In this paper, we focus on a financial market with one riskless and one risky asset, and consider the
asset allocation problem in the form of semi-variable transaction costs. One of the basic ideas of this paper is to
transform the problem of maximizing the expected utility of terminal wealth in a friction market with semi
This document discusses different types of arbitrage opportunities in derivatives markets. It describes deterministic arbitrage as strategies that offer risk-less profits, such as cash and carry arbitrage or reverse cash and carry arbitrage. Statistical arbitrage aims to profit from pricing inefficiencies identified by mathematical models, but carries more risk than deterministic strategies. Pair trading and exploiting corporate actions are provided as examples of statistical arbitrage strategies.
Dissertation template bcu_format_belinda -sampleAssignment Help
Dear student, Warm Greetings of the Day!!! We are a qualified team of consultants and writers who provide support and assistance to students with their Assignments, Essays and Dissertation. If you are having difficulties writing your work, finding it stressful in completing your work or have no time to complete your work yourself, then look no further. We have assisted many students with their projects. Our aim is to help and support students when they need it the most. We oversee your work to be completed from start to end. We specialize in a number of subject areas including, Business, Accounting, Economic, Nursing, Health and Social Care, Criminology, Sociology, English, Law, IT, History, Religious Studies, Social Sciences, Biology, Physic, Chemistry, Psychology and many more. Our consultants are highly qualified in providing the highest quality of work to students. Each work will be unique and not copied like others. You can count on us as we are committed to assist you in producing work of the highest quality. Waiting for your quick response and want to start healthy long term relationship with you. Regards http://www.cheapassignmenthelp.com/ http://www.cheapassignmenthelp.co.uk/
This document discusses pricing CDOs using the intensity gamma approach. Some key points:
1) The intensity gamma approach models default correlation through a business time process, where defaults become conditionally independent given the business time path. This addresses issues with the Gaussian copula model.
2) The approach involves simulating business time paths, then calculating default intensities and times to price CDO tranches.
3) The business time process is modeled as a combination of gamma processes and drift. Efficient simulation techniques are discussed to generate the business time paths.
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.
The document discusses derivatives markets in Nepal. It defines different types of derivatives like futures, options, swaps and forwards. It describes the key parties involved in derivatives trading like clients, brokers, and clearing houses. It provides an overview of the history and development of derivatives exchanges in Nepal. Some challenges of the Nepalese derivatives market mentioned include lack of regulation, expertise and awareness.
The document summarizes a theory that financial markets were previously split into two parts - price risky and price stable. This split market paradigm was disrupted in the 1970s with the introduction of over-the-counter (OTC) instruments. Had the split been maintained, it argues there would have been less risk added to the system and perhaps no financial crisis in 2007. It describes how restoring the split market through new trading technology could reduce risks.
This document provides a syllabus and study material for the Economics class of Class XII in 2014-15. It outlines the course content, which is divided into two parts - Introductory Microeconomics and Introductory Macroeconomics. The Microeconomics section covers topics such as consumer behavior, producer behavior, market structures, and the introduction lesson. The introduction lesson defines key economic concepts such as scarcity, choice, opportunity cost and presents the production possibility curve. It also describes the three basic economic problems of what to produce, how to produce and for whom to produce. The document provides sample questions for students and a test paper with questions from the introduction lesson.
This document discusses different types of arbitrage opportunities in derivatives markets. It describes deterministic arbitrage as strategies that offer risk-less profits, such as cash and carry arbitrage or reverse cash and carry arbitrage. Statistical arbitrage aims to profit from pricing inefficiencies identified by mathematical models, but carries more risk than deterministic strategies. Pair trading and exploiting corporate actions are provided as examples of statistical arbitrage strategies.
Dissertation template bcu_format_belinda -sampleAssignment Help
Dear student, Warm Greetings of the Day!!! We are a qualified team of consultants and writers who provide support and assistance to students with their Assignments, Essays and Dissertation. If you are having difficulties writing your work, finding it stressful in completing your work or have no time to complete your work yourself, then look no further. We have assisted many students with their projects. Our aim is to help and support students when they need it the most. We oversee your work to be completed from start to end. We specialize in a number of subject areas including, Business, Accounting, Economic, Nursing, Health and Social Care, Criminology, Sociology, English, Law, IT, History, Religious Studies, Social Sciences, Biology, Physic, Chemistry, Psychology and many more. Our consultants are highly qualified in providing the highest quality of work to students. Each work will be unique and not copied like others. You can count on us as we are committed to assist you in producing work of the highest quality. Waiting for your quick response and want to start healthy long term relationship with you. Regards http://www.cheapassignmenthelp.com/ http://www.cheapassignmenthelp.co.uk/
This document discusses pricing CDOs using the intensity gamma approach. Some key points:
1) The intensity gamma approach models default correlation through a business time process, where defaults become conditionally independent given the business time path. This addresses issues with the Gaussian copula model.
2) The approach involves simulating business time paths, then calculating default intensities and times to price CDO tranches.
3) The business time process is modeled as a combination of gamma processes and drift. Efficient simulation techniques are discussed to generate the business time paths.
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.
The document discusses derivatives markets in Nepal. It defines different types of derivatives like futures, options, swaps and forwards. It describes the key parties involved in derivatives trading like clients, brokers, and clearing houses. It provides an overview of the history and development of derivatives exchanges in Nepal. Some challenges of the Nepalese derivatives market mentioned include lack of regulation, expertise and awareness.
The document summarizes a theory that financial markets were previously split into two parts - price risky and price stable. This split market paradigm was disrupted in the 1970s with the introduction of over-the-counter (OTC) instruments. Had the split been maintained, it argues there would have been less risk added to the system and perhaps no financial crisis in 2007. It describes how restoring the split market through new trading technology could reduce risks.
This document provides a syllabus and study material for the Economics class of Class XII in 2014-15. It outlines the course content, which is divided into two parts - Introductory Microeconomics and Introductory Macroeconomics. The Microeconomics section covers topics such as consumer behavior, producer behavior, market structures, and the introduction lesson. The introduction lesson defines key economic concepts such as scarcity, choice, opportunity cost and presents the production possibility curve. It also describes the three basic economic problems of what to produce, how to produce and for whom to produce. The document provides sample questions for students and a test paper with questions from the introduction lesson.
The document provides an overview of commodities markets and derivatives trading. It discusses:
1) The origins of commodity futures markets in addressing price risks for farmers and merchants and the establishment of the Chicago Board of Trade in 1848.
2) The definition of derivatives and how their value is derived from underlying assets, as well as the key participants in derivatives markets.
