# Stochastic Modelling and Forecasting

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Stochastic Modelling and Forecasting Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH RSE/NNSFC Workshop on Management Science and Engineering and Public Policy Edinburgh, March 2008

2 Outline Stochastic Modelling 1 Stochastic Modelling EM method EM method for financial quantities

3 Outline Stochastic Modelling 1 Stochastic Modelling EM method EM method for financial quantities

4 Outline Stochastic Modelling 1 Stochastic Modelling EM method EM method for financial quantities

5 Outline Stochastic Modelling 1 Stochastic Modelling EM method EM method for financial quantities

6 One of the important problems in many branches of science and industry, e.g. engineering, management, finance, social science, is the specification of the stochastic process governing the behaviour of an underlying quantity. We here use the term underlying quantity to describe any interested object whose value is known at present but is liable to change in the future. Typical examples are shares in a company, commodities such as gold, oil or electricity, number of working people, number of pupils in primary schools.

7 One of the important problems in many branches of science and industry, e.g. engineering, management, finance, social science, is the specification of the stochastic process governing the behaviour of an underlying quantity. We here use the term underlying quantity to describe any interested object whose value is known at present but is liable to change in the future. Typical examples are shares in a company, commodities such as gold, oil or electricity, number of working people, number of pupils in primary schools.

8 One of the important problems in many branches of science and industry, e.g. engineering, management, finance, social science, is the specification of the stochastic process governing the behaviour of an underlying quantity. We here use the term underlying quantity to describe any interested object whose value is known at present but is liable to change in the future. Typical examples are shares in a company, commodities such as gold, oil or electricity, number of working people, number of pupils in primary schools.

9 One of the important problems in many branches of science and industry, e.g. engineering, management, finance, social science, is the specification of the stochastic process governing the behaviour of an underlying quantity. We here use the term underlying quantity to describe any interested object whose value is known at present but is liable to change in the future. Typical examples are shares in a company, commodities such as gold, oil or electricity, number of working people, number of pupils in primary schools.

10 Now suppose that at time t the underlying quantity is x(t). Let us consider a small subsequent time interval dt, during which x(t) changes to x(t) + dx(t). (We use the notation d for the small change in any quantity over this time interval when we intend to consider it as an infinitesimal change.) By definition, the intrinsic growth rate at t is dx(t)/x(t). How might we model this rate?

11 If, given x(t) at time t, the rate of change is deterministic, say R = R(x(t), t), then dx(t) x(t) = R(x(t), t)dt. This gives the ordinary differential equation (ODE) dx(t) dt = R(x(t), t)x(t).

12 However the rate of change is in general not deterministic as it is often subjective to many factors and uncertainties e.g. system uncertainty, environmental disturbances. To model the uncertainty, we may decompose dx(t) x(t) = deterministic change + random change.

13 The deterministic change may be modeled by Rdt = R(x(t), t)dt where R = r(x(t), t) is the average rate of change given x(t) at time t. So dx(t) x(t) = R(x(t), t)dt + random change. How may we model the random change?

14 In general, the random change is affected by many factors independently. By the well-known central limit theorem this change can be represented by a normal distribution with mean zero and and variance V 2 dt, namely random change = N(0, V 2 dt) = V N(0, dt), where V = V (x(t), t) is the standard deviation of the rate of change given x(t) at time t, and N(0, dt) is a normal distribution with mean zero and and variance dt. Hence dx(t) x(t) = R(x(t), t)dt + V (x(t), t)n(0, dt).

15 A convenient way to model N(0, dt) as a process is to use the Brownian motion B(t) (t 0) which has the following properties: B(0) = 0, db(t) = B(t + dt) B(t) is independent of B(t), db(t) follows N(0, dt).

16 The stochastic model can therefore be written as or dx(t) x(t) = R(x(t), t)dt + V (x(t), t)db(t), dx(t) = R(x(t), t)x(t)dt + V (x(t), t)x(t)db(t) which is a stochastic differential equation (SDE).

