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.
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.
The document summarizes Ahmed Ashmawy's M.Sc defense presentation on mixed integer conditional value-at-risk portfolio optimization. It outlines the problem of optimally allocating an investment across multiple stocks while accounting for risk, describes approaches that combine constraint programming and linear programming to solve the mixed integer optimization problem more efficiently, and presents a greedy algorithm that provides near-optimal solutions with improved time performance by iteratively solving relaxed linear programming subproblems. Experimental results demonstrate the greedy approach achieves solutions of similar quality to traditional mixed integer programming solvers but with significantly better scalability to large problem instances.
This document describes research into applying control theory and feedback strategies to stock trading. It tests several feedback-based trading strategies on stock market portfolios to investigate their ability to generate profits. The strategies tested include a classical linear feedback strategy, a strategy with feedback delay, a "trigger" strategy, and a moving average strategy. Transaction costs are later added to the models. The results show that three of the four strategies were able to produce net profits after accounting for transaction costs over the testing periods.
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.
Lecture slides for Auction Theory (for graduate students) at Osaka University in 2016, 2nd semester. Complementary materials and related information can be obtained from the course website below:
https://sites.google.com/site/yosukeyasuda2/home/lecture/auction16
The dangers of policy experiments Initial beliefs under adaptive learningGRAPE
The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
This document discusses estimating covariance matrices for portfolio selection. It introduces a shrinkage estimator that is an optimally weighted average of the sample covariance matrix and single-index covariance matrix. The empirical part compares these estimators to determine which produces the most efficient portfolio with smallest return variability. The sample covariance matrix has problems when the number of assets is large, as it has high variance and its inverse is a poor estimator. Shrinkage aims to improve upon the sample covariance matrix by combining it with a factor model-based estimator.
While investment management can be readily measured against known benchmarks like the S&P500, wealth management, with its personalized, multi-decade scope, lacks such straightforward comparisons. This study recommends a shift in focus from portfolio to strategy indexing, much like target date funds, using Monte Carlo simulations for evaluation. A proposed methodology enables the construction of personalized benchmarks, combining a client's initial financial status with a selection of specific goals. These benchmarks facilitate a realistic appraisal of a client's financial situation, providing a foundation for tailored financial advice.
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.
The document summarizes Ahmed Ashmawy's M.Sc defense presentation on mixed integer conditional value-at-risk portfolio optimization. It outlines the problem of optimally allocating an investment across multiple stocks while accounting for risk, describes approaches that combine constraint programming and linear programming to solve the mixed integer optimization problem more efficiently, and presents a greedy algorithm that provides near-optimal solutions with improved time performance by iteratively solving relaxed linear programming subproblems. Experimental results demonstrate the greedy approach achieves solutions of similar quality to traditional mixed integer programming solvers but with significantly better scalability to large problem instances.
This document describes research into applying control theory and feedback strategies to stock trading. It tests several feedback-based trading strategies on stock market portfolios to investigate their ability to generate profits. The strategies tested include a classical linear feedback strategy, a strategy with feedback delay, a "trigger" strategy, and a moving average strategy. Transaction costs are later added to the models. The results show that three of the four strategies were able to produce net profits after accounting for transaction costs over the testing periods.
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.
Lecture slides for Auction Theory (for graduate students) at Osaka University in 2016, 2nd semester. Complementary materials and related information can be obtained from the course website below:
https://sites.google.com/site/yosukeyasuda2/home/lecture/auction16
The dangers of policy experiments Initial beliefs under adaptive learningGRAPE
The paper studies the implication of initial beliefs and associated confidence on the system’s
dynamics under adaptive learning. We first illustrate how prior beliefs determine learning dynamics
and the evolution of endogenous variables in a small DSGE model with credit-constrained agents,
in which rational expectations are replaced by constant-gain adaptive learning. We then examine
how discretionary experimenting with new macroeconomic policies is affected by expectations that
agents have in relation to these policies. More specifically, we show that a newly introduced macroprudential policy that aims at making leverage counter-cyclical can lead to substantial increase in
fluctuations under learning, when the economy is hit by financial shocks, if beliefs reflect imperfect
information about the policy experiment. This is in the stark contrast to the effects of such policy
under rational expectations.
This document discusses estimating covariance matrices for portfolio selection. It introduces a shrinkage estimator that is an optimally weighted average of the sample covariance matrix and single-index covariance matrix. The empirical part compares these estimators to determine which produces the most efficient portfolio with smallest return variability. The sample covariance matrix has problems when the number of assets is large, as it has high variance and its inverse is a poor estimator. Shrinkage aims to improve upon the sample covariance matrix by combining it with a factor model-based estimator.
While investment management can be readily measured against known benchmarks like the S&P500, wealth management, with its personalized, multi-decade scope, lacks such straightforward comparisons. This study recommends a shift in focus from portfolio to strategy indexing, much like target date funds, using Monte Carlo simulations for evaluation. A proposed methodology enables the construction of personalized benchmarks, combining a client's initial financial status with a selection of specific goals. These benchmarks facilitate a realistic appraisal of a client's financial situation, providing a foundation for tailored financial advice.
This document discusses valuing and hedging the extrinsic value of a natural gas storage facility using a basket-of-options approach. It presents a formula for calculating the intrinsic value of the storage by maximizing the spread between purchase and sale prices of gas over time. The storage value includes both the intrinsic value and an extrinsic value based on future opportunities. It models the storage as a portfolio of options on spreads between monthly gas prices. Delta hedging with these options provides a lower bound for the storage value and a way to monetize the extrinsic value. The methodology is tested on a six-month period using daily gas price data.
:In this paper, we consider the equity premium puzzle under a general utility function. We derive that
the optimal strategy under a general utility function approximate the optimal strategy under the special utility
function. This result posed in the present paper can be regarded as a generalization of the work by Gong and
Zou [13]
Bid and Ask Prices Tailored to Traders' Risk Aversion and Gain Propension: a ...Waqas Tariq
Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necessary conditions for successful trading are also discussed.
The document discusses key concepts in financial theory including:
1. The postulate of yield of money over time, and capitalization and discounting operations which allow moving money forward or backward in time.
2. Economic laws like the arbitrage principle, law of one price, law of linearity of amounts, and law of monotonicity of amounts which are necessary for an efficient market without arbitrage opportunities.
3. Financial risks including market risk, credit risk, and liquidity risk.
4. Financial engineering and how derivatives instruments derive their value from underlying assets or indices.
The document describes three problems related to constructing optimal self-financing portfolios consisting of stocks and options. Problem 1 involves a portfolio with one stock and one option, Problem 2 involves one stock and two options, and Problem 3 prices an up-and-out option using a finite difference method. For each problem, the document outlines the portfolio composition, parameters, and numerical approach taken to determine optimal quantities of options to minimize portfolio variance.
