The equity premium: a deeper puzzle

Traditional pre-1929 consumption measures understate the extent of serial correlation in the US annual real growth rate of per capita consumption of non-durables and services due to measurement limitations in the construction of their major components. Under alternative measures proposed in this study, the serial correlation of consumption growth is \(0.42\) for the \(1899\) – \(2012\) , contrary to the estimate of \(-0.15\) under the traditional measures. This new evidence implies that the class of economies studied by Mehra and Prescott (J Monet Econ 15(2):145–161, 1985) generates a negative equity premium for reasonable risk aversion levels, thus, further exacerbating the equity premium puzzle.     

Optimal dividend policies for a class of growth-restricted diffusion processes under transaction costs and solvency constraints

In this paper, we consider a company whose surplus follows a rather general diffusion process and whose objective is to maximize expected discounted dividend payments. With each dividend payment, there are transaction costs and taxes, and it is shown in Paulsen (Adv. Appl. Probab. 39:669?C689, 2007) that under some reasonable assumptions, optimality is achieved by using a lump sum dividend barrier strategy, i.e., there is an upper barrier $\bar{u}^{*}$ and a lower barrier $\underline{u}^{*}$ so that whenever the surplus reaches $\bar{u}^{*}$ , it is reduced to $\underline{u}^{*}$ through a dividend payment. However, these optimal barriers may be unacceptably low from a solvency point of view. It is argued that, in that case, one should still look for a barrier strategy, but with barriers that satisfy a given constraint. We propose a solvency constraint similar to that in Paulsen (Finance Stoch. 4:457?C474, 2003); whenever dividends are paid out, the probability of ruin within a fixed time T and with the same strategy in the future should not exceed a predetermined level ??. It is shown how optimality can be achieved under this constraint, and numerical examples are given.     

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 $[\underline{S}, \overline{S}]$ . Such a process S ^? is called a shadow price if it provides the same optimal utility value as in the original model with bid-ask spread. We call S ^? a generalized shadow price if the above property is true for the 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 $[\underline{S}, \overline{S}]$ , 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.     

Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment

We price a contingent claim liability (claim for short) using a 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 a proportional transaction cost ε>0 in two cases: with and without a claim. Using the heuristic computations of Whalley and Wilmott (Math. Finance 7:307–324, 1997), under suitable technical conditions, we provide a rigorous derivation of the asymptotic expansion of the value function in powers of \(\varepsilon^{\frac{1}{3}}\) in both cases with and without a claim. Additionally, using the utility indifference method, we derive the price of the claim at the leading order of \(\varepsilon^{\frac{2}{3}}\) . In both cases, we also obtain a “nearly optimal” strategy, whose expected utility asymptotically matches the leading terms of the value function. We also present an example of how this methodology can be used to price more exotic barrier-type contingent claims.     

Multi-firm voluntary disclosures for correlated operations

We study the no-arbitrage theory of voluntary disclosure (Dye, J Account Res 23:123–145, 1985, and Ostaszewski and Gietzmann, Rev Quant Financ Account 31: 1–27, 2008), generalized to the setting of $n$ firms, simultaneously and voluntarily, releasing at the interim-report date ‘partial’ information concerning their ‘common operating conditions’. Each of the firms has, as in the Dye model, some (known) probability of observing a signal of their end of period performance, but here this signal includes noise determined by a firm-specific precision parameter. The co-dependency of the firms results entirely from their common operating conditions. Each firm has a disclosure cutoff, which is a best response to the cutoffs employed by the remaining firms. To characterize these equilibrium cutoffs explicitly, we introduce $n$ new hypothetical firms, related to the corresponding actual firms, which are operationally independent, but are assigned refined precision parameters and amended means. This impounds all existing correlations arising from conditioning on the other potentially available sources of information. In the model the actual firms’ equilibrium cutoffs are geometric weighted averages of these hypothetical firms. We uncover two countervailing effects. Firstly, there is a bandwagon effect, whereby the presence of other firms raises each individual cutoff relative to what it would have been in the absence of other firms. Secondly, there is an estimator-quality effect, whereby individual cutoffs are lowered, unless the individual precision is above average.     

