Introduction to Algorithmic Trading Strategies Lecture 3
|
|
- Earl Mason
- 5 years ago
- Views:
Transcription
1 Introduction to Algorithmic Trading Strategies Lecture 3 Trend Following Haksun Li haksun.li@numericalmethod.com
2 References Introduction to Stochastic Calculus with Applications. Fima C Klebaner. nd Edition. Estimating continuous time transition matrices from discretely observed data. Inamura, Yasunari. April 006. Optimal Trend Following Trading Rules. Dai, Min and Zhang, Qing and Zhu, Qiji Jim. 011.
3 3 Stochastic Calculus
4 Brownian Motion Independence of increments. dd = B t B s is independent of any history up to s. Normality of increments. dd is normally distributed with mean 0 and variance t s. Continuity of paths. B t is a continuous function of t. 4
5 Stochastic vs. Newtonian Calculus Newtonian calculus: when you zoom in a function enough, the little segment looks like a straight line, hence the approximation: dd = f dd Derivative exists. The function is differentiable. Stochastic calculus: no matter how much you zoom in or how small dd is, the function still looks very zig-zag and random. It is nothing like a straight line. For Brownian motion, however much you zoom in, how small the segment is, it still just looks very much like a Brownian motion. The function is therefore no where differentiable. 5
6 Examples not a trend not mean reversion 6
7 dbdb dd = dddd = dd t 0 db t n i=1 Z n,i Z n,i is of N 0, t n t db 0 t for all i. sum of variances of Z n,i t Making n or dd 0, we have t db 0 t = t, convergence in probability db t = dd, in differential form dddt = 0 dtdd = 0 7
8 Asset Price Model Want to model asset price movement. Change in price = dd Change in price is not too meaningful as $1 change in a penny stock is more significant than $1 change in GOOG. Return = dd S Model return using two parts. Deterministic: μdd, the predictable part. E.g., fixed deposit interest rate. Random/stochastic: σdd, where σ is the volatility of returns and db is a sample from a probability distribution, e.g., Normal. 8
9 Geometric Brownian Motion Asset price: dd S = μμμ + σσb db: normally distributed E db = 0 Variance = dd. It is intuitive that db should be scaled by dd otherwise the (random) return drawn would be too big from any Normal distribution for dd 0. db = φ dd, where φ is a standard Normal distribution. Reasonably good model for stocks and indices. Real data have more big rises and falls than this model predicts, i.e., extreme events. 9
10 GBM Properties Markov property: the distribution of the next price S + dd depend only on the current price S. E ds = E μsdd + σsdb = E μμμμ + E σσσb = μμμμ + σσ E db = μμμμ Var dd = E dd dbdd = 0 dddd = 0 dbdb = dd E dd = σ S dd 10
11 Stochastic Differential Equation Both μ and σ can be as simple as constants, deterministic functions of t and S, or as complicated as stochastic functions adapted to the filtration generated by S t. ds t = μμμ + σσb t ds t = μ t, S t dd + σ t, S t db t 11
12 Univariate Ito s Lemma Assume dx t = μ t dd + σ t db t f t, X t is twice differentiable of two real variables We have dd t, X t = + μ t + σ t f x dd + σ t db t 1
13 Proof Taylor series df t, X = f t dd + f X dx + 1 f ttdddd + f tx dddx + f XX dxdt + f XX dddx = f t dd + f X dd + 1 f XXdddd = f t dd + f X μdd + σdb + 1 f XX μdd + σdd = f t dd + f X μdd + σdd + 1 f XX μdd + σdd = f t dd + f X μdd + σdd + 1 f XX μ dd + σ db + μddσdd = f t dd + f X μdd + σdd + 1 f XXσ dd = f t dd + f X μdd + σdd + 1 f XXσ dd = f t + μf X + 1 σ f XX dd + σf X dd 13
14 Log Example For G.B.M., dx t = μx t dd + σx t dz t, d log X t =? f x = log x f t = 0 f x = 1 x f x = 1 x d log X t = μx t 1 + σx t X t 1 X t dd + σx t 1 X t db t = μ σ dd + σdb t 14
15 Exp Example dx t = μdd + σdz t, de X t =? f x = e x f t = 0 f x = ex f x = ex de X t = μe X t + σ ex t dd + σe X tdb t = e X t dex t e X t μ + σ = μ + σ dd + σdb t dd + σdb t 15
16 Stopping Time A stopping time is a random time that some event is known to have occurred. Examples: when a stock hits a target price when you run out of money when a moving average crossover occurs When a stock has bottomed out is NOT a stopping time because you cannot tell when it has reached its minimum without future information. 16
17 17 Optimal Trend Following Strategy
18 Asset Model Two state Markov model for a stock s prices: BULL and BEAR. ds r = S r μ αr dd + σσb r, t r T < The trading period is between time t, T. α r = 1, are the two Markov states that indicates the BULL and BEAR markets. μ 1 > 0 μ < 0 Q = λ 1 λ 1, the generator matrix for the Markov λ λ chain. 18
19 Generator Matrix From transition matrix to generator matrix: P t, t + Δt = I + Δtt + o Δt We can estimate Q by estimating P. P t, s = P t, t + mδt I + Δtt m = P t, t + Δt = exp Δtt P t exp ΔtQ Q 1 Δt log P t E.g., Δt = 1 5 I + s t m Q m = exp s t Q 19
20 Optimal Stopping Times i = 0, Λ 0 = τ 1, ν 1, τ, ν, no position, bank the $ no position, LONG LONG LONG bank the $ no position, bank the $ t τ 1 ν 1 τ ν τ n ν n BUY SELL BUY SELL BUY SELL T i = 0, Λ 0 = ν 1, τ, ν, τ 3, 0
21 Parameters ρ 0, interest free rate S t = S, the initial stock price i = 0,1, the net position K b, the transaction cost for BUYs in percentage K s, the transaction cost for SELLs in percentage 1
22 Expected Return (starting flat) When i = 0, E 0,t R t = n=1 E t e ρ τ 1 t S νn S τ n 1 K s 1+K b I τ n<t e ρ τ n+1 υ n You are long between τ n and ν n and the return is determined by the price change discounted by the commissions. You are flat between ν n and τ n+1 and the money grows at the risk free rate.