17 Linear model If both R and V are constants, say then the SDE becomes This is R(x(t), t) = µ, V (x(t), t) = σ, dx(t) = µx(t)dt + σx(t)db(t). the Black Scholes geometric Brownian motion in finance, the exponential growth model in engineering and population.

18 Linear model If both R and V are constants, say then the SDE becomes This is R(x(t), t) = µ, V (x(t), t) = σ, dx(t) = µx(t)dt + σx(t)db(t). the Black Scholes geometric Brownian motion in finance, the exponential growth model in engineering and population.

19 Logistic model If R(x(t), t) = b + ax(t), V (x(t), t) = σx(t), then the SDE becomes ( dx(t) = x(t) ) [b + ax(t)]dt + σx(t)db(t). This is the well-known Logistic model in population.

20 Square root process If R(x(t), t) = µ, V (x(t), t) = σ x(t), then the SDE becomes the well-known square root process dx(t) = µx(t)dt + σ x(t)db(t). This is used widely in engineering and finance.

21 Mean-reverting model If R(x(t), t) = then the SDE becomes This is α(µ x(t)), V (x(t), t) = σ, x(t) x(t) dx(t) = α(µ x(t))dt + σ x(t)db(t). the mean-reverting square root process in population, the CIR model for interest rate in finance.

22 Mean-reverting model If R(x(t), t) = then the SDE becomes This is α(µ x(t)), V (x(t), t) = σ, x(t) x(t) dx(t) = α(µ x(t))dt + σ x(t)db(t). the mean-reverting square root process in population, the CIR model for interest rate in finance.

23 Theta process If R(x(t), t) = µ, V (x(t), t) = σ(x(t)) θ 1, then the SDE becomes dx(t) = µx(t)dt + σ(x(t)) θ db(t), which is known as the theta process.

24 Outline Stochastic Modelling 1 Stochastic Modelling EM method EM method for financial quantities

25 In the classical theory of population dynamics, it is assumed that the grow rate is constant µ. Thus dx(t) x(t) = µdt, which is often written as the familiar ordinary differential equation (ODE) dx(t) = µx(t). dt This linear ODE can be solved exactly to give exponential growth (or decay) in the population, i.e. x(t) = x(0)e µt, where x(0) is the initial population at time t = 0.

26 We observe: If µ > 0, x(t) exponentially, i.e. the population will grow exponentially fast. If µ < 0, x(t) 0 exponentially, that is the population will become extinct. If µ = 0, x(t) = x(0) for all t, namely the population is stationary.

27 We observe: If µ > 0, x(t) exponentially, i.e. the population will grow exponentially fast. If µ < 0, x(t) 0 exponentially, that is the population will become extinct. If µ = 0, x(t) = x(0) for all t, namely the population is stationary.

28 We observe: If µ > 0, x(t) exponentially, i.e. the population will grow exponentially fast. If µ < 0, x(t) 0 exponentially, that is the population will become extinct. If µ = 0, x(t) = x(0) for all t, namely the population is stationary.

29 However, if we take the uncertainty into account as explained before, we may have dx(t) x(t) = rdt + σdb(t). This is often written as the linear SDE It has the explicit solution dx(t) = µx(t)dt + σx(t)db(t). x(t) = x(0) exp [ ] (µ 0.5σ 2 )t + σb(t).

30 Recall the properties of the Brownian motion and lim sup t lim inf t B(t) 2t log log t = 1 B(t) 2t log log t = 1 a.s. a.s.

31 If µ > 0.5σ 2, x(t) exponentially with probability 1, i.e. the population will grow exponentially fast. If µ < 0.5σ 2, x(t) 0 exponentially with probability 1, that is the population will become extinct. In particular, this includes the case of 0 < µ < 0.5σ 2 where the population will grow exponentially in the corresponding ODE model but it will become extinct in the SDE model. This reveals the important feature - noise may make a population extinct. If µ = 0.5σ 2, lim sup t x(t) = while lim inf t x(t) = 0 with probability 1.