3) The cost of carry model which states that the futures price is equal to the expected future spot price plus the cost of carrying the underlying asset.
4) Features of the National Commodity & Derivatives Exchange in India and how it collects and disseminates spot price data through polling participants across commodity markets.
This document analyzes the validity of the Capital Asset Pricing Model (CAPM) using data from the Karachi Stock Exchange (KSE) from 2011-2014. It finds that CAPM does not fully hold for the KSE. While expected and actual returns were slightly different for some stocks in some years, indicating CAPM may be applicable, there were large differences for many observations. Overall, CAPM did not accurately convey results for the KSE. The document reviews previous literature on CAPM which also found it does not always hold and may be more accurate for some stocks and time periods. The methodology used beta calculations to estimate expected returns for 30 KSE stocks which were then compared to actual returns.
Mercantile Exchange Nepal Limited: A Commodity Derivative ExchangeKarna Lama
This document discusses Mercantile Exchange Nepal Limited (MEX), a commodity derivative exchange established in 2007. MEX facilitates futures and spot trading in precious metals, energy, base metals, and agricultural commodities. The strategic plan of MEX is to promote local commodities, improve supply chain management, introduce options, focus on customer interests, and adopt sophisticated technology. MEX performs key functions like price discovery, risk transfer, margining, monitoring, and providing an efficient trading platform. However, Nepal's derivative market faces challenges like lack of regulation, low participant knowledge, technological difficulties, and lack of capital and resources.
In the world of the commodity futures markets, no subject generates more interest and questions than spread trading. A spread combines both a long and a short position put on at the same time in related futures contracts.
A synopsis of Final research Report ON Investors' preference on various Investment Avenues in India.
A research report will be generated at the end of the final period evaluating the hypothesis of the reasearch
The document provides an overview of derivatives and their development in the Indian market. It discusses various types of derivative instruments traded in India, including futures, forwards, options, and swaps. It outlines the key users of derivatives in India, with retail investors being major participants in equity derivatives markets while financial institutions are more active in over-the-counter fixed income markets. The document also provides definitions and examples of common derivative contract types such as forwards, futures, options, and swaps.
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.
The document provides instructions for students appearing for CBSE Board Examinations 2011-2012. It mentions that the code number on the question paper should be written on the answer booklet. It also notes that the paper contains 8 printed pages and students have 15 minutes to read the paper before writing answers. The paper then provides instructions and questions for the Economics exam, including short answer questions worth 1-4 marks and long answer questions worth 6 marks.
The document summarizes the capital asset pricing model (CAPM) and reviews early empirical tests of the model. It begins by outlining the logic and key assumptions of the CAPM, including that the market portfolio must be mean-variance efficient. However, empirical tests found problems with the CAPM's predictions about the relationship between expected returns and market betas. Specifically, cross-sectional regressions did not find intercepts equal to the risk-free rate or slopes equal to the expected market premium. To address measurement error, later tests examined portfolios rather than individual assets. In general, the early empirical evidence revealed shortcomings in the CAPM's ability to explain returns.
Risk valuation for securities with limited liquidityJack Sarkissian
Everything seems simple with liquid securities - price is known, risks are more or less known too. It becomes a lot harder when we get illiquid instruments in the book. This is why we developed this model to enable modeling of securities with low liquidity and evaluate impact of risk sources associated with liquidity. And in order to do that we had to demonstrate that price formation has quantum chaotic character.
The document introduces the binomial option pricing model, which uses a binomial tree to represent the possible paths an underlying asset's price may take over the life of an option. It assumes a risk-neutral world where expected returns are equal to the risk-free rate. The model prices options by constructing hedge portfolios that eliminate risk, with the option price being the value that makes the portfolio worth the same whether the asset price rises or falls. For a single time period, if the asset price can rise by u or fall by d, with hedge ratio h, the option price C is derived as p(1-p)P, where p is (r-d)/(u-d) and P is the payoff function.
This document summarizes the Capital Asset Pricing Model (CAPM). It begins by outlining the key assumptions and logic behind the CAPM. The CAPM builds on Harry Markowitz's portfolio choice model by adding assumptions of a risk-free rate and market clearing prices. This implies that the market portfolio must be mean-variance efficient. The CAPM then predicts that an asset's expected return is determined by its beta, or non-diversifiable risk relative to the market. However, the document notes that empirical tests have found the CAPM performs poorly in validating these predictions. It concludes that while theoretical or implementation issues may be to blame, the CAPM's failure in empirical tests means its applications are generally invalid.
Arbitrage pricing theory & Efficient market hypothesisHari Ram
Arbitrage pricing theory (APT) is a multi-factor asset pricing model based on the idea that an asset's returns can be predicted using the linear relationship between the asset's expected return and a number of macroeconomic variables that capture systematic risk.
This document summarizes the capital asset pricing model (CAPM). It begins by outlining the logic and key assumptions of the CAPM, including that all investors hold the same market portfolio which must lie on the efficient frontier. It then states that the CAPM predicts the expected return of an asset is determined by its beta, or non-diversifiable risk relative to the market. However, the document notes that empirical tests have found the CAPM performs poorly in applications. It concludes the CAPM's failings indicate applications based on the model are invalid, challenging researchers to develop alternative models.
Questions and Answers At Least 75 Words each.Please answer th.docxmakdul
Questions and Answers: At Least 75 Words each.
Please answer the following questions.
1. What are the differences and similarities between samples and populations?
2. What are the measures of Central Tendency assumptions?
3. What are measures of Dispersion used for and what are the assumptions for each?
4. Define collaboration and how you will apply it in Statistics? (100 Words)
The Capital Asset Pricing Model:
Theory and Evidence
Eugene F. Fama and Kenneth R. French
T he capital asset pricing model (CAPM) of William Sharpe (1964) and JohnLintner (1965) marks the birth of asset pricing theory (resulting in aNobel Prize for Sharpe in 1990). Four decades later, the CAPM is still
widely used in applications, such as estimating the cost of capital for firms and
evaluating the performance of managed portfolios. It is the centerpiece of MBA
investment courses. Indeed, it is often the only asset pricing model taught in these
courses.1
The attraction of the CAPM is that it offers powerful and intuitively pleasing
predictions about how to measure risk and the relation between expected return
and risk. Unfortunately, the empirical record of the model is poor—poor enough
to invalidate the way it is used in applications. The CAPM’s empirical problems may
reflect theoretical failings, the result of many simplifying assumptions. But they may
also be caused by difficulties in implementing valid tests of the model. For example,
the CAPM says that the risk of a stock should be measured relative to a compre-
hensive “market portfolio” that in principle can include not just traded financial
assets, but also consumer durables, real estate and human capital. Even if we take
a narrow view of the model and limit its purview to traded financial assets, is it
1 Although every asset pricing model is a capital asset pricing model, the finance profession reserves the
acronym CAPM for the specific model of Sharpe (1964), Lintner (1965) and Black (1972) discussed
here. Thus, throughout the paper we refer to the Sharpe-Lintner-Black model as the CAPM.
y Eugene F. Fama is Robert R. McCormick Distinguished Service Professor of Finance,
Graduate School of Business, University of Chicago, Chicago, Illinois. Kenneth R. French is
Carl E. and Catherine M. Heidt Professor of Finance, Tuck School of Business, Dartmouth
College, Hanover, New Hampshire. Their e-mail addresses are �[email protected]
edu� and �[email protected]�, respectively.