23 Expected Return (starting long) When i = 1, E 1,t R t = E t S ν 1 S eρ τ ν 1 1 K s S νn n= You sell the long position at ν 1. S τ n 1 K s 1+K b I τ n<t e ρ τ n+1 υ n 3
24 Value Functions It is easier to work with the log of the returns. n=1 J 0 S, α, t, Λ 0 = E t ρ τ 1 t + log S νn S τ n + I τn <T log 1 K s 1+K b + ρ τ n+1 υ n J 1 S, α, t, Λ 1 = E t log S ν1 + ρ τ S ν 1 + log 1 K s + log S νn n= S τ n + I τn <T log 1 K s 1+K b + ρ τ n+1 υ n 4
25 Conditional Probability of Bull Market p r = P α r = 1 σ S u 0 u r p r is therefore a random process driven by the same Brownian motion that drives S u. dp r = λ 1 + λ p r + λ dd + μ 1 μ p r 1 p r db r = d log S r μ 1 μ p r +μ σ σ dd p t+1 = p t + g p t Δt + μ 1 μ p t 1 p t σ log S t+1 S t σ db r g p = λ 1 + λ p + λ μ 1 μ p 1 p μ 1 μ p+μ σ σ Make p t+1 between 0 and 1 if it falls outside the bounds. 5
26 Objective Find an optimal trading sequence (the stopping times) so that the value functions are maximized. V i p, t = sup Λi J i S, p, t, Λ i V i : the maximum amount of expected returns 6
27 Realistic Expectation of Returns Lower bounds: V 0 p, t ρ T t V 1 p, t ρ T t + log 1 K s Upper bound: V i p, t μ 1 σ No trading zones: T t One should never buy when ρ μ 1 σ 7
28 Principal of Optimality Given an optimal trading sequence, Λ i, starting from time t, the truncated sequence will also be optimal for the same trading problem starting from any stopping times τ n or υ n. 8
29 Coupled Value Functions V 0 p, t = sup E t ρ τ 1 t log 1 + K b + V 1 p τ1, τ 1 τ 1 V 1 p, t = sup ν 1 E t log S ν1 S t + log 1 K s + V 0 p ν1, ν 1 9
30 Hamilton-Jacobi-Bellman Equations min LV 0 ρ, V 0 V 1 + log 1 + K b = 0 min LV 1 f ρ, V 1 V 0 log 1 K s = 0 V with terminal conditions: 0 p, T = 0 V 1 p, T = log 1 K s L = t + 1 μ 1 μ p 1 p σ pp + λ 1 + λ p + λ p 30
31 Penalty Formulation Approach The HJB formulation is equivalent to this system of PDEs. LV 0 h ρ = ξ V 1 V 0 log 1 + K + b LV 1 f ρ = ξ V 0 V 1 + log 1 K + s ξ is a penalization factor such that ξ. h ρ = ρ 31
32 Trading Boundaries BB = p 0,1 [0, T) p p b t SR = p 0,1 [0, T) p p s t 3
33 SP500 33
34 SSE 34
35 Problems The buy entry signals are always delayed (as expected). The market therefore needs to bull long enough for the strategy to make profit. If the bear market comes in too fast and hard, the exit entry may not come in soon enough. A proper stoploss from max drawdown may help. 35
36 Finding the Boundaries At any time t, we need to find p b and p s to determine whether we buy or sell or go flat. p b is determined by V 1 p b, t V 0 p b, t p s is determined by = log 1 + K b V 1 p s, t V 0 p s, t = log 1 K s Need to compute V 0 p, t and V 1 p, t for all p and t. Solve the coupled PDEs. 36
37 Discretization of L ρ, t 0,1 [0,1) ρ, t as ρ j, t n j 0,, M ρ 0 = 0 ρ M = 0 n 1,, N t 1 = 0 t N = 1 V i V n+1 n i,j Vi,j t t 37
38 The Grid ρ M =1 ρ j V i ρ j, t n ρ 0 =0 t 1 =0 t 1 t n t N =1 38
39 Crank-Nicolson Scheme V i ρ 1 n+1 n+1 V i,j+1 Vi,j 1 ρ + V n n i,j+1 V i,j 1 ρ V i ρ 1 n+1 n+1 n+1 V i,j+1 Vi,j +Vi,j 1 ρ + V i,j+1 n V i,j n +V i,j 1 n ρ 39
40 RHS ξ V 1 V 0 log 1 + K b + = n+1 n+1 n+1 V 1,j V0,j ξ 0,j k b = log 1 + K b + V 1,j n n V 0,j ξ V 0 V 1 + log 1 K s + = n+1 V 0,j ξ 1,j n+1 n+1 V1,j k s = log 1 K s + V 0,j n n V 1,j k b + k s 40
41 Penalty n+1 ξ 0,j = ξ t, if V 1,j n+1 ξ 1,j = ξ t, if V 0,j n+1 n+1 V0,j n+1 n+1 V1,j + V 1,j n n V 0,j 0, otherwise n n V 1,j + V 0,j 0, otherwise k b > 0 + k s > 0 41
42 The Discretized PDEs a 1,j V n+1 0,j+1 + a 0,j V n+1 0,j + a 1,j V n+1 0,j 1 = c n 0,j n+1 n+1 n+1 V 1,j V0,j + ξ 0,j + V 1,j n n V 0,j k b a 1,j V n+1 1,j+1 + a 0,j V n+1 1,j + a 1,j V n+1 1,j 1 = c n 1,j n+1 n+1 n+1 V 0,j V1,j + ξ 1,j j 1,, M 1 p = 0, p = 1 are excluded. + V 0,j n n V 1,j + k s 4
