Math 526: Brownian Motion Notes

Size: px
Start display at page:

Download "Math 526: Brownian Motion Notes"

Transcription

1 Math 526: Brownian Motion Notes Definition. Mike Ludkovski, 27, all rights reserved. A stochastic process (X t ) is called Brownian motion if:. The map t X t (ω) is continuous for every ω. 2. (X t X t ), (X t2 X t ),..., (X tn X tn ) are independent for any collection of times t t... t n. 3. The distribution of X t X s depends only on (t s). Property is called continuity of sample paths. Property 2 is called independent increments. Property 3 is called stationarity of increments. Here are some immediate conclusions and comments: Property 2 and 3 imply that (X t ) is time-homogeneous (one may shift the time-axis) and space-homogeneous (since distribution of increments does not depend on current X s ). Property 2 and 3 also imply that (X t ) is Markov, since the future increments are conditionally independent of the past given the present. Properties 2 and 3 are also satisfied by the Poisson process. process certainly does not have continuous sample paths. However the Poisson It is not at all clear that such a process (X t ) exists, since the requirement of continuous paths is very stringent. The underlying probability space here is Ω {continuous functions on [, )}. Brownian Motion has Gaussian Marginals. We have the following Theorem. If (X t ) is a Brownian motion, then there exist constants µ, σ 2 such that X t X s N ((t s)/µ, (t s)σ 2 ). Proof. Without loss of generality take s and pick some n Z and write X t X (X t/n X ) + (X 2t/n X t/n ) (X t X (n )t/n ). Each term on the right hand side is independent and identically distributed. Letting n, by the central limit theorem, the right hand side must converge to some normally distributed random variable (the continuity of (X t ) makes sure that one can apply the CLT, as the increments must become smaller and smaller as n increases).

2 We conclude that X t X N (a(t), b(t)) for some functions a, b. However, since X t+s X (X t X ) + (X t+s X t ) and the two are independent, we have that X t+s X N (a(t), b(t)) + N (a(s), b(s)) d N (a(t) + a(s), b(t) + b(s)). On the other hand, we directly have X t+s X N (a(t + s), b(t + s)). It follows that a(t + s) a(t) + a(s) and similarly for b. But this means that a, b are linear functions, i.e. a(t) µt, b(t) σ 2 t (clearly b must have positive slope), for some µ, σ 2. The parameter µ is called drift of the Brownian motion and σ is called the volatility. To summarize, if (X t ) is a BM then the marginal distribution is Wiener Process X t X N (µt, σ 2 t). The special case µ, σ 2, X is called the Wiener process. We write (W t ) in that case. Here are some computations for the Wiener process: E[W t ]. V ar(w t ) t. Cov(W s, W t ) E[W s W t ] E[W s ]E[W t ] E [W s (W s + W t W s )] E[Ws 2 ] + E[W s ]E[W t W s ] V ar(w s ) + s. s < t The last statement is written as Cov(W s, W t ) min(s, t). Here is another important computation: we find the conditional distribution of W s given W t z, s < t: P(W s dx W t z) P(W s dx, W t dz) P(W t dz) P(W s dx, (W t W s ) (dz x)) P(W t dz) Using the Gaussian density and independent increments: ( 2πs e dx) ( ) x2 /(2s) e (z x)2 /(2(t s)) dz 2π(t s) 2πt e z2 /(2t) dz ( exp ( x 2 /s + (z x) 2 /(t s) z 2 /t )) 2 exp( (x s t z)2 /(2s(t s)/t)). 2πs(t s)/t 2πs(t s)/t 2