Optimal dividend policies with transaction costs for a class of jump-diffusion processes

This paper addresses the problem of finding an optimal dividend policy for a class of jump-diffusion processes. The jump component is a compound Poisson process with negative jumps, and the drift and diffusion components are assumed to satisfy some regularity and growth restrictions. Each dividend payment is changed by a fixed and a proportional cost, meaning that if ?? is paid out by the company, the shareholders receive k???K, where k and K are positive. The aim is to maximize expected discounted dividends until ruin. It is proved that when the jumps belong to a certain class of light-tailed distributions, the optimal policy is a simple lump sum policy, that is, when assets are equal to or larger than an upper barrier $\bar{u}^{*}$ , they are immediately reduced to a lower barrier $\underline{u}^{*}$ through a dividend payment. The case with K=0 is also investigated briefly, and the optimal policy is shown to be a reflecting barrier policy for the same light-tailed class. Methods to numerically verify whether a simple lump sum barrier strategy is optimal for any jump distribution are provided at the end of the paper, and some numerical examples are given.     

q-Optimal Martingale Measures for Discrete Time Models

We focus on a backward induction of the q-optimal martingale measure for discrete-time models, where 1  <  q  <  ∞. As for the bounded asset price process case, the same backward induction has been obtained by Grandits (Bernoulli, 5:225–247, 1999). To remove the boundedness, we shall discuss a sufficient condition under which there exists a signed martingale measure whose density is in the ${\mathcal {L}^q}$ -space, which topic is our second aim.     

Non-monotonic pricing kernel and an extended class of mixture of distributions for option pricing

We derive closed form European option pricing formulae under the general equilibrium framework for underlying assets that have an \(N\) -mixture of transformed normal distributions. The component distributions need not belong to the same class but must all be transformed normal. An important implication of our results is that the mixture of distributions is consistent with a “what appears to be abnormal” non-monotonic (asset specific) pricing kernel for the S&P 500 and that the representative agent has a “logical” monotonic decreasing marginal utility. We show that a mixture of two lognormal distributions is sufficient to produce this result and also implied volatility smiles of a wide variety of shapes.     

Quantitative error estimates for a least-squares Monte Carlo algorithm for American option pricing

We prove new error estimates for the Longstaff–Schwartz algorithm. We establish an $O(\log^{\frac{1}{2}}(N)N^{-\frac{1}{2}})$ convergence rate for the expected L ² sample error of this algorithm (where N is the number of Monte Carlo sample paths), whenever the approximation architecture of the algorithm is an arbitrary set of L ² functions with finite Vapnik–Chervonenkis dimension. Incorporating bounds on the approximation error as well, we then apply these results to the case of approximation schemes defined by finite-dimensional vector spaces of polynomials as well as that of certain nonlinear sets of neural networks. We obtain corresponding estimates even when the underlying and payoff processes are not necessarily almost surely bounded. These results extend and strengthen those of Egloff (Ann. Appl. Probab. 15, 1396–1432, 2005), Egloff et al. (Ann. Appl. Probab. 17, 1138–1171, 2007), Kohler et al. (Math. Finance 20, 383–410, 2010), Glasserman and Yu (Ann. Appl. Probab. 14, 2090–2119, 2004), Clément et al. (Finance Stoch. 6, 449–471, 2002) as well as others.     

Multivariate risk aversion and intertemporal substitution

Researchers often assume that preferences over uncertain consumption streams are representable by $$E\left[ {\left( {{1 \mathord{\left/ {\vphantom {1 \gamma }} \right. \kern-\nulldelimiterspace} \gamma }} \right)\sum\limits_{t = 0}^x {\delta ^t \tilde c_t^\gamma } } \right]$$ , where \(\tilde c_t \) , is (random) period t consumption. It is moreover often asserted that estimates of γ cannot be unambiguously interpreted, since the quantity 1 ? γ measures both relative risk aversion and the reciprocal of the elasticity of substitution. Clearly, this ambiguity arises only if 1 ? γ indeed measures risk aversion. Although changes in γ cannot reflect changes in risk aversion according to standard definitions of comparative multivariate risk aversion, we show that γ is rationalizable as a risk aversion measure provided that the “acceptance set” of sure prospects is restricted. We also show, however, that there is essentially no relationship between changes in γ and optimal consumption, even in a simple two period model; this finding casts doubt upon the interpretation of γ as a risk aversion measure.     

