The document discusses stochastic control for optimal dynamic trading strategies. It examines Merton's portfolio problem in various market models using dynamic programming. Specifically:
- It applies dynamic programming to solve Merton's portfolio problem in the Black-Scholes model under different utility functions, showing the optimal strategy is to hold a constant proportion of wealth in the risky asset.
- It also examines the problem with stochastic volatility, finding the problem can still be solved explicitly through a non-stochastic function of time.
- A brief overview presents the difficulties introduced by incorporating transaction costs into the model.
Affine Term Structure Model with Stochastic Market Price of RiskSwati Mital
- The document proposes a new affine term structure model that combines principal components analysis with a stochastic market price of risk.
- Principal components provide useful information about yield curves and only three components explain over 95% of yield variation.
- Previous models linked risk premium deterministically to return-predicting factors like slope, but this could result in unrealistic risk premium levels.
- The new model introduces an additional state variable to capture the stochastic market price of risk and break the deterministic link between risk premium and return-predicting factors.
The document discusses pricing interest rate derivatives using the one factor Hull-White short rate model. It begins with an introduction to short rate models and the Hull-White model specifically. It describes how the Hull-White model can be calibrated to market prices by relating its parameter θ to the market term structure. The document then discusses implementing the Hull-White model using trinomial trees and pricing constant maturity swaps.
- The document analyzes forecasting volatility for the MSCI Emerging Markets Index using a Stochastic Volatility model solved with Kalman Filtering. It derives the Stochastic Differential Equations for the model and puts them into State Space form solved with a Kalman Filter.
- Descriptive statistics on the daily returns of the MSCI Emerging Markets Index ETF from 2011-2016 show a mean close to 0, standard deviation of 0.01428, negative skewness, and kurtosis close to a normal distribution. The model will be evaluated against a GARCH model.
This document discusses various mathematical models used in finance to model stock prices and returns. It introduces random walk models, the lognormal model, general equilibrium theories, the Capital Asset Pricing Model (CAPM), and the Arbitrage Pricing Theory (ATP). The CAPM and ATP are equilibrium asset pricing models based on assumptions like rational investors seeking to maximize returns while minimizing risk.
This document describes an uncertain volatility model for pricing equity option trading strategies when the volatilities are uncertain. It uses the Black-Scholes Barenblatt equation developed by Avellaneda et al. to derive price bounds. The model is implemented in C++ using recombining trinomial trees to discretize the asset prices over time and space. The code computes the upper and lower price bounds by solving the Black-Scholes Barenblatt PDE using numerical techniques, with the volatility set based on the sign of the option gamma.
The comparative study of finite difference method and monte carlo method for ...Alexander Decker
This document compares the finite difference method and Monte Carlo method for pricing European options. The finite difference method solves the Black-Scholes partial differential equation by approximating it on a grid, while the Monte Carlo method simulates asset price paths and averages discounted payoffs. The study finds that while both methods agree with the Black-Scholes price, the finite difference method converges faster and is more accurate for standard European options, whereas Monte Carlo is better suited for exotic options due to its flexibility.
IRJET- Analytic Evaluation of the Head Injury Criterion (HIC) within the Fram...IRJET Journal
This document presents an analytic evaluation of the Head Injury Criterion (HIC) within the framework of constrained optimization theory. The HIC is a weighted impulse function used to predict the probability of closed head injury based on measured head acceleration. Previous work analyzed the unclipped HIC function, but the clipped HIC formulation used in practice limits the evaluation window duration. The author develops analytic relationships for determining the window initiation and termination points to maximize the clipped HIC function. Example applications illustrate the general solutions for when head acceleration is defined by a single function or composite functions over the evaluation domain.
Affine Term Structure Model with Stochastic Market Price of RiskSwati Mital
- The document proposes a new affine term structure model that combines principal components analysis with a stochastic market price of risk.
- Principal components provide useful information about yield curves and only three components explain over 95% of yield variation.
- Previous models linked risk premium deterministically to return-predicting factors like slope, but this could result in unrealistic risk premium levels.
- The new model introduces an additional state variable to capture the stochastic market price of risk and break the deterministic link between risk premium and return-predicting factors.
The document discusses pricing interest rate derivatives using the one factor Hull-White short rate model. It begins with an introduction to short rate models and the Hull-White model specifically. It describes how the Hull-White model can be calibrated to market prices by relating its parameter θ to the market term structure. The document then discusses implementing the Hull-White model using trinomial trees and pricing constant maturity swaps.
- The document analyzes forecasting volatility for the MSCI Emerging Markets Index using a Stochastic Volatility model solved with Kalman Filtering. It derives the Stochastic Differential Equations for the model and puts them into State Space form solved with a Kalman Filter.
- Descriptive statistics on the daily returns of the MSCI Emerging Markets Index ETF from 2011-2016 show a mean close to 0, standard deviation of 0.01428, negative skewness, and kurtosis close to a normal distribution. The model will be evaluated against a GARCH model.
This document discusses various mathematical models used in finance to model stock prices and returns. It introduces random walk models, the lognormal model, general equilibrium theories, the Capital Asset Pricing Model (CAPM), and the Arbitrage Pricing Theory (ATP). The CAPM and ATP are equilibrium asset pricing models based on assumptions like rational investors seeking to maximize returns while minimizing risk.
This document describes an uncertain volatility model for pricing equity option trading strategies when the volatilities are uncertain. It uses the Black-Scholes Barenblatt equation developed by Avellaneda et al. to derive price bounds. The model is implemented in C++ using recombining trinomial trees to discretize the asset prices over time and space. The code computes the upper and lower price bounds by solving the Black-Scholes Barenblatt PDE using numerical techniques, with the volatility set based on the sign of the option gamma.
The comparative study of finite difference method and monte carlo method for ...Alexander Decker
This document compares the finite difference method and Monte Carlo method for pricing European options. The finite difference method solves the Black-Scholes partial differential equation by approximating it on a grid, while the Monte Carlo method simulates asset price paths and averages discounted payoffs. The study finds that while both methods agree with the Black-Scholes price, the finite difference method converges faster and is more accurate for standard European options, whereas Monte Carlo is better suited for exotic options due to its flexibility.
IRJET- Analytic Evaluation of the Head Injury Criterion (HIC) within the Fram...IRJET Journal
This document presents an analytic evaluation of the Head Injury Criterion (HIC) within the framework of constrained optimization theory. The HIC is a weighted impulse function used to predict the probability of closed head injury based on measured head acceleration. Previous work analyzed the unclipped HIC function, but the clipped HIC formulation used in practice limits the evaluation window duration. The author develops analytic relationships for determining the window initiation and termination points to maximize the clipped HIC function. Example applications illustrate the general solutions for when head acceleration is defined by a single function or composite functions over the evaluation domain.
Market risk and liquidity of the risky bondsIlya Gikhman
This document discusses modeling the effect of liquidity on risky bond pricing using a reduced form approach. It begins by presenting a simplified model where default can only occur at maturity. It then extends this to a discrete time approximation for default occurrence. The key concepts discussed are:
- Defining bid and ask prices for risk-free and corporate bonds to model liquidity spread
- Using a single price framework and extending it to account for liquidity spread
- Modeling the corporate bond price as a random variable based on default/no default scenarios
- Defining market and spot prices of bonds and the associated market risks for buyers and sellers
- Estimating the recovery rate and default probability given observations of spot prices over time
This document provides an introduction to algorithms and data structures. It discusses algorithm design and analysis tools like Big O notation and recurrence relations. Selecting the smallest element from a list, sorting a list using selection sort and merge sort, and merging two sorted lists are used as examples. Key points made are that merge sort has better time complexity than selection sort, and any sorting algorithm requires at least O(n log n) comparisons. The document also introduces data structures like arrays and linked lists, and how the organization of data impacts algorithm performance.
This document provides an overview of a course on the design and analysis of computer algorithms taught by Professor David Mount at the University of Maryland in Fall 2003. The course will cover algorithm design techniques like dynamic programming and greedy algorithms. Major topics will include graph algorithms, minimum spanning trees, shortest paths, and computational geometry. Later sections will discuss intractable problems and approximation algorithms. When designing algorithms, students are expected to provide a description, proof of correctness, and analysis of time and space efficiency. Mathematical background on algorithm analysis, including asymptotic notation and recurrences, will be reviewed.
The document summarizes models and tools for portfolio planning, including:
1) It discusses key modelling issues like cardinality constraints, discrete decision variables, and trade scheduling algorithms.
2) It provides computational results for various portfolio models solved using different solvers, showing performance metrics like iterations and time.
3) It outlines different portfolio models like mean-variance, factor models, and index trackers that are commonly used in practice.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
help.mbaassignments@gmail.com
or
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Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Seminar talk at École des Ponts ParisTech about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model". - Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
This document discusses the economic-theoretic approach to index numbers. It begins by outlining how output and input price indices and output and input quantity indices can be defined using revenue, cost, and distance functions based on microeconomic production theory. It then shows that under certain conditions, such as a translog functional form, the Törnqvist and Fisher indices provide good approximations to the theoretical indices and can be computed directly from price and quantity data without full knowledge of production functions.
1. An algorithm is a sequence of unambiguous instructions to solve a problem within a finite amount of time. It takes an input, processes it, and produces an output.
2. Designing an algorithm involves understanding the problem, choosing a computational model and problem-solving approach, designing and proving the algorithm's correctness, analyzing its efficiency, coding it, and testing it.
3. Important algorithm design techniques include brute force, divide and conquer, decrease and conquer, transform and conquer, dynamic programming, and greedy algorithms.
This document provides an overview of data envelopment analysis (DEA) for measuring efficiency. It begins with introducing efficiency measurement concepts and the constant returns to scale DEA model. It then discusses input and output orientations, and how to estimate technical and allocative efficiency using DEA. The document provides examples to illustrate DEA models and efficiency measures, including presenting results from a simple numerical example. It summarizes key aspects of standard DEA models and their applications.
This document is the MSc project of Mohamed Raagi submitted to Brunel University London in October 2015. It examines excess rates of return from jump risks using geometric Lévy models for asset pricing. The project reviews recent developments in these models and simulates price processes involving jumps to analyze excess rate of return behavior and impact. It introduces Lévy processes and geometric Lévy martingale models as tools for derivative pricing. Specific models discussed include Brownian motion, Poisson, compound Poisson, and geometric gamma. The document also covers option pricing and simulations for each model.
Basics of Algorithms and Analysis of algorithm is in there, which includes Time complexity , space complexity, three cases ( best, average, worst) and analysis of Insertion sort.
*For knowledge purpose only*
*Hope you'll come up with better one*
Affine cascade models for term structure dynamics of sovereign yield curvesLAURAMICHAELA
Rafael Serrano profesor de la Universidad del Rosario
Resumen:
In the first part of the talk, I will present an introduction to stochastic affine short rate models for term structure of yield curves In the second part, I will focus on a recursive affine cascade with persistent factors for which the number of parameters, under specifications, is invariant to the size of the state space and converges to a stochastic limit as the number of factors goes to infinity. The cascade construction thereby overcomes dimensionality difficulties associated with general affine models. We contrast two specfifications of the model using linear Kalman filter for a panel of Colombian sovereign yields.
This document outlines a lecture on linear factor models and event studies in financial econometrics. It discusses the motivation for linear factor pricing models and outlines two main econometric approaches: time-series regression based tests that focus on intercepts when factors are excess returns, and cross-sectional regression based residual tests when factors are not excess returns. It also provides details on specific methods for time-series approach including the joint intercept test of Black, Jensen and Scholes (1972) and extensions by Gibbons, Ross and Shanken (1989) and MacKinlay and Richardson (1991).
This document provides an outline and overview of linear factor models and event studies. For linear factor models, it discusses the time-series approach of regressing returns on factors to test pricing restrictions, the cross-sectional approach of regressing average returns on factor betas, and compares the two approaches. It also discusses formulating linear factor models and tests in a generalized method of moments framework. For event studies, it briefly outlines the motivation and basic methodology.
This document provides an overview of econometric methods for estimating economic relationships such as production, cost, and profit functions. It discusses estimating parameters for different functional forms using ordinary least squares and maximum likelihood methods. It also covers imposing equality constraints to satisfy properties implied by economic theory and testing these constraints using statistical tests.
