Monte Carlo Methods. Prof. Mike Giles. Oxford University Mathematical Institute. Lecture 1 p. 1.
|
|
- Felix Bryan
- 5 years ago
- Views:
Transcription
1 Monte Carlo Methods Prof. Mike Giles Oxford University Mathematical Institute Lecture 1 p. 1
2 Geometric Brownian Motion In the case of Geometric Brownian Motion ds t = rs t dt+σs t dw t the use of Iˆto calculus gives d(logs t ) = (r 1 2 σ2 )dt+σdw t which can be integrated to give S T = S 0 exp ( (r 1 2 σ2 )T +σw T ) so we are able to directly simulate S T to perform Monte Carlo estimation for European options with a payoff f(s T ). Lecture 1 p. 2
3 Euler-Maruyama path simulation In more general cases, the scalar SDE ds t = a(s t,t) dt+b(s t,t) dw t can be approximated using the Euler-Maruyama discretisation Ŝ n+1 = Ŝ n +a(ŝ n,t n )h+b(ŝ n,t n ) W n Here h is the timestep, Ŝ n is the approximation to S nh and the W n are i.i.d. N(0,h) Brownian increments. Lecture 1 p. 3
4 Euler-Maruyama method For ODEs, the forward Euler method has O(h) accuracy, and other more accurate methods are usually preferred. However, SDEs are very much harder to approximate so the Euler-Maruyama method is used widely in practice. Numerical analysis is also very difficult and even the definition of accuracy is tricky. Lecture 1 p. 4
5 Weak convergence In finance applications, we are mostly concerned with weak errors, the error in the expected payoff due to using a finite timestep h. For a European payoff f(s T ), the weak error is E[f(S T )] E[f(Ŝ M )] where M = T/h, and for a path-dependent option it is E[f(S)] E[ f(ŝ)] where f(s) is a function of the entire path S t, and f(ŝ) is a corresponding approximation using the whole discrete path. Lecture 1 p. 5
6 Weak convergence Key theoretical result (Bally and Talay, 1995): If p(s) is the p.d.f. for S T and p(s) is the p.d.f. for Ŝ T/h computed using the Euler-Maruyama approximation, then under certain conditions on a(s,t) and b(s,t) p(s) p(s) = O(h) and hence E[f(S T )] E[f(Ŝ T/h )] = O(h) This holds even for digital options with discontinuous payoffs f(s) earlier theory covered only European options such as put and call options with Lipschitz payoffs. Lecture 1 p. 6
7 Weak convergence Numerical demonstration: Geometric Brownian Motion r = 0.05, σ = 0.5, T = 1 ds = rsdt+σsdw European call: S 0 = 100,K = 110. Plot shows weak error versus analytic expectation when using 10 8 paths, and also Monte Carlo error (3 standard deviations) Lecture 1 p. 7
8 Weak convergence Weak convergence -- comparison to exact solution 10-1 Weak error MC error Error h Lecture 1 p. 8
9 Weak convergence Previous plot showed difference between exact expectation and numerical approximation. What if the exact solution is unknown? Compare approximations with timesteps h and 2h. If then and so E[f(S T )] E[f(Ŝ h T/h )] a h E[f(S T )] E[f(ŜT/2h 2h )] 2ah E[f(ŜT/h h )] E[f(Ŝ2h T/2h )] ah Lecture 1 p. 9
10 Weak convergence To minimise the number of paths that need to be simulated, we use same driving Brownian path when doing 2h and h approximations. i.e. take Brownian increments for h simulation and sum in pairs to get Brownian increments for 2h simulation. The variance is lower because the h and 2h paths are close to each other (strong convergence). (We won t cover this, but this forms the basis for the Multilevel Monte Carlo method (Giles, 2006)) Lecture 1 p. 10
11 Weak convergence Weak convergence -- difference from 2h approximation 10-1 Weak error MC error Error h Lecture 1 p. 11
12 Mean Square Error Question: how do we choose the number of timesteps (to reduce the weak error) the number of paths (to reduce the Monte Carlo sampling error) If the true option value is and the discrete approximation is V = E[f] V = E[ f] and the Monte Carlo estimate is Ŷ = 1 N N f (i) i=1 then... Lecture 1 p. 12
13 Mean Square Error... the Mean Square Error is [ (Ŷ ) ] 2 ] E V = V [Ŷ V = V[Ŷ]+ + ( E[Ŷ] V ( ) 2 E[Ŷ V] ) 2 = N 1 V[ f]+ ( E[ f] E[f] ) 2 first term is due to the variance of estimator second term is square of bias due to weak error Lecture 1 p. 13
14 Mean Square Error If there are M timesteps, the computational cost is proportional to C = MN and the MSE is approximately an 1 +bm 2 = an 1 +bc 2 N 2. For a fixed computational cost, this is a minimum when N = ( ) ac 2 1/3, M = 2b ( 2bC a ) 1/3, and hence an 1 = ( 2a 2 ) 1/3 b, bm 2 = C 2 ( a 2 ) 1/3 b 4C 2, so the MC term is twice as big as the bias term. Lecture 1 p. 14