Exercise boundary of the American put near maturity in an exponential Lévy model

We study the behavior of the critical price of an American put option near maturity in an exponential Lévy model. In particular, we prove that in situations where the limit of the critical price is equal to the strike price, the rate of convergence to the limit is linear if and only if the underlying Lévy process has finite variation. In the case of infinite variation, a variety of rates of convergence can be observed: we prove that when the negative part of the Lévy measure exhibits an α-stable density near the origin, with 1<α<2, the convergence rate is ruled by $\theta^{1/\alpha}|\ln \theta|^{1-\frac{1}{\alpha}}$ , where θ is the time until maturity.     

New solvable stochastic volatility models for pricing volatility derivatives

In this paper we discuss a new approach to extend a class of solvable stochastic volatility models (SVM). Usually, classical SVM adopt a CEV process for instantaneous variance where the CEV parameter γ takes just few values: 0—the Ornstein–Uhlenbeck process, 1/2—the Heston (or square root) process, 1—GARCH, and 3/2—the 3/2 model. Some other models, e.g. with γ = 2 were discovered in Henry-Labordére (Analysis, geometry, and modeling in finance: advanced methods in option pricing. Chapman & Hall/CRC Financial Mathematics Series, London, 2009) by making connection between stochastic volatility and solvable diffusion processes in quantum mechanics. In particular, he used to build a bridge between solvable superpotentials (the Natanzon superpotentials, which allow reduction of a Schrödinger equation to a Gauss confluent hypergeometric equation) and existing SVM. Here we propose some new models with ${\gamma \in \mathbb{R}}$ and demonstrate that using Lie’s symmetries they could be priced in closed form in terms of hypergeometric functions. Thus obtained new models could be useful for pricing volatility derivatives (variance and volatility swaps, moment swaps).     

The impact of quantitative easing on the US term structure of interest rates

This paper estimates the impact of the Federal Reserve’s 2008–2011 quantitative easing (QE) program on the US term structure of interest rates. We estimate an arbitrage-free term structure model that explicitly includes the quantity impact of the Fed’s trades on Treasury market prices. As such, we are able to estimate both the magnitude and duration of the QE price effects. We show that the Fed’s QE program affected forward rates without introducing arbitrage opportunities into the Treasury security markets. Short- to medium- term forward rates were reduced ( \(<\) 12 years), but the QE had little if any impact on long-term forward rates. This is in contrast to the Fed’s stated intentions for the QE program. The persistence of the rate impacts increased with maturity up to 6 years then declined, with half-lives lasting approximately 4, 6, 12, 8 and 4 months for the 1, 2, 5, 10 and 12 years forwards, respectively. Since bond yields are averages of forward rates over a bond’s maturity, QE affected long-term bond yields. The average impacts on bond yields were 327, 26, 50, 70, and 76 basis points for 1, 2, 5, 10 and 30 years, respectively.     

On irregular functionals of SDEs and the Euler scheme

We prove a sharp upper bound for the error $\mathbb {E}|g(X)-g(\hat{X})|^{p}We prove a sharp upper bound for the error in terms of moments of , where X and are random variables and the function g is a function of bounded variation. We apply the results to the approximation of a solution to a stochastic differential equation at time T by the Euler scheme, and show that the approximation of the payoff of the binary option has asymptotically sharp strong convergence rate 1/2. This has consequences for multilevel Monte Carlo methods. The author was supported by the Finnish Graduate School in Stochastics and Statistics, the Ellen and Artturi Nyyss?nen Foundation, and the Academy of Finland, project #110599.     

Generalized stochastic target problems for pricing and partial hedging under loss constraints��application in optimal book liquidation

We consider a singular version with state constraints of the stochastic target problems studied in Soner and Touzi (SIAM J. Control Optim. 41:404?C424, 2002; J. Eur. Math. Soc. 4:201?C236, 2002) and more recently Bouchard et al. (SIAM J. Control Optim. 48:3123?C3150, 2009), among others. This provides a general framework for the pricing of contingent claims under risk constraints. Our extended version perfectly fits the market models with proportional transaction costs and the order book liquidation issues. Our main result is a direct PDE characterization of the associated pricing function. As an example application, we discuss the valuation of VWAP-guaranteed-type book liquidation contracts, for a general class of risk functions.     