32 If µ > 0.5σ 2, x(t) exponentially with probability 1, i.e. the population will grow exponentially fast. If µ < 0.5σ 2, x(t) 0 exponentially with probability 1, that is the population will become extinct. In particular, this includes the case of 0 < µ < 0.5σ 2 where the population will grow exponentially in the corresponding ODE model but it will become extinct in the SDE model. This reveals the important feature - noise may make a population extinct. If µ = 0.5σ 2, lim sup t x(t) = while lim inf t x(t) = 0 with probability 1.

33 If µ > 0.5σ 2, x(t) exponentially with probability 1, i.e. the population will grow exponentially fast. If µ < 0.5σ 2, x(t) 0 exponentially with probability 1, that is the population will become extinct. In particular, this includes the case of 0 < µ < 0.5σ 2 where the population will grow exponentially in the corresponding ODE model but it will become extinct in the SDE model. This reveals the important feature - noise may make a population extinct. If µ = 0.5σ 2, lim sup t x(t) = while lim inf t x(t) = 0 with probability 1.

34 If µ > 0.5σ 2, x(t) exponentially with probability 1, i.e. the population will grow exponentially fast. If µ < 0.5σ 2, x(t) 0 exponentially with probability 1, that is the population will become extinct. In particular, this includes the case of 0 < µ < 0.5σ 2 where the population will grow exponentially in the corresponding ODE model but it will become extinct in the SDE model. This reveals the important feature - noise may make a population extinct. If µ = 0.5σ 2, lim sup t x(t) = while lim inf t x(t) = 0 with probability 1.

35 If µ > 0.5σ 2, x(t) exponentially with probability 1, i.e. the population will grow exponentially fast. If µ < 0.5σ 2, x(t) 0 exponentially with probability 1, that is the population will become extinct. In particular, this includes the case of 0 < µ < 0.5σ 2 where the population will grow exponentially in the corresponding ODE model but it will become extinct in the SDE model. This reveals the important feature - noise may make a population extinct. If µ = 0.5σ 2, lim sup t x(t) = while lim inf t x(t) = 0 with probability 1.

36 Outline Stochastic Modelling 1 Stochastic Modelling EM method EM method for financial quantities

37 The logistic model for single-species population dynamics is given by the ODE dx(t) dt = x(t)[b + ax(t)]. (3.1)

38 If a < 0 and b > 0, the equation has the global solution x(t) = b a + e bt (b + ax 0 )/x 0 (t 0), which is not only positive and bounded but also lim x(t) = b t a. If a > 0, whilst retaining b > 0, then the equation has only the local solution b x(t) = a + e bt (0 t < T ), (b + ax 0 )/x 0 which explodes to infinity at the finite time T = 1 ( ) b log ax0. b + ax 0

39 If a < 0 and b > 0, the equation has the global solution x(t) = b a + e bt (b + ax 0 )/x 0 (t 0), which is not only positive and bounded but also lim x(t) = b t a. If a > 0, whilst retaining b > 0, then the equation has only the local solution b x(t) = a + e bt (0 t < T ), (b + ax 0 )/x 0 which explodes to infinity at the finite time T = 1 ( ) b log ax0. b + ax 0

40 Once again, the growth rate b here is not a constant but a stochastic process. Therefore, bdt should be replaced by bdt + N(0, v 2 dt) = bdt + vn(0, dt) = bdt + vdb(t), where v 2 is the variance of the noise intensity. Hence the ODE evolves to an SDE ( dx(t) = x(t) ) [b + ax(t)]dt + vdb(t). (3.2)

41 The variance may or may not depend on the state x(t). We consider the latter, say v = σx(t). Then the SDE (3.2) becomes ( dx(t) = x(t) ) [b + ax(t)]dt + σx(t)db(t). (3.3) How is this SDE different from its corresponding ODE?

