. 1 Exercises. (c) Let X = {X n } n=0 be a {F n }-supermartingale. Show that. 1. EXERCISES RMSC 45 Stochastic Calculus for Finance and Risk Exercises 1 Exercises 1. (a) Let X = {X n } n= be a {F n }-martingale. Show that E(X n ) = E(X ) n N (b) Let X = {X n } n= be a {F n }-submartingale.

More information

Risk Neutral Measures

Risk Neutral Measures CHPTER 4 Risk Neutral Measures Our aim in this section is to show how risk neutral measures can be used to price derivative securities. The key advantage is that under a risk neutral measure the discounted

More information

Stochastic Modelling Unit 3: Brownian Motion and Diffusions

Stochastic Modelling Unit 3: Brownian Motion and Diffusions Stochastic Modelling Unit 3: Brownian Motion and Diffusions Russell Gerrard and Douglas Wright Cass Business School, City University, London June 2004 Contents of Unit 3 1 Introduction 2 Brownian Motion

More information

How to hedge Asian options in fractional Black-Scholes model

How to hedge Asian options in fractional Black-Scholes model How to hedge Asian options in fractional Black-Scholes model Heikki ikanmäki Jena, March 29, 211 Fractional Lévy processes 1/36 Outline of the talk 1. Introduction 2. Main results 3. Methodology 4. Conclusions

More information

Equivalence between Semimartingales and Itô Processes

Equivalence between Semimartingales and Itô Processes International Journal of Mathematical Analysis Vol. 9, 215, no. 16, 787-791 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.411358 Equivalence between Semimartingales and Itô Processes

More information

Continuous Processes. Brownian motion Stochastic calculus Ito calculus

Continuous Processes. Brownian motion Stochastic calculus Ito calculus Continuous Processes Brownian motion Stochastic calculus Ito calculus Continuous Processes The binomial models are the building block for our realistic models. Three small-scale principles in continuous

More information

Lecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.

Lecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r. Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous

More information

Last Time. Martingale inequalities Martingale convergence theorem Uniformly integrable martingales. Today s lecture: Sections 4.4.1, 5.

Last Time. Martingale inequalities Martingale convergence theorem Uniformly integrable martingales. Today s lecture: Sections 4.4.1, 5. MATH136/STAT219 Lecture 21, November 12, 2008 p. 1/11 Last Time Martingale inequalities Martingale convergence theorem Uniformly integrable martingales Today s lecture: Sections 4.4.1, 5.3 MATH136/STAT219

More information

1 Mathematics in a Pill 1.1 PROBABILITY SPACE AND RANDOM VARIABLES. A probability triple P consists of the following components:

1 Mathematics in a Pill 1.1 PROBABILITY SPACE AND RANDOM VARIABLES. A probability triple P consists of the following components: 1 Mathematics in a Pill The purpose of this chapter is to give a brief outline of the probability theory underlying the mathematics inside the book, and to introduce necessary notation and conventions

More information

Lecture 11: Ito Calculus. Tuesday, October 23, 12

Lecture 11: Ito Calculus. Tuesday, October 23, 12 Lecture 11: Ito Calculus Continuous time models We start with the model from Chapter 3 log S j log S j 1 = µ t + p tz j Sum it over j: log S N log S 0 = NX µ t + NX p tzj j=1 j=1 Can we take the limit

More information

Stochastic Differential equations as applied to pricing of options

Stochastic Differential equations as applied to pricing of options Stochastic Differential equations as applied to pricing of options By Yasin LUT Supevisor:Prof. Tuomo Kauranne December 2010 Introduction Pricing an European call option Conclusion INTRODUCTION A stochastic

More information

Martingale representation theorem

Martingale representation theorem Martingale representation theorem Ω = C[, T ], F T = smallest σ-field with respect to which B s are all measurable, s T, P the Wiener measure, B t = Brownian motion M t square integrable martingale with