43 The Coefficients a 1,j = 1 σ p a 0,j = 1 + σ p t p + 1 μ p t p t t p a 1,j = 1 σ p 1 μ p p p c n n 0,j = a 1,j V 0,j+1 + a 0,j V n n 0,j a 1,j V 0,j 1 + h j p t c n n 1,j = a 1,j V 1,j+1 + a 0,j V n n 1,j a 1,j V 1,j 1 + f j p t t 43
44 Boundary Conditions j = 0, p = 0 a 0,0 =1 a 1,0 = 0 c n 0,0 = V n 0,1 c n 1,0 = V n 1,1 j = M, p = 1 a 0,M =1 a 1,M = 0 n c 0,M n c 1,M n = V 0,M 1 n = V 1,M 1 + h 0 t + f 0 t + h 1 t + f 1 t n = 0, t = 0, for all j 0,, M V 0,j = 0 V 1,j = k s 44
45 System in Matrix Form An V 0 n+1 = c 0 n A n V 1 n+1 = c 1 n A n = a 0,0 a 1, a 0,1 0 a 1,1 a 1,M 1 0 a 0,M a 1,M 1 a 0,M 45
46 Solving the System of Equations We already know the values of V 0 0 and V 1 0. Starting from n = 0, using A n, we iteratively solve for V 0 n+1 and V 1 n+1 until n = M 1. There are totally M systems of equations to solve. The equations are linear in the unknowns V n+1 0,j and V n+1 n+1 1,j if and only if ξ 0,j = 0. When the equations are linear, we can use Thomas algorithm to solve for a tri-diagonal system of linear equations. Otherwise, we use an iterative scheme. 46
47 Iterative Scheme 1 st step, initialization, k = 0 n+1 Assume ξ i,j k th step 0 = 0. Solve for V i n+1 0. Using V n+1 n+1 i k 1, update ξ i,j n+1 When ξ i,j Repeat until convergence. k. k > 0, adjust the A n k and c i n k. 47
48 Adjustments (A) A n k = a 0,0 + ξδt a 1, a 0,1 + ξδt 0 a 1,1 a 1,M 1 0 a 0,M 1 + ξδt a 1,M 1 a 0,M + ξδt 48
49 Adjustments (c) c 0 n k = n c 0,M ξδt V 1,0 n+1 k 1 c n 0,1 + ξδt V 1,1 n+1 k 1 + ξδt n+1 V 1,M 1 k 1 + V 1,0 n n V 0,0 k b + V 1,1 n n V 0,1 k b + V n n 1,M 1 V 0,M 1 k b 1 + ξδt V 1,M n+1 k 1 + V 1,M n n V 0,M k b c 1 n k = n c 1,M ξδt V 0,0 n+1 k 1 c n 1,1 + ξδt V 0,1 n+1 k 1 + ξδt n+1 V 0,M 1 k 1 + V 0,0 n n V 1,0 + k s + V 0,1 n n V 1,1 + k s + V n n 0,M 1 V 1,M 1 + k s 1 + ξδt n+1 V 0,M k 1 + V 0,M n n V 1,M + k s 49
50 Convergence V i n+1 k V i n+1 k 1 1 < ε 50
51 51 Miscellaenous
52 Model Estimation Estimate the transition probabilities in the HMM by EM. Estimate μ and σ. The log of the prices are Gaussian. σ : sample variance σ = σ Δt σ = σ 1 Δt m i : sample mean m i = μ i σ μ i = 1 Δt m i + σ Δt 5
Introduction to Algorithmic Trading Strategies Lecture 4
Introduction to Algorithmic Trading Strategies Lecture 4 Optimal Pairs Trading by Stochastic Control Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Problem formulation Ito s lemma
More informationOptimal portfolio strategies under partial information with expert opinions
1 / 35 Optimal portfolio strategies under partial information with expert opinions Ralf Wunderlich Brandenburg University of Technology Cottbus, Germany Joint work with Rüdiger Frey Research Seminar WU
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationA Barrier Version of the Russian Option
A Barrier Version of the Russian Option L. A. Shepp, A. N. Shiryaev, A. Sulem Rutgers University; shepp@stat.rutgers.edu Steklov Mathematical Institute; shiryaev@mi.ras.ru INRIA- Rocquencourt; agnes.sulem@inria.fr
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationControlled Diffusions and Hamilton-Jacobi Bellman Equations
Controlled Diffusions and Hamilton-Jacobi Bellman Equations Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 2014 Emo Todorov (UW) AMATH/CSE 579, Winter
More informationOptimal exit strategies for investment projects. 7th AMaMeF and Swissquote Conference
Optimal exit strategies for investment projects Simone Scotti Université Paris Diderot Laboratoire de Probabilité et Modèles Aléatories Joint work with : Etienne Chevalier, Université d Evry Vathana Ly
More informationIntroduction to Algorithmic Trading Strategies Lecture 3
