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.     

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.     

Two price economies in continuous time

Static and discrete time pricing operators for two price economies are reviewed and then generalized to the continuous time setting of an underlying Hunt process. The continuous time operators define nonlinear partial integro–differential equations that are solved numerically for the three valuations of bid, ask and expectation. The operators employ concave distortions by inducing a probability into the infinitesimal generator of a Hunt process. This probability is then distorted. Two nonlinear operators based on different approaches to truncating small jumps are developed and termed $QV$ for quadratic variation and $NL$ for normalized Lévy. Examples illustrate the resulting valuations. A sample book of derivatives on a single underlier is employed to display the gap between the bid and ask values for the book and the sum of comparable values for the components of the book.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Erratum to: Utility maximization in incomplete markets with random endowment

K. Larsen, M. Soner and G. ?itkovi? kindly pointed out to us an error in our paper (Cvitani? et al. in Finance Stoch. 5:259–272, 2001) which appeared in 2001 in this journal. They also provide an explicit counterexample in Larsen et al. (https://arxiv.org/abs/1702.02087, 2017).In Theorem 3.1 of Cvitani? et al. (Finance Stoch. 5:259–272, 2001), it was incorrectly claimed (among several other correct assertions) that the value function \(u(x)\) is continuously differentiable. The erroneous argument for this assertion is contained in Remark 4.2 of Cvitani? et al. (Finance Stoch. 5:259–272, 2001), where it was claimed that the dual value function \(v(y)\) is strictly concave. As the functions \(u\) and \(v\) are mutually conjugate, the continuous differentiability of \(u\) is equivalent to the strict convexity of \(v\). By the same token, in Remark 4.3 of Cvitani? et al. (Finance Stoch. 5:259–272, 2001), the assertion on the uniqueness of the element \(\hat{y}\) in the supergradient of \(u(x)\) is also incorrect.Similarly, the assertion in Theorem 3.1(ii) that \(\hat{y}\) and \(x\) are related via \(\hat{y}=u'(x)\) is incorrect. It should be replaced by the relation \(x=-v'(\hat{y})\) or, equivalently, by requiring that \(\hat{y}\) is in the supergradient of \(u(x)\).To the best of our knowledge, all the other statements in Cvitani? et al. (Finance Stoch. 5:259–272, 2001) are correct.As we believe that the counterexample in Larsen et al. (https://arxiv.org/abs/1702.02087, 2017) is beautiful and instructive in its own right, we take the opportunity to present it in some detail.     

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.     

The exact Taylor formula of the implied volatility

In a model driven by a multidimensional local diffusion, we study the behavior of the implied volatility \({\sigma}\) and its derivatives with respect to log-strike \(k\) and maturity \(T\) near expiry and at the money. We recover explicit limits of the derivatives \({\partial_{T}^{q}} \partial_{k}^{m} \sigma\) for \((T,x-k)\) approaching the origin within the parabolic region \(|x-k|\leq\lambda\sqrt{T}\), with \(x\) denoting the spot log-price of the underlying asset and where \(\lambda\) is a positive and arbitrarily large constant. Such limits yield the exact Taylor formula for the implied volatility within the parabola \(|x-k|\leq\lambda\sqrt{T}\). In order to include important models of interest in mathematical finance, e.g. Heston, CEV, SABR, the analysis is carried out under the weak assumption that the infinitesimal generator of the diffusion is only locally elliptic.     

Risk bounds for factor models

Recent literature has investigated the risk aggregation of a portfolio \(X=(X_{i})_{1\leq i\leq n}\) under the sole assumption that the marginal distributions of the risks \(X_{i} \) are specified, but not their dependence structure. There exists a range of possible values for any risk measure of \(S=\sum_{i=1}^{n}X_{i}\), and the dependence uncertainty spread, as measured by the difference between the upper and the lower bound on these values, is typically very wide. Obtaining bounds that are more practically useful requires additional information on dependence.Here, we study a partially specified factor model in which each risk \(X_{i}\) has a known joint distribution with the common risk factor \(Z\), but we dispense with the conditional independence assumption that is typically made in fully specified factor models. We derive easy-to-compute bounds on risk measures such as Value-at-Risk (\(\mathrm{VaR}\)) and law-invariant convex risk measures (e.g. Tail Value-at-Risk (\(\mathrm{TVaR}\))) and demonstrate their asymptotic sharpness. We show that the dependence uncertainty spread is typically reduced substantially and that, contrary to the case in which only marginal information is used, it is not necessarily larger for \(\mathrm{VaR}\) than for \(\mathrm{TVaR}\).     