42 Significant Difference between ODE (3.1) and SDE (3.3) ODE (3.1): The solution explodes to infinity at a finite time if a > 0 and b > 0. SDE (3.3): With probability one, the solution will no longer explode in a finite time, even in the case when a > 0 and b > 0, as long as σ 0.

43 Significant Difference between ODE (3.1) and SDE (3.3) ODE (3.1): The solution explodes to infinity at a finite time if a > 0 and b > 0. SDE (3.3): With probability one, the solution will no longer explode in a finite time, even in the case when a > 0 and b > 0, as long as σ 0.

44 Example Stochastic Modelling dx(t) = x(t)[1 + x(t)], t 0, x(0) = x 0 > 0 dt has the solution x(t) = e t (1 + x 0 )/x 0 (0 t < T ), which explodes to infinity at the finite time ( ) 1 + x0 T = log. However, the SDE x 0 dx(t) = x(t)[(1 + x(t))dt + σx(t)dw(t)] will never explode as long as σ 0.

45 (a) (b)

46 Note on the graphs: In graph (a) the solid curve shows a stochastic trajectory generated by the Euler scheme for time step t = 10 7 and σ = 0.25 for a one-dimensional system (3.3) with a = b = 1. The corresponding deterministic trajectory is shown by the dot-dashed curve. In Graph (b) σ = 1.0.

47 Key Point: When a > 0 and ε = 0 the solution explodes at the finite time t = T ; whilst conversely, no matter how small ε > 0, the solution will not explode in a finite time. In other words, stochastic environmental noise suppresses deterministic explosion.

48 EM method EM method for financial quantities Most of SDEs used in practice do not have explicit solutions. How can we use these SDEs to forecast? One of the important techniques is the method of Monte Carlo simulations. There are two main motivations for such simulations: using a Monte Carlo approach to compute the expected value of a function of the underlying underlying quantity, for example to value a bond or to find the expected payoff of an option; generating time series in order to test parameter estimation algorithms.

49 EM method EM method for financial quantities Most of SDEs used in practice do not have explicit solutions. How can we use these SDEs to forecast? One of the important techniques is the method of Monte Carlo simulations. There are two main motivations for such simulations: using a Monte Carlo approach to compute the expected value of a function of the underlying underlying quantity, for example to value a bond or to find the expected payoff of an option; generating time series in order to test parameter estimation algorithms.

50 EM method EM method for financial quantities Most of SDEs used in practice do not have explicit solutions. How can we use these SDEs to forecast? One of the important techniques is the method of Monte Carlo simulations. There are two main motivations for such simulations: using a Monte Carlo approach to compute the expected value of a function of the underlying underlying quantity, for example to value a bond or to find the expected payoff of an option; generating time series in order to test parameter estimation algorithms.

51 EM method EM method for financial quantities Typically, let us consider the square root process ds(t) = rs(t)dt + σ S(t)dB(t), 0 t T. A numerical method, e.g. the Euler Maruyama (EM) method applied to it may break down due to negative values being supplied to the square root function. A natural fix is to replace the SDE by the equivalent, but computationally safer, problem ds(t) = rs(t)dt + σ S(t) db(t), 0 t T.

52 Outline Stochastic Modelling EM method EM method for financial quantities 1 Stochastic Modelling EM method EM method for financial quantities

53 Discrete EM approximation EM method EM method for financial quantities Given a stepsize > 0, the EM method applied to the SDE sets s 0 = S(0) and computes approximations s n S(t n ), where t n = n, according to where B n = B(t n+1 ) B(t n ). s n+1 = s n (1 + r ) + σ s n B n,

54 EM method EM method for financial quantities Continuous-time EM approximation s(t) := s 0 + r t 0 t s(u))du + σ s(u) db(u), 0 where the step function s(t) is defined by s(t) := s n, for t [t n, t n+1 ). Note that s(t) and s(t) coincide with the discrete solution at the gridpoints; s(t n ) = s(t n ) = s n.