Introduction to Algorithmic Trading Strategies Lecture 3 Pairs Trading by Cointegration Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Distance method Cointegration Stationarity
More informationStochastic Calculus. Kevin Sinclair. August 2, 2016
Stochastic Calculus Kevin Sinclair August, 16 1 Background Suppose we have a Brownian motion W. This is a process, and the value of W at a particular time T (which we write W T ) is a normally distributed
More informationMulti-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form
Multi-dimensional Stochastic Singular Control Via Dynkin Game and Dirichlet Form Yipeng Yang * Under the supervision of Dr. Michael Taksar Department of Mathematics University of Missouri-Columbia Oct
More informationSolution for Problem 7.1. We argue by contradiction. If the limit were not infinite, then since τ M (ω) is nondecreasing we would have
362 Problem Hints and Solutions sup g n (ω, t) g(ω, t) sup g(ω, s) g(ω, t) µ n (ω). t T s,t: s t 1/n By the uniform continuity of t g(ω, t) on [, T], one has for each ω that µ n (ω) as n. Two applications
More informationMA8109 Stochastic Processes in Systems Theory Autumn 2013
Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form
More informationOrder book modeling and market making under uncertainty.
Order book modeling and market making under uncertainty. Sidi Mohamed ALY Lund University sidi@maths.lth.se (Joint work with K. Nyström and C. Zhang, Uppsala University) Le Mans, June 29, 2016 1 / 22 Outline
More informationCNH3C3 Persamaan Diferensial Parsial (The art of Modeling PDEs) DR. PUTU HARRY GUNAWAN
CNH3C3 Persamaan Diferensial Parsial (The art of Modeling PDEs) DR. PUTU HARRY GUNAWAN Partial Differential Equations Content 1. Part II: Derivation of PDE in Brownian Motion PART II DERIVATION OF PDE
More informationp 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
More informationBrownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion
Brownian Motion An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Background We have already seen that the limiting behavior of a discrete random walk yields a derivation of
More informationUtility Maximization in Hidden Regime-Switching Markets with Default Risk
Utility Maximization in Hidden Regime-Switching Markets with Default Risk José E. Figueroa-López Department of Mathematics and Statistics Washington University in St. Louis figueroa-lopez@wustl.edu pages.wustl.edu/figueroa
More informationIntroduction Optimality and Asset Pricing
Introduction Optimality and Asset Pricing Andrea Buraschi Imperial College Business School October 2010 The Euler Equation Take an economy where price is given with respect to the numéraire, which is our
More informationResearch Article Existence and Uniqueness Theorem for Stochastic Differential Equations with Self-Exciting Switching
Discrete Dynamics in Nature and Society Volume 211, Article ID 549651, 12 pages doi:1.1155/211/549651 Research Article Existence and Uniqueness Theorem for Stochastic Differential Equations with Self-Exciting
More informationStochastic Calculus February 11, / 33
Martingale Transform M n martingale with respect to F n, n =, 1, 2,... σ n F n (σ M) n = n 1 i= σ i(m i+1 M i ) is a Martingale E[(σ M) n F n 1 ] n 1 = E[ σ i (M i+1 M i ) F n 1 ] i= n 2 = σ i (M i+1 M
More informationA Model of Optimal Portfolio Selection under. Liquidity Risk and Price Impact
A Model of Optimal Portfolio Selection under Liquidity Risk and Price Impact Huyên PHAM Workshop on PDE and Mathematical Finance KTH, Stockholm, August 15, 2005 Laboratoire de Probabilités et Modèles Aléatoires
More informationBernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012
1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.