Robust pricing–hedging dualities in continuous time

We pursue a robust approach to pricing and hedging in mathematical finance. We consider a continuous-time setting in which some underlying assets and options, with continuous price paths, are available for dynamic trading and a further set of European options, possibly with varying maturities, is available for static trading. Motivated by the notion of prediction set in Mykland (Ann. Stat. 31:1413–1438, 2003), we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. superreplication of a contingent claim is required only for paths falling in the given set. Our framework thus interpolates between model-independent and model-specific settings and allows us to quantify the impact of making assumptions or gaining information. We obtain a general pricing–hedging duality result: the infimum over superhedging prices of an exotic option with payoff \(G\) is equal to the supremum of expectations of \(G\) under calibrated martingale measures. Our results include in particular the martingale optimal transport duality of Dolinsky and Soner (Probab. Theory Relat. Fields 160:391–427, 2014) and extend it to multiple dimensions, multiple maturities and beliefs which are invariant under time-changes. In a general setting with arbitrary beliefs and for a uniformly continuous \(G\), the asserted duality holds between limiting values of perturbed problems.     

Tukey’s transformational ladder for portfolio management

Over the past half-century, the empirical finance community has produced vast literature on the advantages of the equally weighted Standard and Poor (S&P 500) portfolio as well as the often overlooked disadvantages of the market capitalization weighted S&P 500’s portfolio (see Bloomfield et al. in J Financ Econ 5:201–218, 1977; DeMiguel et al. in Rev Financ Stud 22(5):1915–1953, 2009; Jacobs et al. in J Financ Mark 19:62–85, 2014; Treynor in Financ Anal J 61(5):65–69, 2005). However, portfolio allocation based on Tukey’s transformational ladder has, rather surprisingly, remained absent from the literature. In this work, we consider the S&P 500 portfolio over the 1958–2015 time horizon weighted by Tukey’s transformational ladder (Tukey in Exploratory data analysis, Addison-Wesley, Boston, 1977): \(1/x^2,\,\, 1/x,\,\, 1/\sqrt{x},\,\, \text {log}(x),\,\, \sqrt{x},\,\, x,\,\, \text {and} \,\, x^2\), where x is defined as the market capitalization weighted S&P 500 portfolio. Accounting for dividends and transaction fees, we find that the 1/\(x^2\) weighting strategy produces cumulative returns that significantly dominate all other portfolio returns, achieving a compound annual growth rate of 18% over the 1958–2015 horizon. Our story is furthered by a startling phenomenon: both the cumulative and annual returns of the \(1/x^2\) weighting strategy are superior to those of the 1 / x weighting strategy, which are in turn superior to those of the \(1/\sqrt{x}\) weighted portfolio, and so forth, ending with the \(x^2\) transformation, whose cumulative returns are the lowest of the seven transformations of Tukey’s transformational ladder. The order of cumulative returns precisely follows that of Tukey’s transformational ladder. To the best of our knowledge, we are the first to discover this phenomenon.     

Aggregate idiosyncratic volatility,dynamic aspects of loss aversion,and narrow framing

We consider the economy in which an agent faces, in addition to market risk, an additive independent background risk in consumption. In contrast to the Lucas (Econometrica 46:1429–1445, 1978) complete consumption insurance model, under plausible assumptions about the unconditional mean and variance of the agent’s subjective distribution of background risk the model with the additive independent background risk fits the historical average excess return on the US stock market with the coefficient of relative risk aversion (RRA) below five for the subsets of households designated as assetholders. The greater the size and/or the lower the expected value of background risk, the lower (compared to the Lucas (Econometrica 46:1429–1445, 1978) model) the value of the RRA coefficient needed for the model with background risk to match the historical average equity premium. Allowing for an extremely unlike large decrease in the agent’s consumption considerably decreases the required coefficient of RRA. It is concluded that the presence of the additive independent background risk in the consumption of assetholders can account for nearly 60 % of the historical average equity premium, hence rationalizing the equity premium puzzle of Mehra and Prescott (J Monet Econ 15:145–162, 1985). With RRA below five, the model with background risk is consistent with the historical average real interest rate if the agent has the subjective time discount factor lower than, but close to, 1. The findings are robust to the assumed type of background risk, the proxy for the market portfolio, and the threshold value in the definition of assetholders.     

Market completion with derivative securities

Let \(S^{F}\) be a ?-martingale representing the price of a primitive asset in an incomplete market framework. We present easily verifiable conditions on the model coefficients which guarantee the completeness of the market in which in addition to the primitive asset, one may also trade a derivative contract \(S^{B}\). Both \(S^{F}\) and \(S^{B}\) are defined in terms of the solution \(X\) to a two-dimensional stochastic differential equation: \(S^{F}_{t} = f(X_{t})\) and \(S^{B}_{t}:=\mathbb{E}[g(X_{1}) | \mathcal{F}_{t}]\). From a purely mathematical point of view, we prove that every local martingale under ? can be represented as a stochastic integral with respect to the ?-martingale \(S :=(S^{F}, S^{B})\). Notably, in contrast to recent results on the endogenous completeness of equilibria markets, our conditions allow the Jacobian matrix of \((f,g)\) to be singular everywhere on \(\mathbb{R}^{2}\). Hence they cover as a special case the prominent example of a stochastic volatility model being completed with a European call (or put) option.