15 Path-dependent Options For European options, Euler-Maruyama method has O(h) weak convergence. However, for some path-dependent options it may give only O( h) weak convergence, unless the numerical payoff is constructed carefully. Lecture 1 p. 15
16 Barrier option A down-and-out call option has discounted payoff exp( rt) (S T K) + 1 mint S(t)>B i.e. it is like a standard call option except that it pays nothing if the minimum value drops below the barrier B. The natural numerical discretisation of this is f = exp( rt) (Ŝ M K) + 1 minnŝn>b Lecture 1 p. 16
17 Barrier option Numerical demonstration: Geometric Brownian Motion r = 0.05, σ = 0.5, T = 1 ds t = rs t dt+σs t dw t Down-and-out call: S 0 = 100,K = 110,B = 90. Plots shows weak error versus analytic expectation using 10 6 paths, and difference from 2h approximation using 10 5 paths. (We don t need as many paths as before because the weak errors are much larger in this case.) Lecture 1 p. 17
18 Barrier option Barrier weak convergence -- comparison to exact solution 10 1 Weak error MC error Error h Lecture 1 p. 18
19 Barrier option Barrier weak convergence -- difference from 2h approximation Weak error MC error 10 0 Error h Lecture 1 p. 19
20 Lookback option A floating-strike lookback call option has discounted payoff ( ) exp( rt) S T min [0,T] S t The natural numerical discretisation of this is ) f = exp( rt) (ŜM minŝ n n Lecture 1 p. 20
21 Lookback option Lookback weak convergence -- comparison to exact solution Weak error MC error 10 1 Error h Lecture 1 p. 21
22 Lookback option weak convergence -- difference from 2h approximation 1Lookback 10 Weak error MC error 10 0 Error h Lecture 1 p. 22
23 Brownian Bridge To recover O(h) weak convergence we first need some theory. Consider simple Brownian motion ds t = a dt+b dw t with constant a, b and initial data S 0 =0. Question: given S T, what is conditional probability density for S T/2? Lecture 1 p. 23
24 Conditional probability With discrete probabilities, P(A B) = P(A B) P(B) Similarly, with probability density functions where p 1 (x y) = p 2(x,y) p 3 (y) p 1 (x y) is the conditional p.d.f. for x, given y p 2 (x,y) is the joint probability density function for x,y p 3 (y) is the probability density function for y Lecture 1 p. 24
25 Brownian bridge In our case, y S T, x S T/2 p 2 (x,y) = p 3 (y) = = p 1 (x y) = ) 1 exp ( (x at/2)2 πt b b 2 T 1 (y x at/2)2 exp ( πt b 1 exp ( 2πT b 1 πt/2 b exp b 2 T ) (y at)2 2b 2 T ) ( (x y/2)2 b 2 T/2 ) Hence, x is Normally distributed with mean y/2 and variance b 2 T/4. Lecture 1 p. 25
26 Brownian bridge Extending this to a particular timestep with endpoints S n and S n+1, conditional on these the mid-point is Normally distributed with mean and variance b 2 h/ (S n +S n+1 ) We can take a sample from this conditional p.d.f. and then repeat the process, recursively bisecting each interval to fill in more and more detail. Note: the drift a is irrelevant, given the two endpoints. Because of this, we will take a = 0 in the next bit of theory. Lecture 1 p. 26
27 Barrier crossing Consider zero drift Brownian motion with S 0 >0. If the path S t hits a barrier at 0, it is equally likely thereafter to go up or down. Hence, by symmetry, for s > 0, the p.d.f. for paths with S T = s after hitting the barrier is equal to the p.d.f. for paths with S T = s. Thus, for S T > 0, P(hit barrier S T ) = = exp ) exp ( ( S T S 0 ) 2 2b 2 T ( ) exp (S T S 0 ) 2 2b 2 T ( 2S T S 0 b 2 T ) Lecture 1 p. 27
28 Barrier crossing For a timestep [t n,t n+1 ] and non-zero barrier B this generalises to ( P(hit barrier S n,s n+1 > B) = exp 2(S ) n+1 B)(S n B) b 2 h This can also be viewed as the cumulative probability P(S min < B) where S min = min [t n,t n+1 ] S(t). Since this is uniformly distributed on [0,1] we can equate this to a uniform [0,1] random variable U n and solve to get S min = 2 (S 1 n+1 +S n ) (S n+1 S n ) 2 2b 2 h logu n Lecture 1 p. 28
29 Barrier crossing For a barrier above, we have P(hit barrier S n,s n+1 < B) = exp ( 2(B S n+1)(b S n ) b 2 h ) and hence S max = 1 2 (S n+1 +S n + (S n+1 S n ) 2 2b 2 h logu n ) where U n is again a uniform [0,1] random variable. Lecture 1 p. 29
30 Barrier option Returning now to the barrier option, how do we define the numerical payoff f(ŝ)? First, calculate Ŝ n as usual using Euler-Maruyama method. Second, two alternatives: use (approximate) probability of crossing the barrier directly sample (approximately) the minimum in each timestep Lecture 1 p. 30