This document provides an overview of the topics covered in Unit V: Linear Programming. It begins with an introduction to operations research and some example problems that can be modeled as linear programs. It then discusses formulations of linear programs, including the standard and slack forms. The document outlines the simplex algorithm for solving linear programs and how to convert between standard and slack forms. It provides examples demonstrating these concepts. The key topics covered are linear programming models, formulations, and the simplex algorithm.
Pricing Exotics using Change of NumeraireSwati Mital
The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
This document summarizes a master's thesis on integrating market views into quantitative portfolio allocation. It introduces the concepts of optimal asset allocation, mean-variance optimization, and dimension reduction using linear factor models. It then discusses incorporating investor views through information sets and the efficient market hypothesis. The Black-Litterman model uses a Gaussian market assumption, CAPM reverse optimization, and Bayesian updating to integrate views. An alternative approach uses f-divergences to measure distortions between probability distributions and translate views into information gain. The thesis concludes by proposing directions for future research.
Statistical Arbitrage
Pairs Trading, Long-Short Strategy
Cyrille BEN LEMRID

1 Pairs Trading Model 5
1.1 Generaldiscussion ................................ 5 1.2 Cointegration ................................... 6 1.3 Spreaddynamics ................................. 7
2 State of the art and model overview 9
2.1 StochasticDependenciesinFinancialTimeSeries . . . . . . . . . . . . . . . 9 2.2 Cointegration-basedtradingstrategies ..................... 10 2.3 FormulationasaStochasticControlProblem. . . . . . . . . . . . . . . . . . 13 2.4 Fundamentalanalysis............................... 16
3 Strategies Analysis 19
3.1 Roadmapforstrategydesign .......................... 19 3.2 Identificationofpotentialpairs ......................... 19 3.3 Testingcointegration ............................... 20 3.4 Riskcontrolandfeasibility............................ 20
4 Results
22
2
Contents

Introduction
This report presents my research work carried out at Credit Suisse from May to September 2012. This study has been pursued in collaboration with the Global Arbitrage Strategies team.
Quantitative analysis strategy developers use sophisticated statistical and optimization techniques to discover and construct new algorithms. These algorithms take advantage of the short term deviation from the ”fair” securities’ prices. Pairs trading is one such quantitative strategy - it is a process of identifying securities that generally move together but are currently ”drifting away”.
Pairs trading is a common strategy among many hedge funds and banks. However, there is not a significant amount of academic literature devoted to it due to its proprietary nature. For a review of some of the existing academic models, see [6], [8], [11] .
Our focus for this analysis is the study of two quantitative approaches to the problem of pairs trading, the first one uses the properties of co-integrated financial time series as a basis for trading strategy, in the second one we model the log-relationship between a pair of stock prices as an Ornstein-Uhlenbeck process and use this to formulate a portfolio optimization based stochastic control problem.
This study was performed to show that under certain assumptions the two approaches are equivalent.
Practitioners most often use a fundamentally driven approach, analyzing the performance of stocks around a market event and implement strategies using back-tested trading levels.
We also study an example of a fundamentally driven strategy, using market reaction to a stock being dropped or added to the MSCI World Standard, as a signal for a pair trading strategy on those stocks once their inclusion/exclusion has been made effective.
This report is organized as follows. Section 1 provides some background on pairs trading strategy. The theoretical results are described in Section 2. Section 3
Market risk and liquidity of the risky bondsIlya Gikhman
This document discusses modeling the effect of liquidity on risky bond pricing using a reduced form approach. It begins by presenting a simplified model where default can only occur at maturity. It then extends this to a discrete time approximation for default occurrence. The key concepts discussed are:
- Defining bid and ask prices for risk-free and corporate bonds to model liquidity spread
- Using a single price framework and extending it to account for liquidity spread
- Modeling the corporate bond price as a random variable based on default/no default scenarios
- Defining market and spot prices of bonds and the associated market risks for buyers and sellers
- Estimating the recovery rate and default probability given observations of spot prices over time
This document provides an introduction to algorithms and data structures. It discusses algorithm design and analysis tools like Big O notation and recurrence relations. Selecting the smallest element from a list, sorting a list using selection sort and merge sort, and merging two sorted lists are used as examples. Key points made are that merge sort has better time complexity than selection sort, and any sorting algorithm requires at least O(n log n) comparisons. The document also introduces data structures like arrays and linked lists, and how the organization of data impacts algorithm performance.
This document provides an overview of a course on the design and analysis of computer algorithms taught by Professor David Mount at the University of Maryland in Fall 2003. The course will cover algorithm design techniques like dynamic programming and greedy algorithms. Major topics will include graph algorithms, minimum spanning trees, shortest paths, and computational geometry. Later sections will discuss intractable problems and approximation algorithms. When designing algorithms, students are expected to provide a description, proof of correctness, and analysis of time and space efficiency. Mathematical background on algorithm analysis, including asymptotic notation and recurrences, will be reviewed.
The document summarizes models and tools for portfolio planning, including:
1) It discusses key modelling issues like cardinality constraints, discrete decision variables, and trade scheduling algorithms.
2) It provides computational results for various portfolio models solved using different solvers, showing performance metrics like iterations and time.
3) It outlines different portfolio models like mean-variance, factor models, and index trackers that are commonly used in practice.
Dear students get fully solved assignments
Send your semester & Specialization name to our mail id :
help.mbaassignments@gmail.com
or
call us at : 08263069601
Hierarchical Deterministic Quadrature Methods for Option Pricing under the Ro...Chiheb Ben Hammouda
Seminar talk at École des Ponts ParisTech about our recently published work "Hierarchical adaptive sparse grids and quasi-Monte Carlo for option pricing under the rough Bergomi model". - Link of the paper: https://www.tandfonline.com/doi/abs/10.1080/14697688.2020.1744700
This document discusses the economic-theoretic approach to index numbers. It begins by outlining how output and input price indices and output and input quantity indices can be defined using revenue, cost, and distance functions based on microeconomic production theory. It then shows that under certain conditions, such as a translog functional form, the Törnqvist and Fisher indices provide good approximations to the theoretical indices and can be computed directly from price and quantity data without full knowledge of production functions.
1. An algorithm is a sequence of unambiguous instructions to solve a problem within a finite amount of time. It takes an input, processes it, and produces an output.
2. Designing an algorithm involves understanding the problem, choosing a computational model and problem-solving approach, designing and proving the algorithm's correctness, analyzing its efficiency, coding it, and testing it.
3. Important algorithm design techniques include brute force, divide and conquer, decrease and conquer, transform and conquer, dynamic programming, and greedy algorithms.
This document provides an overview of data envelopment analysis (DEA) for measuring efficiency. It begins with introducing efficiency measurement concepts and the constant returns to scale DEA model. It then discusses input and output orientations, and how to estimate technical and allocative efficiency using DEA. The document provides examples to illustrate DEA models and efficiency measures, including presenting results from a simple numerical example. It summarizes key aspects of standard DEA models and their applications.
This document is the MSc project of Mohamed Raagi submitted to Brunel University London in October 2015. It examines excess rates of return from jump risks using geometric Lévy models for asset pricing. The project reviews recent developments in these models and simulates price processes involving jumps to analyze excess rate of return behavior and impact. It introduces Lévy processes and geometric Lévy martingale models as tools for derivative pricing. Specific models discussed include Brownian motion, Poisson, compound Poisson, and geometric gamma. The document also covers option pricing and simulations for each model.
Basics of Algorithms and Analysis of algorithm is in there, which includes Time complexity , space complexity, three cases ( best, average, worst) and analysis of Insertion sort.
*For knowledge purpose only*
*Hope you'll come up with better one*
Affine cascade models for term structure dynamics of sovereign yield curvesLAURAMICHAELA
Rafael Serrano profesor de la Universidad del Rosario
Resumen:
In the first part of the talk, I will present an introduction to stochastic affine short rate models for term structure of yield curves In the second part, I will focus on a recursive affine cascade with persistent factors for which the number of parameters, under specifications, is invariant to the size of the state space and converges to a stochastic limit as the number of factors goes to infinity. The cascade construction thereby overcomes dimensionality difficulties associated with general affine models. We contrast two specfifications of the model using linear Kalman filter for a panel of Colombian sovereign yields.
This document outlines a lecture on linear factor models and event studies in financial econometrics. It discusses the motivation for linear factor pricing models and outlines two main econometric approaches: time-series regression based tests that focus on intercepts when factors are excess returns, and cross-sectional regression based residual tests when factors are not excess returns. It also provides details on specific methods for time-series approach including the joint intercept test of Black, Jensen and Scholes (1972) and extensions by Gibbons, Ross and Shanken (1989) and MacKinlay and Richardson (1991).
This document provides an outline and overview of linear factor models and event studies. For linear factor models, it discusses the time-series approach of regressing returns on factors to test pricing restrictions, the cross-sectional approach of regressing average returns on factor betas, and compares the two approaches. It also discusses formulating linear factor models and tests in a generalized method of moments framework. For event studies, it briefly outlines the motivation and basic methodology.
This document provides an overview of econometric methods for estimating economic relationships such as production, cost, and profit functions. It discusses estimating parameters for different functional forms using ordinary least squares and maximum likelihood methods. It also covers imposing equality constraints to satisfy properties implied by economic theory and testing these constraints using statistical tests.
This document provides an overview of the topics covered in Unit V: Linear Programming. It begins with an introduction to operations research and some example problems that can be modeled as linear programs. It then discusses formulations of linear programs, including the standard and slack forms. The document outlines the simplex algorithm for solving linear programs and how to convert between standard and slack forms. It provides examples demonstrating these concepts. The key topics covered are linear programming models, formulations, and the simplex algorithm.
Pricing Exotics using Change of NumeraireSwati Mital
The intention of this essay is to show how change of numeraire technique is used in pricing derivatives with complex payoffs. In the first instance, we apply the technique to pricing European Call Options and then use the same method to price an exotic Power Option.
In this short notice, we present structure of the perfect hedging. Closed form formulas clarify the fact that Black-Scholes (BS) portfolio which provides perfect hedge only at initial moment. Holding portfolio over a certain period implies additional cash flow, which could not be imbedded in BS pricing scheme, and therefore BS option price cannot be derived without additional cash flow which affects BS option price.
There are several significant drawbacks in derivative price modeling which relate to global regulations of the derivatives market. Here we present a unified approach which in stochastic market interprets option price as a random variable. Therefore spot price does not complete characteristic of the price in stochastic environment. Complete derivatives price includes the spot price as well as thevalue of market risk implied by the use of the spot price. This interpretation is similar to the notion of therandom variable in Probability Theory in which an estimate of the random variable completely defined by its cumulative distribution function
This document summarizes a master's thesis on integrating market views into quantitative portfolio allocation. It introduces the concepts of optimal asset allocation, mean-variance optimization, and dimension reduction using linear factor models. It then discusses incorporating investor views through information sets and the efficient market hypothesis. The Black-Litterman model uses a Gaussian market assumption, CAPM reverse optimization, and Bayesian updating to integrate views. An alternative approach uses f-divergences to measure distortions between probability distributions and translate views into information gain. The thesis concludes by proposing directions for future research.
Statistical Arbitrage
Pairs Trading, Long-Short Strategy
Cyrille BEN LEMRID

1 Pairs Trading Model 5
1.1 Generaldiscussion ................................ 5 1.2 Cointegration ................................... 6 1.3 Spreaddynamics ................................. 7
2 State of the art and model overview 9
2.1 StochasticDependenciesinFinancialTimeSeries . . . . . . . . . . . . . . . 9 2.2 Cointegration-basedtradingstrategies ..................... 10 2.3 FormulationasaStochasticControlProblem. . . . . . . . . . . . . . . . . . 13 2.4 Fundamentalanalysis............................... 16
3 Strategies Analysis 19
3.1 Roadmapforstrategydesign .......................... 19 3.2 Identificationofpotentialpairs ......................... 19 3.3 Testingcointegration ............................... 20 3.4 Riskcontrolandfeasibility............................ 20
4 Results
22
2
Contents

Introduction
This report presents my research work carried out at Credit Suisse from May to September 2012. This study has been pursued in collaboration with the Global Arbitrage Strategies team.
Quantitative analysis strategy developers use sophisticated statistical and optimization techniques to discover and construct new algorithms. These algorithms take advantage of the short term deviation from the ”fair” securities’ prices. Pairs trading is one such quantitative strategy - it is a process of identifying securities that generally move together but are currently ”drifting away”.