Journal of Economic Perspectives—Volume 18, Number 3—Summer 2004 —Pages 25– 46
legitimate to limit further the market portfolio to U.S. common stocks (a typical
choice), or should the market be expanded to include bonds, and other financial
assets, perhaps around the world? In the end, we argue that whether the model’s
problems reflect weaknesses in the theory or in its empirical implementation, the
failure of the CAPM in empirical tests implies that most applications of the model
are invalid.
We begin by outlining the logic of t ...
This document provides an overview of an upcoming presentation on asset pricing models. The presentation will cover capital market theory, the capital market line, security market line, capital asset pricing model, and diversification. It will discuss the assumptions and formulas for the capital market line and security market line. The capital market line shows expected returns based on portfolio risk, while the security market line shows expected individual asset returns based on systematic risk. The capital asset pricing model uses the concept of beta to calculate the expected return of an asset based on its risk relative to the market.
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.
This document summarizes a study that examines measures of liquidity and factors that influence liquidity on the Tunisian stock market. The study analyzes various measures of liquidity like bid-ask spread, trading volume, and depth for 40 stocks traded on the Tunisian market from 2011 to 2013. The results show that measures like spread, volume, and information arrival are significantly correlated with changes in liquidity over time. The arrival of new information may be a common factor influencing different liquidity measures across stocks.
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
Modern portfolio theory (MPT) is a theory of finance that aims to construct portfolios that offer the maximum expected return for a given level of risk or the minimum risk for a given level of expected return. MPT uses diversification and asset allocation to reduce portfolio risk. It assumes investors are rational and markets are efficient. MPT models asset returns as normally distributed and defines risk as standard deviation of returns. It seeks to minimize total portfolio variance by combining assets whose returns are not perfectly correlated. The efficient frontier shows the optimal risk-return tradeoff and the capital allocation line incorporates a risk-free asset into the analysis. MPT is widely used but also faces criticisms around its assumptions.
The_Use_and_Abuse_of_Implementation_Shortfall_whitepaperSamuel Zou
The document examines the evolution and limitations of using implementation shortfall as the dominant benchmark for measuring trading performance. It argues that while implementation shortfall is useful for measuring overall costs to a fund, it has been overly extended and reduced to simply measuring slippage. This has resulted in an inability to provide meaningful insights into improving trading performance. The document proposes an alternative framework that provides greater explanatory power and ability to form actionable hypotheses for enhancing various aspects of the trading process.
The document provides an overview of commodities markets and derivatives trading. It discusses:
1) The origins of commodity futures markets in addressing price risks for farmers and merchants and the establishment of the Chicago Board of Trade in 1848.
2) The definition of derivatives and how their value is derived from underlying assets, as well as the key participants in derivatives markets.
3) The cost of carry model which states that the futures price is equal to the expected future spot price plus the cost of carrying the underlying asset.
4) Features of the National Commodity & Derivatives Exchange in India and how it collects and disseminates spot price data through polling participants across commodity markets.
This document analyzes the validity of the Capital Asset Pricing Model (CAPM) using data from the Karachi Stock Exchange (KSE) from 2011-2014. It finds that CAPM does not fully hold for the KSE. While expected and actual returns were slightly different for some stocks in some years, indicating CAPM may be applicable, there were large differences for many observations. Overall, CAPM did not accurately convey results for the KSE. The document reviews previous literature on CAPM which also found it does not always hold and may be more accurate for some stocks and time periods. The methodology used beta calculations to estimate expected returns for 30 KSE stocks which were then compared to actual returns.
Mercantile Exchange Nepal Limited: A Commodity Derivative ExchangeKarna Lama
This document discusses Mercantile Exchange Nepal Limited (MEX), a commodity derivative exchange established in 2007. MEX facilitates futures and spot trading in precious metals, energy, base metals, and agricultural commodities. The strategic plan of MEX is to promote local commodities, improve supply chain management, introduce options, focus on customer interests, and adopt sophisticated technology. MEX performs key functions like price discovery, risk transfer, margining, monitoring, and providing an efficient trading platform. However, Nepal's derivative market faces challenges like lack of regulation, low participant knowledge, technological difficulties, and lack of capital and resources.
In the world of the commodity futures markets, no subject generates more interest and questions than spread trading. A spread combines both a long and a short position put on at the same time in related futures contracts.
A synopsis of Final research Report ON Investors' preference on various Investment Avenues in India.
A research report will be generated at the end of the final period evaluating the hypothesis of the reasearch
The document provides an overview of derivatives and their development in the Indian market. It discusses various types of derivative instruments traded in India, including futures, forwards, options, and swaps. It outlines the key users of derivatives in India, with retail investors being major participants in equity derivatives markets while financial institutions are more active in over-the-counter fixed income markets. The document also provides definitions and examples of common derivative contract types such as forwards, futures, options, and swaps.
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.
The document provides instructions for students appearing for CBSE Board Examinations 2011-2012. It mentions that the code number on the question paper should be written on the answer booklet. It also notes that the paper contains 8 printed pages and students have 15 minutes to read the paper before writing answers. The paper then provides instructions and questions for the Economics exam, including short answer questions worth 1-4 marks and long answer questions worth 6 marks.
The document summarizes the capital asset pricing model (CAPM) and reviews early empirical tests of the model. It begins by outlining the logic and key assumptions of the CAPM, including that the market portfolio must be mean-variance efficient. However, empirical tests found problems with the CAPM's predictions about the relationship between expected returns and market betas. Specifically, cross-sectional regressions did not find intercepts equal to the risk-free rate or slopes equal to the expected market premium. To address measurement error, later tests examined portfolios rather than individual assets. In general, the early empirical evidence revealed shortcomings in the CAPM's ability to explain returns.
