Latest Results for Mathematics and Financial EconomicsThe latest content available from Springer
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Household risk aversion and portfolio choices
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">In practice, stock investment is one of the most important decisions made by households. The primary goal of this paper is to explain family investment decisions under the assumptions of household member’s preferences and efficient risk sharing based on the collective household model. In particular, by examining the absolute (relative) risk aversion of the household welfare function, we demonstrate how household’s portfolio allocation in stocks changes with family wealth. We examine two types of preference heterogeneity between family members: parameter heterogeneity and functional form heterogeneity. This study offers an alternative explanation of household portfolio choice corresponding with the observation that wealthier households tend to hold greater share of their wealth in risky assets. Specifically, if two decision-makers have standard constant relative risk aversion preference with different relative risk aversions in a household, family’s relative risk aversion decreases as household wealth increases (decreasing relative risk aversion).</p>
http://link.springer.com/10.1007/s11579-017-0184-1
2017-06-0110.1007/s11579-017-0184-1Drawdown: from practice to theory and back again
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">Maximum drawdown, the largest cumulative loss from peak to trough, is one of the most widely used indicators of risk in the fund management industry, but one of the least developed in the context of measures of risk. We formalize drawdown risk as Conditional Expected Drawdown (CED), which is the tail mean of maximum drawdown distributions. We show that CED is a degree one positive homogenous risk measure, so that it can be linearly attributed to factors; and convex, so that it can be used in quantitative optimization. We empirically explore the differences in risk attributions based on CED, Expected Shortfall (ES) and volatility. An important feature of CED is its sensitivity to serial correlation. In an empirical study that fits AR(1) models to US Equity and US Bonds, we find substantially higher correlation between the autoregressive parameter and CED than with ES or with volatility.</p>
http://link.springer.com/10.1007/s11579-016-0181-9
2017-06-0110.1007/s11579-016-0181-9Arbitrage without borrowing or short selling?
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">We show that a trader, who starts with no initial wealth and is not allowed to borrow money or short sell assets, is theoretically able to attain positive wealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous self-financing condition, formulated using a pathwise Riemann–Stieltjes integral. Our result exemplifies the potential intricacies of formulating economically meaningful self-financing conditions in continuous time, when one leaves the conventional arbitrage-free framework.</p>
http://link.springer.com/10.1007/s11579-016-0180-x
2017-06-0110.1007/s11579-016-0180-xOption spanning beyond $$L_p$$
L
p -models
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">The aim of this paper is to study the spanning power of options in a static financial market that allows non-integrable assets. Our findings extend and unify the results in Galvani (J Math Econ 45(1):73–79, <span class="a-plus-plus citation-ref citationid-c-r13">2009</span>), Galvani and Troitsky (J Math Econ 46(4):616–619, <span class="a-plus-plus citation-ref citationid-c-r14">2010</span>) and Nachman (Rev Financ Stud 1(3):311–328, <span class="a-plus-plus citation-ref citationid-c-r23">1988</span>) for <span class="a-plus-plus inline-equation id-i-eq4">
<span class="a-plus-plus equation-source format-t-e-x">\(L_p\)</span>
</span>-models. We also apply the spanning power properties to the pricing problem. In particular, we show that prices on call and put options of a limited liability asset can be uniquely extended by arbitrage to all marketed contingent claims written on the asset.</p>
http://link.springer.com/10.1007/s11579-017-0185-0
2017-06-0110.1007/s11579-017-0185-0Optimal investment in markets with over and under-reaction to information
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">In this paper we introduce a jump-diffusion model of shot-noise type for stock prices, taking into account over and under-reaction of the market to incoming news. We work in a partial information setting, by supposing that standard investors do not have access to the market direction, the drift, (modeled via a random variable) after a jump. We focus on the expected (logarithmic) utility maximization problem by providing the optimal investment strategy in explicit form, both under full (i.e., from the insider point of view, aware of the right kind of market reaction at any time) and under partial information (i.e., from the standard investor viewpoint, who needs to infer the kind of market reaction from data). We test our results on market data relative to Enron and Ahold. The three main contributions of this paper are: the introduction of a new market model dealing with over and under-reaction to news, the explicit computation of the optimal filter dynamics using an original approach combining enlargement of filtrations with Innovation Theory and the application of the optimal portfolio allocation rule to market data.</p>
http://link.springer.com/10.1007/s11579-016-0182-8
2017-06-0110.1007/s11579-016-0182-8The effect of market power on risk-sharing