31 Barrier option Alternative 1: treating the drift and volatility as being approximately constant within each timestep, the probability of having crossed the barrier within timestep n is P n = exp ( 2(Ŝ n+1 B) + (Ŝ n B) + b 2 (Ŝ n,t n ) h Probability at end of not having crossed barrier is (1 P n ) and so the payoff is n ) f(ŝ) = exp( rt) (Ŝ M K) + n (1 P n ). I prefer this approach because it is differentiable good for Greeks Lecture 1 p. 31
32 Barrier option Alternative 2: again treating the drift and volatility as being approximately constant within each timestep, define the minimum within timestep n as ( ) M n = 2 1 Ŝ n+1 +Ŝ n (Ŝ n+1 Ŝ n ) 2 2b 2 (Ŝ n,t n )h logu n where the U n are i.i.d. uniform [0,1] random variables. The payoff is then f(ŝ) = exp( rt) (Ŝ M K) + 1 minn M n >B With this approach one can stop the path calculation as soon as one Mn drops below B. Lecture 1 p. 32
33 Weak convergence Barrier: comparison to solution 10 0 Weak error MC error 10-1 Error h Lecture 1 p. 33
34 Weak convergence Barrier: h versus 2h solution 10-1 Weak error MC error Error h Lecture 1 p. 34
35 Lookback option This is treated in a similar way to Alternative 2 for the barrier option. We construct a minimum Mn within each timestep and then the payoff is ) f(ŝ) = exp( rt) (ŜM min Mn n This is differentiable, so good for Greeks unlike Alternative 2 for the barrier option. Lecture 1 p. 35
36 Weak convergence Lookback: comparison to true solution 10 0 Weak error MC error Error h Lecture 1 p. 36
37 Weak convergence Lookback: h versus 2h solution 10 0 Weak error MC error Error h Lecture 1 p. 37
38 Final Words Euler-Maruyama gives O(h) weak convergence for European options Mean Square Error analysis shows how to balance weak errors and Monte Carlo sampling errors natural approximation of barrier and lookback options leads to poor O( h) weak convergence due to O( h) path variation within each timestep improved treatment based on Brownian bridge theory approximates behaviour within timestep as simple Brownian motion with constant drift and volatility gives O(h) weak convergence Lecture 1 p. 38
Module 4: Monte Carlo path simulation
Module 4: Monte Carlo path simulation Prof. Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Module 4: Monte Carlo p. 1 SDE Path Simulation In Module 2, looked at the case
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Mike Giles (Oxford) Monte Carlo methods 2 1 / 24 Lecture outline
More informationMultilevel Monte Carlo for Basket Options
MLMC for basket options p. 1/26 Multilevel Monte Carlo for Basket Options Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Oxford-Man Institute of Quantitative Finance WSC09,
More information"Vibrato" Monte Carlo evaluation of Greeks
"Vibrato" Monte Carlo evaluation of Greeks (Smoking Adjoints: part 3) Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Oxford-Man Institute of Quantitative Finance MCQMC 2008,
More informationMultilevel Monte Carlo Simulation
Multilevel Monte Carlo p. 1/48 Multilevel Monte Carlo Simulation Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Oxford-Man Institute of Quantitative Finance Workshop on Computational
More informationModule 2: Monte Carlo Methods
Module 2: Monte Carlo Methods Prof. Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute MC Lecture 2 p. 1 Greeks In Monte Carlo applications we don t just want to know the expected
More informationParallel Multilevel Monte Carlo Simulation
Parallel Simulation Mathematisches Institut Goethe-Universität Frankfurt am Main Advances in Financial Mathematics Paris January 7-10, 2014 Simulation Outline 1 Monte Carlo 2 3 4 Algorithm Numerical Results
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Abdul-Lateef Haji-Ali Based on slides by: Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Haji-Ali (Oxford)
More informationMultilevel path simulation for jump-diffusion SDEs
Multilevel path simulation for jump-diffusion SDEs Yuan Xia, Michael B. Giles Abstract We investigate the extension of the multilevel Monte Carlo path simulation method to jump-diffusion SDEs. We consider
More informationComputational Finance
Path Dependent Options Computational Finance School of Mathematics 2018 The Random Walk One of the main assumption of the Black-Scholes framework is that the underlying stock price follows a random walk
More informationMonte Carlo Simulations
Monte Carlo Simulations Lecture 1 December 7, 2014 Outline Monte Carlo Methods Monte Carlo methods simulate the random behavior underlying the financial models Remember: When pricing you must simulate
More informationAnalysing multi-level Monte Carlo for options with non-globally Lipschitz payoff