Pairs trading is a common strategy among many hedge funds and banks. However, there is not a significant amount of academic literature devoted to it due to its proprietary nature. For a review of some of the existing academic models, see [6], [8], [11] .
Our focus for this analysis is the study of two quantitative approaches to the problem of pairs trading, the first one uses the properties of co-integrated financial time series as a basis for trading strategy, in the second one we model the log-relationship between a pair of stock prices as an Ornstein-Uhlenbeck process and use this to formulate a portfolio optimization based stochastic control problem.
This study was performed to show that under certain assumptions the two approaches are equivalent.
Practitioners most often use a fundamentally driven approach, analyzing the performance of stocks around a market event and implement strategies using back-tested trading levels.
We also study an example of a fundamentally driven strategy, using market reaction to a stock being dropped or added to the MSCI World Standard, as a signal for a pair trading strategy on those stocks once their inclusion/exclusion has been made effective.
This report is organized as follows. Section 1 provides some background on pairs trading strategy. The theoretical results are described in Section 2. Section 3
On the Optimality of Kelly Strategies (Presentacion).pdfmaikelcorleoni
This document summarizes a presentation on the optimality of Kelly strategies given by Mark Davis and Sébastien Lleo. It begins with an outline of the topics to be covered, including Kelly strategies, insights from the Merton model of lognormal asset prices, and extending Kelly strategies to more complex models. It then reviews Kelly strategies and fractional Kelly investment, showing they are optimal in the Merton model. The presentation uses a change of measure approach to solve the optimal investment problem, demonstrating that fractional Kelly strategies naturally arise from a fund separation theorem. Special cases like the physical measure and dangers of overbetting are also discussed.
Intro to Quant Trading Strategies (Lecture 1 of 10)Adrian Aley
This document provides an overview of a lecture on quantitative trading strategies given by Dr. Haksun Li. It discusses technical analysis from a scientific perspective and outlines Numerical Method's quantitative trading research process. This includes translating trading intuitions into mathematical models, coding the strategies, evaluating their properties through simulation, and live trading. Moving average crossover is presented as an example strategy and approaches to model it quantitatively are described.
Is the Macroeconomy Locally Unstable and Why Should We Care?ADEMU_Project
The document discusses evidence that macroeconomic dynamics may exhibit locally unstable behavior like limit cycles rather than stable convergence to a steady state. It presents a reduced-form model allowing for nonlinearities, estimated on total hours and other labor market variables. The estimates provide intriguing evidence of limit cycle dynamics, with simulations showing attracting cycles. This suggests macroeconomic fluctuations could be endogenous rather than just responses to shocks, with implications for stabilization policy.
PREDICTIVE EVALUATION OF THE STOCK PORTFOLIO PERFORMANCE USING FUZZY CMEANS A...ijfls
The aim of this paper is to investigate the trend of the return of a portfolio formed randomly or for any
specific technique. The approach is made using two techniques fuzzy: fuzzy c-means (FCM) algorithm and
the fuzzy transform, where the rules used at fuzzy transform arise from the application of the FCM
algorithm. The results show that the proposed methodology is able to predict the trend of the return of a
stock portfolio, as well as the tendency of the market index. Real data of the financial market are used from
2004 until 2007.
This paper analyzes the swap rates issued by the China Inter-bank Offered Rate(CHIBOR) and
selects the one-year FR007 daily data from January 1st, 2019 to June 30th, 2019 as a sample. To fit the data,
we conduct Monte Carlo simulation with several typical continuous short-term swap rate models such as the
Merton model, the Vasicek model, the CIR model, etc. These models contain both linear forms and nonlinear
forms and each has both drift terms and diffusion terms. After empirical analysis, we obtain the parameter
values in Euler-Maruyama scheme and relevant statistical characteristics of each model. The results show that
most of the short-term swap rate models can fit the swap rates and reflect the change of trend, while the CKLSO
model performs best.
This document appears to be an assignment submission for a financial engineering course. It includes a plagiarism declaration signed by the student, Andrew Hair. The assignment contains 11 questions addressing interest rate derivatives and modeling using the Vasicek model. Code is provided in MATLAB to generate simulations and analyze interest rate data based on the questions.
Here are the key advantages and disadvantages of arrays:
Advantages of arrays:
- Increased directivity - Arrays allow signals to be reinforced in desired directions while cancelled in others, providing improved directivity over a single antenna.
- Increased gain - The increased directivity of an array leads to higher gain compared to a single antenna. This allows for longer communication ranges.
- Beam steering - By adjusting the phase or time delay of signals to each antenna, the main transmission direction of an array can be steered electronically without moving physical elements.
Disadvantages of arrays:
- Increased complexity - Arrays require additional hardware for power distribution to each antenna element, as well as phase/time delay control circuitry
This document presents the pricing, hedging, and risk management of a portfolio containing two basket Asian options. It first describes the contractual terms of the two options and the stocks in their respective baskets. It then outlines the necessary preliminary computations, including bootstrapping the discount curve, determining reset dates, and analyzing historical stock data. The document models the stock price dynamics under both Normal Inverse Gaussian (NIG) and Geometric Brownian Motion (GBM) and prices the options using Monte Carlo simulation. It further examines the Greeks, hedging strategy, and calculates Value at Risk both with and without hedging using different approximations.
International journal of engineering and mathematical modelling vol2 no2_2015_1IJEMM
Management of the portfolios containing low liquidity assets is a tedious problem. The buyer proposes the price that can differ greatly from the paper value estimated by the seller, so the seller can not liquidate his portfolio instantly and waits for a more favorable offer. To minimize losses and move the theory towards practical needs one
can take into account the time lag of the liquidation of an illiquid asset. Working in the Merton’s optimal consumption framework with continuous time we consider an optimization problem for a portfolio with an illiquid, a risky and a risk-free asset. While a standard Black-Scholes market describes the liquid part of the investment the illiquid asset is sold at an exogenous random moment with prescribed liquidation time distribution. The investor has the logarithmic utility function as a limit case of a HARA-type utility. Different distributions of the liquidation time of the illiquid asset are under consideration - a classical exponential distribution andWeibull distribution that is more practically relevant. Under certain conditions we show the existence
of the viscosity solution in both cases. Applying numerical methods we compare classical Merton’s strategies and the optimal consumption-allocation strategies for portfolios with different liquidation time distributions of an illiquid asset.
This document summarizes a master's thesis that implemented a continuous sequential importance resampling (CSIR) algorithm to estimate predictive densities in stochastic volatility (SV) models. The thesis began with an introduction to relevant econometrics concepts. It then explained SV models and particle filtering approaches. The thesis described implementing and testing functions to develop an R package for CSIR estimation in SV models. Diagnostics and parameter estimates from simulated and real stock return data were reported. The thesis concluded by discussing the package's applications and potential for future development.
COVARIANCE ESTIMATION AND RELATED PROBLEMS IN PORTFOLIO OPTIMICruzIbarra161
COVARIANCE ESTIMATION AND RELATED PROBLEMS IN PORTFOLIO OPTIMIZATION
Ilya Pollak
Purdue University
School of Electrical and Computer Engineering
West Lafayette, IN 47907
USA
ABSTRACT
This overview paper reviews covariance estimation problems and re-
lated issues arising in the context of portfolio optimization. Given
several assets, a portfolio optimizer seeks to allocate a fixed amount
of capital among these assets so as to optimize some cost function.
For example, the classical Markowitz portfolio optimization frame-
work defines portfolio risk as the variance of the portfolio return,
and seeks an allocation which minimizes the risk subject to a target
expected return. If the mean return vector and the return covariance
matrix for the underlying assets are known, the Markowitz problem
has a closed-form solution.
In practice, however, the expected returns and the covariance
matrix of the returns are unknown and are therefore estimated from
historical data. This introduces several problems which render the
Markowitz theory impracticable in real portfolio management appli-
cations. This paper discusses these problems and reviews some of
the existing literature on methods for addressing them.
Index Terms— Covariance, estimation, portfolio, market, fi-
nance, Markowitz
1. INTRODUCTION
The return of a security between trading day t1 and trading day t2
is defined as the change in the closing price over this time period,
divided by the closing price on day t1. For example, the daily (i.e.,
one-day) return on trading day t is defined as (p(t)−p(t−1))/p(t−
1) where p(t) is the closing price on day t and p(t−1) is the closing
price on the previous trading day. Note that if t is a Monday or the
day after a holiday, the previous trading day will not be the same as
the previous calendar day.
Suppose an investment is made into N assets whose return vec-
tor is R, modeled as a random vector with expected return µ =
E[R] and covariance matrix Λ = E[(R − µ)(R − µ)T ]. In other
words, R = (R(1), . . . , R(N))T where R(n) is the return of the n-th
asset. It is assumed throughout the paper that the covariance matrix
Λ is invertible. This assumption is realistic, since it is quite unusual
in practice to have a set of assets whose linear combination has re-
turns exactly equal to zero. Even if an investment universe contained
such a set, the number of assets in the universe could be reduced to
eliminate the linear dependence and make the covariance matrix in-
vertible.
Out of these N assets, a portfolio is formed with allocation
weights w = (w(1), . . . , w(N))T . The n-th weight is defined as the
amount invested into the n-th asset, as a fraction of the overall invest-
ment into the portfolio: if the overall investment into the portfolio is
$D, and $D(n) is invested into the n-th asset, then w(n) = D(n)/D.
Therefore, by definition, the weights sum to one:
w
T
1 = 1, (1)
where 1 is an N -vector of ones. Note that some of the weights may
be negative, ...
Peter Warken - Effective Pricing of Cliquet Options - Masters thesis 122015Peter Warken
This thesis examines pricing methods for cliquet options, a type of path-dependent option linked to equity indices. The author develops a semi-closed form pricing formula for cliquet options in a Black-Scholes market and compares it to Monte Carlo simulation. The impact of stochastic volatility and interest rates on pricing is also examined. Numerical experiments illustrate how market parameters affect cliquet option prices. The pricing approach is also applied to similar products like sum cap contracts.
Complex Event Processing within a Managed Accounts PlatformMarkus Dieckmann
The document describes a complex event processing (CEP) system for analyzing real-time market data against numerous trading strategies within a managed accounts platform. The main challenges are efficiently processing millions of rule checks per second from a broad stream of market events influenced by many trading strategies, and reducing rule checks without missing opportunities. An efficient parallel processing architecture is needed to distribute the large processing load. Significant reduction of calculations is also required when filtering the input events.
Management science uses analytical methods and decision-making techniques to help organizations operate efficiently and manage risk. It draws from fields like applied mathematics, statistics, and computer modeling to solve problems in areas such as production, inventory management, and scheduling. Some common techniques include linear programming, nonlinear programming, integer programming, stochastic programming, queuing theory, and simulation modeling.
Master of Computer Application (MCA) – Semester 4 MC0079Aravind NC
The document describes mathematical models and provides examples of different types of models. It discusses linear vs nonlinear models, deterministic vs probabilistic models, static vs dynamic models, discrete vs continuous models, and deductive vs inductive vs floating models. It also explains the Erlang family of distributions used in queuing systems and provides the probability density function and cumulative distribution function. Finally, it outlines the graphical method algorithm for solving a linear programming problem with two variables in 8 steps.
The Vasicek model is one of the earliest stochastic models for modeling the term structure of interest rates. It represents the movement of interest rates as a function of market risk, time, and the equilibrium value the rate tends to revert to. This document discusses parameter estimation techniques for the Vasicek one-factor model using least squares regression and maximum likelihood estimation on historical interest rate data. It also covers simulating the term structure and pricing zero-coupon bonds under the Vasicek model. The two-factor Vasicek model is introduced as an extension of the one-factor model.
Mc0079 computer based optimization methods--phpapp02Rabby Bhatt
This document discusses mathematical models and provides examples of different types of mathematical models. It begins by defining a mathematical model as a description of a system using mathematical concepts and language. It then classifies mathematical models in several ways, such as linear vs nonlinear, deterministic vs probabilistic, static vs dynamic, discrete vs continuous, and deductive vs inductive vs floating. The document provides examples and explanations of each type of model. It also discusses using finite queuing tables to analyze queuing systems with a finite population size. In summary, the document outlines different ways to classify mathematical models and provides examples of applying various types of models.