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">The paper studies an oligopolistic equilibrium model of financial agents who aim to share their random endowments. The risk-sharing securities and their prices are endogenously determined as the outcome of a strategic game played among all the participating agents. In the complete-market setting, each agent’s set of strategic choices consists of the security payoffs and the pricing kernel that are consistent with the optimal-sharing rules; while in the incomplete setting, agents respond via demand functions on a vector of given tradeable securities. It is shown that at the (Nash) risk-sharing equilibrium, the sharing securities are suboptimal, since agents submit for sharing different risk exposures than their true endowments. On the other hand, the Nash equilibrium prices stay unaffected by the game only in the special case of agents with the same risk aversion. In addition, agents with sufficiently lower risk aversion act as predatory traders, since they absorb utility surplus from the high risk averse agents and reduce the efficiency of sharing. The main results of the paper also hold under the generalized models that allow the presence of noise traders and heterogeneity in agents’ beliefs.</p>
http://link.springer.com/10.1007/s11579-017-0183-2
2017-06-0110.1007/s11579-017-0183-2Optimal investment with transaction costs under cumulative prospect theory in discrete time
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">We study optimal investment problems under the framework of cumulative prospect theory (CPT). A CPT investor makes investment decisions in a single-period financial market with transaction costs. The objective is to seek the optimal investment strategy that maximizes the prospect value of the investor’s final wealth. We obtain the optimal investment strategy explicitly in two examples. An economic analysis is conducted to investigate the impact of the transaction costs and risk aversion on the optimal investment strategy.</p>
http://link.springer.com/10.1007/s11579-017-0186-z
2017-03-2310.1007/s11579-017-0186-zAdditive portfolio improvement and utility-efficient payoffs
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">How can individual financial contracts be improved in an additive manner, such that any portfolio comprising improved contracts is at least as attractive as the portfolio of original contracts? We show that any additive procedure that improves contracts for all expected utility maximizers is a conditional expectation operator. Improved contracts are also attractive under robust Savage preferences. Furthermore, we generalize Bondarenko’s definition of ‘statistical arbitrage’ and show that the improved contracts do not admit this kind of arbitrage.</p>
http://link.springer.com/10.1007/s11579-016-0179-3
2017-03-0110.1007/s11579-016-0179-3Optimal mean-variance portfolio selection
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">Assuming that the wealth process <span class="a-plus-plus inline-equation id-i-eq1">
<span class="a-plus-plus equation-source format-t-e-x">\(X^u\)</span>
</span> is generated self-financially from the given initial wealth by holding its fraction <em class="a-plus-plus">u</em> in a risky stock (whose price follows a geometric Brownian motion with drift <span class="a-plus-plus inline-equation id-i-eq2">
<span class="a-plus-plus equation-source format-t-e-x">\(\mu \in \mathbb {R}\)</span>
</span> and volatility <span class="a-plus-plus inline-equation id-i-eq3">
<span class="a-plus-plus equation-source format-t-e-x">\(\sigma >0\)</span>
</span>) and its remaining fraction <span class="a-plus-plus inline-equation id-i-eq4">
<span class="a-plus-plus equation-source format-t-e-x">\(1 -u\)</span>
</span> in a riskless bond (whose price compounds exponentially with interest rate <span class="a-plus-plus inline-equation id-i-eq5">
<span class="a-plus-plus equation-source format-t-e-x">\(r \in \mathbb {R}\)</span>
</span>), and letting <span class="a-plus-plus inline-equation id-i-eq6">
<span class="a-plus-plus equation-source format-t-e-x">\(\mathsf{P}_{t,x}\)</span>
</span> denote a probability measure under which <span class="a-plus-plus inline-equation id-i-eq7">
<span class="a-plus-plus equation-source format-t-e-x">\(X^u\)</span>
</span> takes value <em class="a-plus-plus">x</em> at time <em class="a-plus-plus">t</em>, we study the dynamic version of the nonlinear mean-variance optimal control problem <span class="a-plus-plus equation id-equ99">
</span>where <em class="a-plus-plus">t</em> runs from 0 to the given terminal time <span class="a-plus-plus inline-equation id-i-eq8">
<span class="a-plus-plus equation-source format-t-e-x">\(T>0\)</span>
</span>, the supremum is taken over admissible controls <em class="a-plus-plus">u</em>, and <span class="a-plus-plus inline-equation id-i-eq9">
<span class="a-plus-plus equation-source format-t-e-x">\(c>0\)</span>
</span> is a given constant. By employing the method of Lagrange multipliers we show that the nonlinear problem can be reduced to a family of linear problems. Solving the latter using a classic Hamilton-Jacobi-Bellman approach we find that the optimal dynamic control is given by <span class="a-plus-plus equation id-equ98">
<span class="a-plus-plus equation-source format-t-e-x">$$\begin{aligned} u_*(t,x) = \frac{\delta }{2\; c\; \sigma }\; \frac{1}{x}\, e^{(\delta ^2-r)(T-t)} \end{aligned}$$</span>
</span>where <span class="a-plus-plus inline-equation id-i-eq10">
<span class="a-plus-plus equation-source format-t-e-x">\(\delta = (\mu -r)/\sigma \)</span>