More information

Enlargement of filtration

Enlargement of filtration Enlargement of filtration Bernardo D Auria email: bernardo.dauria@uc3m.es web: www.est.uc3m.es/bdauria July 6, 2017 ICMAT / UC3M Enlargement of Filtration Enlargement of Filtration ([1] 5.9) If G is a

More information

Binomial model: numerical algorithm

Binomial model: numerical algorithm Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4

More information

Stochastic Calculus - An Introduction

Stochastic Calculus - An Introduction Stochastic Calculus - An Introduction M. Kazim Khan Kent State University. UET, Taxila August 15-16, 17 Outline 1 From R.W. to B.M. B.M. 3 Stochastic Integration 4 Ito s Formula 5 Recap Random Walk Consider

More information

Lecture Notes for Chapter 6. 1 Prototype model: a one-step binomial tree

Lecture Notes for Chapter 6. 1 Prototype model: a one-step binomial tree Lecture Notes for Chapter 6 This is the chapter that brings together the mathematical tools (Brownian motion, Itô calculus) and the financial justifications (no-arbitrage pricing) to produce the derivative

More information

Math 416/516: Stochastic Simulation

Math 416/516: Stochastic Simulation Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation

More information

Stochastic Calculus for Finance Brief Lecture Notes. Gautam Iyer

Stochastic Calculus for Finance Brief Lecture Notes. Gautam Iyer Stochastic Calculus for Finance Brief Lecture Notes Gautam Iyer Gautam Iyer, 17. c 17 by Gautam Iyer. This work is licensed under the Creative Commons Attribution - Non Commercial - Share Alike 4. International

More information

MSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013

MSc Financial Engineering CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL. To be handed in by monday January 28, 2013 MSc Financial Engineering 2012-13 CHRISTMAS ASSIGNMENT: MERTON S JUMP-DIFFUSION MODEL To be handed in by monday January 28, 2013 Department EMS, Birkbeck Introduction The assignment consists of Reading

More information

Hedging under Arbitrage

Hedging under Arbitrage Hedging under Arbitrage Johannes Ruf Columbia University, Department of Statistics Modeling and Managing Financial Risks January 12, 2011 Motivation Given: a frictionless market of stocks with continuous

More information

Optimal stopping problems for a Brownian motion with a disorder on a finite interval

Optimal stopping problems for a Brownian motion with a disorder on a finite interval Optimal stopping problems for a Brownian motion with a disorder on a finite interval A. N. Shiryaev M. V. Zhitlukhin arxiv:1212.379v1 [math.st] 15 Dec 212 December 18, 212 Abstract We consider optimal

More information

1 IEOR 4701: Notes on Brownian Motion

1 IEOR 4701: Notes on Brownian Motion Copyright c 26 by Karl Sigman IEOR 47: Notes on Brownian Motion We present an introduction to Brownian motion, an important continuous-time stochastic process that serves as a continuous-time analog to

More information

The stochastic calculus

The stochastic calculus Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations

More information

An Introduction to Point Processes. from a. Martingale Point of View

An Introduction to Point Processes. from a. Martingale Point of View An Introduction to Point Processes from a Martingale Point of View Tomas Björk KTH, 211 Preliminary, incomplete, and probably with lots of typos 2 Contents I The Mathematics of Counting Processes 5 1 Counting

More information

Valuation of derivative assets Lecture 8

Valuation of derivative assets Lecture 8 Valuation of derivative assets Lecture 8 Magnus Wiktorsson September 27, 2018 Magnus Wiktorsson L8 September 27, 2018 1 / 14 The risk neutral valuation formula Let X be contingent claim with maturity T.