PPT - Deep Hedging OF Derivatives Using Reinforcement LearningJisang Yoon
The document describes using reinforcement learning to implement hedging of derivatives. It discusses:
1) Setting up a hedging model using reinforcement learning, where the state includes the asset price and time to maturity, the action is the hedge position, and the reward minimizes hedging costs.
2) Conducting experiments hedging a short call option using the model, comparing performance under geometric Brownian motion and stochastic volatility.
3) Finding that the reinforcement learning model outperforms delta hedging strategies and achieves lower hedging costs, demonstrating the effectiveness of using reinforcement learning for hedging derivatives.
Content and format requirementsContent· Introduce the articleAlleneMcclendon878
Content and format requirements
Content:
· Introduce the article by title and author early in the analysis
· Briefly summarize the article before addressing the following questions:
· What is the main point/purpose of the article? How does the author support this point or make it convincing? Is the article convincing? Why?
· What sort of character/authority (ethos) does the author create through the text? How does the author establish their credibility?
· Appeals to the audience: Does the author use Pathos (emotion), Logos (logic), or Kairos (timeliness)? How do these appeals further the purpose?
· What tone or style does the author use? Is the language formal or informal? What do you notice about word choice (such as repeated words) and the arrangement of ideas? What effect does this tone and style create? How does it further the author’s purpose?
Format:
· APA paper format
· 350-500 words
Organization requirements
· Analysis is presented in a logical, sequential order
· Transitions link ideas within and between paragraphs to unify the discussion
Source/ Support
requirements
· Adequate and appropriate evidence to support the discussion.
· Include examples, quotations, and paraphrases from the article to support your points.
· All quotations and paraphrases must be cited with APA in-text-citations.
· Include an APA Style References page.
Technical requirements
· Write in third person (objective voice) – avoid I or You statements.
· Use standard conventions of English: punctuation, capitalization, complete sentences
· Spellcheck and proofread your work.
Link to the Video
https://youtu.be/gRD0UA5ME3I
ECON 401
Advanced Macroeconomics
Midterm Exam
Fabio Ghironi
University of Washington
April 29, 2022
Instructions:
You have 5 hours to work on this exam. It is worth 100 points, contributing to your overall
score for the course as described in the Syllabus. You may consult all course materials and standard
Internet resources while working on the exam, but your work must be original and you may not
solicit or obtain assistance from or provide assistance to other people for any specific content of the
exam. Activities considered cheating include copying or closely paraphrasing content from websites
and discussing exam questions with other students. All exams will be checked for originality and
copied content, and anyone found cheating will be assigned a zero score for the exam. Read
carefully each step of each problem before you jump into working on it and do not panic if you
cannot complete everything. The exam is intended also to stretch your knowledge by forcing you
to use the tools and information you have acquired to think about some things we have not talked
about in class. I want to see how you think about those things based on what you learned.
Problem 1: The RBC Model with Endogenous Labor Supply (50 Points)
The figure in the next page shows the responses to a one-percent innovation to technology at time 0
in the b ...
1) The document discusses Bayesian forecasting of financial time series using stop-loss orders. It proposes two models - one without stop-loss orders that yields positive returns but some negative local yields, and one with stop-loss orders that reduces negative yields for a better overall return.
2) The models use high-frequency "tick-by-tick" data and functional clustering to make trend and stopping time predictions. Dynamic state space representations and particle/Kalman filters are used for generalization and forecasting.
3) Experiments on IBM stock data from 1995-1999 show the model with stop-loss orders achieves better cumulative returns by curbing negative local yields, though with greater computational complexity.
Fair valuation of participating life insurance contracts with jump riskAlex Kouam
A C++ based program which prices the fair value of a participating life insurance whereby the underlying follows a Kou process and the insurer's default occurs only at contract's maturity.
This paper develops a framework to analyze how financial intermediation affects the transmission of macroeconomic policies. It models the financial sector as issuing liquid assets to finance illiquid capital investments. The paper shows that aggregate output responses to policies depend on the elasticities of liquid asset supply with respect to returns. It finds that assuming perfectly inelastic or elastic liquidity supply leads to substantially different predictions than empirical estimates. Applying the framework, it concludes that asset purchases may have a modest effect on output through financial markets, while tax cuts directly targeting households are relatively more effective stimulants.
This document presents a theoretical and empirical study of how incentives affect risk taking in hedge funds. The theoretical section develops a model to analyze how incentive fees and a manager's personal investment in the fund impact the manager's risk preferences under prospect theory. It finds that incentive fees reduce implicit loss aversion and increase risk taking, while a manager's own investment increases loss aversion and reduces risk taking. The empirical section analyzes a large hedge fund database and finds funds with incentive fees have higher downside risk and lower average returns than funds without such fees.
In this paper, we consider an AAI with two types of insurance business with p-thinning dependent
claims risk, diversify claims risk by purchasing proportional reinsurance, and invest in a stock with Heston
model price process, a risk-free bond, and a credit bond in the financial market with the objective of maximizing
the expectation of the terminal wealth index effect, and construct the wealth process of AAI as well as the the
model of robust optimal reinsurance-investment problem is obtained, using dynamic programming, the HJB
equation to obtain the pre-default and post-default reinsurance-investment strategies and the display expression
of the value function, respectively, and the sensitivity of the model parameters is analyzed through numerical
experiments to obtain a realistic economic interpretation. The model as well as the results in this paper are a
generalization and extension of the results of existing studies.
This document summarizes research on strong duality analysis for discrete-time constrained portfolio optimization problems. It begins by introducing the mathematical formulation of a discrete-time portfolio selection model with constraints expressed as convex inequalities. It then discusses a risk neutral computational approach based on embedding the primal constrained problem into a family of unconstrained problems in auxiliary markets. Weak duality is shown to hold, relating the optimal values of the primal and auxiliary problems. The document defines a dual problem, known as Pliska's κ dual, that seeks to minimize the optimal values of the auxiliary problems. Conditions for strong duality are presented, under which the optimal solution to the dual problem also solves the primal constrained problem.
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
This document discusses quantitative finance topics including bond duration and immunization. It provides an example showing how duration and convexity can be used to approximate changes in bond prices from changes in yields. The document also discusses how to construct a bond portfolio with a target duration and convexity. Finally, it briefly defines interest rate swaps, bond options, interest rate caps, and floors.
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.
1. The document presents a proposal to investigate optimal portfolio strategies for investors that maximize utility from terminal wealth while meeting bank capital adequacy requirements under conditions of uncertainty in emerging economies.