Systemic Risk Modeling - André Lucas, April 16 2014SYRTO Project
This document discusses challenges in modeling systemic risk and presents a new class of time series models for systemic risk modeling. It introduces a factor copula model that uses a multivariate skewed-t density with time-varying parameters to assess joint and conditional measures of financial sector risk. The model uses a conditional law of large numbers to efficiently compute risk measures without simulation for high-dimensional, non-Gaussian data. It also defines measures to analyze systemic influence and connectedness within the financial system.
Systemic Risk Modeling - André Lucas, April 16 2014
Varun Balupuri - Thesis
1. Stochastic Control for Optimal Dynamic
Trading Strategies
Varun Balupuri
Department of Mathematics
King’s College London
The Strand, London WC2R 2LS
United Kingdom
Email: varun.nair-balupuri@kcl.ac.uk
Tel: +44 (0)583 248 930
19 September 2016
Report submitted in partial fulfillment of the
requirements for the degree of MSc in Finan-
cial Mathematics in the University of London
2. Abstract
In this paper, we apply dynamic programming to solve Merton’s portfolio prob-
lem in the classical Black-Scholes model under the familiar cases of power, ex-
ponential and logarithmic utility, where we show that the optimal strategy is to
keep a constant proportion of wealth in the risky asset.
We also examine the problem in in the presence of stochastic volatility.
The problem is found to be solved by a non-stochastic function of time and
we perform Monte Carlo simulations to numerically verify this. A focus of this
paper is on numerical estimates and analysis to back up theoretical results.
A brief overview of the problem in the presence of transaction costs and the
associated difficulties is presented in Chapter 4.
1
4. Chapter 1
INTRODUCTION
1.1 Background
Stochastic optimisation is concerned with controlling dynamic systems with
stochastic pertubations to maximise (or minimise) some criteria, usually a func-
tion representing value or a stopping time attaining said value. Richard Bellman
pioneered the dynamic programming equation and approach to these types of
problems in a 1954 paper, ’The theory of dynamic programming’ [1]
In the 1970’s the mathematics of dynamic programming and stochastic
control was applied in the context of economics/financial mathematics, first
recieving widespread recognition due to Merton’s 1973 paper [2].
From a mathematical approach, the field of dynamic portfolio choice was
first approached by Merton in his much celebrated, seminal papers ([3] [4]),
building upon the Black-Scholes framework, Merton employed techniques from
from stochastic control to show how to create optimal portfolios consisting of
a risky and riskless asset in a friction-less market with constant volatility and
constant interest rates. Since then, Merton’s ’portfolio problem’ has become a
highly studied topic and there have been many papers addressing improvements
and extensions to this framework.
In this paper we consider maximizing an investors value function using
dynamic programming to solve the associated Heath-Jacobi-Bellman equation
with various utility functions in the HARA1
class, as Merton did and we place
extra emphasis on numerical analysis and verification of these results through
Monte Carlo methods.
In Chapter 1 we look at the topic of stochastic control in the context of
portfolio optimisation, building the initial framework and presenting the relevant
mathematical theorems and results which are important for later chapters.
In Chapter 2 we look at the classical 2-asset Merton portfolio problem in
1hyperbolic absolute risk aversion (HARA) functions are easy to model mathematically
and have properties which make them suited to modeling reality.
3
5. CHAPTER 1. INTRODUCTION 4
the Black-Scholes framework under exponential utility and power utility. In the
most basic sense, the investor wants to decide what proportion of his wealth
to keep in the risky asset to maximise their expected utility over a finite time
horizon. In addition to optimising the investor’s ’risky wealth’ we consider the
case when a consumption parameter is added, seeking to optimise the investors
rate of removal of money as well. We will consider Merton’s problem over
an infinite horizon too, which is useful when modeling how to invest/consume
ones savings until retirement or death. We will show via the use of dynamic
programming, that all of these problems have a closed form solution and can be
explicitly calculated.
Since Merton’s problem is built upon many unrealistic assumptions, such
as the ability to trade continuously without fee and constant interest rates, the
addition of transaction costs and stochastic volatility is a very important step
in building a more realistic model.
Mathematical models have been introduced to add stochastic volatility
and stochastic interest rates to model stock prices. We examine Merton’s prob-
lem with stochastic volatility in Chapter 3, focusing on Heston’s model. Using
Liu and Muhle-Karbe’s approach we show that even in this setting, Merton’s
problem has an explicit solution and verify the theoretical results by discretizing
and using Monte Carlo methods.
The addition of transaction costs are briefly considered in chapter 4. Pro-
portional transaction costs were first incorporated By Davis and Norman in
1990 [6]. Since the investor cannot make a large amount of trades without ac-
cruing large fees, the optimal strategy changes from continuous re-balancing of
the portfolio to one which involves only making trades if the ratio falls outside
of a region, called the ’no trade’ region. Finding the boundaries of this region
is not a trivial task and computational approaches fall beyond the scope of this
thesis.
This paper focuses on models where an explicit solution to Merton’s prob-
lem exists, however this is not generally the case. We can only use the Heath-
Jacobi-Bellman equation when the value function is sufficiently smooth. For
many problems, this is not the case and alternate methods such as the viscos-
ity solution approach introduced by Crandall and Lions (1980) is an effective
method to attack such problems. A modern development in the literature is on
backward stochastic differential equations, which provide a probabilistic repre-
sentation of non-linear PDE’s. These topics are not discussed in this paper.
1.2 Market model and notation
Unless explicitly stated, consider a frictionless market with two assets, a risky
stock, denoted by S, modelled by a stochastic price process (St)t≥0, which
follows the standard Black-Scholes dynamics:
dSt = St(µdt + σdWt)
6. CHAPTER 1. INTRODUCTION 5
and a riskless ’bank account’ Bt, whose dynamics are given by the ODE:
dBt = rBtdt
Definition 1.2.1 (Self Financing Portfolio). In continuous time, consider a
portfolio consisting of ∆t units of the risky asset S and φt units of the riskless
asset B. This portfolio has a time t value of Vt = ∆tSt + φtBt. We say that
the portfolio is self-financing2
if:
Vt = V0 +
t
0
∆wdSw +
t
0
φwdBw
We assume the investor’s actions have no affect on the market and the
investor’s strategy is self-financing.
Unless explicity stated otherwise, let µ, σ, r ∈ R and Wt be a standard one-
dimensional Brownian Motion on complete probability space (Ω, F, P), with
(Ft)t≥0 being a filtration.
Definition 1.2.2 (Adapted Process). A process (Xt)t≥0 is adapted with respect
to a filtration F = (Ft)t≥0 if:
∀t ≥ 0, Xt is measurable.
Definition 1.2.3 (Progressively Measurable Process). A process (Xt)t≥0 is
progressively measurable with respect to F if ∀t ≥ 0:
the map [0, t] × Ω defined by (s, ω) → Xs(ω) is B([0, t]) ⊗ Ft-measurable.
1.3 Preliminaries
Important theorems and results which are applied in latter chapters are pre-
sented here without formal proof. References to derivations are provided for
the interested reader.
Consider stochastic differential equation:
dXt = b(Xt, αt)dt + σ(Xt, αt)dWt (1.1)
with X0 = x , (αt)t≥0 is a progressively measurable process. Let functions a, σ
satisfy the Lipschitz condition.
Finite Horizon Case
For proofs and extensions of the following theorems, see pg. 40-46 of Pham’s
textbook [5]
2Intuitively this means that no money is exogenously added or withdrawn from the portfolio
7. CHAPTER 1. INTRODUCTION 6
Fix T ∈ (0, ∞). Define
A = α : E
T
0
|b(0, αt)|2
+ |σ(0, αt)|2
dt < ∞
For functions f, g define our gain function as
J(t, x, α) = E
t,x
T
t
f(s, Xs, αs)ds + g(XT )
The value function linked to this is v(t, x) = sup
α∈A
J(t, x, a)
Theroem 1.3.1 (Dynamic Programming Principle). For all t1 ∈ [t, T]
v(t, x) = sup
α∈A
E
t,x
t1
t
f(s, Xs, αs)ds + v(t1, Xt1
)
If v is smooth, making use of Itˆo’s formula leads us to the HJB equation
in the finite-horizon case.
Theroem 1.3.2 (Finite Horizon Hamilton-Jacobi-Bellman equation). 3
Let
α = a ∈ R with a arbitrary. The value function v(t, x) satisfies the following
partial differential equation (if the supremum is finite), known as the Hamilton-
Jacobi-Bellman equation:
∂v
∂t
+ sup
a
f(t, x, a) + b(x, a)
∂v
∂x
+
1
2
|σ(x, a)|2 ∂2
v
∂x2
= 0
with boundary condition v(T, x) = g(x)
Infinite Horizon Case
For the infinite horizon class, we consider problems where T = ∞. Xt is time-
homogeneous and we discount the gain function to maintain finiteness of J(x, α)
For β > 0, let A(x) be the set of admissible controls α satisfying
E
∞
0
e−βs
|f(XS, αS)|ds < ∞
Define
J(x, α) = E
∞
0
e−βs
f(Xx
S, αS)ds
and similarly to the finite horizon case, the value function is v(x) = sup
α∈A((§)
J(x, α)
Theroem 1.3.3 (Infinite Horizon Hamilton-Jacobi-Bellman equation). Assum-
ing our SDE follows (1.1) v(t, x) satisfies ∀x ∈ R:
βv(x) + sup
a
f(x, a) + b(x, a)
∂v
∂x
+
1
2
|σ(x, a)|2 ∂2
v
∂x2
= 0
3In Pham’s proof, the infinitesimal generator, La is used. We do not use it throughout
this paper for consistency and to highlight the explicit equations
8. CHAPTER 1. INTRODUCTION 7
A vital result in dynamic programming is the verification theorem, which
ensures that for an optimal control problem, a candidate solution of a non-linear
PDE coincides with the value function.
We do not state the theorem here, but it’s consequences are important for
ensuring our solutions are indeed optimal controls.[5].
Utility Classes
The aim of Merton’s portfolio problem is to optimise the investors (expected)
utility. We must take into account the investors risk aversion. A more risk
averse investor would prefer to place more of his capital in the riskless asset to
ensure guaranteed returns rather than a riskier stock.
Definition 1.3.4 (Hyperbolic Absolute Risk Aversion). A utility function U(x)
is said to be a HARA utility function if and only if it is of the form:
U(x) =
1 − γ
γ
ax
1 − γ
+ b
γ
with a > 0 and ax
1−γ + b > 0.
In this paper we consider three types of utility function:
• Exponential : U(x) = −e−αx
• Power : U(x) = 1
1−γ x1−γ
• Logarithmic : U(x) = log(x)
To see that exponential utility belongs in the class, take the limit as γ tends to
∞ and b = 1 in Definition 1.3.4. Similarly, the logarithmic utility is obtained
by setting a = 1 and observing the limit as γ tends to 0.
These utility functions all fall under the HARA class and for many of the
optimization problems studied in this paper, they lead to closed form solutions.
In the case of power utility which we will consider in this section, we can
simplify calculations by using a result about its homotheticity.
Proposition 1.3.5 (Power Utility Homotheticy). Let U(x) = 1
γ xγ
for γ ∈
(0, 1). Then V (t, x, y) is increasing and concave in both x and y and ∀ρ > 0:
V (t, ρx, ρy) = ργ
V (t, x, y)
For a proof, see Davis and Norman 1990 Theorem 3.1 [6]
9. Chapter 2
Portfolio Allocation
2.1 Optimising Risky Wealth Under Exponen-
tial Utility
In this subsection, we assume interest rate, r = 0. Hence the risk-free account
follows Bt = 1. Let Xt denote the investor’s total wealth at time t. Let πt be
the proportion of total wealth that the investor has invested in the risky asset
S at time t.
Hence, πtXt is the amount of wealth in units of currency invested in the
stock at time t and πtXt
St
is the number of shares the investor owns at t.