Finance Stoch 2009 13: 403 413 DOI 10.1007/s00780-009-0092-1 Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff Michael B. Giles Desmond J. Higham Xuerong Mao Received: 1
More informationMultilevel quasi-monte Carlo path simulation
Multilevel quasi-monte Carlo path simulation Michael B. Giles and Ben J. Waterhouse Lluís Antoni Jiménez Rugama January 22, 2014 Index 1 Introduction to MLMC Stochastic model Multilevel Monte Carlo Milstein
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationEFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS
Commun. Korean Math. Soc. 23 (2008), No. 2, pp. 285 294 EFFICIENT MONTE CARLO ALGORITHM FOR PRICING BARRIER OPTIONS Kyoung-Sook Moon Reprinted from the Communications of the Korean Mathematical Society
More informationComputing Greeks with Multilevel Monte Carlo Methods using Importance Sampling
Computing Greeks with Multilevel Monte Carlo Methods using Importance Sampling Supervisor - Dr Lukas Szpruch Candidate Number - 605148 Dissertation for MSc Mathematical & Computational Finance Trinity
More informationVariance Reduction Through Multilevel Monte Carlo Path Calculations
Variance Reduction Through Mutieve Monte Caro Path Cacuations Mike Gies gies@comab.ox.ac.uk Oxford University Computing Laboratory Mutieve Monte Caro p. 1/30 Mutigrid A powerfu technique for soving PDE
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationEstimating the Greeks
IEOR E4703: Monte-Carlo Simulation Columbia University Estimating the Greeks c 207 by Martin Haugh In these lecture notes we discuss the use of Monte-Carlo simulation for the estimation of sensitivities
More informationGamma. The finite-difference formula for gamma is
Gamma The finite-difference formula for gamma is [ P (S + ɛ) 2 P (S) + P (S ɛ) e rτ E ɛ 2 ]. For a correlation option with multiple underlying assets, the finite-difference formula for the cross gammas
More informationMONTE CARLO EXTENSIONS
MONTE CARLO EXTENSIONS School of Mathematics 2013 OUTLINE 1 REVIEW OUTLINE 1 REVIEW 2 EXTENSION TO MONTE CARLO OUTLINE 1 REVIEW 2 EXTENSION TO MONTE CARLO 3 SUMMARY MONTE CARLO SO FAR... Simple to program
More informationAD in Monte Carlo for finance
AD in Monte Carlo for finance Mike Giles giles@comlab.ox.ac.uk Oxford University Computing Laboratory AD & Monte Carlo p. 1/30 Overview overview of computational finance stochastic o.d.e. s Monte Carlo
More information3.1 Itô s Lemma for Continuous Stochastic Variables
Lecture 3 Log Normal Distribution 3.1 Itô s Lemma for Continuous Stochastic Variables Mathematical Finance is about pricing (or valuing) financial contracts, and in particular those contracts which depend
More informationMonte Carlo Methods for Uncertainty Quantification
Monte Carlo Methods for Uncertainty Quantification Abdul-Lateef Haji-Ali Based on slides by: Mike Giles Mathematical Institute, University of Oxford Contemporary Numerical Techniques Haji-Ali (Oxford)
More informationNumerical schemes for SDEs
Lecture 5 Numerical schemes for SDEs Lecture Notes by Jan Palczewski Computational Finance p. 1 A Stochastic Differential Equation (SDE) is an object of the following type dx t = a(t,x t )dt + b(t,x t
More informationMultilevel Monte Carlo for VaR
Multilevel Monte Carlo for VaR Mike Giles, Wenhui Gou, Abdul-Lateef Haji-Ali Mathematical Institute, University of Oxford (BNP Paribas, Hong Kong) (also discussions with Ralf Korn, Klaus Ritter) Advances
More informationComputational Finance Improving Monte Carlo
Computational Finance Improving Monte Carlo School of Mathematics 2018 Monte Carlo so far... Simple to program and to understand Convergence is slow, extrapolation impossible. Forward looking method ideal
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Simulating Stochastic Differential Equations Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com
More information"Pricing Exotic Options using Strong Convergence Properties
Fourth Oxford / Princeton Workshop on Financial Mathematics "Pricing Exotic Options using Strong Convergence Properties Klaus E. Schmitz Abe schmitz@maths.ox.ac.uk www.maths.ox.ac.uk/~schmitz Prof. Mike
More informationOptimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing
Optimal Search for Parameters in Monte Carlo Simulation for Derivative Pricing Prof. Chuan-Ju Wang Department of Computer Science University of Taipei Joint work with Prof. Ming-Yang Kao March 28, 2014
More informationMath 416/516: Stochastic Simulation
Math 416/516: Stochastic Simulation Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 13 Haijun Li Math 416/516: Stochastic Simulation Week 13 1 / 28 Outline 1 Simulation