2. The methodology will use modeling, simulations, and analytical approaches to study robust optimal portfolios and the effects of low quality data and ambiguity on portfolio choices in African markets. Dynamic programming and Choquet expectations will be applied.
3. The results are expected to provide insights on emerging market behavior and calibrate the models using Kenyan market data to derive impacts of risk and ambiguity on portfolios.
This document introduces the concept of "ultimate profitability" to evaluate the effectiveness of market research. Ultimate profitability measures the maximum possible annual return from perfectly timing entry and exit from a market based on its price extremes. The document outlines a methodology to calculate ultimate profitability for different markets and indexes based on varying the scale of price movements considered. It presents an example calculation of ultimate profitability for the Russian equity index RUIX under different scales and finds an inverse power law relationship between profitability and scale.
Numerical method for pricing american options under regime Alexander Decker
This document presents a numerical method for pricing American options under regime-switching jump-diffusion models. It begins with an abstract that describes using a cubic spline collocation method to solve a set of coupled partial integro-differential equations (PIDEs) with the free boundary feature. The document then provides background on regime-switching Lévy processes and derives the PIDEs that describe the American option price under different regimes. It presents the time and spatial discretization methods, using Crank-Nicolson for time stepping and cubic spline collocation for the spatial variable. The method is shown to exhibit second order convergence in space and time.
Managerial economics applies economic theory and decision science tools to help organizations achieve their objectives efficiently. It uses microeconomics, macroeconomics, mathematical economics, and econometrics to solve managerial problems and make optimal decisions. The goal is to maximize the value of the firm over time by finding the optimal solutions to managerial problems through the application of economic theory and analysis.
More Related Content
Similar to On the Optimality of Kelly Strategies (Presentacion).pdf
This document discusses valuing and hedging the extrinsic value of a natural gas storage facility using a basket-of-options approach. It presents a formula for calculating the intrinsic value of the storage by maximizing the spread between purchase and sale prices of gas over time. The storage value includes both the intrinsic value and an extrinsic value based on future opportunities. It models the storage as a portfolio of options on spreads between monthly gas prices. Delta hedging with these options provides a lower bound for the storage value and a way to monetize the extrinsic value. The methodology is tested on a six-month period using daily gas price data.
:In this paper, we consider the equity premium puzzle under a general utility function. We derive that
the optimal strategy under a general utility function approximate the optimal strategy under the special utility
function. This result posed in the present paper can be regarded as a generalization of the work by Gong and
Zou [13]
Bid and Ask Prices Tailored to Traders' Risk Aversion and Gain Propension: a ...Waqas Tariq
Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necessary conditions for successful trading are also discussed.
The document discusses key concepts in financial theory including:
1. The postulate of yield of money over time, and capitalization and discounting operations which allow moving money forward or backward in time.
2. Economic laws like the arbitrage principle, law of one price, law of linearity of amounts, and law of monotonicity of amounts which are necessary for an efficient market without arbitrage opportunities.
3. Financial risks including market risk, credit risk, and liquidity risk.
4. Financial engineering and how derivatives instruments derive their value from underlying assets or indices.
The document describes three problems related to constructing optimal self-financing portfolios consisting of stocks and options. Problem 1 involves a portfolio with one stock and one option, Problem 2 involves one stock and two options, and Problem 3 prices an up-and-out option using a finite difference method. For each problem, the document outlines the portfolio composition, parameters, and numerical approach taken to determine optimal quantities of options to minimize portfolio variance.
PPT - Deep Hedging OF Derivatives Using Reinforcement LearningJisang Yoon
The document describes using reinforcement learning to implement hedging of derivatives. It discusses:
1) Setting up a hedging model using reinforcement learning, where the state includes the asset price and time to maturity, the action is the hedge position, and the reward minimizes hedging costs.
2) Conducting experiments hedging a short call option using the model, comparing performance under geometric Brownian motion and stochastic volatility.
3) Finding that the reinforcement learning model outperforms delta hedging strategies and achieves lower hedging costs, demonstrating the effectiveness of using reinforcement learning for hedging derivatives.
Content and format requirementsContent· Introduce the articleAlleneMcclendon878
Content and format requirements
Content:
· Introduce the article by title and author early in the analysis
· Briefly summarize the article before addressing the following questions:
· What is the main point/purpose of the article? How does the author support this point or make it convincing? Is the article convincing? Why?
· What sort of character/authority (ethos) does the author create through the text? How does the author establish their credibility?
· Appeals to the audience: Does the author use Pathos (emotion), Logos (logic), or Kairos (timeliness)? How do these appeals further the purpose?
· What tone or style does the author use? Is the language formal or informal? What do you notice about word choice (such as repeated words) and the arrangement of ideas? What effect does this tone and style create? How does it further the author’s purpose?
Format:
· APA paper format
· 350-500 words
Organization requirements
· Analysis is presented in a logical, sequential order
· Transitions link ideas within and between paragraphs to unify the discussion
Source/ Support
requirements
· Adequate and appropriate evidence to support the discussion.
· Include examples, quotations, and paraphrases from the article to support your points.
· All quotations and paraphrases must be cited with APA in-text-citations.
· Include an APA Style References page.
Technical requirements
· Write in third person (objective voice) – avoid I or You statements.
· Use standard conventions of English: punctuation, capitalization, complete sentences
· Spellcheck and proofread your work.
Link to the Video
https://youtu.be/gRD0UA5ME3I
ECON 401
Advanced Macroeconomics
Midterm Exam
Fabio Ghironi
University of Washington
April 29, 2022
Instructions:
You have 5 hours to work on this exam. It is worth 100 points, contributing to your overall
score for the course as described in the Syllabus. You may consult all course materials and standard
Internet resources while working on the exam, but your work must be original and you may not
solicit or obtain assistance from or provide assistance to other people for any specific content of the
exam. Activities considered cheating include copying or closely paraphrasing content from websites
and discussing exam questions with other students. All exams will be checked for originality and
copied content, and anyone found cheating will be assigned a zero score for the exam. Read
carefully each step of each problem before you jump into working on it and do not panic if you
cannot complete everything. The exam is intended also to stretch your knowledge by forcing you
to use the tools and information you have acquired to think about some things we have not talked
about in class. I want to see how you think about those things based on what you learned.
Problem 1: The RBC Model with Endogenous Labor Supply (50 Points)
The figure in the next page shows the responses to a one-percent innovation to technology at time 0
in the b ...
1) The document discusses Bayesian forecasting of financial time series using stop-loss orders. It proposes two models - one without stop-loss orders that yields positive returns but some negative local yields, and one with stop-loss orders that reduces negative yields for a better overall return.
2) The models use high-frequency "tick-by-tick" data and functional clustering to make trend and stopping time predictions. Dynamic state space representations and particle/Kalman filters are used for generalization and forecasting.