</span>. The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself.</p>
http://link.springer.com/10.1007/s11579-016-0174-8
2017-03-0110.1007/s11579-016-0174-8Optimal placement in a limit order book: an analytical approach
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">This paper proposes and studies an optimal placement problem in a limit order book. Under a correlated random walk model with mean-reversion for the best ask/bid price, optimal placement strategies for both static and dynamic cases are derived. In the static case, the optimal strategy involves only the market order, the best bid, and the second best bid; the optimal strategy for the dynamic case is shown to be of a threshold type depending on the remaining trading time, the market momentum, and the price mean-reversion factor. Critical to the analysis is a generalized reflection principle for correlated random walks, which enables a significant dimension reduction.</p>
http://link.springer.com/10.1007/s11579-016-0177-5
2017-03-0110.1007/s11579-016-0177-5On uniqueness of equilibrium in the Kyle model
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">A longstanding unresolved question is whether the one-period Kyle model of an informed trader and a noisily informed market maker has an equilibrium that is different from the closed-form solution derived by Kyle (Econometrica 53:1315–1335, <span class="a-plus-plus citation-ref citationid-c-r9">1985</span>). This note advances what is known about this open problem.</p>
http://link.springer.com/10.1007/s11579-016-0175-7
2017-03-0110.1007/s11579-016-0175-7Hedging with temporary price impact
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">We consider the problem of hedging a European contingent claim in a Bachelier model with temporary price impact as proposed by Almgren and Chriss (J Risk 3:5–39, <span class="a-plus-plus citation-ref citationid-c-r2">2001</span>). Following the approach of Rogers and Singh (Math Financ 20:597–615, <span class="a-plus-plus citation-ref citationid-c-r24">2010</span>) and Naujokat and Westray (Math Financ Econ 4(4):299–335, <span class="a-plus-plus citation-ref citationid-c-r21">2011</span>), the hedging problem can be regarded as a cost optimal tracking problem of the frictionless hedging strategy. We solve this problem explicitly for general predictable target hedging strategies. It turns out that, rather than towards the current target position, the optimal policy trades towards a weighted average of expected future target positions. This generalizes an observation of Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, <span class="a-plus-plus citation-ref citationid-c-r12">2013b</span>) from their homogenous Markovian optimal investment problem to a general hedging problem. Our findings complement a number of previous studies in the literature on optimal strategies in illiquid markets as, e.g., Gârleanu and Pedersen (Dynamic portfolio choice with frictions. Preprint, <span class="a-plus-plus citation-ref citationid-c-r12">2013b</span>), Naujokat and Westray (Math Financ Econ 4(4):299–335, <span class="a-plus-plus citation-ref citationid-c-r21">2011</span>), Rogers and Singh (Math Financ 20:597–615, <span class="a-plus-plus citation-ref citationid-c-r24">2010</span>), Almgren and Li (Option hedging with smooth market impact. Preprint, <span class="a-plus-plus citation-ref citationid-c-r3">2015</span>), Moreau et al. (Math Financ. doi:<span class="a-plus-plus non-url-ref">10.1111/mafi.12098</span>, <span class="a-plus-plus citation-ref citationid-c-r20">2015</span>), Kallsen and Muhle-Karbe (High-resilience limits of block-shaped order books. Preprint, <span class="a-plus-plus citation-ref citationid-c-r17">2014</span>), Guasoni and Weber (Mathematical Financ. doi:<span class="a-plus-plus non-url-ref">10.1111/mafi.12099</span>, <span class="a-plus-plus citation-ref citationid-c-r14">2015a</span>; Nonlinear price impact and portfolio choice. Preprint, <span class="a-plus-plus citation-ref citationid-c-r15">2015b</span>), where the frictionless hedging strategy is confined to diffusions. The consideration of general predictable reference strategies is made possible by the use of a convex analysis approach instead of the more common dynamic programming methods.</p>
http://link.springer.com/10.1007/s11579-016-0178-4
2017-03-0110.1007/s11579-016-0178-4Existence of solutions in non-convex dynamic programming and optimal investment
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">We establish the existence of minimizers in a rather general setting of dynamic stochastic optimization in finite discrete time without assuming either convexity or coercivity of the objective function. We apply this to prove the existence of optimal investment strategies for non-concave utility maximization problems in financial market models with frictions, a first result of its kind. The proofs are based on the dynamic programming principle whose validity is established under quite general assumptions.</p>
http://link.springer.com/10.1007/s11579-016-0176-6
2017-03-0110.1007/s11579-016-0176-6The lifetime of a financial bubble
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">We combine both a mathematical analysis of financial bubbles and a statistical procedure for determining when a given stock is in a bubble, with an analysis of a large data set, in order to compute the empirical distribution of the lifetime of financial bubbles. We find that it follows a generalized gamma distribution, and we provide estimates for its parameters. We also perform goodness of fit tests, and we provide a derivation, within the context of bubbles, that explains why the generalized gamma distribution might be the natural one to expect for the lifetimes of financial bubbles.</p>