More informationKing s College London
King s College London University Of London This paper is part of an examination of the College counting towards the award of a degree. Examinations are governed by the College Regulations under the authority
More informationSimulating Stochastic Differential Equations
IEOR E4603: Monte-Carlo Simulation c 2017 by Martin Haugh Columbia University Simulating Stochastic Differential Equations In these lecture notes we discuss the simulation of stochastic differential equations
More informationCS 774 Project: Fall 2009 Version: November 27, 2009
CS 774 Project: Fall 2009 Version: November 27, 2009 Instructors: Peter Forsyth, paforsyt@uwaterloo.ca Office Hours: Tues: 4:00-5:00; Thurs: 11:00-12:00 Lectures:MWF 3:30-4:20 MC2036 Office: DC3631 CS
More informationMultilevel Monte Carlo Path Simulation
Mutieve Monte Caro Path Simuation Mike Gies gies@comab.ox.ac.uk Oxford University Computing Laboratory First IMA Conference on Computationa Finance Mutieve Monte Caro p. 1/34 Generic Probem Stochastic
More informationMultilevel Change of Measure for Complex Digital Options
Multilevel Change of Measure for Complex Digital Options Jiaxing Wang Somerville College University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance Trinity 2014 This
More informationLecture outline. Monte Carlo Methods for Uncertainty Quantification. Importance Sampling. Importance Sampling
Lecture outline Monte Carlo Methods for Uncertainty Quantification Mike Giles Mathematical Institute, University of Oxford KU Leuven Summer School on Uncertainty Quantification Lecture 2: Variance reduction
More informationDefinition Pricing Risk management Second generation barrier options. Barrier Options. Arfima Financial Solutions
Arfima Financial Solutions Contents Definition 1 Definition 2 3 4 Contenido Definition 1 Definition 2 3 4 Definition Definition: A barrier option is an option on the underlying asset that is activated
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationStochastic Differential Equations in Finance and Monte Carlo Simulations
Stochastic Differential Equations in Finance and Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH China 2009 Outline Stochastic Modelling in Asset Prices 1 Stochastic
More informationComputer labs. May 10, A list of matlab tutorials can be found under
Computer labs May 10, 2018 A list of matlab tutorials can be found under http://snovit.math.umu.se/personal/cohen_david/teachlinks.html Task 1: The following MATLAB code generates (pseudo) uniform random
More informationMonte Carlo Methods in Structuring and Derivatives Pricing
Monte Carlo Methods in Structuring and Derivatives Pricing Prof. Manuela Pedio (guest) 20263 Advanced Tools for Risk Management and Pricing Spring 2017 Outline and objectives The basic Monte Carlo algorithm
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationContinuous Processes. Brownian motion Stochastic calculus Ito calculus
Continuous Processes Brownian motion Stochastic calculus Ito calculus Continuous Processes The binomial models are the building block for our realistic models. Three small-scale principles in continuous
More informationLecture 3. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 6
Lecture 3 Sergei Fedotov 091 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 091 010 1 / 6 Lecture 3 1 Distribution for lns(t) Solution to Stochastic Differential Equation
More informationNEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 MAS3904. Stochastic Financial Modelling. Time allowed: 2 hours
NEWCASTLE UNIVERSITY SCHOOL OF MATHEMATICS, STATISTICS & PHYSICS SEMESTER 1 SPECIMEN 2 Stochastic Financial Modelling Time allowed: 2 hours Candidates should attempt all questions. Marks for each question
More informationA Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option
A Moment Matching Approach To The Valuation Of A Volume Weighted Average Price Option Antony Stace Department of Mathematics and MASCOS University of Queensland 15th October 2004 AUSTRALIAN RESEARCH COUNCIL
More informationTheory and practice of option pricing
Theory and practice of option pricing Juliusz Jabłecki Department of Quantitative Finance Faculty of Economic Sciences University of Warsaw jjablecki@wne.uw.edu.pl and Head of Monetary Policy Analysis
More informationModule 10:Application of stochastic processes in areas like finance Lecture 36:Black-Scholes Model. Stochastic Differential Equation.