More informationLecture 22 Girsanov s Theorem
Lecture 22: Girsanov s Theorem of 8 Course: Theory of Probability II Term: Spring 25 Instructor: Gordan Zitkovic Lecture 22 Girsanov s Theorem An example Consider a finite Gaussian random walk X n = n
More informationFrom Random Variables to Random Processes. From Random Variables to Random Processes
Random Processes In probability theory we study spaces (Ω, F, P) where Ω is the space, F are all the sets to which we can measure its probability and P is the probability. Example: Toss a die twice. Ω
More informationMaximum Process Problems in Optimal Control Theory
J. Appl. Math. Stochastic Anal. Vol. 25, No., 25, (77-88) Research Report No. 423, 2, Dept. Theoret. Statist. Aarhus (2 pp) Maximum Process Problems in Optimal Control Theory GORAN PESKIR 3 Given a standard
More informationAn Uncertain Control Model with Application to. Production-Inventory System
An Uncertain Control Model with Application to Production-Inventory System Kai Yao 1, Zhongfeng Qin 2 1 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 2 School of Economics
More informationVast Volatility Matrix Estimation for High Frequency Data
Vast Volatility Matrix Estimation for High Frequency Data Yazhen Wang National Science Foundation Yale Workshop, May 14-17, 2009 Disclaimer: My opinion, not the views of NSF Y. Wang (at NSF) 1 / 36 Outline
More informationDiscretization of SDEs: Euler Methods and Beyond
Discretization of SDEs: Euler Methods and Beyond 09-26-2006 / PRisMa 2006 Workshop Outline Introduction 1 Introduction Motivation Stochastic Differential Equations 2 The Time Discretization of SDEs Monte-Carlo
More informationIntroduction to Algorithmic Trading Strategies Lecture 10
Introduction to Algorithmic Trading Strategies Lecture 10 Risk Management Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Value at Risk (VaR) Extreme Value Theory (EVT) References
More informationTrading and Investment As A Science. Technical Analysis: A Scientific Perspective. Haksun Li
Trading and Investment As A Science Technical Analysis: A Scientific Perspective Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Speaker Profile Dr. Haksun Li CEO, Numerical Method Inc.
More informationEconomics 2010c: Lectures 9-10 Bellman Equation in Continuous Time
Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time David Laibson 9/30/2014 Outline Lectures 9-10: 9.1 Continuous-time Bellman Equation 9.2 Application: Merton s Problem 9.3 Application:
More informationStochastic Integration and Stochastic Differential Equations: a gentle introduction
Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process
More informationMarch 16, Abstract. We study the problem of portfolio optimization under the \drawdown constraint" that the
ON PORTFOLIO OPTIMIZATION UNDER \DRAWDOWN" CONSTRAINTS JAKSA CVITANIC IOANNIS KARATZAS y March 6, 994 Abstract We study the problem of portfolio optimization under the \drawdown constraint" that the wealth
More informationStochastic Differential Equations
CHAPTER 1 Stochastic Differential Equations Consider a stochastic process X t satisfying dx t = bt, X t,w t dt + σt, X t,w t dw t. 1.1 Question. 1 Can we obtain the existence and uniqueness theorem for
More informationSTOCHASTIC PERRON S METHOD AND VERIFICATION WITHOUT SMOOTHNESS USING VISCOSITY COMPARISON: OBSTACLE PROBLEMS AND DYNKIN GAMES
STOCHASTIC PERRON S METHOD AND VERIFICATION WITHOUT SMOOTHNESS USING VISCOSITY COMPARISON: OBSTACLE PROBLEMS AND DYNKIN GAMES ERHAN BAYRAKTAR AND MIHAI SÎRBU Abstract. We adapt the Stochastic Perron s
More informationStochastic Modelling in Climate Science
Stochastic Modelling in Climate Science David Kelly Mathematics Department UNC Chapel Hill dtbkelly@gmail.com November 16, 2013 David Kelly (UNC) Stochastic Climate November 16, 2013 1 / 36 Why use stochastic
More informationThe Mabinogion Sheep Problem
The Mabinogion Sheep Problem Kun Dong Cornell University April 22, 2015 K. Dong (Cornell University) The Mabinogion Sheep Problem April 22, 2015 1 / 18 Introduction (Williams 1991) we are given a herd
More informationFE 5204 Stochastic Differential Equations
Instructor: Jim Zhu e-mail:zhu@wmich.edu http://homepages.wmich.edu/ zhu/ January 20, 2009 Preliminaries for dealing with continuous random processes. Brownian motions. Our main reference for this lecture
More informationOrder book resilience, price manipulation, and the positive portfolio problem
Order book resilience, price manipulation, and the positive portfolio problem Alexander Schied Mannheim University Workshop on New Directions in Financial Mathematics Institute for Pure and Applied Mathematics,
More informationRECURSION EQUATION FOR
Math 46 Lecture 8 Infinite Horizon discounted reward problem From the last lecture: The value function of policy u for the infinite horizon problem with discount factor a and initial state i is W i, u
More informationStochastic optimal control with rough paths
Stochastic optimal control with rough paths Paul Gassiat TU Berlin Stochastic processes and their statistics in Finance, Okinawa, October 28, 2013 Joint work with Joscha Diehl and Peter Friz Introduction
More informationLecture 21 Representations of Martingales
Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let
More informationInformation and Credit Risk
Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information
More informationBrownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539
Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory
More informationBernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010
1 Stochastic Calculus Notes Abril 13 th, 1 As we have seen in previous lessons, the stochastic integral with respect to the Brownian motion shows a behavior different from the classical Riemann-Stieltjes
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationMortality Surface by Means of Continuous Time Cohort Models
Mortality Surface by Means of Continuous Time Cohort Models Petar Jevtić, Elisa Luciano and Elena Vigna Longevity Eight 2012, Waterloo, Canada, 7-8 September 2012 Outline 1 Introduction Model construction
More informationarxiv: v1 [math.pr] 24 Sep 2018
A short note on Anticipative portfolio optimization B. D Auria a,b,1,, J.-A. Salmerón a,1 a Dpto. Estadística, Universidad Carlos III de Madrid. Avda. de la Universidad 3, 8911, Leganés (Madrid Spain b
More informationKey words. Ambiguous correlation, G-Brownian motion, Hamilton Jacobi Bellman Isaacs equation, Stochastic volatility
PORTFOLIO OPTIMIZATION WITH AMBIGUOUS CORRELATION AND STOCHASTIC VOLATILITIES JEAN-PIERRE FOUQUE, CHI SENG PUN, AND HOI YING WONG Abstract. In a continuous-time economy, we investigate the asset allocation
More informationStochastic Calculus Made Easy
Stochastic Calculus Made Easy Most of us know how standard Calculus works. We know how to differentiate, how to integrate etc. But stochastic calculus is a totally different beast to tackle; we are trying
More informationSolution of Stochastic Optimal Control Problems and Financial Applications
Journal of Mathematical Extension Vol. 11, No. 4, (2017), 27-44 ISSN: 1735-8299 URL: http://www.ijmex.com Solution of Stochastic Optimal Control Problems and Financial Applications 2 Mat B. Kafash 1 Faculty
More informationPartial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula
Partial Differential Equations with Applications to Finance Seminar 1: Proving and applying Dynkin s formula Group 4: Bertan Yilmaz, Richard Oti-Aboagye and Di Liu May, 15 Chapter 1 Proving Dynkin s formula
More informationStochastic Volatility and Correction to the Heat Equation
Stochastic Volatility and Correction to the Heat Equation Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar Abstract. From a probabilist s point of view the Twentieth Century has been a century
More informationAffine Processes. Econometric specifications. Eduardo Rossi. University of Pavia. March 17, 2009
Affine Processes Econometric specifications Eduardo Rossi University of Pavia March 17, 2009 Eduardo Rossi (University of Pavia) Affine Processes March 17, 2009 1 / 40 Outline 1 Affine Processes 2 Affine
More informationHigh-dimensional Problems in Finance and Economics. Thomas M. Mertens
High-dimensional Problems in Finance and Economics Thomas M. Mertens NYU Stern Risk Economics Lab April 17, 2012 1 / 78 Motivation Many problems in finance and economics are high dimensional. Dynamic Optimization:
More informationB8.3 Mathematical Models for Financial Derivatives. Hilary Term Solution Sheet 2
B8.3 Mathematical Models for Financial Derivatives Hilary Term 18 Solution Sheet In the following W t ) t denotes a standard Brownian motion and t > denotes time. A partition π of the interval, t is a
More informationRisk-Minimality and Orthogonality of Martingales
Risk-Minimality and Orthogonality of Martingales Martin Schweizer Universität Bonn Institut für Angewandte Mathematik Wegelerstraße 6 D 53 Bonn 1 (Stochastics and Stochastics Reports 3 (199, 123 131 2
More informationNonlinear representation, backward SDEs, and application to the Principal-Agent problem
Nonlinear representation, backward SDEs, and application to the Principal-Agent problem Ecole Polytechnique, France April 4, 218 Outline The Principal-Agent problem Formulation 1 The Principal-Agent problem
More informationPathwise volatility in a long-memory pricing model: estimation and asymptotic behavior
Pathwise volatility in a long-memory pricing model: estimation and asymptotic behavior Ehsan Azmoodeh University of Vaasa Finland 7th General AMaMeF and Swissquote Conference September 7 1, 215 Outline
More informationLecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University
Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationStochastic Calculus and Black-Scholes Theory MTH772P Exercises Sheet 1
Stochastic Calculus and Black-Scholes Theory MTH772P Exercises Sheet. For ξ, ξ 2, i.i.d. with P(ξ i = ± = /2 define the discrete-time random walk W =, W n = ξ +... + ξ n. (i Formulate and prove the property
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More information(a) Write down the Hamilton-Jacobi-Bellman (HJB) Equation in the dynamic programming
1. Government Purchases and Endogenous Growth Consider the following endogenous growth model with government purchases (G) in continuous time. Government purchases enhance production, and the production
More informationNotes on Random Variables, Expectations, Probability Densities, and Martingales
Eco 315.2 Spring 2006 C.Sims Notes on Random Variables, Expectations, Probability Densities, and Martingales Includes Exercise Due Tuesday, April 4. For many or most of you, parts of these notes will be
More informationLecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.
Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal
More informationWorst Case Portfolio Optimization and HJB-Systems
Worst Case Portfolio Optimization and HJB-Systems Ralf Korn and Mogens Steffensen Abstract We formulate a portfolio optimization problem as a game where the investor chooses a portfolio and his opponent,
More informationRobust Markowitz portfolio selection. ambiguous covariance matrix
under ambiguous covariance matrix University Paris Diderot, LPMA Sorbonne Paris Cité Based on joint work with A. Ismail, Natixis MFO March 2, 2017 Outline Introduction 1 Introduction 2 3 and Sharpe ratio
More informationContinuous Time Finance
Continuous Time Finance Lisbon 2013 Tomas Björk Stockholm School of Economics Tomas Björk, 2013 Contents Stochastic Calculus (Ch 4-5). Black-Scholes (Ch 6-7. Completeness and hedging (Ch 8-9. The martingale
More informationFiltrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition
Filtrations, Markov Processes and Martingales Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition David pplebaum Probability and Statistics Department,
More informationPRIMAL-DUAL METHODS FOR THE COMPUTATION OF TRADING REGIONS UNDER PROPORTIONAL TRANSACTION COSTS
PRIMAL-DUAL METHODS FOR THE COMPUTATION OF TRADING REGIONS UNDER PROPORTIONAL TRANSACTION COSTS ROLAND HERZOG, KARL KUNISCH *, AND JÖRN SASS Abstract. Portfolio optimization problems on a finite time horizon
More information1. Stochastic Process
HETERGENEITY IN QUANTITATIVE MACROECONOMICS @ TSE OCTOBER 17, 216 STOCHASTIC CALCULUS BASICS SANG YOON (TIM) LEE Very simple notes (need to add references). It is NOT meant to be a substitute for a real
More informationStochastic Modelling Unit 1: Markov chain models
Stochastic Modelling Unit 1: Markov chain models Russell Gerrard and Douglas Wright Cass Business School, City University, London June 2004 Contents of Unit 1 1 Stochastic Processes 2 Markov Chains 3 Poisson
More informationLAN property for ergodic jump-diffusion processes with discrete observations
LAN property for ergodic jump-diffusion processes with discrete observations Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Arturo Kohatsu-Higa (Ritsumeikan University, Japan) &
More informationHamilton-Jacobi-Bellman Equation of an Optimal Consumption Problem
Hamilton-Jacobi-Bellman Equation of an Optimal Consumption Problem Shuenn-Jyi Sheu Institute of Mathematics, Academia Sinica WSAF, CityU HK June 29-July 3, 2009 1. Introduction X c,π t is the wealth with
More informationModeling Ultra-High-Frequency Multivariate Financial Data by Monte Carlo Simulation Methods
Outline Modeling Ultra-High-Frequency Multivariate Financial Data by Monte Carlo Simulation Methods Ph.D. Student: Supervisor: Marco Minozzo Dipartimento di Scienze Economiche Università degli Studi di
More informationON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME
ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME SAUL D. JACKA AND ALEKSANDAR MIJATOVIĆ Abstract. We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems
More informationNon-linear Supervised High Frequency Trading Strategies with Applications in US Equity Markets
Non-linear Supervised High Frequency Trading Strategies with Applications in US Equity Markets Nan Zhou, Wen Cheng, Ph.D. Associate, Quantitative Research, J.P. Morgan nan.zhou@jpmorgan.com The 4th Annual
More informationSystemic Risk and the Mathematics of Falling Dominoes
Systemic Risk and the Mathematics of Falling Dominoes Reimer Kühn Disordered Systems Group Department of Mathematics, King s College London http://www.mth.kcl.ac.uk/ kuehn/riskmodeling.html Teachers Conference,
More informationDynamic Risk Measures and Nonlinear Expectations with Markov Chain noise
Dynamic Risk Measures and Nonlinear Expectations with Markov Chain noise Robert J. Elliott 1 Samuel N. Cohen 2 1 Department of Commerce, University of South Australia 2 Mathematical Insitute, University
More informationStochastic Calculus for Finance II - some Solutions to Chapter VII
Stochastic Calculus for Finance II - some Solutions to Chapter VII Matthias hul Last Update: June 9, 25 Exercise 7 Black-Scholes-Merton Equation for the up-and-out Call) i) We have ii) We first compute
More informationBackward martingale representation and endogenous completeness in finance
Backward martingale representation and endogenous completeness in finance Dmitry Kramkov (with Silviu Predoiu) Carnegie Mellon University 1 / 19 Bibliography Robert M. Anderson and Roberto C. Raimondo.
