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arXiv2011

A simplified Capital Asset Pricing Model

We consider a Black-Scholes market in which a number of stocks and an index are traded. The simplified Capital Asset Pricing Model is the conjunction of the usual Capital Asset Pricing Model, or CAPM, and the statement that the appreciation rate of the index is equal to its squared volatility plus the interest rate. (The mathematical statement of the conjunction is simpler than that of the usual CAPM.) Our main result is that either we can outperform the index or the simplified CAPM holds.

q-fin.PM↗
arXiv2013

On the Existence of Shadow Prices

For utility maximization problems under proportional transaction costs, it has been observed that the original market with transaction costs can sometimes be replaced by a frictionless "shadow market" that yields the same optimal strategy and utility. However, the question of whether or not this indeed holds in generality has remained elusive so far. In this paper we present a counterexample which shows that shadow prices may fail to exist. On the other hand, we prove that short selling constraints are a sufficient condition to warrant their existence, even in very general multi-currency market models with possibly discontinuous bid-ask-spreads.

q-fin.PM↗
arXiv2011

The Nature of Alpha

We suggest an empirical model of investment strategy returns which elucidates the importance of non-Gaussian features, such as time-varying volatility, asymmetry and fat tails, in explaining the level of expected returns. Estimating the model on the (former) Lehman Brothers Hedge Fund Index data, we demonstrate that the volatility compensation is a significant component of the expected returns for most strategy styles, suggesting that many of these strategies should be thought of as being `short vol'. We present some fundamental and technical reasons why this should indeed be the case, and suggest explanation for exception cases exhibiting `long vol' characteristics. We conclude by drawing some lessons for hedge fund portfolio construction.

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arXiv2011

Common persistence in conditional variance: A reconsideration

This paper demonstrates the flaws of co-persistence theory proposed by Bollerslev and Engle (1993) which cause the theory can hardly be applied. With the introduction of the half-life of decay coefficient as the measure of the persistence, and both the weak definition of persistence and co-persistence in variance, this study attempts to solve the problems by using exhaustive search algorithm for obtaining co-persistent vector. In addition, this method is illustrated to research the co-persistence of stock return volatility in 10 European countries.

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arXiv2011

On the game interpretation of a shadow price process in utility maximization problems under transaction costs

To any utility maximization problem under transaction costs one can assign a frictionless model with a price process $S^*$, lying in the bid/ask price interval $[\underline S, \bar{S}]$. Such process $S^*$ is called a \emph{shadow price} if it provides the same optimal utility value as in the original model with bid-ask spread. We call $S^*$ a \emph{generalized shadow price} if the above property is true for the \emph{relaxed} utility function in the frictionless model. This relaxation is defined as the lower semicontinuous envelope of the original utility, considered as a function on the set $[\underline S, \bar{S}]$, equipped with some natural weak topology. We prove the existence of a generalized shadow price under rather weak assumptions and mark its relation to a saddle point of the trader/market zero-sum game, determined by the relaxed utility function. The relation of the notion of a shadow price to its generalization is illustrated by several examples. Also, we briefly discuss the interpretation of shadow prices via Lagrange duality.

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arXiv2011

Asymptotic Analysis for Optimal Investment in Finite Time with Transaction Costs

We consider an agent who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of a proportional transaction cost. The utility function considered is power utility. We provide a heuristic and a rigorous derivation of the asymptotic expansion of the value function in powers of transaction cost parameter. We also obtain a "nearly optimal" strategy, whose utility asymptotically matches the leading terms in the value function.

q-fin.PM↗
arXiv2014

An Explicit Example Of Optimal Portfolio-Consumption Choices With Habit Formation And Partial Observations

We consider a model of optimal investment and consumption with both habit formation and partial observations in incomplete Itô processes market. The investor chooses his consumption under the addictive habits constraint while only observing the market stock prices but not the instantaneous rate of return. Applying the Kalman-Bucy filtering theorem and the Dynamic Programming arguments, we solve the associated Hamilton-Jacobi-Bellman (HJB) equation explicitly for the path dependent stochastic control problem in the case of power utilities. We provide the optimal investment and consumption policies in explicit feedback forms using rigorous verification arguments.