Stochastic Differential Equation Consider. Moreover partition the interval into and define, where. Now by Rieman Integral we know that, where. Moreover. Using the fundamentals mentioned above we can easily
More informationFE610 Stochastic Calculus for Financial Engineers. Stevens Institute of Technology
FE610 Stochastic Calculus for Financial Engineers Lecture 13. The Black-Scholes PDE Steve Yang Stevens Institute of Technology 04/25/2013 Outline 1 The Black-Scholes PDE 2 PDEs in Asset Pricing 3 Exotic
More information23 Stochastic Ordinary Differential Equations with Examples from Finance
23 Stochastic Ordinary Differential Equations with Examples from Finance Scraping Financial Data from the Web The MATLAB/Octave yahoo function below returns daily open, high, low, close, and adjusted close
More informationResults for option pricing
Results for option pricing [o,v,b]=optimal(rand(1,100000 Estimators = 0.4619 0.4617 0.4618 0.4613 0.4619 o = 0.46151 % best linear combination (true value=0.46150 v = 1.1183e-005 %variance per uniform
More informationWrite legibly. Unreadable answers are worthless.
MMF 2021 Final Exam 1 December 2016. This is a closed-book exam: no books, no notes, no calculators, no phones, no tablets, no computers (of any kind) allowed. Do NOT turn this page over until you are
More informationMulti-Asset Options. A Numerical Study VILHELM NIKLASSON FRIDA TIVEDAL. Master s thesis in Engineering Mathematics and Computational Science
Multi-Asset Options A Numerical Study Master s thesis in Engineering Mathematics and Computational Science VILHELM NIKLASSON FRIDA TIVEDAL Department of Mathematical Sciences Chalmers University of Technology
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationComputer Exercise 2 Simulation
Lund University with Lund Institute of Technology Valuation of Derivative Assets Centre for Mathematical Sciences, Mathematical Statistics Spring 2010 Computer Exercise 2 Simulation This lab deals with
More informationExotic Derivatives & Structured Products. Zénó Farkas (MSCI)
Exotic Derivatives & Structured Products Zénó Farkas (MSCI) Part 1: Exotic Derivatives Over the counter products Generally more profitable (and more risky) than vanilla derivatives Why do they exist? Possible
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
More informationNumerical Simulation of Stochastic Differential Equations: Lecture 1, Part 1. Overview of Lecture 1, Part 1: Background Mater.