More information

Non-semimartingales in finance

Non-semimartingales in finance Non-semimartingales in finance Pricing and Hedging Options with Quadratic Variation Tommi Sottinen University of Vaasa 1st Northern Triangular Seminar 9-11 March 2009, Helsinki University of Technology

More information

Remarks: 1. Often we shall be sloppy about specifying the ltration. In all of our examples there will be a Brownian motion around and it will be impli

Remarks: 1. Often we shall be sloppy about specifying the ltration. In all of our examples there will be a Brownian motion around and it will be impli 6 Martingales in continuous time Just as in discrete time, the notion of a martingale will play a key r^ole in our continuous time models. Recall that in discrete time, a sequence ; 1 ;::: ; n for which

More information

Reading: You should read Hull chapter 12 and perhaps the very first part of chapter 13.

Reading: You should read Hull chapter 12 and perhaps the very first part of chapter 13. FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Asset Price Dynamics Introduction These notes give assumptions of asset price returns that are derived from the efficient markets hypothesis. Although a hypothesis,

More information

Martingale Pricing Theory in Discrete-Time and Discrete-Space Models

Martingale Pricing Theory in Discrete-Time and Discrete-Space Models IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,

More information

Asymptotic results discrete time martingales and stochastic algorithms

Asymptotic results discrete time martingales and stochastic algorithms Asymptotic results discrete time martingales and stochastic algorithms Bernard Bercu Bordeaux University, France IFCAM Summer School Bangalore, India, July 2015 Bernard Bercu Asymptotic results for discrete

More information

Tangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.

Tangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford. Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey

More information

Lévy models in finance

Lévy models in finance Lévy models in finance Ernesto Mordecki Universidad de la República, Montevideo, Uruguay PASI - Guanajuato - June 2010 Summary General aim: describe jummp modelling in finace through some relevant issues.

More information

Dr. Maddah ENMG 625 Financial Eng g II 10/16/06

Dr. Maddah ENMG 625 Financial Eng g II 10/16/06 Dr. Maddah ENMG 65 Financial Eng g II 10/16/06 Chapter 11 Models of Asset Dynamics () Random Walk A random process, z, is an additive process defined over times t 0, t 1,, t k, t k+1,, such that z( t )

More information

Brownian Motion. Richard Lockhart. Simon Fraser University. STAT 870 Summer 2011

Brownian Motion. Richard Lockhart. Simon Fraser University. STAT 870 Summer 2011 Brownian Motion Richard Lockhart Simon Fraser University STAT 870 Summer 2011 Richard Lockhart (Simon Fraser University) Brownian Motion STAT 870 Summer 2011 1 / 33 Purposes of Today s Lecture Describe

More information

Derivatives Pricing and Stochastic Calculus

Derivatives Pricing and Stochastic Calculus Derivatives Pricing and Stochastic Calculus Romuald Elie LAMA, CNRS UMR 85 Université Paris-Est Marne-La-Vallée elie @ ensae.fr Idris Kharroubi CEREMADE, CNRS UMR 7534, Université Paris Dauphine kharroubi

More information

Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models

Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016 Overview 1 Self-financing portfolio 2 Complete

More information

PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS

PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS MATHEMATICAL TRIPOS Part III Thursday, 5 June, 214 1:3 pm to 4:3 pm PAPER 27 STOCHASTIC CALCULUS AND APPLICATIONS Attempt no more than FOUR questions. There are SIX questions in total. The questions carry

More information

The value of foresight

The value of foresight Philip Ernst Department of Statistics, Rice University Support from NSF-DMS-1811936 (co-pi F. Viens) and ONR-N00014-18-1-2192 gratefully acknowledged. IMA Financial and Economic Applications June 11, 2018

More information

Introduction Taylor s Theorem Einstein s Theory Bachelier s Probability Law Brownian Motion Itô s Calculus. Itô s Calculus.