Risk valuation for securities with limited liquidityJack Sarkissian
Everything seems simple with liquid securities - price is known, risks are more or less known too. It becomes a lot harder when we get illiquid instruments in the book. This is why we developed this model to enable modeling of securities with low liquidity and evaluate impact of risk sources associated with liquidity. And in order to do that we had to demonstrate that price formation has quantum chaotic character.
The document introduces the binomial option pricing model, which uses a binomial tree to represent the possible paths an underlying asset's price may take over the life of an option. It assumes a risk-neutral world where expected returns are equal to the risk-free rate. The model prices options by constructing hedge portfolios that eliminate risk, with the option price being the value that makes the portfolio worth the same whether the asset price rises or falls. For a single time period, if the asset price can rise by u or fall by d, with hedge ratio h, the option price C is derived as p(1-p)P, where p is (r-d)/(u-d) and P is the payoff function.
This document summarizes the Capital Asset Pricing Model (CAPM). It begins by outlining the key assumptions and logic behind the CAPM. The CAPM builds on Harry Markowitz's portfolio choice model by adding assumptions of a risk-free rate and market clearing prices. This implies that the market portfolio must be mean-variance efficient. The CAPM then predicts that an asset's expected return is determined by its beta, or non-diversifiable risk relative to the market. However, the document notes that empirical tests have found the CAPM performs poorly in validating these predictions. It concludes that while theoretical or implementation issues may be to blame, the CAPM's failure in empirical tests means its applications are generally invalid.
Arbitrage pricing theory & Efficient market hypothesisHari Ram
Arbitrage pricing theory (APT) is a multi-factor asset pricing model based on the idea that an asset's returns can be predicted using the linear relationship between the asset's expected return and a number of macroeconomic variables that capture systematic risk.
This document summarizes the capital asset pricing model (CAPM). It begins by outlining the logic and key assumptions of the CAPM, including that all investors hold the same market portfolio which must lie on the efficient frontier. It then states that the CAPM predicts the expected return of an asset is determined by its beta, or non-diversifiable risk relative to the market. However, the document notes that empirical tests have found the CAPM performs poorly in applications. It concludes the CAPM's failings indicate applications based on the model are invalid, challenging researchers to develop alternative models.
Questions and Answers At Least 75 Words each.Please answer th.docxmakdul
Questions and Answers: At Least 75 Words each.
Please answer the following questions.
1. What are the differences and similarities between samples and populations?
2. What are the measures of Central Tendency assumptions?
3. What are measures of Dispersion used for and what are the assumptions for each?
4. Define collaboration and how you will apply it in Statistics? (100 Words)
The Capital Asset Pricing Model:
Theory and Evidence
Eugene F. Fama and Kenneth R. French
T he capital asset pricing model (CAPM) of William Sharpe (1964) and JohnLintner (1965) marks the birth of asset pricing theory (resulting in aNobel Prize for Sharpe in 1990). Four decades later, the CAPM is still
widely used in applications, such as estimating the cost of capital for firms and
evaluating the performance of managed portfolios. It is the centerpiece of MBA
investment courses. Indeed, it is often the only asset pricing model taught in these
courses.1
The attraction of the CAPM is that it offers powerful and intuitively pleasing
predictions about how to measure risk and the relation between expected return
and risk. Unfortunately, the empirical record of the model is poor—poor enough
to invalidate the way it is used in applications. The CAPM’s empirical problems may
reflect theoretical failings, the result of many simplifying assumptions. But they may
also be caused by difficulties in implementing valid tests of the model. For example,
the CAPM says that the risk of a stock should be measured relative to a compre-
hensive “market portfolio” that in principle can include not just traded financial
assets, but also consumer durables, real estate and human capital. Even if we take
a narrow view of the model and limit its purview to traded financial assets, is it
1 Although every asset pricing model is a capital asset pricing model, the finance profession reserves the
acronym CAPM for the specific model of Sharpe (1964), Lintner (1965) and Black (1972) discussed
here. Thus, throughout the paper we refer to the Sharpe-Lintner-Black model as the CAPM.
y Eugene F. Fama is Robert R. McCormick Distinguished Service Professor of Finance,
Graduate School of Business, University of Chicago, Chicago, Illinois. Kenneth R. French is
Carl E. and Catherine M. Heidt Professor of Finance, Tuck School of Business, Dartmouth
College, Hanover, New Hampshire. Their e-mail addresses are �[email protected]
edu� and �[email protected]�, respectively.
Journal of Economic Perspectives—Volume 18, Number 3—Summer 2004 —Pages 25– 46
legitimate to limit further the market portfolio to U.S. common stocks (a typical
choice), or should the market be expanded to include bonds, and other financial
assets, perhaps around the world? In the end, we argue that whether the model’s
problems reflect weaknesses in the theory or in its empirical implementation, the
failure of the CAPM in empirical tests implies that most applications of the model
are invalid.
We begin by outlining the logic of t ...
This document provides an overview of an upcoming presentation on asset pricing models. The presentation will cover capital market theory, the capital market line, security market line, capital asset pricing model, and diversification. It will discuss the assumptions and formulas for the capital market line and security market line. The capital market line shows expected returns based on portfolio risk, while the security market line shows expected individual asset returns based on systematic risk. The capital asset pricing model uses the concept of beta to calculate the expected return of an asset based on its risk relative to the market.
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.
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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
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Optimal investment strategy based on semi-variable transaction costs
1. International Journal of Business Marketing and Management (IJBMM)
Volume 6 Issue 4 April 2021, P.P. 15-27
ISSN: 2456-4559
www.ijbmm.com
International Journal of Business Marketing and Management (IJBMM) Page 15
Optimal investment strategy based on semi-variable transaction
costs
Shifeng Shen1
, PeibiaoZhao2
1
(School of Science, Nanjing University of Science and Technology, China)
2
(School of Science, Nanjing University of Science and Technology, China)
Abstract:In this paper, we focus on a financial market with one riskless and one risky asset, and consider the
asset allocation problem in the form of semi-variable transaction costs. One of the basic ideas of this paper is to
transform the problem of maximizing the expected utility of terminal wealth in a friction market with semi-
variable transaction costs into a frictionless market which can produce the same maximum utility, and then give
the analytical formula for the original problem. Generally, the price process of risky assets in such frictionless
market is called "shadow price", and the corresponding problem after conversion is called "shadow problem".
Keywords –Utility maximization, Semi-variable cost, Shadow price
I. Introduction
In the field of financial mathematics, asset allocation is a very hot and difficult problem. Generally speaking,
investors invest their money in different types of assets in financial market in order to maximize the expected
utility of their wealth at some point in the future. This kind of problem is usually regarded as the utility
maximization problem with constraints.