3 The last line is the density of a Gaussian random variable and we conclude that ( s ) W s W t z N t z, s(t s)/t. This is quite intuitive: E[W s W t z] (s/t)z and V ar(w s W t z) (s/t) 2 (t s) 2. Constructing Brownian Motion. Here is the physicist s construction of BM. This idea was first advanced by Einstein in 96 in connection with molecular/atomic forces. Consider a particle of pollen suspended in liquid. The particle is rather large but light and is subject to being hit at random by the liquid molecules (which are small but very many). Let us focus on the movement of the particle along the x axis, and record its position at time t as Xt ɛ. Each time a particle is hit on the left, it moves to the right by a distance ɛ, and each time it is hit on the right, it moves left by an ɛ. Let L t, M t be the number of hits by time t on the left and right respectively. Then Xt ɛ ɛl t ɛm t (for simplicity we took X ). It is natural to assume that L t, M t are two independent stationary Markov processes with iid increments. That is, L t, M t are two Poisson processes with same intensity λ ɛ. We immediately get E[Xt ɛ ] and V ar(x ɛ t ) ɛ 2 E[(L t M t ) 2 ] ɛ 2 (2(λ ɛ t + (λ ɛ t) 2 ) 2E[L t M t ] 2ɛ 2 λ ɛ t. So far ɛ was a dummy parameter and of course we are planning on taking ɛ. To obtain an interesting (X t ) in the limit, we should make sure that the variance above converges to some non-zero finite limit. This would happen if we take λ ɛ C/(2ɛ 2 ). Thus, as the displacement of the pollen particle from each hit is diminished we naturally increase the frequency of hits (and the correct scaling is quadratic). To figure out what is the limiting (X t ) we compute the mgf (recall the mgf of P oisson(λ) is e λ(es ) ): E[e sxɛ t ] exp (λɛ t(e sɛ )) exp ( λ ɛ t(e sɛ ) ) Since by L Hopital rule we have e sɛ + e sɛ 2 lim ɛ ɛ 2 e tc/(2ɛ2 )(e sɛ +e sɛ 2). lim ɛ se sɛ se sɛ 2ɛ lim ɛ s 2 e sɛ + s 2 e sɛ 2 s 2 lim E[e sxɛ t ] e tcs 2 /2. ɛ The latter is the moment generating function of a N (, tc) random variable. Since for every ɛ >, (X ɛ t ) had stationary and independent increments (because L t, M t do), so does the 3

4 limiting (X t ). Moreover, since the displacement ɛ, (X t ) should be continuous. Putting it all together we conclude that (X t ) is a Brownian motion with zero drift and volatility C. If C then we get the Wiener process. The name Brownian motion comes from the botanist Robert Brown who first observed the irregular motion of pollen particles suspended in water in 828. As you can see it took 8 years until Einstein could provide a good mathematical model for these observations. Einstein s model was a crucial step in convincing physicists that molecules really exist and move around randomly when in liquid or gaseous form. Since then, this model has indicated to physicists that any random movement in physics should be based on Brownian motion, at least on the atomic/molecular level where external forces, such as gravity, are minimal. Remark: note that (X ɛ t ) above is a scaled simple birth-and-death process. Recognizing Brownian Motion. We first observe that if (X t ) is a Brownian motion with µ then (X t ) is a martingale with respect to itself. Indeed, let F t {X u : u t} be the history of X up to time t. Then for s > t E[X s F t ] E[(X s X t ) + X t F t ] E[X s X t ] + X t X t, since the expected value of any increment is zero. More generally, Y t X t µt is a martingale for any Brownian motion with drift µ. The above property has far-reaching consequences. In fact, it turns out that the Wiener process is the canonical continuous martingale. This fact forms the basis for stochastic calculus and underlines the importance of understanding the behavior of BM. One basic application is the following Levy characterization of Wiener process: Theorem 2. Suppose that (Z t ) is a continuous-time stochastic process such that: The paths of Z are continuous. (Z t ) is a martingale with respect to its own history. V ar(z t Z s ) (t s) for any t > s >. Then (Z t Z ) is a Wiener process. As a corollary, if (Z t ) is a continuous process such that Z t µt is a martingale and V ar(z t ) σ 2 t then (Z t ) is a Brownian motion with drift µ and volatility σ. From Random Walk to Brownian Motion. Here is another construction of Brownian motion. Let (S δ t ) be a simple symmetric random walk that makes steps of size ±δ at times t /n, 2/n,.... We know that S( δ t) is a time- and space-stationary discrete-time martingale. In particular, E[S δ t ], and V ar(s δ t ) δ 2 nt. To pass to the limit δ we therefore select n /δ 2. As δ, we obtain a continuoustime, continuous-space stochastic process that has stationary and independent increments, 4

5 continuous-paths and the martingale property. Moreover, V ar(st ) t, so (St ) must be the Wiener process. One can also pick other random walks to make the convergence faster. For example a good choice is to take the step distribution as X n +δ with prob. /6 with prob. 2/3 δ with prob. /6 and n 3/δ 2. The corresponding random walk S n X X n can go up, go down, or stay at the same level. It is often helpful to think of Brownian motion as the limit of symmetric random walks. As we will see, most of the general results about random walks (including recurrence, reflection principle, ballot theorem, ruin probabilities) carry over nicely to Brownian motion. Hitting Time Distribution. Let (W t ) be the Wiener process and T b (ω) min{t : W t (ω) b} be the first time (W t ) hits level b. We are interested in computing the distribution of T b. Since the behavior of the Wiener process is symmetric about the x-axis, we take b >. Define Ŵt W Tb +t W Tb to be the future of W after time T b. Note that T b is random, so it is not clear how Ŵ looks like. However, it is possible to verify all the conditions of Levy s Theorem and conclude that Ŵ is again a Wiener process. This is because (W t) is a strong Markov process, meaning that the Markov property of (W t ) continues to hold when applied at random (stopping!) times, such as T b. Thus, the future of (W t ) after T b is independent of its history up to T b. From this fact we obtain P(W t > b T b < t) P(Ŵt T b > ) /2, since P(Ŵs > ) /2 for any time s by symmetry. However, the LHS above can also be written as P(W t > b T b < t) P(W t > b, T b < t) P(T b < t) P(W t > b) P(T b < t), since the only way that W is above b at t is if the hitting time of b has already occurred. Comparing we conclude that P(T b < t) 2P(W t > b) 2 b/ t 2π e x2 /2 dx. An immediate corollary is that if we take t then the RHS goes to 2 2π e x2 /2 dx, which means that P(T b < ), irrespective of value of b. Therefore, the Wiener process hits any level b with probability. 5