Introduction Taylor s Theorem Einstein s Theory Bachelier s Probability Law Brownian Motion Itô s Calculus. Itô s Calculus. Itô s Calculus Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October 21, 2016 Christopher Ting QF 101 Week 10 October

More information

THE MARTINGALE METHOD DEMYSTIFIED

THE MARTINGALE METHOD DEMYSTIFIED THE MARTINGALE METHOD DEMYSTIFIED SIMON ELLERSGAARD NIELSEN Abstract. We consider the nitty gritty of the martingale approach to option pricing. These notes are largely based upon Björk s Arbitrage Theory

More information

Chapter 3: Black-Scholes Equation and Its Numerical Evaluation

Chapter 3: Black-Scholes Equation and Its Numerical Evaluation Chapter 3: Black-Scholes Equation and Its Numerical Evaluation 3.1 Itô Integral 3.1.1 Convergence in the Mean and Stieltjes Integral Definition 3.1 (Convergence in the Mean) A sequence {X n } n ln of random

More information

CONVERGENCE OF OPTION REWARDS FOR MARKOV TYPE PRICE PROCESSES MODULATED BY STOCHASTIC INDICES

CONVERGENCE OF OPTION REWARDS FOR MARKOV TYPE PRICE PROCESSES MODULATED BY STOCHASTIC INDICES CONVERGENCE OF OPTION REWARDS FOR MARKOV TYPE PRICE PROCESSES MODULATED BY STOCHASTIC INDICES D. S. SILVESTROV, H. JÖNSSON, AND F. STENBERG Abstract. A general price process represented by a two-component

More information

From Discrete Time to Continuous Time Modeling

From Discrete Time to Continuous Time Modeling From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy

More information

Weierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions

Weierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions Weierstrass Institute for Applied Analysis and Stochastics Maximum likelihood estimation for jump diffusions Hilmar Mai Mohrenstrasse 39 1117 Berlin Germany Tel. +49 3 2372 www.wias-berlin.de Haindorf

More information

Convergence. Any submartingale or supermartingale (Y, F) converges almost surely if it satisfies E Y n <. STAT2004 Martingale Convergence

Convergence. Any submartingale or supermartingale (Y, F) converges almost surely if it satisfies E Y n <. STAT2004 Martingale Convergence Convergence Martingale convergence theorem Let (Y, F) be a submartingale and suppose that for all n there exist a real value M such that E(Y + n ) M. Then there exist a random variable Y such that Y n

More information

A No-Arbitrage Theorem for Uncertain Stock Model

A No-Arbitrage Theorem for Uncertain Stock Model Fuzzy Optim Decis Making manuscript No (will be inserted by the editor) A No-Arbitrage Theorem for Uncertain Stock Model Kai Yao Received: date / Accepted: date Abstract Stock model is used to describe

More information

4 Martingales in Discrete-Time

4 Martingales in Discrete-Time 4 Martingales in Discrete-Time Suppose that (Ω, F, P is a probability space. Definition 4.1. A sequence F = {F n, n = 0, 1,...} is called a filtration if each F n is a sub-σ-algebra of F, and F n F n+1

More information

Introduction to Stochastic Calculus With Applications

Introduction to Stochastic Calculus With Applications Introduction to Stochastic Calculus With Applications Fima C Klebaner University of Melbourne \ Imperial College Press Contents Preliminaries From Calculus 1 1.1 Continuous and Differentiable Functions.

More information

IEOR E4703: Monte-Carlo Simulation

IEOR E4703: Monte-Carlo Simulation IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 11 10/9/2013. Martingales and stopping times II

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 11 10/9/2013. Martingales and stopping times II MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 11 10/9/013 Martingales and stopping times II Content. 1. Second stopping theorem.. Doob-Kolmogorov inequality. 3. Applications of stopping

More information

IEOR E4703: Monte-Carlo Simulation

IEOR E4703: Monte-Carlo Simulation IEOR E4703: Monte-Carlo Simulation Generating Random Variables and Stochastic Processes Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com

More information

STOCHASTIC INTEGRALS

STOCHASTIC INTEGRALS Stat 391/FinMath 346 Lecture 8 STOCHASTIC INTEGRALS X t = CONTINUOUS PROCESS θ t = PORTFOLIO: #X t HELD AT t { St : STOCK PRICE M t : MG W t : BROWNIAN MOTION DISCRETE TIME: = t < t 1

More information

1 Geometric Brownian motion

1 Geometric Brownian motion Copyright c 05 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is

More information

Are the Azéma-Yor processes truly remarkable?