3) Experiments on IBM stock data from 1995-1999 show the model with stop-loss orders achieves better cumulative returns by curbing negative local yields, though with greater computational complexity.
Fair valuation of participating life insurance contracts with jump riskAlex Kouam
A C++ based program which prices the fair value of a participating life insurance whereby the underlying follows a Kou process and the insurer's default occurs only at contract's maturity.
This paper develops a framework to analyze how financial intermediation affects the transmission of macroeconomic policies. It models the financial sector as issuing liquid assets to finance illiquid capital investments. The paper shows that aggregate output responses to policies depend on the elasticities of liquid asset supply with respect to returns. It finds that assuming perfectly inelastic or elastic liquidity supply leads to substantially different predictions than empirical estimates. Applying the framework, it concludes that asset purchases may have a modest effect on output through financial markets, while tax cuts directly targeting households are relatively more effective stimulants.
This document presents a theoretical and empirical study of how incentives affect risk taking in hedge funds. The theoretical section develops a model to analyze how incentive fees and a manager's personal investment in the fund impact the manager's risk preferences under prospect theory. It finds that incentive fees reduce implicit loss aversion and increase risk taking, while a manager's own investment increases loss aversion and reduces risk taking. The empirical section analyzes a large hedge fund database and finds funds with incentive fees have higher downside risk and lower average returns than funds without such fees.
In this paper, we consider an AAI with two types of insurance business with p-thinning dependent
claims risk, diversify claims risk by purchasing proportional reinsurance, and invest in a stock with Heston
model price process, a risk-free bond, and a credit bond in the financial market with the objective of maximizing
the expectation of the terminal wealth index effect, and construct the wealth process of AAI as well as the the
model of robust optimal reinsurance-investment problem is obtained, using dynamic programming, the HJB
equation to obtain the pre-default and post-default reinsurance-investment strategies and the display expression
of the value function, respectively, and the sensitivity of the model parameters is analyzed through numerical
experiments to obtain a realistic economic interpretation. The model as well as the results in this paper are a
generalization and extension of the results of existing studies.
This document summarizes research on strong duality analysis for discrete-time constrained portfolio optimization problems. It begins by introducing the mathematical formulation of a discrete-time portfolio selection model with constraints expressed as convex inequalities. It then discusses a risk neutral computational approach based on embedding the primal constrained problem into a family of unconstrained problems in auxiliary markets. Weak duality is shown to hold, relating the optimal values of the primal and auxiliary problems. The document defines a dual problem, known as Pliska's κ dual, that seeks to minimize the optimal values of the auxiliary problems. Conditions for strong duality are presented, under which the optimal solution to the dual problem also solves the primal constrained problem.
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
This document discusses quantitative finance topics including bond duration and immunization. It provides an example showing how duration and convexity can be used to approximate changes in bond prices from changes in yields. The document also discusses how to construct a bond portfolio with a target duration and convexity. Finally, it briefly defines interest rate swaps, bond options, interest rate caps, and floors.
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.
1. The document presents a proposal to investigate optimal portfolio strategies for investors that maximize utility from terminal wealth while meeting bank capital adequacy requirements under conditions of uncertainty in emerging economies.
2. The methodology will use modeling, simulations, and analytical approaches to study robust optimal portfolios and the effects of low quality data and ambiguity on portfolio choices in African markets. Dynamic programming and Choquet expectations will be applied.
3. The results are expected to provide insights on emerging market behavior and calibrate the models using Kenyan market data to derive impacts of risk and ambiguity on portfolios.
This document introduces the concept of "ultimate profitability" to evaluate the effectiveness of market research. Ultimate profitability measures the maximum possible annual return from perfectly timing entry and exit from a market based on its price extremes. The document outlines a methodology to calculate ultimate profitability for different markets and indexes based on varying the scale of price movements considered. It presents an example calculation of ultimate profitability for the Russian equity index RUIX under different scales and finds an inverse power law relationship between profitability and scale.
Numerical method for pricing american options under regime Alexander Decker
This document presents a numerical method for pricing American options under regime-switching jump-diffusion models. It begins with an abstract that describes using a cubic spline collocation method to solve a set of coupled partial integro-differential equations (PIDEs) with the free boundary feature. The document then provides background on regime-switching Lévy processes and derives the PIDEs that describe the American option price under different regimes. It presents the time and spatial discretization methods, using Crank-Nicolson for time stepping and cubic spline collocation for the spatial variable. The method is shown to exhibit second order convergence in space and time.
Managerial economics applies economic theory and decision science tools to help organizations achieve their objectives efficiently. It uses microeconomics, macroeconomics, mathematical economics, and econometrics to solve managerial problems and make optimal decisions. The goal is to maximize the value of the firm over time by finding the optimal solutions to managerial problems through the application of economic theory and analysis.
Similar to On the Optimality of Kelly Strategies (Presentacion).pdf (20)
On the Optimality of Kelly Strategies (Presentacion).pdf
1. On the Optimality of Kelly Strategies
On the Optimality of Kelly Strategies
Mark Davis1 and Sébastien Lleo2
Workshop on Stochastic Models and Control
Bad Herrenalb, April 1, 2011
1
Department of Mathematics, Imperial College London, London SW7 2AZ,
England, Email: mark.davis@imperial.ac.uk
2
Finance Department, Reims Management School, 59 rue Pierre Taittinger,
51100 Reims, France, Email: sebastien.lleo@reims-ms.fr
2. On the Optimality of Kelly Strategies
Outline
Outline
Where to invest?
Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
Insights from the Merton Model: Change of Measure and Duality
Further Insights from the Merton Model...
Getting Kelly Strategies to Work for Factor Models
Beyond Factor Models: Random Coefficients
Conclusion and Next Steps
3. On the Optimality of Kelly Strategies
Where to invest?
Where to invest?
The central question for investment management practitioners is:
where should I put my money?
Ideas such as the utility of wealth, and even expected portfolio
returns (and risks) take a backseat.
Yet, in finance theory, the optimal asset allocation appears as less
important than the utility of wealth. This is particularly true in the
duality/martingale approach, where the optimal asset allocation is
often obtained last.
4. On the Optimality of Kelly Strategies
Kelly Strategies
Kelly Strategies
As a result of the differences between investment management
theories and practice, a number of techniques and tools have been
created to speed up the derivation of an optimal or near-optimal
asset allocation.
Fractional Kelly strategies are one such technique.
5. On the Optimality of Kelly Strategies
Kelly Strategies
The Kelly (criterion) portfolio invest in the growth maximizing
portfolio.
I In continuous time, this represents an investment in the
optimal portfolio under the logarithmic utility function.