A complete financial market is a kind of very ideal investment environment, where any assets in this
market can be bought and sold without transaction costs. In a complete financial market, no arbitrage is
equivalent to the existence of a unique equivalent martingale measure Q, such that the security price under this
measure is a martingale, and this property can be used to price a contingent claim. At the same time, all assets
can be replicated with the underlying assets. However, the assumption of the complete financial market is not
consistent with the real investment environment faced by investors. Therefore, it is of more practical
significance to consider the optimization of the portfolio in the case of incomplete market.
Investment in incomplete market is faced with transaction costs. Specifically, investors buy and sell assets
in different prices. The ask price is higher, while the bid price is lower because of transaction costs. Due to the
existence of transaction costs, investors have to balance between transaction profits and payments.
The research on incomplete financial market still started from the research on complete market. By
applying convex analysis and martingale properties, Pliska[1] solved the problem of maximizing the expected
utility of wealth at a terminal planning horizon by selecting portfolio of securities. Karatzas et al.[2] studied that
when the number of stocks is less than the dimension of multi-dimensional Brownian motion, the incomplete
market can be transformed into a complete market by introducing "virtual" stocks, and they proved that the
optimal portfolio obtained in this method is consistent with that in the original incomplete market. Kramkov and
Schachermayer[3] studied the problem of maximizing the expected utility of terminal wealth in the framework
of a general incomplete semi-martingale model of a financial market. They showed that the necessary and
sufficient condition on a utility function for the validity of the theory to hold true is that asymptotic elasticity of
the utility function should be strictly less than 1.
Liu and Loewenstein[4] studied the optimal trading strategy for a CRRA investor who wants to maximize
the expected utility of wealth on a finite date when facing transaction costs. They showed that even small
transaction costs can have large impact on the optimal portfolio. Hence, an interesting question is that if this
impact can be replaced in a frictionless market which yields the same optimal strategy and utility. If so, this
frictionless market is called “shadow market”. The concept of such shadow market was first proposed by
Cvitanić and Karatzas[5]. In their pioneering work, they found that if a dual problem was solved by a suitable
solution, then the optimal portfolio is the one that hedges the inverse of marginal utility evaluated at the shadow
2. Optimal investment strategy based on semi-variable transaction costs
International Journal of Business Marketing and Management (IJBMM) Page 16
price density solving the corresponding dual problem. Later, Kallsen and Muhle-Karbe[6] found that shadow
price always exists in finite space. Benedettiet al.[7] showed that if short selling is not allowed in the financial
market, then shadow price can always exist. As for càdlàg security price process S, Czichowsky and
Schachermayer[8] proved that shadow price can be defined by means of a “sandwiched” process which consists
of a predictable and an optional strong super-martingale. This conclusion was then extended by Bayraktar and
Yu[9] to a similar problem with random endowment.
Instead of constructing the shadow price from dual problem, Loewenstein[10] assumed that short sales
was not allowed and investor faced transactions costs, then he proved that shadow price can be constructed from
the derivatives of dynamic primal value functions. This result was then extended by Benedetti and Campi[7] to a
similar problem with Kabanov’s muti-currency model.
In addition, shadow price plays an important role in optimization. Under the geometric Brownian motion
model, the optimal investment and consumption problem with logarithmic utility function is studied by Kallsen
and Muhle-Karbe[11] using the results of stochastic control theory, then shadow price was constructed by
solving the free boundary problem. Some researchers have also studied in the form of logarithmic utility
function[12] and power utility function[13].
In view of establishing and solving the utility maximization model simply, most of researches on shadow
market focus on the assumption that there are only proportional transaction costs for the trade of risky assets in
financial market. Most of the changes in the research only focus on the form of utility function, the form of the
price process for risky assets, and the description for the financial market. Although the hypothesis of
proportional transaction costs can be used to prove the existence of the solution to the utility maximization
problem and to establish the duality problem simply, there is obviously a big difference between the hypothesis
and the trading market in our real life. Therefore, it is of great significance to extend the proportional transaction
costs to match our real market.
The remainder of this article goes as follows. In Section 2 we formulate the utility maximization problem
with semi-variable cost and prove the existence of its solution. In Section 3 we first present that shadow price
can be constructed under semi-variable cost, then give the recurrence formula of the optimal strategy under the
friction market. Section 4 is a case analysis in order to prove our conclusion.
II. Utility maximization problem with semi-variable cost
This section is to consider the problem of securities investment with semi variable cost. By using the basic
theory of stochastic process, especially the properties and conclusions of martingale method, we prove that there
is a unique optimal trading strategy for the problem of maximizing the expected utility of terminal wealth in this
friction market through the estimation of total variation of self financing trading strategy.
2.1 Construction of the wealth expected utility maximization model
For the sake of simplicity, we consider a market only consisting of a riskless asset and a risky asset. The
riskless asset has a constant price 1, and the trade of risky asset needs transaction costs. For example, an investor
needs to pay a higher ask price S when buying, but only receives a lower bid price ,( ) - when selling.
Here ( ) is proportional transaction cost rate, and C is a constant on behalf of the commission every time
the investor trade assets in market.
Assumption 2.1.1 The price of risky asset ( ) is positive and Right Continuous with Left Limits, and
is adapted to probability space( ( ) ). Moreover, and .
Remark 2.1.2 Throughout this paper, we assume that the price of risky asset S cannot jump at terminal time T,
thus we have and , which means that the investor can also liquidate his position in risky assets
at terminal time T.
Definition 2.1.3 Trading strategy with semi-variable cost is an R2
-valued, predictable, finite variation
process ( ) , where describes the holdings in the riskless asset and describes the holdings
in the risky asset.
During the whole paper, there is no additional capital in the process of investment except for initial
investment, and no capital is moved out of the market. Thus the following definition of self-financing trading
strategy can be well-defined:
3. Optimal investment strategy based on semi-variable transaction costs
International Journal of Business Marketing and Management (IJBMM) Page 17
Definition 2.1.4 A finite variation process trading strategy ( ) for all is called
self-financing trading strategy which satisfies
∫ ∫ ,( ) - ∫ ∫
where and denote the holding of increase and decrease in riskless asset by investors, while and
denote the holding of increase and decrease in risky asset.
In addition, we assume that every time the investor in this market cannot short any position, then we
consider a utility maximization problem in this constraint.
Assumption 2.1.5 Suppose that the investor’s preferences are modeled by a standard utility function
, which also satisfy the Inada conditions
( ) ( ) n ( ) ( )
Assumption 2.1.6 The utility function satisfies the Reasonable Asymptotic Elasticity, i.e.