Numerical Simulation of Stochastic Differential Equations: Lecture, Part Des Higham Department of Mathematics University of Strathclyde Course Aim: Give an accessible intro. to SDEs and their numerical
More informationMultilevel Monte Carlo for multi-dimensional SDEs
Mutieve Monte Caro for muti-dimensiona SDEs Mike Gies mike.gies@maths.ox.ac.uk Oxford University Mathematica Institute Oxford-Man Institute of Quantitative Finance MCQMC, Warsaw, August 16-20, 2010 Mutieve
More informationMFE/3F Questions Answer Key
MFE/3F Questions Download free full solutions from www.actuarialbrew.com, or purchase a hard copy from www.actexmadriver.com, or www.actuarialbookstore.com. Chapter 1 Put-Call Parity and Replication 1.01
More informationResearch on Monte Carlo Methods
Monte Carlo research p. 1/87 Research on Monte Carlo Methods Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute Mathematical and Computational Finance Group Nomura, Tokyo, August
More informationFast and accurate pricing of discretely monitored barrier options by numerical path integration
Comput Econ (27 3:143 151 DOI 1.17/s1614-7-991-5 Fast and accurate pricing of discretely monitored barrier options by numerical path integration Christian Skaug Arvid Naess Received: 23 December 25 / Accepted:
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationMultilevel Monte Carlo Methods for American Options
Multilevel Monte Carlo Methods for American Options Simon Gemmrich, PhD Kellog College University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance November 19, 2012
More informationAn Introduction to Stochastic Calculus
An Introduction to Stochastic Calculus Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 2-3 Haijun Li An Introduction to Stochastic Calculus Week 2-3 1 / 24 Outline
More informationMAFS Computational Methods for Pricing Structured Products
MAFS550 - Computational Methods for Pricing Structured Products Solution to Homework Two Course instructor: Prof YK Kwok 1 Expand f(x 0 ) and f(x 0 x) at x 0 into Taylor series, where f(x 0 ) = f(x 0 )
More informationValuation of Asian Option. Qi An Jingjing Guo
Valuation of Asian Option Qi An Jingjing Guo CONTENT Asian option Pricing Monte Carlo simulation Conclusion ASIAN OPTION Definition of Asian option always emphasizes the gist that the payoff depends on
More informationELEMENTS OF MONTE CARLO SIMULATION
APPENDIX B ELEMENTS OF MONTE CARLO SIMULATION B. GENERAL CONCEPT The basic idea of Monte Carlo simulation is to create a series of experimental samples using a random number sequence. According to the
More informationMATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS
MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material.
More informationComputational Finance Least Squares Monte Carlo
Computational Finance Least Squares Monte Carlo School of Mathematics 2019 Monte Carlo and Binomial Methods In the last two lectures we discussed the binomial tree method and convergence problems. One
More informationMULTILEVEL MONTE CARLO FOR BASKET OPTIONS. Michael B. Giles
Proceedings of the 29 Winter Simuation Conference M. D. Rossetti, R. R. Hi, B. Johansson, A. Dunkin, and R. G. Ingas, eds. MULTILEVEL MONTE CARLO FOR BASKET OPTIONS Michae B. Gies Oxford-Man Institute
More informationStochastic Modelling in Finance
in Finance Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH April 2010 Outline and Probability 1 and Probability 2 Linear modelling Nonlinear modelling 3 The Black Scholes
More informationThe Use of Importance Sampling to Speed Up Stochastic Volatility Simulations
The Use of Importance Sampling to Speed Up Stochastic Volatility Simulations Stan Stilger June 6, 1 Fouque and Tullie use importance sampling for variance reduction in stochastic volatility simulations.
More informationMFE/3F Questions Answer Key
MFE/3F Questions Download free full solutions from www.actuarialbrew.com, or purchase a hard copy from www.actexmadriver.com, or www.actuarialbookstore.com. Chapter 1 Put-Call Parity and Replication 1.01
More information2.1 Mathematical Basis: Risk-Neutral Pricing
Chapter Monte-Carlo Simulation.1 Mathematical Basis: Risk-Neutral Pricing Suppose that F T is the payoff at T for a European-type derivative f. Then the price at times t before T is given by f t = e r(t
More informationNumerical Methods II
Numerical Methods II Prof. Mike Giles mike.giles@maths.ox.ac.uk Oxford University Mathematical Institute MC Lecture 3 p. 1 Variance Reduction Monte Carlo starts as a very simple method; much of the complexity
More information- 1 - **** d(lns) = (µ (1/2)σ 2 )dt + σdw t
- 1 - **** These answers indicate the solutions to the 2014 exam questions. Obviously you should plot graphs where I have simply described the key features. It is important when plotting graphs to label
More informationUniversity of Oxford. Robust hedging of digital double touch barrier options. Ni Hao
University of Oxford Robust hedging of digital double touch barrier options Ni Hao Lady Margaret Hall MSc in Mathematical and Computational Finance Supervisor: Dr Jan Ob lój Oxford, June of 2009 Contents
More informationExact Sampling of Jump-Diffusion Processes
1 Exact Sampling of Jump-Diffusion Processes and Dmitry Smelov Management Science & Engineering Stanford University Exact Sampling of Jump-Diffusion Processes 2 Jump-Diffusion Processes Ubiquitous in finance
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/33 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/33 Outline The Binomial Lattice Model (BLM) as a Model
More informationA Continuity Correction under Jump-Diffusion Models with Applications in Finance
A Continuity Correction under Jump-Diffusion Models with Applications in Finance Cheng-Der Fuh 1, Sheng-Feng Luo 2 and Ju-Fang Yen 3 1 Institute of Statistical Science, Academia Sinica, and Graduate Institute
More informationImportance Sampling for Option Pricing. Steven R. Dunbar. Put Options. Monte Carlo Method. Importance. Sampling. Examples.