I The Kelly criterion comes from the signal
processing/gambling literature (see Kelly [7]).
I See MacLean, Thorpe and Ziemba [11] for a thorough view of
the Kelly criterion.
A number of “great” investors from Keynes to Buffet can be
viewed as Kelly investors. Others, such as Bill Gross probably are.
The Kelly criterion portfolio has a number of interesting properties
but it is inherently risky.
6. On the Optimality of Kelly Strategies
Kelly Strategies
To reduce the risk while keeping some of the nice properties,
MacLean and Ziemba proposed the concept of fractional Kelly
investment:
I Invest a fraction k of the wealth in the Kelly portfolio;
I invest a fraction 1 − k in the risk-free asset.
Fractional Kelly strategies are
I optimal in the Merton world of lognormal asset prices and
with a globally risk-free asset;
I not optimal when the lognormality assumption is removed.
7. On the Optimality of Kelly Strategies
Kelly Strategies
Three questions:
I Why are fractional Kelly strategies not optimal outside of the
Merton model?
I How is an optimal strategy constructed?
I Can we improve the definition of fractional Kelly strategies to
guarantee optimality?
8. On the Optimality of Kelly Strategies
Kelly Strategies
In this talk, we will use the term Kelly strategies to refer to:
I The Kelly portfolio.
I Fractional Kelly investment.
9. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
Back to Basics: Kelly Strategies in the Merton Model
Key ingredients:
I Probability space (Ω, {Ft} , F, P);
I Rm-valued (Ft)-Brownian motion W (t);
I Price at time t of the ith security is Si (t), i = 0, . . . , m;
The dynamics of the money market account and of the m risky
securities are respectively given by:
dS0(t)
S0(t)
= rdt, S0(0) = s0 (1)
dSi (t)
Si (t)
= µi dt +
N
X
k=1
σikdWk(t), Si (0) = si , i = 1, . . . , m
(2)
where r, µ ∈ Rm and Σ := [σij ] is a m × m matrix.
10. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
Assumption
The matrix Σ is positive definite.
Let Gt := σ(S(s), 0 ≤ s ≤ t) be the sigma-field generated by the
security process up to time t.
An investment strategy or control process is an Rm-valued process
with the interpretation that hi (t) is the fraction of current portfolio
value invested in the ith asset, i = 1, . . . , m.
I Fraction invested in the money market account is
h0(t) = 1 −
Pm
i=1 hi (t).;
11. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
Definition
An Rm-valued control process h(t) is in class A(T) if the following
conditions are satisfied:
1. h(t) is progressively measurable with respect to
{B([0, t]) ⊗ Gt}t≥0 and is càdlàg;
2. P
R T
0 |h(s)|2
ds +∞
= 1, ∀T 0;
3. the Doléans exponential χh
t , given by
χh
t := exp
γ
Z t
0
h(s)0
ΣdWs −
1
2
γ2
Z t
0
h(s)0
ΣΣ0
h(s)ds
(3)
is an exponential martingale, i.e. E
χh
T
= 1
We say that a control process h(t) is admissible if h(t) ∈ A(T).
12. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
Taking the budget equation into consideration, the wealth, V (t),
of the asset in response to an investment strategy h ∈ H follows
the dynamics
dV (t)
V (t)
= rdt + h0
(t) (µ − r1) dt + h0
(t)ΣdWt
(4)
with initial endowment V (0) = v and where 1 ∈ Rm is the
m-element unit column vector.
13. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
The objective of an investor with a fixed time horizon T, no
consumption and a power utility function is to maximize the
expected utility of terminal wealth:
J(t, h; T, γ) = E [U(VT )] = E
V γ
T
γ
= E
eγ ln VT
γ
with risk aversion coefficient γ ∈ (−∞, 0) ∪ (0, 1).
14. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
Define the value function Φ corresponding to the maximization of
the auxiliary criterion function J(t, x, h; θ; T; v) as
Φ(t) = sup
h∈A
J(t, h; T, γ) (5)
By Itô’s lemma,
eγ ln V (t)
= vγ
exp
γ
Z t
0
g(h(s); γ)ds
χh
t (6)
where
g(h; γ) = −
1
2
(1 − γ) h0
ΣΣ0
h + h0
(µ − r1) + r
and the Doléans exponential χh
t is defined in (3)
15. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
We can solve the stochastic control problem associated with (5) by
a change of measure argument (see exercise 8.18 in [8]).
Define a measure Ph on (Ω, FT ) via the Radon-Nikodým derivative
dPh
dP
:= χh
T (7)
For h ∈ A(T),
W h
t = Wt − γ
Z t
0
Σ0
h(s)ds
is a standard Ph-Brownian motion.
The control criterion under this new measure is
I(t, h; T, γ) =
vγ
γ
Eh
t
exp
γ
Z T
t
g(h(s); θ)ds
(8)
where Eh
t [·] denotes the expectation taken with respect to the
measure Ph at an initial time t.
16. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
Under the measure Ph, the control problem can be solved through
a pointwise maximisation of the auxiliary criterion function
I(v, x; h; t, T).
The optimal control h∗ is simply the maximizer of the function
g(x; h; t, T) given by
h∗
=
1
1 − γ
(ΣΣ0
)−1
(µ − r1)
which represents a position of 1
1−γ in the Kelly criterion portfolio.
17. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
The value function Φ(t), or optimal utility of wealth, equals
Φ(t) =
vγ
γ
exp
γ
r +
1
2(1 − γ)
(µ − r1) (ΣΣ0
)−1
(µ − r1)
(T − t)
Substituting (9) into (3), we obtain an exact form for the Doléans
exponential χ∗
t associated with the control h∗:
χ∗
t := exp
γ
1 − γ
(µ − r1)0
Σ−1
W (t)
−
1
2
γ
1 − γ
2
(µ − r1)0
(ΣΣ0
)−1
(µ − r1) t
)
(9)
We can easily check that χ∗
t is indeed an exponential martingale.
Therefore h∗ is an admissible control.
18. On the Optimality of Kelly Strategies
Back to Basics: Kelly Strategies in the Merton Model
In the Merton model, fractional Kelly strategies appear naturally as
a result of a classical Fund Separation Theorem:
Theorem (Fund Separation Theorem)
Any portfolio can be expressed as a linear combination of
investments in the Kelly (log-utility) portfolio
hK
(t) = (ΣΣ0
)−1
(µ − r) (10)
and the risk-free rate. Moreover, if an investor has a risk aversion
γ, then the proportion of the Kelly portfolio will equal 1
1−γ .
19. On the Optimality of Kelly Strategies
Insights from the Merton Model: Change of Measure and Duality
Insights from the Merton Model: Change of Measure and
Duality
The measure Ph defined in (7) is also used in the
martingale/duality approach to dynamic portfolio selection (see for
example [14] and references therein).