http://link.springer.com/10.1007/s11579-016-0170-z
2017-01-0110.1007/s11579-016-0170-zDiversification, protection of liability holders and regulatory arbitrage
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">Any solvency regime for financial institutions should be aligned with the fundamental objectives of regulation: protecting liability holders and securing the stability of the financial system. The first objective leads to consider surplus-invariant capital adequacy tests, i.e. tests that do not depend on the surplus of a financial institution. We provide a complete characterization of closed, convex, surplus-invariant capital adequacy tests that highlights an inherent tension between surplus-invariance and the desire to give credit for diversification. The second objective leads to requiring consistency of capital adequacy tests across jurisdictions. Of particular importance in this respect are capital adequacy tests that remain invariant under a change of numéraire. We establish an intimate link between surplus- and numéraire invariant tests.</p>
http://link.springer.com/10.1007/s11579-016-0171-y
2017-01-0110.1007/s11579-016-0171-yThe robust Merton problem of an ambiguity averse investor
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. Confidence is here represented using ellipsoidal uncertainty sets for the drift, given a (compact valued) volatility realization. This specification affords a simple and concise analysis, as the agent becomes observationally equivalent to one with constant, worst case parameters. The result is based on a max–min Hamilton–Jacobi–Bellman–Isaacs PDE, which extends the classical Merton problem and reverts to it for an ambiguity-neutral investor.</p>
http://link.springer.com/10.1007/s11579-016-0168-6
2017-01-0110.1007/s11579-016-0168-6Liquidity risk and optimal dividend/investment strategies
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">In this paper, we study the problem of determining an optimal control on the dividend and investment policy of a firm operating under uncertain environment and risk constraints. We allow the company to make investment decisions by acquiring or selling producing assets whose value is governed by a stochastic process. The firm may face liquidity costs when it decides to buy or sell assets. We formulate this problem as a multi-dimensional mixed singular and multi-switching control problem and use a viscosity solution approach. We numerically compute our optimal strategies and enrich our studies with numerical results and illustrations.</p>
http://link.springer.com/10.1007/s11579-016-0173-9
2017-01-0110.1007/s11579-016-0173-9On optimal partitions, individual values and cooperative games: Does a wiser agent always produce a higher value?
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">We consider an optimal partition of resources (e.g. consumers) between several agents, given utility functions (“wisdoms”) for the agents and their capacities. This problem is a variant of optimal transport (Monge–Kantorovich) between two measure spaces where one of the measures is discrete (capacities) and the costs of transport are the wisdoms of the agents. We concentrate on the individual value for each agent under optimal partition and show that, counter-intuitively, this value may decrease if the agent’s wisdom is increased. Sufficient and necessary conditions for the monotonicity with respect to the wisdom functions of the individual values will be given, independently of the other agents. The sharpness of these conditions is also discussed. Motivated by the above we define a cooperative game based on optimal partition and investigate conditions for stability of the grand coalition.</p>
http://link.springer.com/10.1007/s11579-016-0172-x
2017-01-0110.1007/s11579-016-0172-xOn the equivalence of financial structures with long-term assets
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">In a stochastic financial exchange economy, two financial structures are equivalent if, for each given state price, the marketable payoffs are identical for the associated asset prices. The key property of two equivalent financial structures is that, when associated with any standard exchange economy, they lead to the same financial equilibrium. We exhibit a sufficient condition for the equivalence of two financial structures without re-trading with possibly long-term assets. We then apply this result to financial structures built upon primitive assets and their re-trading. We also borrow an assumption from Bonnisseau and Chéry (Ann Financ 10:523–552, <span class="a-plus-plus citation-ref citationid-c-r6">2014</span>) to prove the equivalence between a financial structure and its reduced forms.</p>
http://link.springer.com/10.1007/s11579-016-0169-5
2017-01-0110.1007/s11579-016-0169-5Existence and uniqueness of a steady state for an OTC market with several assets
<h3 class="a-plus-plus">Abstract</h3>
<p class="a-plus-plus">We introduce and study a class of over-the-counter market models specified by systems of Ordinary Differential Equations (ODE’s), in the spirit of Duffie-Gârleanu-Pedersen Duffie et al. (Econometrica 73(1):1815–1847, <span class="a-plus-plus citation-ref citationid-c-r5">2005</span>). The key innovation is allowing for multiple assets. We show the existence and uniqueness of a steady state for these ODE’s.</p>
http://link.springer.com/10.1007/s11579-016-0167-7
2016-09-0110.1007/s11579-016-0167-7