55 EM method EM method for financial quantities The ability of the EM method to approximate the true solution is guaranteed by the ability of either s(t) or s(t) to approximate S(t) which is described by: Theorem ( lim E sup 0 0 t T s(t) S(t) 2) ( = lim E 0 sup 0 t T s(t) S(t) 2) = 0.

56 EM method EM method for financial quantities The ability of the EM method to approximate the true solution is guaranteed by the ability of either s(t) or s(t) to approximate S(t) which is described by: Theorem ( lim E sup 0 0 t T s(t) S(t) 2) ( = lim E 0 sup 0 t T s(t) S(t) 2) = 0.

57 Outline Stochastic Modelling EM method EM method for financial quantities 1 Stochastic Modelling EM method EM method for financial quantities

58 Bond Stochastic Modelling EM method EM method for financial quantities If S(t) models short-term interest rate dynamics, it is pertinent to consider the expected payoff ( ) β := E exp T 0 S(t)dt from a bond. A natural approximation based on the EM method is ( ) Theorem β := E exp N 1 n=0 s n lim β β = 0. 0, where N = T /.

59 Bond Stochastic Modelling EM method EM method for financial quantities If S(t) models short-term interest rate dynamics, it is pertinent to consider the expected payoff ( ) β := E exp T 0 S(t)dt from a bond. A natural approximation based on the EM method is ( ) Theorem β := E exp N 1 n=0 s n lim β β = 0. 0, where N = T /.

60 Up-and-out call option EM method EM method for financial quantities An up-and-out call option at expiry time T pays the European value with the exercise price K if S(t) never exceeded the fixed barrier, c, and pays zero otherwise. Theorem Define V = E [ (S(T ) K ) + I {0 S(t) c, 0 t T } ], V = E [ ( s(t ) K ) + I {0 s(t) B, 0 t T } ]. Then lim V V = 0. 0

61 Up-and-out call option EM method EM method for financial quantities An up-and-out call option at expiry time T pays the European value with the exercise price K if S(t) never exceeded the fixed barrier, c, and pays zero otherwise. Theorem Define V = E [ (S(T ) K ) + I {0 S(t) c, 0 t T } ], V = E [ ( s(t ) K ) + I {0 s(t) B, 0 t T } ]. Then lim V V = 0. 0

### 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

### Simulating Stochastic Differential Equations

Monte Carlo Simulation: IEOR E473 Fall 24 c 24 by Martin Haugh Simulating Stochastic Differential Equations 1 Brief Review of Stochastic Calculus and Itô s Lemma Let S t be the time t price of a particular

### 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

### 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

### The Black-Scholes Model

The Black-Scholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The Black-Scholes Model Options Markets 1 / 19 The Black-Scholes-Merton

### Finite Differences Schemes for Pricing of European and American Options

Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the Black-Scholes

### Applications of Stochastic Processes in Asset Price Modeling

Applications of Stochastic Processes in Asset Price Modeling TJHSST Computer Systems Lab Senior Research Project 2008-2009 Preetam D Souza November 11, 2008 Abstract Stock market forecasting and asset

### 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

### The SIS Epidemic Model with Markovian Switching

The SIS Epidemic with Markovian Switching Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH (Joint work with A. Gray, D. Greenhalgh and J. Pan) Outline Motivation 1 Motivation

### 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,

### CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

### 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

### Financial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan

Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction

### Numerical Methods for Option Pricing

Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly

### 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

### 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 }

### European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

### Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes

Bias in the Estimation of Mean Reversion in Continuous-Time Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA

### 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

### The Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014

Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA 1.

### Market Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk

Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day

### Hedging Options In The Incomplete Market With Stochastic Volatility. Rituparna Sen Sunday, Nov 15

Hedging Options In The Incomplete Market With Stochastic Volatility Rituparna Sen Sunday, Nov 15 1. Motivation This is a pure jump model and hence avoids the theoretical drawbacks of continuous path models.