In complete market, the change of measure technique is equivalent
to the martingale approach: the change of measure approach relies
on the optimal asset allocation to identify the equivalent martingale
measure, the martingale approach works in the opposite direction.
20. On the Optimality of Kelly Strategies
Further Insights from the Merton Model...
Further Insights from the Merton Model...
The level of risk aversion γ dictates the choice of measure Ph.
Two special cases are worth mentioning.
Case 1: The Physical Measure
The measures Ph is the physical measure P in the limit as γ → 0,
that is in the log utility or Kelly criterion case. This observation
forms the basis for the ‘benchmark approach to finance’ (see [13]).
21. On the Optimality of Kelly Strategies
Further Insights from the Merton Model...
Case 2: The Dangers of Overbetting
A well established and somewhat surprising folk theorem holds that
“when an agent overbets by investing in twice the Kelly portfolio,
his/her expected return will be equal to the risk-free rate.” (see for
instance [15] and [11]).
Why is that???
22. On the Optimality of Kelly Strategies
Further Insights from the Merton Model...
The risk aversion of an agent who invest into twice the Kelly
portfolio must be γ = 1
2.
I Hence, the measures Ph coincides with the equivalent
martingale measure Q!
I Under this measure, the portfolio value discounted at the
risk-free rate is a martingale.
23. On the Optimality of Kelly Strategies
Further Insights from the Merton Model...
In the setting of the Merton model and its lognormally distributed
asset prices, the definition of Fractional Kelly allocations
guarantees optimality of the strategy.
However, Fractional Kelly strategies are no longer optimal as soon
as the lognormality assumption is removed (see Thorpe in [11]).
This situation suggests that the definition of Fractional Kelly
strategies could be broadened in order to guarantee optimality. We
can take a first step in this direction by revisiting the ICAPM (see
Merton[12]) in which the drift rate of the asset prices depend on a
number of Normally-distributed factors.
24. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
Getting Kelly Strategies to Work for Factor Models
Key ingredients:
I Probability space (Ω, {Ft} , F, P);
I RN-valued (Ft)-Brownian motion W (t), N := n + m;
I Si (t) denotes the price at time t of the ith security, with
i = 0, . . . , m;
I Xj (t) denotes the level at time t of the jth factor, with
j = 1, . . . , n.
For the time being, we assume that the factors are observable.
25. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
The dynamics of the money market account is given by:
dS0(t)
S0(t)
= a0 + A0
0X(t)
dt, S0(0) = s0 (11)
The dynamics of the m risky securities and n factors are
dS(t) = D (S(t)) (a + AX(t))dt + D (S(t)) ΣdW (t),
Si (0) = si , i = 1, . . . , m (12)
and
dX(t) = (b + BX(t))dt + ΛdW (t), X(0) = x (13)
where X(t) is the Rn-valued factor process with components Xj (t)
and the market parameters a, A, b, B, Σ := [σij ], Λ := [Λij ] are
vectors and matrices of appropriate dimensions.
26. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
Assumption
The matrices ΣΣ0 and ΛΛ0 are positive definite.
I The objective of an investor remains the maximization of
criterion (15) at a fixed time horizon T;
I The wealth V (t) of the portfolio in response to an investment
strategy h ∈ A(T) is now factor-dependent with dynamics:
dV (t)
V (t)
= a0 + A0
0X(t)
dt + h0
(t)(â + ÂX(t))dt + h0
(t)ΣdWt
(14)
with â := a − a01, Â := A − 1A0
0, and initial endowment V (0) = v.
27. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
The expected utility of terminal wealth J(t, x, h; T, γ) si factor
dependent:
J(t, x, h; T, γ) = E [U(VT )] = E
V γ
T
γ
= E
eγ ln VT
γ
By Itô’s lemma,
eγ ln V (t)
= vγ
exp
γ
Z t
0
g(Xs, h(s); θ)ds
χh
t (15)
where
g(x, h; γ) = −
1
2
(1 − γ) h0
ΣΣ0
h + h0
(â + Âx) + a0 + A0
0x
(16)
and the exponential martingale χh
t is still given by (3).
28. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
Applying the change of measure argument, we obtain the control
criterion under the measure Ph
I(t, x, h; T, γ) =
vγ
γ
Eh
t,x
exp
γ
Z T
t
g(Xs, h(s); θ)ds
(17)
where Eh
t,x [·] denotes the expectation taken with respect to the
measure Ph and with initial conditions (t, x).
The Ph-dynamics of the state variable X(t) is
dX(t) = b + BX(t) + γΛΣ0
h(t)
dt + ΛdW h
t , t ∈ [0, T]
(18)
29. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
Then value function Φ associated with the auxiliary criterion
function I(t, x; h; T, γ) is defined as
Φ(t, x) = sup
h∈A(T)
I(t, x; h; T, γ) (19)
We use the Feynman-Kač formula to write down the HJB PDE:
∂Φ
∂t
(t, x) +
1
2
tr ΛΛ0
D2
Φ(t, x)
+ H(t, x, Φ, DΦ) = 0 (20)
subject to terminal condition
Φ(T, x) =
vγ
γ
(21)
and where
H(s, x, r, p) = sup
h∈R
n
b + Bx + γΛΣ0
h
0
p − γg(x, h; γ)r
o
(22)
for r ∈ R and p ∈ Rn.
30. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
To obtain the optimal control in a more convenient format and
derive the value function Φ(t, x) more easily, we find it convenient
to consider the logarithmically transformed value function
Φ̃(t, x) := 1
γ ln γΦ(t, x) with associated HJB PDE
∂Φ̃
∂t
(t, x) + inf
h∈Rm
Lh
t (t, x, DΦ, D2
Φ) = 0 (23)
where
Lh
t (s, x, p, M) = b + Bx + γΛΣ0
h(s)
0
p +
1
2
tr ΛΛ0
M
+
γ
2
p0
ΛΛ0
p − g(x, h; γ) (24)
for r ∈ R and p ∈ Rn and subject to terminal condition
Φ̃(T, x) = ln v (25)
This is in fact a risk-sensitive asset management problem
(see [1], [9], [6]).
31. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
Solving the optimization problem in (24) gives the optimal
investment policy h∗(t)
h∗
(t) =
1
1 − γ
ΣΣ0
−1
h
â + ÂX(t) + γΣΛ0
DΦ̃(t, X(t))
i
(26)
The solution to HJB PDE (23) is
Φ̃(t, x) =
1
2
x0
Q(t)x + x0
q(t) + k(t)
where Q(t) satisfies a matrix Riccati equation, q(t) satisfies a
vector linear ODE and k(t) is an integral. As a result,
h∗
(t) =
1
1 − γ
ΣΣ0
−1
h
â + ÂX(t) + γΣΛ0
(Q(t)X(t) + q(t))
i
(27)
32. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
Substituting (26) into (3), we obtain an exact form for the Doléans
exponential χ∗
t associated with the control h∗:
χ∗
t := exp
γ
1 − γ
Z t
0
h
â + ÂX(s) + γΣΛ0
(Q(s)X(s) + q(s))
i0
×(ΣΣ0
)−1
ΣdW (s)
−
1
2
γ
1 − γ
2 Z t
0
â + ÂX(s) + γΣΛ0
(Q(s)X(s) + q(s))
0
×(ΣΣ0
)−1
â + ÂX(s) + γΣΛ0
(Q(s)X(s) + q(s))
0
ds
(28)
33. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
We can interpret the Girsanov kernel
γ
1 − γ
Σ0
(ΣΣ0
)−1
h
â + ÂX(s) + γΣΛ0
(Q(s)X(s) + q(s))
i
as the projection of the factor-dependent returns
â + ÂX(s) + γΣΛ0
(Q(s)X(s) + q(s))
on the subspace spanned by the asset volatilities Σ, scaled by γ
1−γ .
This observation also implies that the appropriate Girsanov kernel
in a complete (factor-free) setting is
γ
1 − γ
Σ0
(ΣΣ0
)−1
(µ − r1)
with a similar interpretation.
34. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
In the ICAPM, a new view of Fractional Kelly investing emerges:
Theorem (ICAPM Fund Separation Theorem)
Any portfolio can be expressed as a linear combination of
investments into two funds’ with respective risky asset allocations:
hK
(t) = (ΣΣ0
)−1
â + ÂX(t)
hI
(t) = (ΣΣ0
)−1
ΣΛ0
(Q(t)X(t) + q(t)) (29)
and respective allocation to the money market account given by
hK
0 (t) = 1 − 10
(ΣΣ0
)−1
â + ÂX(t)
hI
0(t) = 1 − 10
(ΣΣ0
)−1
ΣΛ0
(Q(t)X(t) + q(t))
Moreover, if an investor has a risk aversion γ, then the respective
weights of each mutual fund in the investor’s portfolio equal 1
1−γ
and γ
, respectively.
35. On the Optimality of Kelly Strategies
Getting Kelly Strategies to Work for Factor Models
In the factor-based ICAPM,
ΣΣ0
−1
h
â + ÂX(t)
i
(30)
represents the Kelly (log utility) portfolio and
ΣΣ0
−1
ΣΛ0
(Q(t)X(t) + q(t)) (31)
is the ‘intertemporal hedging porfolio’ identified by Merton.
This mutual fund theorem raises some questions as to the
practicality of the intertemporal hedging portfolio as an investment
option.
36. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Beyond Factor Models: Random Coefficients
The change of measure approach still works well in a partial
observation case where we would need to estimate the coefficients
of the factor process through a Kalman filter.
However, this presupposes that we know the form of the factor
process.
A more general approach would be to assume that the coefficients
of the asset price dynamics are random (See Bjørk, Davis and
Landén [14] and references therein).
37. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Key ingredients:
I The “full” underlying probability space (Ω, {Ft} , F, P);
I The filtration FW
t := σ(W (s)), 0 ≤ s ≤ t) generated by an
m-dimensional Brownian motions driving the asset returns
(augmented by the P-null sets);
I The filtration FS
t := σ(S(s)), 0 ≤ s ≤ t) generated by the m
asset price (also augmented by the P-null sets);
The dynamics of the money market account is given by:
dS0(t)
S0(t)
= r(t)dt, S0(0) = s0 (32)
where r ∈ R+ is a bounded FS
t -adapted process. We will denote
by Z(t, T) the discount factor:
Z(t, T) = exp
−
Z T
t
r(s)ds
(33)
38. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
The dynamics of the m risky securities and n factors can be
expressed as:
dSi (t)
Si (t)
= µi (t)dt+
m
X
k=1
σik(t)dWk(t), Si (0) = si , i = 1, . . . , m
(34)
where µ(t) = (µ1(t), . . . , µm(t))0 is an Ft-adapted process and the
volatility Σ(t) := [σij (t)] , i = 1, . . . , m, j = 1, . . . , m is an
FS
t -adapted process. More synthetically,
dS(t) = D(S(t))µ(t)dt + D(S(t))Σ(t)dW (t) (35)
Note that no Markovian structure is either assumed or required.
Assumption
We assume that the matrix Σ(t) is positive definite ∀t.
39. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
The critical step in this approach is to establish a projection onto
the observable filtration.
Once this is done, the partial observation problem related to the
filtration F can be rewritten as an equivalent complete observation
problem with respect to the filtration FS .
This effectively changes a utility maximizaton problem set in an
incomplete market into a standard utility maximization problem set
in a complete market.
40. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Define the process Y (t) by:
dY (t) = Σ−1
t D(St)−1
dSt = Σ−1
t µtdt + ΣtdW (t) (36)
For any F-adapted process X, define the filter estimate process X̂
as the optional projection of X onto the filtration FS :
Ŷt = EP
h
Yt|FS
t
i
41. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Next, define the innovation process U by
dU(t) = dZ(t) −
(Σ−1µ(t))dt = dZ(t) − Σ−1
µ̂(t)dt (37)
where the second equality follows from the observability of Σt.
Note that the innovation process U(t) is a standard FS -Brownian
motion (see for example Lipster and Shiryaev [10]).
As a result, we can rewrite (35) using the filter estimate for α and
the innovation process U obtained with respect to the filtration
FS :
dS(t) = D(S(t))µ̂(t)dt + D(S(t))Σ(t)dU(t) (38)
The remainder follows from a standard martingale argument.
42. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Definition
The Girsanov kernel is the vector process ϕt given by
ϕ(t) = Σ−1
(t)(µ̂(t) − r1) (39)
for all t.
Note that he Girsanov kernel ϕ(t) and the market price of risk
vector λ(t) are related by λ(t) = −ϕ(t).
43. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Assumption
We assume that the Girsanov kernel satisfies the Novikov condition
E
h
e
1
2
R T
0 kϕ(t)k2dt
i
∞ (40)
and the integrability condition
E
e
1
2
R T
0
γ
1−γ
2
kϕ(t)k2dt
∞ (41)
44. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Definition
The equivalent martingale measure Q is defined by the
Radon-Nikodým derivative
dQ
dP
= Lt := exp
−
Z t
0
ϕ0
(s)dU(s) −
1
2
Z t
0
ϕ0
(s)ϕ(s)ds
,
on FS
t .
I It follows from Assumption 4 that Lt is a true martingale.