We wish to optimize over all possible dynamic strategies πt, the proportion
of the investor’s total wealth which is ’risky wealth’ in order to maximize their
expected utility. Xt follows dynamics1
dXt =
πtXt
St
dSt +
(1 − πt)Xt
Bt
dBt
dXt =
πtXt
St
St(µdt + σdWt)
= πtXt(µdt + σdWt)
(2.1)
The sum of proportions invested over all assets in the market model must equal
1. In this two asset market this of course means the investor places (1 − πt)Xt
units of wealth into the riskless asset. If πt > 1, this implies the investor is short
in the riskless asset. Similarly if πt < 0 the investor is short in the risky asset.
Define utility function U : R → R with risk aversion parameter α by
U(x) = −e−αx
This falls under the class of constant absolute risk aversion (CARA) utility
1We may also impose the additional restriction that Xt ≥ 0 depending on whether the
investor is allowed to continue trading after bankruptcy.
8
10. CHAPTER 2. PORTFOLIO ALLOCATION 9
functions and is concave and increasing on R implying ∀t ∈ [0, T], V (t, x) is also
increasing and concave. ([5] see pg.52 for proof)
Define the set of all admissible trading strategies
A = {π(·) : bounded, adapted stochastic process s.t: Xπ
x ≥ 0 P-a.s }
Where Xπ
x (t) corresponds to the wealth of the investor following strategy π at
time t with initial wealth x. We want to maximise expected utility over all
π ∈ A
Define our value function:
V (t, x) = sup
π∈A
E
t,x
[U(XT )] (2.2)
The Hamilton-Jacobi-Bellman equation corresponding to (2.2) is given by The-
orem 1.3.22
∂V
∂t
+ sup
π∈R
πxµ
∂V
∂x
+
1
2
π2
σ2
x2 ∂2
V
∂x2
= 0 (2.3)
With boundary condition V (T, x) = U(x) Denote optimal π by π∗
. Differenti-
ating the sumpremum part in (2.3) with respect to π and equating to zero to
find optimal π yields:
d
dπ
πxµ
∂V
∂x
+
1
2
π2
σ2
x2 ∂2
V
∂x2
= 0
xµ
∂V
∂x
+ x2
σ2
π∗ ∂2
V
∂x2
= 0
=⇒ π∗
=
− µ∂V
∂x
xσ2 ∂2V
∂x2
(2.4)
Substituting this optimal value of π∗
into (2.3) gives rise to a non-linear PDE :
∂V
∂t
−
µ2
2σ2
∂V
∂x
2
∂2V
∂x
= 0 (2.5)
This is a separable partial differential equation, so we can find a solution using
an ansatz of the form V (t, x) = −e−αx−αβ(T −t)
We now have:
∂V
∂t
= −αβe−αx−αβ(T −t)
∂V
∂x
= αe−αx−αβ(T −t)
∂2
V
∂x2
= −α2
e−αx−αβ(T −t)
(2.6)
2In this case the corresponding variable are f(t, x, π) = 0, b(x, π)dt+σ(x, π)dWt = πµxdt+
σπxdWt
11. CHAPTER 2. PORTFOLIO ALLOCATION 10
Substituting (2.6) into (2.5) gives us a value for β
β =
µ2
2ασ2
(2.7)
Also, π∗
in (2.4) becomes
π∗
=
µ
xσ2α
(2.8)
It is important to note that π∗
is constant (x is fixed as X0 = x) in agreement
to Merton’s discovery, meaning the optimal strategy is for the investor to keep
a constant fraction of his wealth, π∗
in the risky asset S and rebalance his
portfolio continuously to maintain this proportion.
The analytic exponential utility V (t, x) is
V (t, x) = − exp(−αx − α(
µ2
2ασ2
)(T − t)) (2.9)
The dynamics of Xt derived in (2.1) now reduce to Arithmetic Brownian Motion,
which by direct integration gives:
dXt =
µ2
σ2α
dt +
µ
σα
dWt =⇒ Xt = X0 +
µ2
σ2α
t +
µ
σα
Wt (2.10)
Hence Xt
Dist
∼ N(X0 + µ2
σ2α t, µ2
σ2α2 t). We see that if no modification to the
model is made to prevent the investor from stopping when he hits bankruptcy,
then Xt can become negative.
2.2 Extension to non-zero interest rates
We extend our market model from the previous section by now adding a constant
interest rate r to the riskless asset.
Let St follow:
dSt = St[(r + µ)dt + σdWt]
and let the riskless asset obey dBt = rBtdt as usual.
Consider a ’power utility’/constant relative risk aversion (CRRA) utility
function:
U(x) =
1
1 − γ
x1−γ
, γ ∈ (0, 1)
Consider as before all admissible dynamic trading strategies, πt. Again we want
to optimise V (t, x) as defined in (2.2) for our new utility function on a finite
time horizon.
The investor’s total wealth obeys
12. CHAPTER 2. PORTFOLIO ALLOCATION 11
dXt =
πtXt
St
dSt +
Xt(1 − πt)
Bt
dBt
= πtXt[(r + µ)dt + σdWt] + Xt(1 − πt)rdt
= Xt[πtµ + r]dt + XtπtσdWt
(2.11)
By applying Theorem 1.3.2, the corresponding HJB equation:
∂V
∂t
+ sup
π∈A
(πxµ + rx)
∂V
∂x
+
1
2
π2
σ2
x2 ∂2
V
∂x2
= 0 (2.12)
With boundary condition V (T, x) = U(x)
By using ansatz V (t, x) = 1
1−γ x1−γ
f(t) for some function f and following
the same procedure as in the exponential utility case to find a solution for π∗
and β results in
∂V
∂t
=
1
1 − γ
x1−γ
f (t)
∂V
∂x
= x−γ
f(t)
∂2
V
∂x2
= −γx−γ−1
f(t)
(2.13)
Substituting (2.13) in to (2.12) for optimal π∗
and unknown f(t) gives us
explicit solutions:
π∗
=
µ
γσ2
(2.14)
1
1 − γ
x1−γ
f (t) = (
−µ2
2γσ2
− r)x1−γ
f(t)
=⇒ f (t) = −(1 − γ)(
µ2
2γσ2
+ r)f(t)
(2.15)
The solution of the ODE in (2.15) with the initial condition f(T) = 1 is given
by3
f(t) = e
(1−γ)( µ2
2γσ2 +r)(T −t)
3This is obvious as at time T we have V (T, x) = U(x) = 1
1−γ
x1−γ
13. CHAPTER 2. PORTFOLIO ALLOCATION 12
So the previously unknown function f(t) is:
f(t) = e(T −t)(1−γ)β
Where we have used:
β =
1
2
µ2
γσ2
+ r
We say beta represents a ’fictitious safe rate’. If the investor was to place his
wealth in a safe asset with compound interest rate β, they would attain the
same utility as following trading strategy π∗
.
This gives us an explicit representation for V (t, x)
V (t, x) =
1
1 − γ
x1−γ
exp (T − t)(1 − γ) + (
1
2
µ2
γσ2
+ r) (2.16)
Optimal π∗
is constant and is given by:
π∗
=
µ
γσ2
In this case, the dynamics of Xt in (2.11) reduce to Geometric Brownian
Motion
dXt = Xt[πtµ + r]dt + XtπtσdWt
= (
µ2
γσ2
+ r)Xtdt +
µ
γσ
dWt
=⇒ Xt = X0 exp (
µ2
γσ2
+ r) −
µ2
2γ2σ2
t +
µ
γσ
Wt
(2.17)
2.3 Numerical Implementation/Monte Carlo Es-
timates
In this section we wish to verify previous results. We must first use a method
to disretize from the continuous time setting in the previous section so we can
perform numerical analysis.
Once we have a model to simulate stock prices and the dynamics of the
wealth processes, we can perform thousands of simulations and use these paths
to find an estimate for the value function in (2.2).
Simulation Model
The stock price process and wealth process with constant π∗
follow geometric
brownian motion and can be discretized by various methods such as the Euler
14. CHAPTER 2. PORTFOLIO ALLOCATION 13
Maruyama method or the Milstein method, but since GBM has a closed form
solution it is possible to simulate the log-price process then exponentiate.
Assume dSt = St(µdt + σdWt). Let us divide the interval [0, T] into N
equal time steps such that T = Nδ then the following code simulates a price
path, making use of the property Wt
Dist
∼
√
tN(0, 1) 4
#GenStockPrice.py
def gen_BS_pricepath(mu,sigma,S0,N,T):
delta = T/float(N)
s = np.zeros((N+1)
s[0] = np.log(S0)
for i in range(1,N+1):
s[i] = s[i-1] + (mu-0.5*sigma**2)*delta +
sigma*np.sqrt(delta)*np.random.normal()
return np.exp(s)
The riskless asset is not influenced by any stochastic element, so to simu-
late the riskless balance at each δ5
we simply loop over:
#Bank Balance logic
b[i] = b[i-1]*np.exp(r*delta)
Optimal pi
For exponential utility U(x) = −e−αx
, using our derived V (t, x) = −e−αx−αβ(T −t)
in (2.9) with β = −µ2
2ασ2 with test parameters: t = 0, T = 1, x = 1, µ = 0.05, σ =
0.2, α = 1 gives π∗
= 1.25.
Comparing the theoretically derived value function, V (t, x) to a Monte
Carlo simulation of the expected utility of the wealth process with optimal π∗
and 10,000 scenarios gives us a result in very close agreement:
• Analytic V (t, x) = −0.35678
• Simulated V (t, x) = −0.35656
To verify that this value of π∗
is indeed the optimal choice for these pa-
rameters, we show numerically that the expected utility decreases when π∗
is
perturbed.
For the power utility case with γ = 1
2 and σ = 0.2, a Monte Carlo estimate
with 10000 simulations of V (t, x) lead to results in very close agreement to the
theoretical values of V (t, x) shown in the table below
4N(µ, σ2) is the CDF of a normal random variable with mean µ and variance σ2
5Interest is continuously compounded in these simulations.
15. CHAPTER 2. PORTFOLIO ALLOCATION 14
Figure 2.1: Proportion Risky Wealth vs. Expected Utility for π-values at in-
crements of 0.02 with 1,000 simulations each. Optimal π∗
= 1.25 shown by red
line
Figure 2.2: A sample stock price path with optimal π∗
and optimal number of
shares π∗
Xt
St
for µ = 0.05, σ = 0.2, α = 1 and T = 1 year. (Exponential Utility)
Parameters Monte Carlo V (t, x) Theoretical V (t, x)
µ = 0.02, r = 0 6.383373 6.388118
µ = 0.05, r = 0 6.712548 6.732454
µ = 0, r = 0 6.324555 6.324555
µ = 0.02, r = 0.02 6.524122 6.517167
µ = 0.05, r = 0.02 6.884561 6.868459
µ = −0.02, r = 0.02 6.514090 6.517167
16. CHAPTER 2. PORTFOLIO ALLOCATION 15
We turn our attention now to the investor’s risk aversion parameter. In-
tuitively, increasing α (or γ in the power utility case) means the investor is
less prone to taking on risk via the risky asset. Consequentially, their expected
wealth should be lower, but standard deviation should also be smaller. Con-
versely a low risk aversion means the investor is willing to invest a greater
proportion of his/her wealth in the risky asset and (in the case of these parame-
ters) will result in a higher expected terminal wealth but with a higher standard
deviation. This can be seen in Figure 2.3 for exponential utility, with the final
wealth distribution Xt tending to N(X0 + µ2
σ2α t, µ2
σ2α2 t) as expected due to the
ABM behaviour of Xt.
In the case of power utility function, the wealth dynamics are given by
the Geometric Brownian Motion in (2.17), and so in this case Xt will be log-
normally distributed and this behaviour can be clearly seen in Figure 2.4 with
the lack of symmetry and longer right tail.
17. CHAPTER 2. PORTFOLIO ALLOCATION 16
Figure 2.3: Histogram of terminal wealth distribution for different risk aversions
for exonential case utility case. µ = 0.05, σ = 0.2, T = 1 year.
18. CHAPTER 2. PORTFOLIO ALLOCATION 17
Figure 2.4: Histogram of terminal wealth distribution for different risk aversions
for power utility case. µ = 0.05, σ = 0.2, T = 1 year
19. CHAPTER 2. PORTFOLIO ALLOCATION 18
2.3.1 Re-balancing the portfolio
The results we have so far derived rely on the investor to be able to rebalance
his portfolio continuously at time. However, in practice it is impossible for
an to rebalance in such a manner. In this subsection, we look at rebalancing
our portfolio weights at various frequencies to see how this affects our terminal
wealth and utility.