6 Furthermore, differentiating the above with respect to t we find that the density of T b is given by: /(2t) P(T b dt) b e b2, b R. () 2πt 3 The latter belongs to the family of stable (or inverse Gamma) distributions (we say that T b has stable(/2)-distribution). In fact, here is a curious connection: let Z N (, ) and Y b 2 /Z 2. Then T b Y. Indeed, Differentiating, P(Y t) P(b 2 /Z 2 t) P( Z b / t) 2P(Z b / t). P(Y dt) 2 b 2t 3/2 2π e (b/ t) 2 /2 which matches with (). Using () one also shows that E[T b ] + for any b. So the conclusion is: No matter how large b is, P(T b < ). No matter how small b is, E[T b ]. Maximum Process. Let M t max s t W s be the running maximum of Wiener process W. Then (M t ) is a non-decreasing process that goes to + since we know that (W t ) will eventually hit any positive level. Moreover, P(M t > x) P(T x < t) 2P(W t > x) P( t Z > x) based on properties of normal r.v. s. We conclude that the marginal distribution of M t is the same as that of an absolute value of a Gaussian random variable. In particular, E[M t ] 2t/π, V ar(m t ) ( 2/π)t. Note that while (M t ) is increasing, it does so in an imperceptible creep: for any fixed t, (M t ) is flat in the neighborhood of t with probability. If you draw a typical path of (M t ) on an interval of say [, ], the length of flat areas will add up to (even though M > M with probability ). For those of you who know some analysis, the set of points where (M t ) icnreases is a Cantor set of measure zero. By symmetry the minimum m t min s t W s has the same distribution as M t. Since both m t, M t + it must be that (W t ) recrosses zero an infinite number of times. Thus, the zero level (and by space-homogeneity, any level) is recurrent for (W t ). The next computation shows that much more is true: 6

7 Probability of a Zero. We are interested in computing the probability that the Wiener process W hits zero on the time interval [s, t]. Observe that if W s a, then the probability that W hits zero on [s, t] is just t s a P(T a < (t s)) /(2y) 2πy 3 e a2 dy. However, W s N (, s), so putting it together (ie conditioning on W s ) we have: P(T a < (t s)) e a2 /(2s) da 2πs t s t s 2 π t s t s a /(2y) 2πy 3 e a2 dy e a2 /(2s) da 2πs ( ) a e a2 /(2y) a 2 /(2s) da dy 2πs /2 y 3/2 ( ) a e a2 (y+s)/(2sy) da dy πs /2 y 3/2 sy πs /2 y 3/2 s + y dy s t s dy π (s + y) y (t s)/s dx by substitution y sx 2 + x 2 2 π arctan( (t s)/s) 2 π arccos s t. Corollary: if s then probability of a zero on [, t] is 2 arccos for any t. Therefore, π W returns to zero immediately after t. After some thinking and using the strong Markov property of W, it follows that W has infinitely many zeros on any interval [, t]. This is quite counterintuitive given that E[T ɛ ] + even for very small ɛ. Thus W really oscillates wildly around zero. This is one indication that W is nowhere differentiable: it oscillates around every point and the scale of oscillations over time interval of length h is on the order of O( W h), which means that lim t+h W t h cannot exist. h Time of Last Zero. Let L t be the time of last zero of W before time t. Let D t be the time of first zero of W after time t. We observe that {L t < s} {D s > t} {W has no zeros on [s, t]}. From the last section, we therefore find that meaning that the density of L t is: P(L t < s) 2 π arccos s t 2 π arcsin s t, P(L t ds) π s(t s), s t. 7