( ) ( )
( )
( )
Remark 2.1.7 For the utility function U, ( ) is a necessary and sufficient condition for the existence of
the optimal trading strategy in the problem (4). In pr ctic l sense, the el stic function π (x) represents the r tio
of marginal utility ( ) to average utility ( ) .
Then the problem faced by an investor is to find an optimal trading strategŷ (̂ ̂ ) to maximize the
expected utility of terminal wealth
, ( ,( ) -)- ( )
where ( ) denotes the set of all self-financing trading strategies starting from initial endowment ( )
( ).
If we define
,( ) -
then we can rewrite 4 as
, ( )- ( )
where
( ) * ,( ) -| ( )+ ( )
denotes the set of terminal wealth at time T after the investor liquidates his position in risky assets to riskless
assets.
So the primal problem for the investor is to maximize the expected utility of terminal wealth in the sense
of semi-variable transaction costs
( ) , ( )- ( )
2.2Existence and uniqueness of solutions for the primal problem
This subsection aims to prove the existence and uniqueness of solution for the problem 7 .
First of all, we give the definition and property of option strong super-martingale in the sense of semi-
variable transaction costs:
Definition 2.2.1 An option process ( ) is called an option strong super-martingale, if for all stopping
time it satisfies
[ ]
in which we suppose that is integrable.
According to Doob-Meyer decomposition, an option process X is called an option strong super-martingale
if and only if it can be decomposed into
where M is a local martingale as well as a super-martingale and A is an increasing predictable process.
Lemma 2.2.2 Assume that ( ) satisfied self-financing trading strategies in the sense of semi-variable
costs. Fix risky asset process S as above, suppose that there is a price process ̃ which satisfies ̃
,( ) -, and there exists a probability measure Q such that ̃ is a martingale under Q. Then for all
stopping time the process
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̃( ) ̃
is an option strong super-martingale under Q.
Proof: As what we have discussed, we should proof that ̃( ) can be decomposed as in 9 .
According to the definition, we have
̃( ) ( ̃ ) ̃
so that
̃( ) ∫ ( ̃ ) ∫ ̃
We may use self-financial trading strategies
,
,( ) -
for the first term of 12 so that there exists ( ) , which is a decrease process. And
for the second term of 12 , it defines a martingale under Q measure as ̃ is so. Hence 12 is an optional
strong super-martingale. □
In Definition 2.1.3, we assume that trading strategies ( ) have finite variation. Our next
lemma 2.2.3 proves that this assumption goes true.
Lemma 2.2.3 Assume that ( ) satisfied self-financing trading strategies in the sense of semi-variable
costs. Fix risky asset process S as above, suppose that there is a price process ̃ which satisfies ̃
,( ) -, and there exists a probability measure Q such that ̃ is a martingale under Q. Then the total
variation of remains bounded in ( ).
Proof: We can rewrite and as the sum of two increasing functions,
respectively. Fix , we define a new process by
(( ) ( ) ) (
( )
( )
)
Obviously, is also a self-financing process under transaction costs( ), and
( )
( )
is the
amount that the investor can get more under transaction costs( ) than in( ).
By Lemma 2.2.2 we can find that
(( ) ( ) ̃ ) (
( )̃
( )̃
̃ )
is an optional strong super-martingale. Hence at terminal time T we have
[ ̃ ] *
( )̃
( )̃
+
in which
*
( )̃
( )̃
+
( )
[ *
( )̃
+]
Because the investor liquidate his position in risky assets at terminal time T, then we have in 13 that
. So 13 can be rewritten as
*
( )̃
( )̃
+
hence we have
*
( )̃
+ ( )
( ) ( )
( )
On the other hand, 15 can be rewritten as
*
( )̃
+ *
( )̃
+ [ ]
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where and denote the variance of( )̃
and , respectively. If we denote [̃ ] and [̃ ]
, then by Taylor series expansion at [̃ ], we have
*
( )̃
+
( ) [̃ ]
( )
[( ) [̃ ] ]
For 15 and the properties of we have
0( )̃
1 [ ]
where denotes the correlation coefficient. Therefore
[ ]
0( )̃
1
Note that , we just have [ ] [ ]. Then
[ ]
0( )̃
1
Finally if we use Chebyshev's inequality, we can easily get that
[ ]
As for the total variation of , from the self-financial strategies we have
By the assumption that is strictly positive, we can control by 18 and estimate by 17 . Finally
we can control just using . □
Our next lemma proves the set of trading strategies is closed.
Lemma 2.2.4 Under Assumption 2.1.1, the set ( ) is closed in .
Proof: As what has been showed in Lemma2.2.3and the fact that investor cannot short any position at any time
in this financial market, and are bounded in .
Let { } be a nonnegative sequence in ( ) converging to some nonnegative . We have to
show ( ). We can find self-financing strategies ( ) , which start at ( ) and
end with ( ) at terminal time T. Then these processes can be decomposed into
. Just as what we have showed in Lemma 2.2.3, ( ) and ( ) as well as their convex
combinations are bounded in . By Lemma A1.1a in [14] we can find convex combinations converging a.s. to
elements and . Therefore we have
( )
In addition, for each rational time , ), assume that ( ) , ( ) , ( ) and
( ) converge to some elements ̃ , ̃ , ̃ and ̃ . By passing to a diagonal subsequence, we can
suppose that this convergence holds true for all rational time , ), and the four limit processes are
increasing and bounded in .
So if we define the process ( ) as ( ) , it is easy to see that this is
predictable, nonnegative and satisfies the self-financing conditions because the processes ( ) is
convergence for all , -. □
Finally we give a proof of existence and uniqueness of solutions for the primal problem.
Theorem 2.2.5 Assume that the utility function satisfies Assumption 2.1.5, the price process of risky asset
satisfies Assumption 2.1.1. Moreover, we assume
( )
then the solution of utility maximization problem 7 ̂ ( )exists and uniqueness.
Proof: The uniqueness of the solution is given by contradiction.
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If there are two solutions for the utility maximization problem, suppose they are ̂ and ̂ which belong to
( ). So for every , -, [ ̂ ( )̂], which is the convex combination of ̂ and ̂, should belong to
( ). However, since utility function U is concave, we have
[ ( ̂ ( )̂)] [ (̂)] ( ) , (̂)-
[ (̂)] , (̂)- , (̂)- , (̂)-
which is contradiction to the optimality of , (̂)-.
Next we prove the existence of the solution to the utility maximization problem.