We have derived a closed form solution for the wealth process Xt and
shown that in the absence of transaction costs or fees, the optimal control pit
is constant. This gives us a theoretical value in a continuous time setting with
instant and continuous rebalancing given by (2.17) in the case of power utility
and (2.10) in the case of exponential utility.
We shall consider the power utility case, with Xt behaving as in (2.17).
Recall, the stock price obeys Geometric Brownian Motion and the riskless ac-
count is continuously compounded at a constant interest rate of r.
Figure 2.5: Sample path of theoretical total wealth process, with amount of
wealth in stock and in riskless if continuous rebalancing is performed.
Figure 2.5 demonstrates how Xt, wealth in stock and wealth in bank evolve
in the case of continuous portfolio rebalancing.
As before, divide [0, T] into N equal time steps such that T = Nδ. Let
Xiδ be the investor’s total wealth at discrete time point iδ for i ∈ [0, N]. Let
XS
iδ denote the amount of wealth invested in the stock S at iδ and XB
iδ be the
amount of wealth invested in the riskless asset B.
At certain regular time-points kiδ for some k ∈ N the investor rebalances
20. CHAPTER 2. PORTFOLIO ALLOCATION 19
their portfolio6
, buying or selling the risky asset (stock) such that the proportion
of wealth they have invested in the stock is again π∗
.
At each rebalance step kiδ, the investor want to re-attain the propotion
πt = π∗
so he must transfer
Rkiδ = (1 − π∗
)XS
kiδ − π∗
XB
kiδ
from the stock to the riskless asset.
At these rebalancing steps, the usual evolution process for the stock is
replaced by XS
kiδ − Rkiδ. Similarly for the riskless asset, we replace XB
kiδ with
XB
kiδ + Rkiδ.
Figure 2.6: Rebalancing πt with k = 50. πt evolves as usual according to the
evolution of St and Bt until every 50th day, where the investor buys or sells
shares to bring πt back to π∗
, in this case π∗
= 1.25
Figure 2.6 demonstrates how the proportion of wealth held in the stock S
ie. πt varies with time. The same parameters are used as at the start of Section
2.3.
When rebalancing is performed less than daily (k = 5) as in figure 2.7
the investor’s total actual wealth is very close to the theoretical wealth process
given in (2.17), but begins to deviate from optimal.
Figure 2.8 shows the discrepancy between the theoretical wealth pro-
cess and the investors actual wealth process with different rebalancing peri-
ods, demonstrating how actual wealth deviates from the optimal wealth path if
perfect ratio π∗
is obeyed continuously.
6For simplicity assume there are 250 trading days in a year, 20 trading days in a month
and 5 in a week (in reality there are approximately 252 trading days in a year).
21. CHAPTER 2. PORTFOLIO ALLOCATION 20
Figure 2.7: Wealth in stock, riskless account and total wealth with continuous
time result over 5 years (T = 5, X0 = 1)
22. CHAPTER 2. PORTFOLIO ALLOCATION 21
Figure 2.8: Optimal stock wealth (red) vs. actual stock wealth (green) for
various rebalancing frequencies.(a) = Daily, (b) = Weekly, (c) = Fortnightly,
(d) = Monthly, (e) = Bi-monthly, (f) = No rebalancing
23. CHAPTER 2. PORTFOLIO ALLOCATION 22
2.4 Addition of Consumption and Infinite Hori-
zons
In this section we add an additional variable to our model. Now, let us assume
the investor wants to optimize how they withdraw wealth from their portfolio
in order to spend on goods and services.
Let us assume the stock price evolves as dSt = St[(r + µ)dt + σdWt] as in
section 2.2. Let πt be the proportion of wealth invested in the stock at time t
and ct be the ’consumption rate’.
The SDE for Xt is:
dXt =
πtXt
St
dSt +
Xt(1 − πt)
Bt
dBt − ctdt
= πtXt[(r + µ)dt + σdWt] + Xt(1 − πt)rdt − ctdt
= (πtµXt + rXt − ct)dt + XtπtσdWt
(2.18)
Note in the case of zero-interest rate r = 0, which we will consider from now on
for simplicity, the SDE for Xt becomes
dXt = (µπtXt − ct)dt + σπtXtdWt (2.19)
We consider an infinite time horizon and we want to maximise over (π, c) ∈ A×C
where A is the set of admissible control processes α and C is the set of control
processes for consumption c.7
In this problem, the investor is trying to maximise log-utility of his con-
sumption. Our value function is given by:
V (x) = sup
(π,c)∈A×C
E
x
∞
0
e−δt
log(ct)dt (2.20)
Since we are working in the infinite horizon case, the corresponding Hamilton-
Jacobi-Bellman equation is given by Theorem 2.21. The coefficients b(x, π) and
σ(x, π) in Theorem 2.21 correspond to (πµx − c) and σπx as in (2.19). We also
have terminal condition f = log(c).
− δV + sup
(π,c)∈R2
(πµx − c)
∂V
∂x
+
1
2
(πσx)2 ∂2
V
∂x2
+ log(c) = 0 (2.21)
Let π∗
, c∗
denote optimal values of π, c respectively.
To find maximum values of π and c, we can take partial derivatives of the
supremum part of (2.21), ∂
∂c (· · · ) = 0 and ∂
∂π (· · · ) = 0.
7c has the restrictions that ct ≥ 0 and ∀t, ∞
0 |ct| < ∞
24. CHAPTER 2. PORTFOLIO ALLOCATION 23
=⇒ −
∂V
∂x
+
1
c∗
= 0
c∗
= 1/
∂V
∂x
Similarly,
π∗
=
− µ∂V
∂x
xσ2 ∂2V
∂x2
=
µ
σ2
Hence we see that the optimal consumption is proportional to the investor’s
wealth at time t. Using ansatz V (x) = 1
δ log(x) + C1 we calculate derivatives
∂V
∂x
=
1
δx
∂2
V
∂x2
=
−1
δx2
We see that c∗
= δx8
and the HJB equation (2.21) becomes
0 = µπ∗
−
c∗
x
−
1
2
(π∗
σ)2
+ δ log(c∗
) − δ log(x) − δ2
C1
0 =
µ2
σ
− δ −
µ2
2σ2
+ δ log(
δx
x
) − δ2
C1
After re-arranging,
C1 =
µ2
− 2σ2
δ + 2σ2
δlog(δ)
2δ2σ2
and for optimal (π∗
, c∗
), the SDE for Xt as in (2.19) becomes
dXt = (
µ2
σ2
− δ)Xtdt +
µ
σ
XtdWt
As in previous sections, the solution is in the form of Geometric Brownian
Motion
Xt = X0 exp (−δ +
µ2
σ2
−
µ2
2σ2
)t +
µ
σ
Wt (2.22)
V (x) is given by:
V (x) =
1
δ
log(x) +
µ2
− 2σ2
δ + 2σ2
δlog(δ)
2δ2σ2
(2.23)
8c∗
t is a constant fraction of Xt, meaning that the investors consumption rate is linearly
linked to his current wealth.
25. CHAPTER 2. PORTFOLIO ALLOCATION 24
Numerical Results
From the value function (2.23), we aim to find an approximation to the V (x) by
simulating many wealth paths obeying (2.22). Since this problem is stated in
the context of an infinite horizon, we take a large value for T for approximation
purposes. Let T = 100 years.
For parameters µ = 0.1, σ = 0.4, X0 = S0 = 1, δ = 0.5, T = 100 years,
assuming 250 trading days per year and running 1000 simulations yields an
estimation ˆV (x) = −3.28909282. This is in close agreement to the analytic
expression for the value function, V (x) = −3.26129436
Figure 2.9: Optimal consumption, bank and stock wealth with stock price path
and total wealth shown. Note how consumption is a constant proportion of
total wealth. µ = 0.1, σ = 0.4, X0 = S0 = 1, δ = 0.5, T = 1 year.
26. Chapter 3
Extension to Stochastic Volatil-
ity
It is unrealistic over longer time horizons to assume that interest rates and
volatility will be constant. In this chapter we examine Merton’s problem in the
presence of stochastic volatility.
3.1 Merton’s Problem in the Heston Model
In this section we address solving Merton’s problem in the presence of stochastic
volatility. We examine the problem in the framework of the much celebrated
and studied Heston model pioneered by Steven Heston in his 1993 paper [11] .
Assume St follows
dSt = (µYt + r)Stdt + YtStdWS
t
µ is the rate of return of the stock, as in the Black-Scholes framework
and we assume fixed constant interest rate r, but the main difference being the
volatility in the Heston model is itself a stochastic process, following dynamics
dYt = κ(θ − Yt)dt + ξ YtdWY
t
We say κ is the rate of reversion, ie: the rate at which Yt returns to the
long term mean given by θ. ξ is the volatility of the volatility.
In this model there are two driving Brownian Motions, WS
t and WY
t .
These are correlated with correlation ρ ∈ [0, 1].1
1In order to keep the volatility positive, we enforce that 2κθ > ξ2. This is known at the
Feller Condition.
25
27. CHAPTER 3. EXTENSION TO STOCHASTIC VOLATILITY 26
Let us consider a finite horizon problem, where the investor seeks to op-
timise proportion of wealth πt invested in the risky-asset. Let Xt denote the
investors wealth at time t. Xt follows
dXt =
Xtπt
St
dSt +
Xt(1 − πt)
Bt
dBt (3.1)
= Xt[πtµYt + r]dt + πtXt YtdWS
t (3.2)
The investor’s optimisation problem is to maximise
V (t, x, y) = sup
π
E
t,x,y
[U(XT )]
Where the utility function is a power utility of the form U(x) = 1
1−γ x1−γ
The HJB equation in the finite horizon case is:
∂V
∂t
+sup
π
x(πµy + r)
∂V
∂x
+ κ(θ − y)
∂V
∂y
+
1
2
x2
π2
y
∂2
V
∂x2
+ xπρξy
∂V
∂xy
+
1
2
ξy
1
2
ξ2
y
∂2
V
∂y2
In a similar style to our constant volatility 1-dimensional derivation, to find
optimal πt, we differentiate the supremum part and equate to zero.
d
dπ
x(πµy + r)
∂V
∂x
+ κ(θ − y)
∂V
∂y
+
1
2
x2
π2
y
∂2
V
∂x2
+ xπρξy
∂V
∂xy
+
1
2
ξy
1
2
ξ2
y
∂2
V
∂y2
= 0 =⇒
µxy
∂V
∂x
+ x2
yπ∗ ∂2
V
∂x2
+ xρyξ
∂2
V
∂xy
= 0
=⇒ π∗
=
− µx∂V
∂x − ρξx∂2
V
∂xy
x2 ∂2V
∂x2
π∗
=
− µ∂V
∂x − ρξ ∂2
V
∂xy
x∂2V
∂x2
(3.3)
Substituting this into our HJB equation reduces it to:
∂V
∂t
=
µ
2
(∂V
∂x )2
∂2V
∂x2
+ µρξy
∂V
∂x
∂2
V
∂xy
∂2V
∂x2
+
ρ2
ξ2
y
2
∂2
V
∂xy
∂2V
∂x2
− rv
∂V
∂x
− κ(θ − y)
∂V
∂y
−
ξy
2
∂2
V
∂y2
For the purposes of our numerical analysis, we will consider only the power
utility case. We can exploit results on homotheticity as in [9], defining a reduced
value function2
v(t, y) = (1−γ)V (t, 1, y), where V (t, x, y) = x1−γ
V (t, 1, y). The
partial derivatives are
2See Proposition 1.3.5
28. CHAPTER 3. EXTENSION TO STOCHASTIC VOLATILITY 27
∂V
∂x
= (1 − γ)x−γ
V (t, 1, y)
∂2
V
∂x2
= −γ(1 − γ)x−γ−1
V (t, 1, y)
∂2
V
∂xy
= (1 − γ)x−γ ∂V (t, 1, y)
∂y
∂2
V
∂y2
= x1−γ ∂V (t, 1, y)
∂2y
The problem is simplified to solving a corresponding ’reduced’ HJB equation:
∂v
∂t
=
1 − γ
γ
[
−1
2
µ2
y − γr v − µyρξ
∂v
∂y
−
1
2v
ρ2
ξ2
y(
∂v
∂y
)2
]
− κ(θ − y)
∂v
∂y
−
1
2
ξ2
y
∂2
v
∂y2
with v(T, y) = 1
(3.4)
When these partial derivatives are substituted in (??) we find that the optimal
ratio π∗
t becomes
π∗
t =
µ
γ
+
ρξ
γ
∂v
∂y
v
(3.5)
One possible approach to attacking this problem under power utility and ex-
ponential utility is to use Boguslavskaya and Muravey’s (2015) result, reduc-
ing the optimal control to a linear parabolic boundary problem. Kallsen and
Muhle-Karbe showed that in a general stochastic volatility setting, the coeffi-
cients in the reduced PDE are affine linear functions, meaning we can use ansatz
v = eA(t)+B(t)y
with some smooth functions A and B[13].