8 These facts are known as the Arc-sine Law and show that the last zero of W is likely to be either close to zero or to t. Here is another derivation of the same result using the Gaussian-Gamma connection. Note that conditional on W t a, the distribution of D t is D t t + T a t + a2 γ where γ Gamma(/2, /2). Therefore, P(D t (t + u)) P(D t t u W t a)f Wt (a) da P(a 2 /γ u)f Wt (a) da P(Wt 2 /γ u) P(tγ /γ u) ( ) tγ P u, γ where γ Gamma(/2, /2) is another Gamma rv, independent of γ (since T a is indep of W ). So distribution of D t t is same as tγ γ ie distribution of D t is equal to that of t( + γ /γ ) t/ γ γ +γ. The latter expression β γ γ +γ is known to have a Beta(/2, /2) distribution with density f β (u), u. So P(D t < (t + u)) P(t/β < π u( u) (t + u)) P(β > t/(t + u)). However, as mentioned before, s/t P(L t < s) P(D s > s + (t s)) P(β < s/t) π u( u) du 2 s π arcsin t, using the substitution x u. In particular note that the distribution of L is Beta(/2, /2). Planar BM Let (W, W 2 ) be two independent Wiener processes, and take X t X + Wt, Y t Y + Wt 2. Then viewing (X t, Y t ) as the (x, y)-coordinates of a moving particle, we obtain what is called a planar Brownian motion, starting at the initial location (X, Y ). Here is one curious computation: Suppose (X, Y ) (, b) so the particle starts somewhere on the positive y-axis. Let T be the first time that Y t the particle hits the x-axis. We are interested in the distribution of X T. Recall that from the remark after (), T b 2 /Z 2. It follows that X T T Z where (Z, Z) are two independent standard normal random variables. In other words, X T b Z Z which we recall has a Cauchy distribution, namely P(X T dx) b (b 2 + x 2 )π. Like the one-dimensional case, two-dimensional Brownian motion is recurrent and will hit every point in the plane with probability. 8

1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM)

1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM) Copyright c 2013 by Karl Sigman 1 Simulating Brownian motion (BM) and geometric Brownian motion (GBM) For an introduction to how one can construct BM, see the Appendix at the end of these notes A stochastic

More information

1 Geometric Brownian motion

1 Geometric Brownian motion Copyright c 006 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

More information

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

More information

Stochastic Processes and Advanced Mathematical Finance. The Definition of Brownian Motion and the Wiener Process

Stochastic Processes and Advanced Mathematical Finance. The Definition of Brownian Motion and the Wiener Process Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebraska-Lincoln Lincoln, NE 68588-0130 http://www.math.unl.edu Voice: 402-472-3731 Fax: 402-472-8466 Stochastic Processes and Advanced

More information

IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS

IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)

More information

THE CENTRAL LIMIT THEOREM TORONTO

THE CENTRAL LIMIT THEOREM TORONTO THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................

More information

The Exponential Distribution

The Exponential Distribution 21 The Exponential Distribution From Discrete-Time to Continuous-Time: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

Transformations and Expectations of random variables

Transformations and Expectations of random variables Transformations and Epectations of random variables X F X (): a random variable X distributed with CDF F X. Any function Y = g(x) is also a random variable. If both X, and Y are continuous random variables,

More information

The Black-Scholes-Merton Approach to Pricing Options

The Black-Scholes-Merton Approach to Pricing Options he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining

More information

A spot price model feasible for electricity forward pricing Part II

A spot price model feasible for electricity forward pricing Part II A spot price model feasible for electricity forward pricing Part II Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Wolfgang Pauli Institute, Wien January 17-18

More information

Mathematical Finance

Mathematical Finance Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European

More information

Probability and Statistics

Probability and Statistics CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

More information

The Black-Scholes Formula

The Black-Scholes Formula FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

More information

Definition 6.1.1. A r.v. X has a normal distribution with mean µ and variance σ 2, where µ R, and σ > 0, if its density is f(x) = 1. 2σ 2.

Definition 6.1.1. A r.v. X has a normal distribution with mean µ and variance σ 2, where µ R, and σ > 0, if its density is f(x) = 1. 2σ 2. Chapter 6 Brownian Motion 6. Normal Distribution Definition 6... A r.v. X has a normal distribution with mean µ and variance σ, where µ R, and σ > 0, if its density is fx = πσ e x µ σ. The previous definition

More information

Stochastic Processes LECTURE 5

Stochastic Processes LECTURE 5 128 LECTURE 5 Stochastic Processes We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that

More information

Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

More information

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS

COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS COMPLETE MARKETS DO NOT ALLOW FREE CASH FLOW STREAMS NICOLE BÄUERLE AND STEFANIE GRETHER Abstract. In this short note we prove a conjecture posed in Cui et al. 2012): Dynamic mean-variance problems in

More information

Marshall-Olkin distributions and portfolio credit risk

Marshall-Olkin distributions and portfolio credit risk Marshall-Olkin distributions and portfolio credit risk Moderne Finanzmathematik und ihre Anwendungen für Banken und Versicherungen, Fraunhofer ITWM, Kaiserslautern, in Kooperation mit der TU München und