First of all, since ( ) , we can find a maximizing sequence * + ( ), in other words,
( ) , ( )-
If we pass to a sequence of convex combinations ( ), by Lemma A1.1a in [14] and
Lemma 2.2.4, we can suppose that converges to ̂ ( ).
So we can also prove that ̂ is the solution to 7 by contradiction. If not, there exists a ( - such
that
( ) , (̂)-
Assume that and are two disjoint sets. By Lemma 3.16 in [15], there exists
1 ( ) in and ( ) in ;
2 [ ( ) ] and [ ( ) ] ;
3 [ ( ) ( )] , (̂)- and [ ( ) ( )] , (̂)-
Then we have
[ ( )] [ ( ) ( )] [ ( ) ( )]
On the other hand, from Assumption 2.1.6 there exists that . / ( ). Then
[ ( ) ( )] [ ( ) ( )] ( )
In addition, since U is concave, then
[ ( ) ( )] { [ ( ) ( )] [ ( ) ( )]}
, (̂)- 20
From 19 and 20 we have
[ ( )] , (̂)- , ( ) ( )-
Since can be arbitrarily small, we can assume , ( ) ( )- is positive. But is a maximizing
sequence and ( ) , (̂)- is supremum, so it appears a contradiction, which means we have proved ̂ is
the solution to 7 . □
III. Utility maximization problem in shadow market
In this section, we first introduce the notion and then establish the shadow market which can produce the same
maximum utility with the market in semi-variable cost. From the property of the shadow market and the friction
market, we finally give the analytical formula for the original problem with semi-variable cost.
3.1 Modeling of shadow problem
In the market with transaction cost, consistent price system plays an important role. In order to establish
the shadow problem and to solve the utility maximization problem with semi-variable cost, we first give the
definition of consistent price systems as follows:
Definition 3.1.1 Assume that the price process of risky asset satisfies Assumption 2.1.1. Under semi-variable
cost, a pair ̃ is called a consistent price system if it satisfies
1 ̃ takes value in the bid and ask spread ,( ) -
2 ̃ is a martingale under Qmeasure.
In addition, ̃ can be also written as the ratio of two super-martingales. In other words, for all , -,
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̃ ,( ) -
and the set of all consistent price systems is denoted by .
Assumption 3.1.2 Under transaction costs ( ) in trading one risky asset, for some , we have
.
Next we consider constructing a market with one zero interest rate bond and an risky asset whose price
process is ̃:
Definition 3.1.3 An R2
-valued, predictable, finite variation process̃ (̃ ̃ ) is called a self-financing
trading strategy in frictionless market, if for all , ̃ is integrable under ̃ and
̃ ̃ ̃ ∫ ̃ ̃
We also assume in frictionless market that no assets can be shorted, so the shadow problem is established
by all acceptable strategies defined as follows:
Definition 3.1.4 Aprocess̃is called acceptable,if for all , we have
̃ ̃
Therefore, the shadow problem can be written as
, (̃)- ̃ ̃( ) 22
where
̃( ) *̃|̃ (̃ ̃ ) ̃( )+ ( )
denotes the terminal wealth for the investor in frictionless market at time T, and ̃ ̃ ̃ ̃ . In addition,
̃( ) denotes the set of all acceptable strategies starting from initial wealth (̃ ̃ ) ( ).
Lemma 3.1.5 Assume . Then for all we have
̃ ̃ ̃ ̃
Proof: For every ( ) ( ), using the integration by parts formula we have
̃ ∫ ∫ ̃ ∫ ̃ ∫ ̃
if we define
{
̃ ∫ ̃ ̃
̃
then
̃ ̃ ̃ ∫ ̃ ̃
which means Lemma 3.1.5 has been proved. □
Note that for every , we have ( ) ̃( ). Thus if we define
̃( ) , (̃)-̃ ̃( )
and compare with Lemma 3.1.5, then
( ) ̃( )
which means that transactions in frictionless market are always better than that in friction market. Hence we
consider a proper price process ̂ which can let inequality in 26 becomes equality. If so, the process is called
shadow price.
Definition 3.1.6 Assume that consistent price system exists. Fixed the initial wealth x, the price process ̂
̂
̂
is called shadow price, if there exists
( )
, ( )-
̃ ̃( )
, (̃)-
Next we shall first prove that shadow price exists and then construct it under semi-variable transaction
costs.
Let V denotes the convex conjugate function of U defined by
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( ) * ( ) +
then V is strictly decreasing, convex and continuously differentiable and satisfies
( ) ( ) ( ) ( ) 。
By definition,
( ) * (̃) ̃+
For every ̃ ̃( ) under consistent price system,
, (̃)- , ( )- ,( ̃)-
By Lemma 4.1 in [7], is a super-martingale process, thus we have
[̂ ̂] ̂
So if there exists shadow price, then Lemma 4.1.10 should exist first:
Lemma 3.1.7 Let Assumption 2.1.1, 2.1.5, 2.1.6 and 3.1.2 hold. Then there exists ̂ such that
1 ̂ (̂);
2 [̂ ̂] ̂ .
Proof: For each , we have ( ) , ( ̂)-. Note that u is concave, so
̂ ( ) ( ) ( ) , (̂)- , ( ̂)-
Then by the property of utility function U, we have
̂ , (̂)̂- [̂ ̂]
Compared 29 with 30 we prove Lemma 3.1.7. □
Theorem 3.1.8 The consistent price system which satisfies Lemma 3.1.7 defines the shadow price ̂.
Proof: By Lemma 3.1.5 and Lemma 3.1.7 we have
̃( ) ( ) , (̂)- [ (̂ ) ̂ ̂] [ (̂ )] ̂
By 29 we also have
[ (̂ )] ̂ , (̃)- ̃( )
Compared 31 with 32 we complete the proof of Theorem 3.1.8. □
Remark 3.1.9 As has been proved above, if shadow price indeed exists, then the optimal strategy ̃ for the
utility maximization problem in frictionless market is also the optimal strategy for the problem in friction
market. Hence shadow price ̂ is the least favorable price in frictionless market. Thus the optimal strategy ̂ in
friction market only trades when ̂ is at bid or ask price, in other words,
* ̂ + {̂ } n * ̂ + {̂ ( ) }
3.2The expression of shadow price
This subsection gives the expression of shadow price. First we give the definition of ̃ (̃ ̃ ) , then
because the shadow price should satisfy the consistent price system, we finally prove this conclusion.
Firstly, we give the following dynamic programming principle similar to Section 7 in [10].
Definition 3.2.1 For every self-financing trading strategy ( ), define its value function as
( ) [ ( ) ]
where is the set of all self-financing trading strategies under semi-variable cost and no short selling constraint
for each asset in , -.