I Moreover, Q is unique: this is a direct consequence of the use
of filtered estimates in (38).
45. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
The objective is to maximize the expected utility of terminal
wealth:
J(t, h; T, γ) = EP
V γ
T
γ
subject to the budget constraint
EQ
[K0,T VT ] = v = EP
[K0,T LT VT ] (42)
Next, form the Lagrangian
L(h, λ; T) =
1
γ
EP
V γ
T
− λ
EP
[K0,T LT VT ] − v
=
Z
Ω
1
γ
V γ
T − λ (K0,T (ω)LT (ω)VT (ω) − v) dP(ω)
46. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
This separable problem can be maximized for each ω. The first
order condition leads to
V ∗
T = [λK0,T LT ]
−1
1−γ
Substituting in the budget equation (42), we get
λ
−1
1−γ EP
h
(K0,T LT )
−γ
1−γ
i
= v
and as a result,
V ∗
T =
v
H0
K
−1
1−γ
0,T L
−1
1−γ
T (43)
with
H0 = EP
h
(K0,T LT )
−γ
1−γ
i
47. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Therefore, the optimal expected utility is equal to:
U0 = EP
(V ∗
T )γ
γ
= H1−γ
0
v
γ
(44)
Observe that H0 can be expressed as
H0 = EP
L0
T exp
Z T
0
rt +
1
2(1 − γ)
kϕ(t)k2
dt
where
L0
t := exp
(
γ
1 − γ
Z t
0
ϕ0
(s)dU(s) −
1
2
γ
1 − γ
2 Z t
0
ϕ0
(s)ϕ(s)ds
)
(45)
is a true martingale. This leads us to the following definition:
48. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Definition
(i). The measure Q0 is defined by the Radon-Nikodým derivative
dQ0
dP
= L0
t , on FS
t
(ii). The process H is defined by
Ht = E0
exp
γ
1 − γ
Z T
t
rs +
1
2(1 − γ)
kϕ(s)k2
ds
53. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
Finally, we recall Proposition 4.1 in Bjørk, Davis and Landén [14]:
Proposition
The following hold:
(i). The optimal wealth process V ∗(t) is given by
V ∗
(t) =
H(t)
H(0)
K0,tL̄t
−1
1−γ
x
(ii). The optimal weight vector h∗ is given by
h∗
(t) =
1
1 − γ
(ΣtΣ0
t)−1
(µt − r1) + σH(t) Σ−1
t
0
(47)
where σH is the volatility term of H, i.e. H has dynamics of
the form
H(t) = µH(t)dt + σH(t)dU(t)
54. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
In the limit as γ → 0, the optimal wealth process V̄ ∗(t) is given by
V ∗
(t) = K0,tL̄t
−1
1−γ
x
and the optimal investment is the Kelly portfolio,
h∗
(t) = (ΣtΣ0
t)−1
(µt − r1) (48)
55. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
With the choice γ = 1
2, and
h∗
(t) = 2(ΣtΣ0
t)−1
(µt − r1) + σH(t) Σ−1
t
0
(49)
As a result,
V ∗
(t) =
H(t)
H(0)
K0,tL̄t
2
x = x exp
Z T
t
r(s)ds
as expected.
56. On the Optimality of Kelly Strategies
Beyond Factor Models: Random Coefficients
However, there is still something unsatisfactory about this result:
the optimal strategy depends on the volatility of the process H(t):
Ht = E0
exp
γ
1 − γ
Z T
t
rs +
1
2(1 − γ)
kϕ(s)k2
ds
57.
58.
59.
60. Ft
As a result, getting some intuition on the intertemporal hedging
term is ‘difficult’.
61. On the Optimality of Kelly Strategies
Conclusion and Next Steps
Conclusion and Next Steps
I Get stronger statements of equivalence between the
martingale/duality approach and the change of measure
technique in incomplete markets;
I Study the practicality of intertemporal hedging portfolio as an
investment option;
I Add jumps.
62. On the Optimality of Kelly Strategies
Conclusion and Next Steps
Thank you!
3
Any question?
3
Copyright: W. Krawcewicz, University of Alberta
63. On the Optimality of Kelly Strategies
Conclusion and Next Steps
T.R. Bielecki and S.R. Pliska.
Risk-sensitive dynamic asset management.
Applied Mathematics and Optimization, 39:337–360, 1999.
T.R. Bielecki and S.R. Pliska.
Risk sensitive intertemporal CAPM.
IEEE Transactions on Automatic Control, 49(3):420–432,
March 2004.
M.H.A. Davis and S. Lleo.
Risk-sensitive benchmarked asset management.
Quantitative Finance, 8(4):415–426, June 2008.
M.H.A. Davis and S. Lleo.
On the optimality of kelly strategies ii: Applications.
working paper, 2010.
M.H.A. Davis and S. Lleo.
64. On the Optimality of Kelly Strategies
Conclusion and Next Steps
Jump-diffusion risk-sensitive asset management i: Diffusion
factor model.
SIAM Journal on Financial Mathematics (to appear), 2011.
M.H.A. Davis and S. Lleo.
The Kelly Capital Growth Investment Criterion: Theory and
Practice, chapter Fractional Kelly Strategies for Benchmarked
Asset Management.
World Scientific, 2011.
J. Kelly.
A new interpretation of information rate.
Bell System Technology, 35:917–926, 1956.
B. Øksendal.
Stochastic Differential Equations.
Universitext. Springer-Verlag, 6 edition, 2003.
K. Kuroda and H. Nagai.
65. On the Optimality of Kelly Strategies
Conclusion and Next Steps
Risk-sensitive portfolio optimization on infinite time horizon.
Stochastics and Stochastics Reports, 73:309–331, 2002.
R. Lipster and A. Shiryaev.
Statistics of Random Processes: I. General Theory.
Probability and Its Applications. Springer-Verlag, 2 edition,
2004.
L.C. MacLean, E. Throp, and W.T. Ziemba, editors.
The Kelly Capital Growth Investment Criterion: Theory and
Practice.
World Scientific, 2011.
R.C. Merton.
An intertemporal capital asset pricing model.
Econometrica, 41(866–887), 1973.
E. Platen and D. Heath.
A Benchmark Approach to Quantitative Finance.
66. On the Optimality of Kelly Strategies
Conclusion and Next Steps
Springer Finance. Springer-Verlag, 2006.
T. Bjørk, M. Davis, and C. Landén.
Optimal investment under partial information.
Mathematical Methods of Operations Research, 2010.
E. Thorp.
Handbook of Asset and Liability Management, chapter The
Kelly Criterion in Blackjack, Sports Betting and the Stock
Market.
Handbook in Finance. North Holland, 2006.