### Pricing and calibration in local volatility models via fast quantization

Pricing and calibration in local volatility models via fast quantization Parma, 29 th January 2015. Joint work with Giorgia Callegaro and Martino Grasselli Quantization: a brief history Birth: back to

### More Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options

More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path

### The Black-Scholes pricing formulas

The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

### Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

### The Black-Scholes Formula

ECO-30004 OPTIONS AND FUTURES SPRING 2008 The Black-Scholes Formula The Black-Scholes Formula We next examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they

### Numerical methods for American options

Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

### 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.

### Generation Asset Valuation with Operational Constraints A Trinomial Tree Approach

Generation Asset Valuation with Operational Constraints A Trinomial Tree Approach Andrew L. Liu ICF International September 17, 2008 1 Outline Power Plants Optionality -- Intrinsic vs. Extrinsic Values

### Private Equity Fund Valuation and Systematic Risk

An Equilibrium Approach and Empirical Evidence Axel Buchner 1, Christoph Kaserer 2, Niklas Wagner 3 Santa Clara University, March 3th 29 1 Munich University of Technology 2 Munich University of Technology

### OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION

OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION NITESH AIDASANI KHYAMI Abstract. Option contracts are used by all major financial institutions and investors, either to speculate on stock market

### Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last

### 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?

### Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation

Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Yoon W. Kwon CIMS 1, Math. Finance Suzanne A. Lewis CIMS, Math. Finance May 9, 000 1 Courant Institue of Mathematical Science,

### Betting on Volatility: A Delta Hedging Approach. Liang Zhong

Betting on Volatility: A Delta Hedging Approach Liang Zhong Department of Mathematics, KTH, Stockholm, Sweden April, 211 Abstract In the financial market, investors prefer to estimate the stock price

### 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

### 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

### 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

### General theory of stochastic processes

CHAPTER 1 General theory of stochastic processes 1.1. Definition of stochastic process First let us recall the definition of a random variable. A random variable is a random number appearing as a result

### Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering

Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014

### 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

### Numerical Methods for Pricing Exotic Options

Imperial College London Department of Computing Numerical Methods for Pricing Exotic Options by Hardik Dave - 00517958 Supervised by Dr. Daniel Kuhn Second Marker: Professor Berç Rustem Submitted in partial

### 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

### Pricing of an Exotic Forward Contract

Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### 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

### Markov modeling of Gas Futures

Markov modeling of Gas Futures p.1/31 Markov modeling of Gas Futures Leif Andersen Banc of America Securities February 2008 Agenda Markov modeling of Gas Futures p.2/31 This talk is based on a working

### 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

### 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

### Simple approximations for option pricing under mean reversion and stochastic volatility

Simple approximations for option pricing under mean reversion and stochastic volatility Christian M. Hafner Econometric Institute Report EI 2003 20 April 2003 Abstract This paper provides simple approximations

### 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

### LogNormal stock-price models in Exams MFE/3 and C/4

Making sense of... LogNormal stock-price models in Exams MFE/3 and C/4 James W. Daniel Austin Actuarial Seminars http://www.actuarialseminars.com June 26, 2008 c Copyright 2007 by James W. Daniel; reproduction

### Monte Carlo simulations in the case of several risk factors: Cholesky decomposition and copulas

wwwijcsiorg 233 Monte Carlo simulations in the case of several risk factors: Cholesky decomposition and copulas Naima SOKHER 1, Boubker DAAFI 2, Jamal BOYAGHROMNI 1, Abdelwahed NAMIR 1 1 Department of

### 3. Monte Carlo Simulations. Math6911 S08, HM Zhu

3. Monte Carlo Simulations Math6911 S08, HM Zhu References 1. Chapters 4 and 8, Numerical Methods in Finance. Chapters 17.6-17.7, Options, Futures and Other Derivatives 3. George S. Fishman, Monte Carlo:

### Numerical PDE methods for exotic options

Lecture 8 Numerical PDE methods for exotic options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Barrier options For barrier option part of the option contract is triggered if the asset

### 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

### 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

### The Evaluation of Barrier Option Prices Under Stochastic Volatility. BFS 2010 Hilton, Toronto June 24, 2010

The Evaluation of Barrier Option Prices Under Stochastic Volatility Carl Chiarella, Boda Kang and Gunter H. Meyer School of Finance and Economics University of Technology, Sydney School of Mathematics

### The Standard Normal distribution

The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance

### FAST SIMULATION FOR COMPUTING CREDIT VALUE ADJUSTMENTS OF INTEREST RATE PORTFOLIOS

FAST SIMULATION FOR COMPUTING CREDIT VALUE ADJUSTMENTS OF INTEREST RATE PORTFOLIOS Team members Jonatan Eriksson (Swedbank) Jonas Hallgren (KTH) Ola Hammarlid (Swedbank) Henrik Hult (KTH, team leader)

### 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

### Using simulation to calculate the NPV of a project

Using simulation to calculate the NPV of a project Marius Holtan Onward Inc. 5/31/2002 Monte Carlo simulation is fast becoming the technology of choice for evaluating and analyzing assets, be it pure financial

### A State Space Model for Wind Forecast Correction

A State Space Model for Wind Forecast Correction Valérie Monbe, Pierre Ailliot 2, and Anne Cuzol 1 1 Lab-STICC, Université Européenne de Bretagne, France (e-mail: valerie.monbet@univ-ubs.fr, anne.cuzol@univ-ubs.fr)

### Generating Random Numbers Variance Reduction Quasi-Monte Carlo. Simulation Methods. Leonid Kogan. MIT, Sloan. 15.450, Fall 2010

Simulation Methods Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Simulation Methods 15.450, Fall 2010 1 / 35 Outline 1 Generating Random Numbers 2 Variance Reduction 3 Quasi-Monte

### 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

### Lévy-Vasicek Models and the Long-Bond Return Process

Lévy-Vasicek Models and the Long-Bond Return Process Lane P. Hughston Department of Mathematics Brunel University London Uxbridge UB8 3PH, United Kingdom lane.hughston@brunel.ac.uk Third London-Paris Bachelier

### 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

### Jung-Soon Hyun and Young-Hee Kim

J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL Jung-Soon Hyun and Young-Hee Kim Abstract. We present two approaches of the stochastic interest

### Master of Mathematical Finance: Course Descriptions

Master of Mathematical Finance: Course Descriptions CS 522 Data Mining Computer Science This course provides continued exploration of data mining algorithms. More sophisticated algorithms such as support

### General Sampling Methods

General Sampling Methods Reference: Glasserman, 2.2 and 2.3 Claudio Pacati academic year 2016 17 1 Inverse Transform Method Assume U U(0, 1) and let F be the cumulative distribution function of a distribution

### Lecture 7: Finding Lyapunov Functions 1

Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1

### Ito Excursion Theory. Calum G. Turvey Cornell University

Ito Excursion Theory Calum G. Turvey Cornell University Problem Overview Times series and dynamics have been the mainstay of agricultural economic and agricultural finance for over 20 years. Much of the

### Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com

Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the

### Slides for Risk Management

Slides for Risk Management Introduction to the modeling of assets Groll Seminar für Finanzökonometrie Prof. Mittnik, PhD Groll (Seminar für Finanzökonometrie) Slides for Risk Management Prof. Mittnik,

### Forecasting in supply chains

1 Forecasting in supply chains Role of demand forecasting Effective transportation system or supply chain design is predicated on the availability of accurate inputs to the modeling process. One of the

### 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

### A Regime-Switching Model for Electricity Spot Prices. Gero Schindlmayr EnBW Trading GmbH g.schindlmayr@enbw.com

A Regime-Switching Model for Electricity Spot Prices Gero Schindlmayr EnBW Trading GmbH g.schindlmayr@enbw.com May 31, 25 A Regime-Switching Model for Electricity Spot Prices Abstract Electricity markets