Lemma 3.2.2 For every optimal strategy ̂in , the value function is a martingale, i.e.
(̂ ) [ (̂ ) ]
Proposition 3.2.3 Define
{
̃
(̂ ̂ ) (̂ ̂ )
̃
(̂ ̂ ) (̂ ̂ )
on , and
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,
̃ (̂ )
̃ (̂ ),( ) -
at terminal time T.
Then for all , ̃ is Right Continuous with Left Limits and is a martingale. In addition, ̃ satisfies
,( ) -
̃
̃
which means ̃ ̃ is the shadow price under semi-variable cost.
Proof: Firstly, assume that . By Definition 3.2.1 and the property of U as a concave function, we
have
([ (̂ ) ( )] ̂ ̂ ) ((̂ ) ̂ ) ( ) (̂ ̂ )
which can be also written as
(̂ ̂ ) (̂ ̂ ) (̂ ̂ ) (̂ ̂ )
Since ̃ is the limit of an increasing sequence, it is well-defined, and so on with ̃ .
Next, since the set of trading strategies in 33 is directed upwards, then 33 can be rewritten by
(̂ )
̂
[ ( ) ] [ ( ) ]
where ( ) is the increasing sequences in
̂
, and for simplicity
̃
(̂ ) (̂ )
, -
Then, we may prove that ̃ is super-martingale. Clearly
̂ ̂
for all , so
(̂ )
̂
[ ( ) ]
̂
[ ( ) ] [ ( ) ]
By the monotone convergence theorem,
(̂ ) [ [ ( ) ] ] [ (̂ ) ]
Hence by definition of ̃ ( ), for we have
̃ *
(̂ ) (̂ )
+ *
(̂ ) (̂ )
+ [̃ ]
And we still have to verify that ̃ is also super-martingale when .
By the monotone convergence theorem again, we have
̃
(̂ ̂ ) (̂ ̂ )
*
(̂ ) (̂ )
+
[ (̂ ) ] [̃ ]
̃
(̂ ̂ ) (̂ ̂ )
*
(̂ ,( ) -) (̂ )
+
[ (̂ ),( ) - ] [̃ ]
As for ̃ , because of ( ) , we have
̃
(̂ ̂ ) (̂ ̂ ) ( ) ( )
Hence ̃ is super-martingale.
Finally, we show that the ratio of ̃ ̃ is bounded between bid and ask price.
By definition of ̃, this conclusion is obviously at terminal time T.
For all , ), let ( ) be a partition of, ). For each , on every set
we have
(̂ ̂ ) (̂ ̂ ) (̂ ̂ )
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Therefore
(̂ ̂ ) (̂ ̂ ) .̂ ̂ / (̂ ̂ )
By the monotone convergence theorem,
̃ ̃
and thus
∑ ̃ ∑ ̃
When we have
̃
̃
On the other hand, on every set again we have
(̂ ̂ ) (̂ ,( ) - ̂ )
Following the same steps we finally get
̃
̃
( )
Last but not least, if we define
,
̂ ̃
̂ ̃
then by Proposition 1.3.14(i) in [17] we know that ̂ ̂ is a super-martingale and Right Continuous with Left
Limits process, which is bounded between bid and ask price. Hence it defines the shadow price under semi-
variable transaction cost. □
3.3The optimal strategy for the friction problem
Assume that in frictionless market, the price process of risky asset satisfies Geometric Brownian motion, i.e.
̃ ̃ ̃
where denotes the instantaneous expected rate of return of risky asset, denotes the instantaneous volatility,
and denotes the standard Brownian motion.
Let denotes the proportion of risky asset in total assets at any time t, i.e.
̃ ̃
̃ ̃ ̃
wherẽ denotes the price of risky asset in frictionless market. Then by Lemma 3.1 in [18] the optimal
proportion of risk asset in total assets should be
By 38 and 39 we have
̃
( )̃
̃
Because the risky asset in frictionless market only trades when ̃ ( ) , so we have
̃
( ),( ) -
̃
Finally by 24 we get
̂
,( ) -
( ( ) ∫ ̂ )
which is the analytical formula for the original problem under semi-variable cost.
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IV. Case analysis
This section gives an example in order to verify the rationality and effectiveness of our conclusion about shadow
price under semi-variable cost.
Assume that there are only one riskless asset and one risky asset in the market. The riskless asset is zero
interest rate and the price of risky asset is given as follows:
{
( )
where( ) is a Brownian motion on a filtered probability space( ( ) ).
Fix proportional transaction cost rate ( ) and the constant transaction cost C, assume the initial
wealth of the investor is , which belongs to the riskless asset at .
In addition, from the definition of shadow price, for every , -, we have
̃
{
(
( )
)
( )
By 44 , we have optimal strategy under semi-variable cost
(̂ ̂ )
{
( )
( )
.( ) /
According to 45 , we calculate the utility of terminal wealth, and find that the utility in friction market
and in frictionless market are the same, hence 45 is indeed the optimal strategy under semi-variable cost.
We can also know from 44 that with the increase of the price for risky assets with transaction costs,
the shadow price will also increase; in addition, with the increase of the fixed cost C and the fixed proportional
coefficient , the shadow price will decrease.
Figure 1: The relationship between shadow price ̃ and transaction cost rate
Figure 1 is the change of shadow price for every 10% increase of transactioncost rate. With the increase of
transaction cost rate , shadow pricẽdecreases gradually.
93.00
93.20
93.40
93.60
93.80
94.00
94.20
94.40
94.60
0.50% 0.55% 0.61% 0.67% 0.73% 0.81% 0.89% 0.97% 1.07% 1.18% 1.30% 1.43%
Shadow
price
S
transaction cost rate λ
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International Journal of Business Marketing and Management (IJBMM) Page 26
Figure 2: The relationship between shadow price ̃ and fixed cost C
Figure 2 is the change of shadow price for every 10% increase of fixedcost C. With the increase of fixed
cost C, shadow price ̃ decreases gradually.
Figure 3: The relationship between shadow price ̃ and the price of security in friction market
Figure 3 is the change of shadow price for every 10% increase of the priceof security in friction market .
With the incre se of the price of securityin friction m rket St, sh ow price ˜S lso incre ses
V. Conclusion
Based on real market transactions, we consider an asset allocation problem in the form of semi-variable
transaction costs, and prove the existence of shadow price. Then we give the analytical formula for the problem
with friction under the property of shadow price with the price of security in friction market. Our conclusion
provides another way to solve the expected utility maximization problem with constraints.
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