Optimal execution strategies in limit order books with general shape functions

We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005 Obizhaeva, A and Wang, J. 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009)[Crossref] , [Google Scholar]) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modelling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e. of the volume of the LOB, or the exponential recovery of the bid–ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy of a risk-neutral investor, which explicitly solves the recursive scheme given in Obizhaeva and Wang (2005 Obizhaeva, A and Wang, J. 2005. Optimal trading strategy and supply/demand dynamics, Preprint Available online at: http://www.rhsmith.umd.edu/faculty/obizhaeva/OW060408.pdf (accessed 16 February 2009)[Crossref] , [Google Scholar]). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type.     

Price discovery in the presence of boundedly rational agents

In this paper we propose a sequential model of security trading which, compared to existing models, is extended along the notions of (Simon, H.A., A behavioral model of rational choice. Quart. J. Econ., 1955 Simon, HA. 1955. A behavioral model of rational choice. Quart. J. Econ., 64: 99–118.  [Google Scholar], 64, 99–118; Rubinstein, A., Modeling Bounded Rationality, Zeuthen Lecture Book Series, 1998 (MIT Press: Cambridge, MA), and Odean, T., Do investors trade too much? Am. Econ. Rev., 1999, 89(5), 1279–1298) by adding boundedly rational traders. Our results indicate that both momentum and mean-reversion in asset prices can be attributed to the presence of agents who are subject to systematic errors in the process of forecasting the liquidation value of a risky security. The length of the momentum period is inversely related to both the amount of information-based trading in the market and the rate at which asset specific information is learned by boundedly rational agents. Furthermore, the model allows explicitly to establish a link between the component of the bid–ask spread that can be explained by bounded rationality and both momentum and reversal.     

Company investment announcements and the market value of the firm

This paper examines the stock market reaction to 402 company investment announcements made by UK companies during the 1991–1996 period. The market-adjusted abnormal returns are generally positive but small. Investment announcements are classified according to functional categories, and we find the level of abnormal returns to vary according to the type of capital investment being announced. In particular, we find the market to react more favourably to investments that ‘create’ future investment opportunities, than to investments which can be categorized as ‘exercising’ investment opportunities. The market reaction also varies with firm size, with large companies tending to experience smaller responses to announcements than do smaller firms. Chung et al. (1998 Chung, KH, Wright, P and Charoenwong, C. (1998). Investment opportunities and market reaction to capital expenditure decisions. Journal of Banking and Finance, 22: 41–60.  ) reported that the quality of a company's investment opportunities is the primary determinant of market reactions to capital expenditure decisions. The findings presented here lend some support to a role for investment opportunities in market valuations. Project size is also found to have a significant positive impact on the level of abnormal returns.     

Correction note for ��The large-maturity smile for the Heston model��

The papers (Forde and Jacquier in Finance Stoch. 15:755?C780, 2011; Forde et al. in Finance Stoch. 15:781?C784, 2011) study large-time behaviour of the price process in the Heston model. This note corrects typos in Forde and Jacquier (Finance Stoch. 15:755?C780, 2011), Forde et al. (Finance Stoch. 15:781?C784, 2011) and clarifies the proof of Forde et al. (Finance Stoch. 15:781?C784, 2011, Proposition 2.3).     

Commonalities in the order book

This paper uses data from one of the most important European stock markets and shows that, in line with predictions from theoretical market microstructure, a small number of latent factors captures most of the variation in stock specific order books. We show that these order book commonalities are much stronger than liquidity commonality across stocks. The result that bid and ask side as well as the visible and hidden parts of the order book exhibit quite specific dynamics is interpreted as evidence that open order book markets attract a heterogeneous trader population in terms of asset valuations and impatience. Quantifying the informational content of the extracted factors with respect to the evolution of the asset price, we find that the factor information shares are highest (about 10%) for less frequently traded stocks. We also show that the informational content of hidden orders is limited.