More information

Goal Problems in Gambling and Game Theory. Bill Sudderth. School of Statistics University of Minnesota

Goal Problems in Gambling and Game Theory. Bill Sudderth. School of Statistics University of Minnesota Goal Problems in Gambling and Game Theory Bill Sudderth School of Statistics University of Minnesota 1 Three problems Maximizing the probability of reaching a goal. Maximizing the probability of reaching

More information

Lecture 13: Martingales

Lecture 13: Martingales Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

More information

Exponential Distribution

Exponential Distribution Exponential Distribution Definition: Exponential distribution with parameter λ: { λe λx x 0 f(x) = 0 x < 0 The cdf: F(x) = x Mean E(X) = 1/λ. f(x)dx = Moment generating function: φ(t) = E[e tx ] = { 1

More information

e.g. arrival of a customer to a service station or breakdown of a component in some system.

e.g. arrival of a customer to a service station or breakdown of a component in some system. Poisson process Events occur at random instants of time at an average rate of λ events per second. e.g. arrival of a customer to a service station or breakdown of a component in some system. Let N(t) be

More information

LECTURE 15: AMERICAN OPTIONS

LECTURE 15: AMERICAN OPTIONS LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These

More information

Binomial lattice model for stock prices

Binomial lattice model for stock prices Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }

More information

Monte Carlo Methods in Finance

Monte Carlo Methods in Finance Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction

More information

Lectures 5-6: Taylor Series

Lectures 5-6: Taylor Series Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

More information

Lectures on Stochastic Processes. William G. Faris

Lectures on Stochastic Processes. William G. Faris Lectures on Stochastic Processes William G. Faris November 8, 2001 2 Contents 1 Random walk 7 1.1 Symmetric simple random walk................... 7 1.2 Simple random walk......................... 9 1.3

More information

Wald s Identity. by Jeffery Hein. Dartmouth College, Math 100

Wald s Identity. by Jeffery Hein. Dartmouth College, Math 100 Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,

More information

Maximum likelihood estimation of mean reverting processes

Maximum likelihood estimation of mean reverting processes Maximum likelihood estimation of mean reverting processes José Carlos García Franco Onward, Inc. jcpollo@onwardinc.com Abstract Mean reverting processes are frequently used models in real options. For

More information

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist

More information

CHAPTER IV - BROWNIAN MOTION

CHAPTER IV - BROWNIAN MOTION CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time

More information

Pricing Barrier Options under Local Volatility

Pricing Barrier Options under Local Volatility Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly

More information

WHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION?

WHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION? WHEN DOES A RANDOMLY WEIGHTED SELF NORMALIZED SUM CONVERGE IN DISTRIBUTION? DAVID M MASON 1 Statistics Program, University of Delaware Newark, DE 19717 email: davidm@udeledu JOEL ZINN 2 Department of Mathematics,

More information

A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University

A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses. Michael R. Powers[ 1 ] Temple University and Tsinghua University A Coefficient of Variation for Skewed and Heavy-Tailed Insurance Losses Michael R. Powers[ ] Temple University and Tsinghua University Thomas Y. Powers Yale University [June 2009] Abstract We propose a

More information

LOGNORMAL MODEL FOR STOCK PRICES

LOGNORMAL MODEL FOR STOCK PRICES LOGNORMAL MODEL FOR STOCK PRICES MICHAEL J. SHARPE MATHEMATICS DEPARTMENT, UCSD 1. INTRODUCTION What follows is a simple but important model that will be the basis for a later study of stock prices as

More information

Markovian projection for volatility calibration

Markovian projection for volatility calibration cutting edge. calibration Markovian projection for volatility calibration Vladimir Piterbarg looks at the Markovian projection method, a way of obtaining closed-form approximations of European-style option

More information

Martingale Pricing Applied to Options, Forwards and Futures

Martingale Pricing Applied to Options, Forwards and Futures IEOR E4706: Financial Engineering: Discrete-Time Asset Pricing Fall 2005 c 2005 by Martin Haugh Martingale Pricing Applied to Options, Forwards and Futures We now apply martingale pricing theory to the

More information

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:

correct-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were: Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that

More information

1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let

1. (First passage/hitting times/gambler s ruin problem:) Suppose that X has a discrete state space and let i be a fixed state. Let Copyright c 2009 by Karl Sigman 1 Stopping Times 1.1 Stopping Times: Definition Given a stochastic process X = {X n : n 0}, a random time τ is a discrete random variable on the same probability space as