for for January 25, 2016 1 / 26 Outline for 1 2 3 4 2 / 26 Put Option for A put option is the right to sell an asset at an established price at a certain time. The established price is the strike price,
More informationOutline Brownian Process Continuity of Sample Paths Differentiability of Sample Paths Simulating Sample Paths Hitting times and Maximum
Normal Distribution and Brownian Process Page 1 Outline Brownian Process Continuity of Sample Paths Differentiability of Sample Paths Simulating Sample Paths Hitting times and Maximum Searching for a Continuous-time
More informationMath 239 Homework 1 solutions
Math 239 Homework 1 solutions Question 1. Delta hedging simulation. (a) Means, standard deviations and histograms are found using HW1Q1a.m with 100,000 paths. In the case of weekly rebalancing: mean =
More informationComputer Exercise 2 Simulation
Lund University with Lund Institute of Technology Valuation of Derivative Assets Centre for Mathematical Sciences, Mathematical Statistics Fall 2017 Computer Exercise 2 Simulation This lab deals with pricing
More informationBarrier Option. 2 of 33 3/13/2014
FPGA-based Reconfigurable Computing for Pricing Multi-Asset Barrier Options RAHUL SRIDHARAN, GEORGE COOKE, KENNETH HILL, HERMAN LAM, ALAN GEORGE, SAAHPC '12, PROCEEDINGS OF THE 2012 SYMPOSIUM ON APPLICATION
More informationCredit Risk : Firm Value Model
Credit Risk : Firm Value Model Prof. Dr. Svetlozar Rachev Institute for Statistics and Mathematical Economics University of Karlsruhe and Karlsruhe Institute of Technology (KIT) Prof. Dr. Svetlozar Rachev
More informationMultilevel Monte Carlo methods for finance
Multilevel Monte Carlo methods for finance Mike Giles Mathematical Institute, University of Oxford Oxford-Man Institute of Quantitative Finance HPCFinance Final Conference March 14, 2016 Mike Giles (Oxford)
More informationMarket interest-rate models
Market interest-rate models Marco Marchioro www.marchioro.org November 24 th, 2012 Market interest-rate models 1 Lecture Summary No-arbitrage models Detailed example: Hull-White Monte Carlo simulations
More informationSTOCHASTIC VOLATILITY AND OPTION PRICING
STOCHASTIC VOLATILITY AND OPTION PRICING Daniel Dufresne Centre for Actuarial Studies University of Melbourne November 29 (To appear in Risks and Rewards, the Society of Actuaries Investment Section Newsletter)
More informationMath489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems
Math489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems Steve Dunbar No Due Date: Practice Only. Find the mode (the value of the independent variable with the
More informationSTOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL
STOCHASTIC CALCULUS AND BLACK-SCHOLES MODEL YOUNGGEUN YOO Abstract. Ito s lemma is often used in Ito calculus to find the differentials of a stochastic process that depends on time. This paper will introduce
More informationRisk Neutral Valuation
copyright 2012 Christian Fries 1 / 51 Risk Neutral Valuation Christian Fries Version 2.2 http://www.christian-fries.de/finmath April 19-20, 2012 copyright 2012 Christian Fries 2 / 51 Outline Notation Differential
More informationBluff Your Way Through Black-Scholes
Bluff our Way Through Black-Scholes Saurav Sen December 000 Contents What is Black-Scholes?.............................. 1 The Classical Black-Scholes Model....................... 1 Some Useful Background
More information1 Mathematics in a Pill 1.1 PROBABILITY SPACE AND RANDOM VARIABLES. A probability triple P consists of the following components:
1 Mathematics in a Pill The purpose of this chapter is to give a brief outline of the probability theory underlying the mathematics inside the book, and to introduce necessary notation and conventions
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More information1 Geometric Brownian motion
Copyright c 05 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is
More informationOptimized Least-squares Monte Carlo (OLSM) for Measuring Counterparty Credit Exposure of American-style Options
Optimized Least-squares Monte Carlo (OLSM) for Measuring Counterparty Credit Exposure of American-style Options Kin Hung (Felix) Kan 1 Greg Frank 3 Victor Mozgin 3 Mark Reesor 2 1 Department of Applied
More information