Calculating derivatives of v gives
∂v
∂y
= B(t)eA(t)+B(t)y
∂2
v
∂y2
= B(t)2
eA(t)+B(t)y
Inserting these into (3.4) gives (after canceling exponential terms throughout)
dA(t)
dt
+
dB(t)
dt
y =
1 − γ
γ
[
−1
2
µ2
y − γr − µyρξB(t) −
1
2
ρ2
ξ2
yB(t)2
]
− κ(θ − y)B(t) −
1
2
ξ2
B(t)2
(3.6)
To solve this we use the following result.
29. CHAPTER 3. EXTENSION TO STOCHASTIC VOLATILITY 28
Result 3.1.1 (Liu & Muhle-Karbe’s Representation of A(t) and B(t)). Let:
a =
(γ − 1)µ2
2γ
b =
γ − 1
γ
µρξ + κ
c =
ξ2
2
(
γ − 1
γ
ρ2
− 1)
D = b2
− 4ac
By comparing coefficients in (3.6) we can separate terms to give a system of
ODE’s for A(t) and B(t):
dB(t)
dt
= cB(t)2
+ bB(t) + a B(T) = 0
dA(t)
dt
= (γ − 1)r − κθB(t) A(T) = 0
A(t) is a straightforward integral and B(t) is a Ricatti equation. These are
solved by3
:
B(t) = −2a
e
√
D(T −t)
− 1
e
√
D(T −t)(b +
√
D) − b +
√
D
A(t) = r(1 − γ)(T − t) + κθ
T
t
B(s)ds
Then the theoretical value function is:
V (t, x, y) =
1
1 − γ
x1−γ
eA(t)+B(t)y
For a more detailed explanation, see [9].
Importantly, we now have deterministic expression for π∗
t by using Result
3.1.1 with (3.5):
π∗
t =
µ
γ
+
ρσB(t)
γ
(3.7)
Perhaps quite surprisingly, even in the case of the Heston model, Merton’s
problem has an explicit solution, however unlike the constant volatility Black-
Scholes case, π∗
t is a deterministic function depending on the current time t and
time horizon T.
Numerical Analysis
In this section we wish to explore the properties of the optimal portfolio π∗
t and
confirm that V (t, x, y) explicitly derived in Result 3.1.1 is in line with simulated
estimates.
3Provided D > 0
30. CHAPTER 3. EXTENSION TO STOCHASTIC VOLATILITY 29
For our simulation model, we can no longer model the wealth process as
Geometric Brownian Motion. π∗
t is a function dependent on time t, rather than
constant in the Black-Scholes case. Instead we can simulate the behaviour of
Xt by using a finite difference method.
With N discretization steps of equal size dt such that T = N∗dt,using (3.1)
as the dynamics of Xt, we first generate a π∗
t array by calculating deterministic
functions A(t) and B(t). We then generate a bivartiate Normal distribution
with correlation ρ. We can then simulate St, Yt and Xt by looping the following
code over N time steps and for M paths:
epsilon = np.random.multivariate_normal([0,0], cov)
dW_S = epsilon[0]*np.sqrt(dt)
dW_Y = epsilon[1]*np.sqrt(dt)
pi[j,i] = (mu/gamma + ((rho*sigma)/gamma)*B[j,i])
S_values[j,i] = S_values[j,i-1] +
(Y_values[j,i]*mu+r)*S_values[j,i-1]*dt +
np.sqrt(Y_values[j,i-1])*S_values[j,i-1]*dW_S
X_values[j,i] = X_values[j,i-1] +
(pi[j,i]*Y_values[j,i-1]*mu+r)*X_values[j,i-1]*dt +
pi[j,i]*np.sqrt(Y_values[j,i-1])*X_values[j,i-1]*dW_S
Y_values[j,i] = Y_values[j,i-1] + kappa*(theta - Y_values[j,i-1])*dt +
sigma*np.sqrt( Y_values[j,i-1])* dW_Y
Y_values[j,i] = abs(Y_values[j,i]) #force non-neg volatility
To verify simulated V (t, x, y) agrees with the analytic result, we use example
parameters
31. CHAPTER 3. EXTENSION TO STOCHASTIC VOLATILITY 30
Table 3.1: Simulation Parameters
µ 0.05
r 0
T 1
X0 1
S0 1
Y0 0.1
θ 0.024
κ 5
ρ 0.3
ξ 0.38
As Figure 3.1 shows, the simulated V (t, x, y) is in very close agreement to
the analytic result given in Result 3.1.1.
Figure 3.1: Simulating V (t, x, y) with 100 scenarios for γ ∈ (0, 1) against ana-
lytic V (t, x, y).
Behaviour of π∗
t
From (3.7) we can see that π∗
t is a deterministic function involving B(t). In-
stinctively, by the definition of B(t) this means that it is dependent of both the
time-horizon T and current time t. Since we have terminal condition B(T) = 0
we expect that as t → T, the ρσB(t)
γ term will disappear meaning π∗
t approaches
µ
γ .
32. CHAPTER 3. EXTENSION TO STOCHASTIC VOLATILITY 31
Figure 3.2 demonstrates how π∗
t evolves for different time-horizons, using
the parameters in Table 3.1. It is clear that in a Heston type model, the in-
vestor’s horizon T affects the optimal amount of wealth he should place in the
risky asset S. As expected, π∗
t approaches µ
γ with equality at T. 4
Figure 3.2: Behaviour of π∗
t with γ = 0.5. (a) = 1 Year, (b) = 2 Years, (c) =
5 Years, (d) = 10 Years
4For these parameters µ
γ
= 0.05
0.5
= 0.1
33. Chapter 4
Transaction Costs
In this chapter, we briefly touch on Merton’s problem when transaction costs
are present. This is an important step towards a more realistic model.
In the transaction-cost free environment, an investor would ideally re-
balance his portfolio by trading as close to continuously as possible. When
transaction costs are added however, we see that this is irrational behaviour as
the cost of rebalancing continuously is greater than the benefit in utility gained.
We will show that it is only beneficial to the investor to trade when the ratio
πt is within certain bounds.
We say the investor pays a fixed transaction cost ψ ∈ R+ every time they
buy or sell any amount of the stock.1
4.1 Proportional Transaction Costs
Let us assume the investor trades as in the Chapter 2, starting with a wealth of
$10,000, rebalancing daily, but with the presence of a 1% proportional transac-
tion cost and a small fixed fee of $5.
From Figure 4.1, we see clearly that maintaining optimal constant π∗
as in the frictionless case is far from the rational way to trade and due to the
cumulative fees paid, rebalancing regularly to keep πt at π∗
leads to the investor
losing his wealth. For this reason we must modify our strategy and framework.
Proportional transaction costs were first studied by Magill and Constan-
tinides [7] in 1976 and expanded upon by Davis and Norman in 1990 who in-
troduced the notion of the no-trade region and provided mathematical rigour.
Davis and Norman showed that the optimal strategy is to make the minimal
trade required (if neccesary) to the closest point in the ’wedge’ defined by the
no-trade region.[6]
1Many brokers charge a price to make trades. This can vary from as low as $5 to upwards
of $100.
32
34. CHAPTER 4. TRANSACTION COSTS 33
Figure 4.1: Actual Wealth with daily rebalancing in presence of transaction
costs vs. theoretical transaction cost free wealth
Assume St follows dSt = St(µdt + σdWt) and dBt = rBtdt as usual. We
use a similar notation as in Davis and Norman.
To model a bid/ask spread, we now assume the investor can buy the stock
at ask price SA
t and sell the stock at the bid price SB
t given by:
Investor Sells at SA
t = (1 + λ)St
Investor Buys at SB
t = (1 − )St
with , λ ∈ [0, 1)
Let Dt and Lt be the cumulative wealth from selling and buying stock
respectively. Let Xt and Yt be the amount invested in the riskless asset and stock
respectively (X0 = x, Y0 = y), so our total wealth at time t is now Zt = Xt + Yt
and starting wealth is x + y.
This gives rise to wealth equations:
dXt = (rXt − ct)dt − (1 + λ)dLt + (1 − )dDt, X0 = x
dYt = µYtdt + σYtdWt + dLt − dDt, Y0 = y
(4.1)
where ct is the investor’s consumption rate.
In the infinite horizon case the value function is defined for utility function
U:
35. CHAPTER 4. TRANSACTION COSTS 34
V (x, y) = sup
(c,L,D)
E
∞
0
e−δt
U(c(t)) (4.2)
Davis and Norman showed that the holding’s at time t are within a closed
region, given by
Sλ, = {(x, y) ∈ R2
: x + (1 − )y ≥ 0 and x + (1 + λ)y ≥ 0}
The investor wants find a triplet (c, L, D) ∈ A(x, y) which maximizes V (x, y) =
sup(E[U(ZT )]). By using Proposition 1.3.5 we can factor V (t, x, y) = xγ
V (t, 1, y)
Using the dynamics in (4.1), the HJB equation for this problem becomes:
−δV + sup
c,l,d
[
1
2
σ2
y2 ∂2
V
∂y2
+ (rx − c)
∂V
∂x
+ µy
∂V
∂y
+
1
γ
cγ
+ −(1 + λ)
∂V
∂x
+
∂V
∂y
l + (1 − )
∂V
∂x
−
∂V
∂y
d] = 0
We differentiate with respect to c and set to zero to find maxima. This gives
optimal consumption:
c∗γ−1
=
∂V
∂x
=⇒ c∗
= (
∂V
∂x
)
1
1−γ
In the no-trade region, l and d are zero as the investor does not make any trades,
so the value function satisfies:
−δV + sup
1
2
σ2
y2 ∂2
V
∂y2
+ (rx − c)
∂V
∂x
+ µy
∂V
∂y
+
1
γ
xγ
In the sell region, dL attains it maximum and dD = 0 as the investor sells the
stock to rebalance his portfolio meaning
∂V
∂y
= (1 + λ)
∂V
∂x
(4.3)
Similarly, in the buy region,
∂V
∂y
= (1 − )
∂V
∂x
(4.4)
We know that the value function is concave and by the homotheticity property,
we have reason to believe that the no-trade region is a cone in R2
.[10]
36. CHAPTER 4. TRANSACTION COSTS 35
The main difficulty arises when trying to solve the HJB equation directly,
as unlike in the classical or Heston case, it cannot be solved analytically[?].
By reducing the dimensionality of the problem, exploiting the homoth-
eticity property and solving a free boundary problem, it can be shown that the
optimal strategy consists of a pair of ’local time’ processes. Davis and Norman
showed how to numerically calculate the buy and sell boundaries. This is still
very much an active area of research.
37. Chapter 5
CONCLUSION
In this paper, we have presented solutions to Merton’s portfolio problem in
various different settings using the dynamic programming approach. We have
demonstrated and numerically verified how an investor can optimise his portfolio
when their utility function is in the class of HARA functions and stock prices
are assumed to obey Geometric Brownian Motion.
In Chapter 2, we have shown that the optimal portfolio consists of holding
a constant proportion of wealth in the risky asset in the idealised model when
volatility is constant and there are no transaction costs. This is true both in
the finite and infinite time horizon case.
In the stochastic volatility setting, we showed that rather surprisingly, an
explicit solution exists and the optimal portfolio is characterized by a deter-
ministic function. In Chapter 3 we presented a simulation model to verify this
optimal strategy via Monte Carlo estimations.
The main difficulties arise when proportional transaction costs are present,
resulting in the HJB equation no longer having an explicit solution. Various
approaches to define the boundaries of the trading regions have been proposed,
such as those by Muthuraman and Zha (2008), Budhiraja and Ross(2007).