More information

What is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference

What is Statistics? Lecture 1. Introduction and probability review. Idea of parametric inference 0. 1. Introduction and probability review 1.1. What is Statistics? What is Statistics? Lecture 1. Introduction and probability review There are many definitions: I will use A set of principle and procedures

More information

Asian Option Pricing Formula for Uncertain Financial Market

Asian Option Pricing Formula for Uncertain Financial Market Sun and Chen Journal of Uncertainty Analysis and Applications (215) 3:11 DOI 1.1186/s4467-15-35-7 RESEARCH Open Access Asian Option Pricing Formula for Uncertain Financial Market Jiajun Sun 1 and Xiaowei

More information

Some stability results of parameter identification in a jump diffusion model

Some stability results of parameter identification in a jump diffusion model Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany Abstract In this paper we discuss

More information

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine Aït-Sahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times

More information

1 IEOR 4700: Introduction to stochastic integration

1 IEOR 4700: Introduction to stochastic integration Copyright c 7 by Karl Sigman 1 IEOR 47: Introduction to stochastic integration 1.1 Riemann-Stieltjes integration Recall from calculus how the Riemann integral b a h(t)dt is defined for a continuous function

More information

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for

More information

Metric Spaces. Chapter 7. 7.1. Metrics

Metric Spaces. Chapter 7. 7.1. Metrics Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

More information

Black-Scholes Option Pricing Model

Black-Scholes Option Pricing Model Black-Scholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,

More information

Diusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute

Diusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 Black-Scholes 5 Equity linked life insurance 6 Merton

More information

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS

LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS LECTURES ON REAL OPTIONS: PART II TECHNICAL ANALYSIS Robert S. Pindyck Massachusetts Institute of Technology Cambridge, MA 02142 Robert Pindyck (MIT) LECTURES ON REAL OPTIONS PART II August, 2008 1 / 50

More information

IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem

IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem IEOR 6711: Stochastic Models I Fall 2012, Professor Whitt, Tuesday, September 11 Normal Approximations and the Central Limit Theorem Time on my hands: Coin tosses. Problem Formulation: Suppose that I have

More information

MS3b/MScMCF Lévy Processes and Finance. Matthias Winkel Department of Statistics University of Oxford

MS3b/MScMCF Lévy Processes and Finance. Matthias Winkel Department of Statistics University of Oxford MS3b/MScMCF Lévy Processes and Finance Matthias Winkel Department of Statistics University of Oxford HT 21 Prerequisites MS3b (and MScMCF) Lévy Processes and Finance Matthias Winkel 16 lectures HT 21

More information

The integrating factor method (Sect. 2.1).

The integrating factor method (Sect. 2.1). The integrating factor method (Sect. 2.1). Overview of differential equations. Linear Ordinary Differential Equations. The integrating factor method. Constant coefficients. The Initial Value Problem. Variable

More information

Notes on Elastic and Inelastic Collisions

Notes on Elastic and Inelastic Collisions Notes on Elastic and Inelastic Collisions In any collision of 2 bodies, their net momentus conserved. That is, the net momentum vector of the bodies just after the collision is the same as it was just

More information

Grey Brownian motion and local times

Grey Brownian motion and local times Grey Brownian motion and local times José Luís da Silva 1,2 (Joint work with: M. Erraoui 3 ) 2 CCM - Centro de Ciências Matemáticas, University of Madeira, Portugal 3 University Cadi Ayyad, Faculty of

More information

Notes on Black-Scholes Option Pricing Formula

Notes on Black-Scholes Option Pricing Formula . Notes on Black-Scholes Option Pricing Formula by De-Xing Guan March 2006 These notes are a brief introduction to the Black-Scholes formula, which prices the European call options. The essential reading

More information

Some ergodic theorems of linear systems of interacting diffusions

Some ergodic theorems of linear systems of interacting diffusions Some ergodic theorems of linear systems of interacting diffusions 4], Â_ ŒÆ êæ ÆÆ Nov, 2009, uà ŒÆ liuyong@math.pku.edu.cn yangfx@math.pku.edu.cn 1 1 30 1 The ergodic theory of interacting systems has

More information

Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 126 Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

More information

General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions

More information

LECTURE 9: A MODEL FOR FOREIGN EXCHANGE

LECTURE 9: A MODEL FOR FOREIGN EXCHANGE LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling

More information

Asymptotics of discounted aggregate claims for renewal risk model with risky investment

Asymptotics of discounted aggregate claims for renewal risk model with risky investment Appl. Math. J. Chinese Univ. 21, 25(2: 29-216 Asymptotics of discounted aggregate claims for renewal risk model with risky investment JIANG Tao Abstract. Under the assumption that the claim size is subexponentially