Are the Azéma-Yor processes truly remarkable? Are the Azéma-Yor processes truly remarkable? Jan Obłój j.obloj@imperial.ac.uk based on joint works with L. Carraro, N. El Karoui, A. Meziou and M. Yor Welsh Probability Seminar, 17 Jan 28 Are the Azéma-Yor

More information

M5MF6. Advanced Methods in Derivatives Pricing

M5MF6. Advanced Methods in Derivatives Pricing Course: Setter: M5MF6 Dr Antoine Jacquier MSc EXAMINATIONS IN MATHEMATICS AND FINANCE DEPARTMENT OF MATHEMATICS April 2016 M5MF6 Advanced Methods in Derivatives Pricing Setter s signature...........................................

More information

The Black-Scholes PDE from Scratch

The Black-Scholes PDE from Scratch The Black-Scholes PDE from Scratch chris bemis November 27, 2006 0-0 Goal: Derive the Black-Scholes PDE To do this, we will need to: Come up with some dynamics for the stock returns Discuss Brownian motion

More information

Constructing Markov models for barrier options

Constructing Markov models for barrier options Constructing Markov models for barrier options Gerard Brunick joint work with Steven Shreve Department of Mathematics University of Texas at Austin Nov. 14 th, 2009 3 rd Western Conference on Mathematical

More information

Replication and Absence of Arbitrage in Non-Semimartingale Models

Replication and Absence of Arbitrage in Non-Semimartingale Models Replication and Absence of Arbitrage in Non-Semimartingale Models Matematiikan päivät, Tampere, 4-5. January 2006 Tommi Sottinen University of Helsinki 4.1.2006 Outline 1. The classical pricing model:

More information

A note on the existence of unique equivalent martingale measures in a Markovian setting

A note on the existence of unique equivalent martingale measures in a Markovian setting Finance Stochast. 1, 251 257 1997 c Springer-Verlag 1997 A note on the existence of unique equivalent martingale measures in a Markovian setting Tina Hviid Rydberg University of Aarhus, Department of Theoretical

More information

then for any deterministic f,g and any other random variable

then for any deterministic f,g and any other random variable Martingales Thursday, December 03, 2015 2:01 PM References: Karlin and Taylor Ch. 6 Lawler Sec. 5.1-5.3 Homework 4 due date extended to Wednesday, December 16 at 5 PM. We say that a random variable is

More information

Hedging of Contingent Claims under Incomplete Information

Hedging of Contingent Claims under Incomplete Information Projektbereich B Discussion Paper No. B 166 Hedging of Contingent Claims under Incomplete Information by Hans Föllmer ) Martin Schweizer ) October 199 ) Financial support by Deutsche Forschungsgemeinschaft,

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 19 11/20/2013. Applications of Ito calculus to finance

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 19 11/20/2013. Applications of Ito calculus to finance MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.7J Fall 213 Lecture 19 11/2/213 Applications of Ito calculus to finance Content. 1. Trading strategies 2. Black-Scholes option pricing formula 1 Security

More information

Geometric Brownian Motions

Geometric Brownian Motions Chapter 6 Geometric Brownian Motions 1 Normal Distributions We begin by recalling the normal distribution briefly. Let Z be a random variable distributed as standard normal, i.e., Z N(0, 1). The probability

More information

Are the Azéma-Yor processes truly remarkable?

Are the Azéma-Yor processes truly remarkable? Are the Azéma-Yor processes truly remarkable? Jan Obłój j.obloj@imperial.ac.uk based on joint works with L. Carraro, N. El Karoui, A. Meziou and M. Yor Swiss Probability Seminar, 5 Dec 2007 Are the Azéma-Yor

More information

Probability without Measure!