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arXiv2011

Pricing a Contingent Claim Liability with Transaction Costs Using Asymptotic Analysis for Optimal Investment

We price a contingent claim liability using the utility indifference argument. We consider an agent with exponential utility, who invests in a stock and a money market account with the goal of maximizing the utility of his investment at the final time T in the presence of positive proportional transaction cost in two cases with and without a contingent claim liability. Using the computations from the heuristic argument in Whalley & Wilmott we provide a rigorous derivation of the asymptotic expansion of the value function in powers of the transaction cost parameter around the known value function for the case of zero transaction cost in both cases with and without a contingent claim liability. Additionally, using utility indifference method we derive an asymptotic expansion of the price of the contingent claim liability. In both cases, we also obtain a "nearly optimal" strategy, whose expected utility asymptotically matches the leading terms of the value function.

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arXiv2011

Optimal Constrained Investment in the Cramer-Lundberg model

We consider an insurance company whose surplus is represented by the classical Cramer-Lundberg process. The company can invest its surplus in a risk free asset and in a risky asset, governed by the Black-Scholes equation. There is a constraint that the insurance company can only invest in the risky asset at a limited leveraging level; more precisely, when purchasing, the ratio of the investment amount in the risky asset to the surplus level is no more than a; and when shortselling, the proportion of the proceeds from the short-selling to the surplus level is no more than b. The objective is to find an optimal investment policy that minimizes the probability of ruin. The minimal ruin probability as a function of the initial surplus is characterized by a classical solution to the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We study the optimal control policy and its properties. The interrelation between the parameters of the model plays a crucial role in the qualitative behavior of the optimal policy. E.g., for some ratios between a and b, quite unusual and at first ostensibly counterintuitive policies may appear, like short-selling a stock with a higher rate of return to earn lower interest, or borrowing at a higher rate to invest in a stock with lower rate of return. This is in sharp contrast with the unrestricted case, first studied in Hipp and Plum (2000), or with the case of no shortselling and no borrowing studied in Azcue and Muler (2009).

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arXiv2012

Analysis of hedging based on co-persistence theory

This article analyzes the relationship between co-persistence and hedging which indicates co-persistence ratio is just the long-term hedging ratio. The new method of exhaustive search algorithm for deriving co-persistence ratio is derived in the article. And we also develop a new hedging strategy of combining co-persistence with dynamic hedging which can enhance the hedging effectiveness and reduce the persistence of the hedged portfolio. Finally our strategy is illustrated to study the hedge of JIASHI300 index and HS300 stock index future .

q-fin.PM↗
arXiv2011

Measuring market liquidity: An introductory survey

Asset liquidity in modern financial markets is a key but elusive concept. A market is often said to be liquid when the prevailing structure of transactions provides a prompt and secure link between the demand and supply of assets, thus delivering low costs of transaction. Providing a rigorous and empirically relevant definition of market liquidity has, however, provided to be a difficult task. This paper provides a critical review of the frameworks currently available for modelling and estimating the market liquidity of assets. We consider definitions that stress the role of the bid-ask spread and the estimation of its components that arise from alternative sources of market friction. In this case, intra-daily measures of liquidity appear relevant for capturing the core features of a market, and for their ability to describe the arrival of new information to market participants.

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arXiv2011

Saddlepoint methods in portfolio theory

We discuss the use of saddlepoint methods in the analysis of portfolios, with particular reference to credit portfolios. The objective is to proceed from a model of the loss distribution, given through probabilities, correlations and the like, to an analytical approximation of the distribution. Once this is done we show how to derive the so-called risk contributions which are the derivatives of risk measures, such as a given quantile (VaR) or expected shortfall, to the allocations in the underlying assets. These show, informally, where the risk is coming from, and also indicate how to go about optimising the portfolio.

q-fin.PM↗
arXiv2013

Optimal Portfolio Choice for a Behavioural Investor in Continuous-Time Markets

The aim of this work consists in the study of the optimal investment strategy for a behavioural investor, whose preference towards risk is described by both a probability distortion and an S-shaped utility function. Within a continuous-time financial market framework and assuming that asset prices are modelled by semimartingales, we derive sufficient and necessary conditions for the well-posedness of the optimisation problem in the case of piecewise-power probability distortion and utility functions. Finally, under straightforwardly verifiable conditions, we further demonstrate the existence of an optimal strategy.