More information

BROWNIAN MOTION DEVELOPMENT FOR MONTE CARLO METHOD APPLIED ON EUROPEAN STYLE OPTION PRICE FORECASTING

BROWNIAN MOTION DEVELOPMENT FOR MONTE CARLO METHOD APPLIED ON EUROPEAN STYLE OPTION PRICE FORECASTING International journal of economics & law Vol. 1 (2011), No. 1 (1-170) BROWNIAN MOTION DEVELOPMENT FOR MONTE CARLO METHOD APPLIED ON EUROPEAN STYLE OPTION PRICE FORECASTING Petar Koĉović, Fakultet za obrazovanje

More information

2.2. Instantaneous Velocity

2.2. Instantaneous Velocity 2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical

More information

Probability density function : An arbitrary continuous random variable X is similarly described by its probability density function f x = f X

Probability density function : An arbitrary continuous random variable X is similarly described by its probability density function f x = f X Week 6 notes : Continuous random variables and their probability densities WEEK 6 page 1 uniform, normal, gamma, exponential,chi-squared distributions, normal approx'n to the binomial Uniform [,1] random

More information

Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density

Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density HW MATH 461/561 Lecture Notes 15 1 Definition: Suppose that two random variables, either continuous or discrete, X and Y have joint density and marginal densities f(x, y), (x, y) Λ X,Y f X (x), x Λ X,

More information

1 Short Introduction to Time Series

1 Short Introduction to Time Series ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The

More information

Introduction to Probability

Introduction to Probability Introduction to Probability EE 179, Lecture 15, Handout #24 Probability theory gives a mathematical characterization for experiments with random outcomes. coin toss life of lightbulb binary data sequence

More information

Hydrodynamic Limits of Randomized Load Balancing Networks

Hydrodynamic Limits of Randomized Load Balancing Networks Hydrodynamic Limits of Randomized Load Balancing Networks Kavita Ramanan and Mohammadreza Aghajani Brown University Stochastic Networks and Stochastic Geometry a conference in honour of François Baccelli

More information

Chapter 8 - Power Density Spectrum

Chapter 8 - Power Density Spectrum EE385 Class Notes 8/8/03 John Stensby Chapter 8 - Power Density Spectrum Let X(t) be a WSS random process. X(t) has an average power, given in watts, of E[X(t) ], a constant. his total average power is

More information

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem

Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial

More information

OPTIMAl PREMIUM CONTROl IN A NON-liFE INSURANCE BUSINESS

OPTIMAl PREMIUM CONTROl IN A NON-liFE INSURANCE BUSINESS ONDERZOEKSRAPPORT NR 8904 OPTIMAl PREMIUM CONTROl IN A NON-liFE INSURANCE BUSINESS BY M. VANDEBROEK & J. DHAENE D/1989/2376/5 1 IN A OPTIMAl PREMIUM CONTROl NON-liFE INSURANCE BUSINESS By Martina Vandebroek

More information

Generating Random Variables and Stochastic Processes

Generating Random Variables and Stochastic Processes Monte Carlo Simulation: IEOR E4703 c 2010 by Martin Haugh Generating Random Variables and Stochastic Processes In these lecture notes we describe the principal methods that are used to generate random

More information

BROWNIAN MOTION 1. INTRODUCTION

BROWNIAN MOTION 1. INTRODUCTION BROWNIAN MOTION 1.1. Wiener Process: Definition. 1. INTRODUCTION Definition 1. A standard (one-dimensional) Wiener process (also called Brownian motion) is a stochastic process {W t } t 0+ indexed by nonnegative

More information

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

More information

Final Mathematics 5010, Section 1, Fall 2004 Instructor: D.A. Levin

Final Mathematics 5010, Section 1, Fall 2004 Instructor: D.A. Levin Final Mathematics 51, Section 1, Fall 24 Instructor: D.A. Levin Name YOU MUST SHOW YOUR WORK TO RECEIVE CREDIT. A CORRECT ANSWER WITHOUT SHOWING YOUR REASONING WILL NOT RECEIVE CREDIT. Problem Points Possible

More information

1 The Brownian bridge construction

1 The Brownian bridge construction The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof

More information

t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).

t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d). 1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction

More information

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE

ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.