More informationLiquidity risk and optimal dividend/investment strategies
Liquidity risk and optimal dividend/investment strategies Vathana LY VATH Laboratoire de Mathématiques et Modélisation d Evry ENSIIE and Université d Evry Joint work with E. Chevalier and M. Gaigi ICASQF,
More informationLiquidation in Limit Order Books. LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Liquidation in Limit Order Books with Controlled Intensity Erhan and Mike Ludkovski University of Michigan and UCSB
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 3. Calculaus in Deterministic and Stochastic Environments Steve Yang Stevens Institute of Technology 01/31/2012 Outline 1 Modeling Random Behavior
More informationLecture 5. If we interpret the index n 0 as time, then a Markov chain simply requires that the future depends only on the present and not on the past.
1 Markov chain: definition Lecture 5 Definition 1.1 Markov chain] A sequence of random variables (X n ) n 0 taking values in a measurable state space (S, S) is called a (discrete time) Markov chain, if
More informationStochastic Shortest Path Problems
Chapter 8 Stochastic Shortest Path Problems 1 In this chapter, we study a stochastic version of the shortest path problem of chapter 2, where only probabilities of transitions along different arcs can
More informationShape of the return probability density function and extreme value statistics
Shape of the return probability density function and extreme value statistics 13/09/03 Int. Workshop on Risk and Regulation, Budapest Overview I aim to elucidate a relation between one field of research
More informationEmpirical properties of large covariance matrices in finance
Empirical properties of large covariance matrices in finance Ex: RiskMetrics Group, Geneva Since 2010: Swissquote, Gland December 2009 Covariance and large random matrices Many problems in finance require
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationResearch Article A Necessary Characteristic Equation of Diffusion Processes Having Gaussian Marginals
Abstract and Applied Analysis Volume 01, Article ID 598590, 9 pages doi:10.1155/01/598590 Research Article A Necessary Characteristic Equation of Diffusion Processes Having Gaussian Marginals Syeda Rabab
More information(6, 4) Is there arbitrage in this market? If so, find all arbitrages. If not, find all pricing kernels.
Advanced Financial Models Example sheet - Michaelmas 208 Michael Tehranchi Problem. Consider a two-asset model with prices given by (P, P 2 ) (3, 9) /4 (4, 6) (6, 8) /4 /2 (6, 4) Is there arbitrage in
More informationThe Azéma-Yor Embedding in Non-Singular Diffusions
Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let
More informationLectures 6 7 : Marchenko-Pastur Law
Fall 2009 MATH 833 Random Matrices B. Valkó Lectures 6 7 : Marchenko-Pastur Law Notes prepared by: A. Ganguly We will now turn our attention to rectangular matrices. Let X = (X 1, X 2,..., X n ) R p n
More informationEfficient Solvers for Stochastic Finite Element Saddle Point Problems
Efficient Solvers for Stochastic Finite Element Saddle Point Problems Catherine E. Powell c.powell@manchester.ac.uk School of Mathematics University of Manchester, UK Efficient Solvers for Stochastic Finite
More informationRough paths methods 1: Introduction
Rough paths methods 1: Introduction Samy Tindel Purdue University University of Aarhus 216 Samy T. (Purdue) Rough Paths 1 Aarhus 216 1 / 16 Outline 1 Motivations for rough paths techniques 2 Summary of
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 7 9/25/2013
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 7 9/5/013 The Reflection Principle. The Distribution of the Maximum. Brownian motion with drift Content. 1. Quick intro to stopping times.
More informationEEL 5544 Noise in Linear Systems Lecture 30. X (s) = E [ e sx] f X (x)e sx dx. Moments can be found from the Laplace transform as
L30-1 EEL 5544 Noise in Linear Systems Lecture 30 OTHER TRANSFORMS For a continuous, nonnegative RV X, the Laplace transform of X is X (s) = E [ e sx] = 0 f X (x)e sx dx. For a nonnegative RV, the Laplace
More informationCHAPTER 2 THE MAXIMUM PRINCIPLE: CONTINUOUS TIME. Chapter2 p. 1/67
CHAPTER 2 THE MAXIMUM PRINCIPLE: CONTINUOUS TIME Chapter2 p. 1/67 THE MAXIMUM PRINCIPLE: CONTINUOUS TIME Main Purpose: Introduce the maximum principle as a necessary condition to be satisfied by any optimal
More information