36
38. Bibliography
[1] R. Bellman, ”The theory of dynamic programming”, Bull. Amer. Math.
Soc. 60, no. 6, 503-515, 1954.
[2] R. C. Merton, ”An Intertemporal capital asset pricing model,” Econo-
metrica, vol. 41, no. 5, pp. 867-887, 1973.
[3] R. C. Merton, ”Lifetime portfolio selection under uncertainty: The
continuous-time case”, The Review of Economics and Statistics, vol. 51,
no. 3, pp. 247-257, 1969.
[4] R. C. Merton, ”Optimal consumption and portfolio rules in a continuous
time model”, J. Econom. Theory vol. 3, no. 4, pp. 373-413, 1971.
[5] H. Pham, ”Continuous-time stochastic control and optimization with fi-
nancial applications”. Springer-Verlag, 2009.
[6] M. H. H. Davis and A. R. Norman, ”Portfolio Selection with Trans-
action Costs”, Mathematics of Operations Research vol. 15, no. 4, pp.
676-713, 1990.
[7] M. Magill and G. M. Constantinides, ”Portfolio selection with trans-
actions costs”, J. of Econom. Theory vol. 13, no. 2, pp. 245-263, 1976.
[8] H. Liu and M. Loewenstein, ”Optimal portfolio selection with trans-
action costs and finite horizons”, The Review of Financial Studies vol. 15,
no. 3, pp. 805-835, 2002
[9] R. Liu and J. Muhle-Karbe, ”Portfolio Choice with Stochastic Invest-
ment Opportunities: a Users Guide”, 2013
[10] K. Muthuraman and S. Kumar , ” Solving Free-boundary Problems
with Applications in Finance.”, Now Publishers, 2008.
[11] S. L. Heston, ”A Closed Solution For Options With Stochastic Volatility,
With Application to Bond and Currency Options”, Review of Financial
Studies vol. 6, no. 2, pp. 327-343.
[12] E. Boguslavskaya and D. Muravey, ”An explicit solution for optimal
investment in Heston model”, Teor. Veroyatnost. i Primenen vol 60, no. 4,
pp 811-819, 2015.
37
39. BIBLIOGRAPHY 38
[13] J. Kallsen and J. Muhle-Karbe, ”Utility maximization in affine
stochastic volatility models”, 2008.
[14] M. Monoyios, ”Finite horizon portfolio selection with transaction costs”,
Journal of Economic Dynamics and Control vol 28, pp 889-913, 2004.
40. Appendix A
Simulation and Graphing Code
# All code was run on Python 3.5.1
# Only dependencies: numpy, seaborn.
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
# GBM WEALTH PROCESS
def generateWealthPricePaths(mu,sigma,pi,X0,nSteps,nPaths,T):
delta = T/float(nSteps)
logX0 = np.log(X0)
x_values = np.zeros((nPaths,nSteps+1))
( x_values[:,0] ) = [logX0]*nPaths
for j in range(0,nPaths):
for i in range(1,nSteps+1):
x_values[j,i] = x_values[j,i-1] +
(mu*pi-0.5*(sigma*pi)**2)*delta +
sigma**pi*np.sqrt(delta)*np.random.normal()
return np.exp(x_values)
print generateWealthPricePaths(0.05,0.2,1.25,1,10,10,1)
plt.title(’’)
plt.ylabel(’Asset Price’)
plt.xlabel(’Time Steps’)
fig, ax = plt.subplots()
ax.ticklabel_format(useOffset=False)
mu = 0.05
sigma = 0.2
pi = 1.25
X0 = 1
nSteps = 100
nPaths = 10
T = 1
for scenario in generateWealthPricePaths(mu,sigma,pi,X0,nSteps,nPaths,T):
ax.plot(range(nSteps+1), scenario, alpha = 0.2, color = ’green’)
ax.plot(range(nSteps+1), scenario, alpha = 1., color = ’red’)
sns.plt.show()
def exp_utility(alpha, x):
39
41. APPENDIX A. SIMULATION AND GRAPHING CODE 40
return -np.exp(-alpha*x)
#EXP UTILITY MONTE CARLO AND HISTOGRAM
#Calculates Monte Carlo estimate of value function with N scenarios
def Value_function(N, alpha,X0,pi_star,mu,sigma,t):
value_functions_list = []
for i in range(N):
X_t = X0 + ((mu**2)/(alpha*sigma**2))*t +
(mu/(sigma*alpha))*np.sqrt(t)*np.random.normal()
X_t = np.random.normal(X0 + ((mu**2)/(alpha*sigma**2))*t,
np.sqrt(t)*(mu/(alpha*sigma)))
#value_functions_list.append( exp_utility(alpha,X_t) )
value_functions_list.append( -np.exp(-alpha*X_t) )
return np.mean(value_functions_list)
alpha = 1. # pre-defined
t=1. # pre-defined
X0=1 # pre-defined
mu=0.05 # pre-defined
sigma = 0.2 # pre-defined
analytic_pi_star = mu/(X0*alpha*sigma**2) # optimal risky wealth
beta = (mu**2)/(2*alpha*(sigma)**2) # analytic calculation
def analyticValueFunction(alpha,beta,X0):
return -np.exp((-alpha*X0)-alpha*beta*(t-0))
print( Value_function(10000,alpha,X0,analytic_pi_star,mu,sigma,t))
print( analyticValueFunction(alpha,beta,X0))
# Generates 2M+1 points, M values below pi_star, M values above, equally
spaced with distance epsilon
# with N Monte Carlo scenarios
def errors(M,N,epsilon):
x_axis = [analytic_pi_star]
for i in range(1,M):
x_axis.append(analytic_pi_star-i*epsilon)
x_axis.append(analytic_pi_star+i*epsilon)
x_axis = sorted(x_axis)
y_axis =[]
for pi_val in x_axis:
y_axis.append( Value_function(N, alpha,X0,pi_val,mu,sigma,t) )
fig, ax = plt.subplots()
plt.title(’’)
plt.ylabel(’Value (exponential utility)’)
plt.xlabel(’pi’)
x_position = 1
plt.axvline(analytic_pi_star, color = ’red’, alpha = 0.3)
plt.axhline(analyticValueFunction(alpha,beta,X0),color = ’red’, alpha
= 0.3)
plt.xticks(np.arange(-10, 10, 0.5))
plt.plot(x_axis,y_axis)
sns.plt.show()
fig.savefig(’MonteCarlo_value_function_optimality.png’, format=’png’,
dpi=800)
errors(250,1000,0.05)
#EXPONENTIAL UTILITY HISTOGRAM
def final_wealth(N, alpha,X0,pi_star,mu,sigma,t):
42. APPENDIX A. SIMULATION AND GRAPHING CODE 41
wealth_list = []
for i in range(N):
X_t = np.random.normal(X0 + ((mu**2)/(alpha*sigma**2))*t,
np.sqrt(t)*(mu/(alpha*sigma)))
wealth_list.append( X_t )
return wealth_list
alpha_1 = (final_wealth(1000,5,1,1.25,0.05,0.2,1))
fig, ax = plt.subplots()
plt.hist(alpha_1, bins=100, alpha=0.5, label=’alpha = 5.0’,color=’blue’)
plt.xlabel(’Final wealth’)
plt.ylabel(’Frequency’)
plt.legend(loc=’upper right’,prop={’size’:14})
sns.plt.show()
#OPTIMAL SHARES, S_t AND pi FOR EXPONENTIAL UTILITY
pi_t = 1.25
def optimals(mu,sigma,alpha,S0,X0,nSteps,T):
delta = T/float(nSteps)
logS0 = np.log(S0)
s_values = np.zeros(nSteps+1)
s_values[0] = logS0
X = np.zeros(nSteps+1)
X[0] = X0
optimal_shares = np.zeros(nSteps+1)
optimal_shares[0] = (pi_t*X0)/S0
for i in range(1,nSteps+1):
X[i] = X[i-1]+ (mu**2)/(alpha*sigma**2)*delta +
(mu/(sigma*alpha))*np.sqrt(delta)*np.random.normal()
s_values[i] = s_values[i-1] + (mu-0.5*sigma**2)*delta +
sigma*np.sqrt(delta)*np.random.normal()
optimal_shares[i] = (pi_t* X[i])/np.exp(s_values[i])
return np.exp(s_values),X,optimal_shares
nSteps = 250
listo = optimals(0.05,0.2,1,1,1,nSteps,1)
fig, ax = plt.subplots()
ax.ticklabel_format(useOffset=False)
ax.plot(range(nSteps+1), listo[0], alpha = 1, color = ’red’,label=’Stock
Price’)
ax.plot(range(nSteps+1), listo[2], alpha = 1, color =
’green’,label=’Optimal number of shares’)
ax.plot(range(nSteps+1), [1.25]*(nSteps+1), alpha = 1, color =
’blue’,label=’Optimal pi’)
plt.xlabel(’Time Steps (days)’)
for item in ([ax.title, ax.xaxis.label, ax.yaxis.label] +
ax.get_xticklabels() + ax.get_yticklabels()):
item.set_fontsize(14)
plt.legend(loc=’upper right’, prop={’size’:14})
sns.plt.show()
#MONTE CARLO VERIFICATIONS WITH POWER UTIL AND NON-ZERO INTEREST #RATES
def power_utility(gamma, x):
if gamma >=1 or gamma<=0:
return ’ERROR, GAMMA out of bounds’
return 1/(1-float(gamma))*x**(1-float(gamma))
43. APPENDIX A. SIMULATION AND GRAPHING CODE 42
#Calculates Monte Carlo estimate of value function with N scenarios
def Value_function(N, gamma,X0,pi_star,mu,r,sigma,t):
value_functions_list = []
for i in range(N):
X_t = X0 * np.exp( (pi_star * mu + r - 0.5*(sigma*pi_star)**2)*t +
pi_star*sigma*np.sqrt(t)*np.random.normal() )
#value_functions_list.append( exp_utility(alpha,X_t) )
value_functions_list.append( power_utility(gamma,X_t) )
return np.mean(value_functions_list)
gamma = 0.5 # pre-defined
t=2. # pre-defined
X0=10. # pre-defined
mu=0.02 # pre-defined
sigma = 0.2 # pre-defined
r = 1 # pre-defined
analytic_pi_star = mu/(gamma*sigma**2) # optimal risky wealth
beta = 0.5*(mu**2)/(gamma*sigma**2) + r # analytic calculation
def analyticValueFunction(gamma,beta,X0):
return 1/(1-float(gamma))*X0**(1-float(gamma)) *
np.exp((t-0)*(1-gamma)*beta)
def printer(muu,rr):
analytic_pi_star = muu/(gamma*sigma**2) # optimal risky wealth
beta = 0.5*(muu**2)/(gamma*sigma**2) + rr # analytic calculation
print(’mu is ’+str(muu))
print(’r is ’ + str(rr))
print(Value_function(10000,gamma,X0,analytic_pi_star,muu,rr,sigma,t))
print(analyticValueFunction(gamma,beta,X0))
print(’nn’)
#CODE FOR VARYING REBALANCING FREQUENCIES
def generateWealthPricePaths(mu,sigma,r,pi,X0,S0,nSteps,nPaths,T,freq):
delta = T/float(nSteps)
# stock
logS0 = np.log(S0)
s_values = np.zeros((nPaths,nSteps+1))
( s_values[:,0] ) = [logS0]*nPaths
B0=1
logX0 = np.log(X0)
theo_money_in_stock = np.zeros((nPaths,nSteps+1))
theo_money_in_bank = np.zeros((nPaths,nSteps+1))
theo_money_in_stock[:,0] = (pi * X0)
theo_money_in_bank[:,0] = (1-pi)*X0
x_values = np.zeros((nPaths,nSteps+1))
( x_values[:,0] ) = [logX0]*nPaths
money_in_bank = np.zeros((nPaths,nSteps+1))
( money_in_bank[:,0] ) = [B0]*nPaths
units_of_stock = (pi * X0)/S0
init_units_of_bank = (1-pi)*B0
money_in_stock = np.zeros((nPaths,nSteps+1))
money_in_bank[:,0] = init_units_of_bank*B0
money_in_stock[:,0] = units_of_stock*S0
for j in range(0,nPaths):
units_of_stock = (pi * X0)/S0
for i in range(1,nSteps+1):