More information

Section 5.1 Continuous Random Variables: Introduction

Section 5.1 Continuous Random Variables: Introduction Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,

More information

Introduction to Markov Chain Monte Carlo

Introduction to Markov Chain Monte Carlo Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem

More information

THE BLACK-SCHOLES MODEL AND EXTENSIONS

THE BLACK-SCHOLES MODEL AND EXTENSIONS THE BLAC-SCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the Black-Scholes pricing model of a European option by calculating the expected value of the option. We will assume that

More information

Brownian Motion and Stochastic Flow Systems. J.M Harrison

Brownian Motion and Stochastic Flow Systems. J.M Harrison Brownian Motion and Stochastic Flow Systems 1 J.M Harrison Report written by Siva K. Gorantla I. INTRODUCTION Brownian motion is the seemingly random movement of particles suspended in a fluid or a mathematical

More information

Some Notes on Taylor Polynomials and Taylor Series

Some Notes on Taylor Polynomials and Taylor Series Some Notes on Taylor Polynomials and Taylor Series Mark MacLean October 3, 27 UBC s courses MATH /8 and MATH introduce students to the ideas of Taylor polynomials and Taylor series in a fairly limited

More information

A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model

A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model A Vega-Gamma Relationship for European-Style or Barrier Options in the Black-Scholes Model Fabio Mercurio Financial Models, Banca IMI Abstract In this document we derive some fundamental relationships

More information

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

More information

Generating Random Variables and Stochastic Processes

Generating Random Variables and Stochastic Processes Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Generating Random Variables and Stochastic Processes 1 Generating U(0,1) Random Variables The ability to generate U(0, 1) random variables

More information

Simple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University

Simple Arbitrage. Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila. Christian Bender. Saarland University Simple Arbitrage Motivated by and partly based on a joint work with T. Sottinen and E. Valkeila Saarland University December, 8, 2011 Problem Setting Financial market with two assets (for simplicity) on

More information

Black-Scholes Equation for Option Pricing

Black-Scholes Equation for Option Pricing Black-Scholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there

More information

Dirichlet forms methods for error calculus and sensitivity analysis

Dirichlet forms methods for error calculus and sensitivity analysis Dirichlet forms methods for error calculus and sensitivity analysis Nicolas BOULEAU, Osaka university, november 2004 These lectures propose tools for studying sensitivity of models to scalar or functional

More information

ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES

ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES ANALYZING INVESTMENT RETURN OF ASSET PORTFOLIOS WITH MULTIVARIATE ORNSTEIN-UHLENBECK PROCESSES by Xiaofeng Qian Doctor of Philosophy, Boston University, 27 Bachelor of Science, Peking University, 2 a Project

More information

Flash crashes and order avalanches

Flash crashes and order avalanches Flash crashes and order avalanches Friedrich Hubalek and Thorsten Rheinländer Vienna University of Technology November 29, 2014 Friedrich Hubalek and Thorsten Rheinländer (Vienna Flash University crashes

More information

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks

6.042/18.062J Mathematics for Computer Science December 12, 2006 Tom Leighton and Ronitt Rubinfeld. Random Walks 6.042/8.062J Mathematics for Comuter Science December 2, 2006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Random Walks Gambler s Ruin Today we re going to talk about one-dimensional random walks. In

More information

Important Probability Distributions OPRE 6301

Important Probability Distributions OPRE 6301 Important Probability Distributions OPRE 6301 Important Distributions... Certain probability distributions occur with such regularity in real-life applications that they have been given their own names.

More information

A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS

A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS A SURVEY ON CONTINUOUS ELLIPTICAL VECTOR DISTRIBUTIONS Eusebio GÓMEZ, Miguel A. GÓMEZ-VILLEGAS and J. Miguel MARÍN Abstract In this paper it is taken up a revision and characterization of the class of

More information

LS.6 Solution Matrices

LS.6 Solution Matrices LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions

More information

Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk

Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk Jose Blanchet and Peter Glynn December, 2003. Let (X n : n 1) be a sequence of independent and identically distributed random

More information

Alternative Price Processes for Black-Scholes: Empirical Evidence and Theory

Alternative Price Processes for Black-Scholes: Empirical Evidence and Theory Alternative Price Processes for Black-Scholes: Empirical Evidence and Theory Samuel W. Malone April 19, 2002 This work is supported by NSF VIGRE grant number DMS-9983320. Page 1 of 44 1 Introduction This

More information

Volatility Index: VIX vs. GVIX

Volatility Index: VIX vs. GVIX I. II. III. IV. Volatility Index: VIX vs. GVIX "Does VIX Truly Measure Return Volatility?" by Victor Chow, Wanjun Jiang, and Jingrui Li (214) An Ex-ante (forward-looking) approach based on Market Price

More information

M/M/1 and M/M/m Queueing Systems

M/M/1 and M/M/m Queueing Systems M/M/ and M/M/m Queueing Systems M. Veeraraghavan; March 20, 2004. Preliminaries. Kendall s notation: G/G/n/k queue G: General - can be any distribution. First letter: Arrival process; M: memoryless - exponential

More information

Pricing American Options without Expiry Date

Pricing American Options without Expiry Date Pricing American Options without Expiry Date Carisa K. W. Yu Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom, Hong Kong E-mail: carisa.yu@polyu.edu.hk Abstract This paper

More information