### Monte Carlo simulations and option pricing

Monte Carlo simulations and option pricing by Bingqian Lu Undergraduate Mathematics Department Pennsylvania State University University Park, PA 16802 Project Supervisor: Professor Anna Mazzucato July,

### OPTIMAL TIMING OF THE ANNUITY PURCHASES: A

OPTIMAL TIMING OF THE ANNUITY PURCHASES: A COMBINED STOCHASTIC CONTROL AND OPTIMAL STOPPING PROBLEM Gabriele Stabile 1 1 Dipartimento di Matematica per le Dec. Econ. Finanz. e Assic., Facoltà di Economia

### From CFD to computational finance (and back again?)

computational finance p. 1/21 From CFD to computational finance (and back again?) Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Oxford-Man Institute of Quantitative Finance

### Notes on Continuous Random Variables

Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes

### Barrier Options. Peter Carr

Barrier Options Peter Carr Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU March 14th, 2008 What are Barrier Options?

### 1 Error in Euler s Method

1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of

### 3 Results. σdx. df =[µ 1 2 σ 2 ]dt+ σdx. Integration both sides will form

Appl. Math. Inf. Sci. 8, No. 1, 107-112 (2014) 107 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080112 Forecasting Share Prices of Small Size Companies

### 7 Hypothesis testing - one sample tests

7 Hypothesis testing - one sample tests 7.1 Introduction Definition 7.1 A hypothesis is a statement about a population parameter. Example A hypothesis might be that the mean age of students taking MAS113X

### A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model

Applied Mathematical Sciences, vol 8, 14, no 143, 715-7135 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting

### 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

### P(a X b) = f X (x)dx. A p.d.f. must integrate to one: f X (x)dx = 1. Z b

Continuous Random Variables The probability that a continuous random variable, X, has a value between a and b is computed by integrating its probability density function (p.d.f.) over the interval [a,b]:

### Review for Calculus Rational Functions, Logarithms & Exponentials

Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

### OPTION PRICING FOR WEIGHTED AVERAGE OF ASSET PRICES

OPTION PRICING FOR WEIGHTED AVERAGE OF ASSET PRICES Hiroshi Inoue 1, Masatoshi Miyake 2, Satoru Takahashi 1 1 School of Management, T okyo University of Science, Kuki-shi Saitama 346-8512, Japan 2 Department

### Stephane Crepey. Financial Modeling. A Backward Stochastic Differential Equations Perspective. 4y Springer

Stephane Crepey Financial Modeling A Backward Stochastic Differential Equations Perspective 4y Springer Part I An Introductory Course in Stochastic Processes 1 Some Classes of Discrete-Time Stochastic

### Determining distribution parameters from quantiles

Determining distribution parameters from quantiles John D. Cook Department of Biostatistics The University of Texas M. D. Anderson Cancer Center P. O. Box 301402 Unit 1409 Houston, TX 77230-1402 USA cook@mderson.org

### 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

### Risk Formulæ for Proportional Betting

Risk Formulæ for Proportional Betting William Chin DePaul University, Chicago, Illinois Marc Ingenoso Conger Asset Management LLC, Chicago, Illinois September 9, 2006 Introduction A canon of the theory

### MAT12X Intermediate Algebra

MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions

### Supply Chain Analysis Tools

Supply Chain Analysis Tools MS&E 262 Supply Chain Management April 14, 2004 Inventory/Service Trade-off Curve Motivation High Inventory Low Poor Service Good Analytical tools help lower the curve! 1 Outline

### A Log-Robust Optimization Approach to Portfolio Management

A Log-Robust Optimization Approach to Portfolio Management Dr. Aurélie Thiele Lehigh University Joint work with Ban Kawas Research partially supported by the National Science Foundation Grant CMMI-0757983