Probability without Measure! Probability without Measure! Mark Saroufim University of California San Diego msaroufi@cs.ucsd.edu February 18, 2014 Mark Saroufim (UCSD) It s only a Game! February 18, 2014 1 / 25 Overview 1 History of

More information

No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate

No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate Fuzzy Optim Decis Making 217 16:221 234 DOI 117/s17-16-9246-8 No-arbitrage theorem for multi-factor uncertain stock model with floating interest rate Xiaoyu Ji 1 Hua Ke 2 Published online: 17 May 216 Springer

More information

Module 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation.

Module 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation. Stochastic Differential Equation Consider. Moreover partition the interval into and define, where. Now by Rieman Integral we know that, where. Moreover. Using the fundamentals mentioned above we can easily

More information

Hedging under arbitrage

Hedging under arbitrage Hedging under arbitrage Johannes Ruf Columbia University, Department of Statistics AnStAp10 August 12, 2010 Motivation Usually, there are several trading strategies at one s disposal to obtain a given

More information

Martingale Measure TA

Martingale Measure TA Martingale Measure TA Martingale Measure a) What is a martingale? b) Groundwork c) Definition of a martingale d) Super- and Submartingale e) Example of a martingale Table of Content Connection between

More information

Are stylized facts irrelevant in option-pricing?

Are stylized facts irrelevant in option-pricing? Are stylized facts irrelevant in option-pricing? Kyiv, June 19-23, 2006 Tommi Sottinen, University of Helsinki Based on a joint work No-arbitrage pricing beyond semimartingales with C. Bender, Weierstrass

More information

Beyond the Black-Scholes-Merton model

Beyond the Black-Scholes-Merton model Econophysics Lecture Leiden, November 5, 2009 Overview 1 Limitations of the Black-Scholes model 2 3 4 Limitations of the Black-Scholes model Black-Scholes model Good news: it is a nice, well-behaved model

More information

Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing

Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving

More information

Bluff Your Way Through Black-Scholes

Bluff Your Way Through Black-Scholes Bluff our Way Through Black-Scholes Saurav Sen December 000 Contents What is Black-Scholes?.............................. 1 The Classical Black-Scholes Model....................... 1 Some Useful Background

More information

An overview of some financial models using BSDE with enlarged filtrations

An overview of some financial models using BSDE with enlarged filtrations An overview of some financial models using BSDE with enlarged filtrations Anne EYRAUD-LOISEL Workshop : Enlargement of Filtrations and Applications to Finance and Insurance May 31st - June 4th, 2010, Jena

More information

Equity correlations implied by index options: estimation and model uncertainty analysis

Equity correlations implied by index options: estimation and model uncertainty analysis 1/18 : estimation and model analysis, EDHEC Business School (joint work with Rama COT) Modeling and managing financial risks Paris, 10 13 January 2011 2/18 Outline 1 2 of multi-asset models Solution to

More information

Lecture 4. Finite difference and finite element methods

Lecture 4. Finite difference and finite element methods Finite difference and finite element methods Lecture 4 Outline Black-Scholes equation From expectation to PDE Goal: compute the value of European option with payoff g which is the conditional expectation

More information

The Black-Scholes Model

The Black-Scholes Model The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes

More information

Stochastic modelling of electricity markets Pricing Forwards and Swaps

Stochastic modelling of electricity markets Pricing Forwards and Swaps Stochastic modelling of electricity markets Pricing Forwards and Swaps Jhonny Gonzalez School of Mathematics The University of Manchester Magical books project August 23, 2012 Clip for this slide Pricing

More information

4 Risk-neutral pricing

4 Risk-neutral pricing 4 Risk-neutral pricing We start by discussing the idea of risk-neutral pricing in the framework of the elementary one-stepbinomialmodel. Supposetherearetwotimest = andt = 1. Attimethestock has value S()

More information