q-fin.PM↗
arXiv2013

Homogenization and asymptotics for small transaction costs

We consider the classical Merton problem of lifetime consumption-portfolio optimization problem with small proportional transaction costs. The first order term in the asymptotic expansion is explicitly calculated through a singular ergodic control problem which can be solved in closed form in the one-dimensional case. Unlike the existing literature, we consider a general utility function and general dynamics for the underlying assets. Our arguments are based on ideas from the homogenization theory and use the convergence tools from the theory of viscosity solutions. The multidimensional case is studied in our accompanying paper using the same approach.

math.OC↗
arXiv2012

Incorporating fat tails in financial models using entropic divergence measures

In the existing financial literature, entropy based ideas have been proposed in portfolio optimization, in model calibration for options pricing as well as in ascertaining a pricing measure in incomplete markets. The abstracted problem corresponds to finding a probability measure that minimizes the relative entropy (also called $I$-divergence) with respect to a known measure while it satisfies certain moment constraints on functions of underlying assets. In this paper, we show that under $I$-divergence, the optimal solution may not exist when the underlying assets have fat tailed distributions, ubiquitous in financial practice. We note that this drawback may be corrected if `polynomial-divergence' is used. This divergence can be seen to be equivalent to the well known (relative) Tsallis or (relative) Renyi entropy. We discuss existence and uniqueness issues related to this new optimization problem as well as the nature of the optimal solution under different objectives. We also identify the optimal solution structure under $I$-divergence as well as polynomial-divergence when the associated constraints include those on marginal distribution of functions of underlying assets. These results are applied to a simple problem of model calibration to options prices as well as to portfolio modeling in Markowitz framework, where we note that a reasonable view that a particular portfolio of assets has heavy tailed losses may lead to fatter and more reasonable tail distributions of all assets.

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arXiv2013

Generalized Kuhn-Tucker Conditions for N-Firm Stochastic Irreversible Investment under Limited Resources

In this paper we study a continuous time, optimal stochastic investment problem under limited resources in a market with N firms. The investment processes are subject to a time-dependent stochastic constraint. Rather than using a dynamic programming approach, we exploit the concavity of the profit functional to derive some necessary and sufficient first order conditions for the corresponding Social Planner optimal policy. Our conditions are a stochastic infinite-dimensional generalization of the Kuhn-Tucker Theorem. The Lagrange multiplier takes the form of a nonnegative optional random measure on [0,T] which is flat off the set of times for which the constraint is binding, i.e. when all the fuel is spent. As a subproduct we obtain an enlightening interpretation of the first order conditions for a single firm in Bank (2005). In the infinite-horizon case, with operating profit functions of Cobb-Douglas type, our method allows the explicit calculation of the optimal policy in terms of the `base capacity' process, i.e. the unique solution of the Bank and El Karoui representation problem (2004).

math.OC↗
arXiv2012

Optimal Investment Under Transaction Costs

We investigate how and when to diversify capital over assets, i.e., the portfolio selection problem, from a signal processing perspective. To this end, we first construct portfolios that achieve the optimal expected growth in i.i.d. discrete-time two-asset markets under proportional transaction costs. We then extend our analysis to cover markets having more than two stocks. The market is modeled by a sequence of price relative vectors with arbitrary discrete distributions, which can also be used to approximate a wide class of continuous distributions. To achieve the optimal growth, we use threshold portfolios, where we introduce a recursive update to calculate the expected wealth. We then demonstrate that under the threshold rebalancing framework, the achievable set of portfolios elegantly form an irreducible Markov chain under mild technical conditions. We evaluate the corresponding stationary distribution of this Markov chain, which provides a natural and efficient method to calculate the cumulative expected wealth. Subsequently, the corresponding parameters are optimized yielding the growth optimal portfolio under proportional transaction costs in i.i.d. discrete-time two-asset markets. As a widely known financial problem, we next solve optimal portfolio selection in discrete-time markets constructed by sampling continuous-time Brownian markets. For the case that the underlying discrete distributions of the price relative vectors are unknown, we provide a maximum likelihood estimator that is also incorporated in the optimization framework in our simulations.

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