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 \(\mathrm{TVaR}\).     

Alpha-CIR model with branching processes in sovereign interest rate modeling

We introduce a class of interest rate models, called the \(\alpha\)-CIR model, which is a natural extension of the standard CIR model by adding a jump part driven by \(\alpha\)-stable Lévy processes with index \(\alpha\in(1,2]\). We deduce an explicit expression for the bond price by using the fact that the model belongs to the family of CBI and affine processes, and analyze the bond price and bond yield behaviors. The \(\alpha\)-CIR model allows us to describe in a unified and parsimonious way several recent observations on the sovereign bond market such as the persistency of low interest rates together with the presence of large jumps. Finally, we provide a thorough analysis of the jumps, and in particular the large jumps.     

An enlargement of filtration formula with applications to multiple non-ordered default times

In this work, for a reference filtration \(\mathbb {F}\), we develop a method for computing the semimartingale decomposition of \(\mathbb {F}\)-martingales in a specific type of enlargement of \(\mathbb {F}\). As an application, we study the progressive enlargement of \(\mathbb {F}\) with a sequence of non-ordered default times and show how to deduce results concerning the first-to-default, \(k\)th-to-default, k-out-of-n-to-default or all-to-default events. In particular, using this method, we compute explicitly the semimartingale decomposition of \(\mathbb {F}\)-martingales under the absolute continuity condition of Jacod.     

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.     

New methods in the classical economics of uncertainty: comparing risks

Second-order stochastic dominance answers the question “Under what conditions will all risk-averse agents prefer \(\tilde{x}_2\) to \(\tilde{x}_1\)?” Consider the following related question: “Under what conditions will all risk-averse agents who prefer lottery \(\tilde{x}_1\) to a reference lottery \(\tilde{\omega }\) also prefer lottery \(\tilde{x}_2\) to that reference lottery?” Each of these two questions is an example of a broad category of questions of great relevance for the economics of risk. The second question is an example of a contingent risk comparison, while the question behind second-order stochastic dominance is an example of a non-contingent risk comparison. The stochastic order arising from a contingent risk comparison is obviously weaker than that arising from the corresponding non-contingent risk comparison, but we show that the two stochastic orders are closely related, so that the answer to a non-contingent risk comparison problem always provides the answer to the corresponding contingent risk comparison problem. In addition to showing the connection between parallel contingent and non-contingent risk comparison problems, we articulate a method for solving both kinds of problems using the “basis” approach. The basis approach has often been used implicitly, but we argue that there is value in making its use explicit, particularly in indicating which new, previously unsolved problems can readily be solved by the basis approach and which cannot.     

No arbitrage of the first kind and local martingale numéraires

A supermartingale deflator (resp. local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp. local martingales). A supermartingale numéraire (resp. local martingale numéraire) is a wealth process whose reciprocal is a supermartingale deflator (resp. local martingale deflator). It has been established in previous works that absence of arbitrage of the first kind (\(\mbox{NA}_{1}\)) is equivalent to the existence of the (unique) supermartingale numéraire, and further equivalent to the existence of a strictly positive local martingale deflator; however, under \(\mbox{NA}_{1}\), a local martingale numéraire may fail to exist. In this work, we establish that under \(\mbox{NA}_{1}\), a supermartingale numéraire under the original probability \(P\) becomes a local martingale numéraire for equivalent probabilities arbitrarily close to \(P\) in the total variation distance.     

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.     

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.     

Modeling the inconsistency in intertemporal choice: the generalized Weibull discount function and its extension

The aim of this paper is to obtain the family of the so-called generalized Weibull discount functions, introduced by Takeuchi (Game Econ Behav 71:456–478, 2011), by deforming the q-exponential discount function by means of the Stevens’ “power” law. The obtained discount functions exhibit different degrees of inconsistency and so they can be classified according to the value of their characteristic deforming parameters. Moreover, we extend the construction of the generalized Weibull discount function starting from any discount function instead of the q-exponential discounting. In any case, the value of the parameter \(\theta \) of these new discount functions is extended from (0, 1] to the union of the intervals \((-\,\infty ,0) \cup (0,+\,\infty )\).     

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.     

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.     

Multilevel Monte Carlo for exponential Lévy models

We apply the multilevel Monte Carlo method for option pricing problems using exponential Lévy models with a uniform timestep discretisation. For lookback and barrier options, we derive estimates of the convergence rate of the error introduced by the discrete monitoring of the running supremum of a broad class of Lévy processes. We then use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the variance gamma, NIG and \(\alpha\)-stable processes. We also provide an analysis of a trapezoidal approximation for Asian options. Our method is illustrated by numerical experiments.     

Zur Verteilung der Abschlusskosten in deutschen Lebensversicherungen

It will be shown that an amortization spread of the acquisition costs to the first N years of a classical german life insurance policy or lifelong annuity always leads to a non negative technical provision at the end of the first year if \( N\,{=}\,5 \) (and the technical provision starting with zero). For usual combinations of the age at entry and the policy duration even \( N\,{=}\,3 \) is sufficient for this purpose. The legal specification given by the new version of VVG in Germany concerning surrender value (i.e. positive value even in case of early lapse) is therefore consistent with the requirement of a amortization spread to 5 years. Additionally a recursion formula for the technical provision with N years spreaded acquisition costs is proven.     

Investor valuations of Japan’s adoption of a territorial tax regime: quantifying the direct and competitive effects of international tax reform

This paper examines the impact of Japan’s 2009 adoption of a territorial tax regime using event study methods which leverage individual firm characteristics to identify underlying drivers of market reactions. Differences in Japanese firms’ foreign and domestic effective tax rates yield an aggregate capitalization effect of \(\yen \)4.3 trillion, while firms with less prior foreign exposure and fewer opportunities for tax avoidance experienced relatively larger abnormal returns. We attribute these results to tax savings on existing undistributed foreign earnings, enhanced opportunities for international expansion, and cultural biases against tax planning. Spillovers to the US (through tax or firm competition) appear insignificant.     

Dynamically consistent investment under model uncertainty: the robust forward criteria

We combine forward investment performance processes and ambiguity-averse portfolio selection. We introduce robust forward criteria which address ambiguity in the specification of the model, the risk preferences and the investment horizon. They encode the evolution of dynamically consistent ambiguity-averse preferences.We focus on establishing dual characterisations of the robust forward criteria, which is advantageous as the dual problem amounts to the search for an infimum whereas the primal problem features a saddle point. Our approach to duality builds on ideas developed in Schied (Finance Stoch. 11:107–129, 2007) and ?itkovi? (Ann. Appl. Probab. 19:2176–2210, 2009). We also study in detail the so-called time-monotone criteria. We solve explicitly the example of an investor who starts with logarithmic utility and applies a quadratic penalty function. Such an investor builds a dynamic estimate of the market price of risk \(\hat{\lambda}\) and updates her stochastic utility in accordance with the so-perceived elapsed market opportunities. We show that this leads to a time-consistent optimal investment policy given by a fractional Kelly strategy associated with \(\hat{\lambda}\) and with the leverage being proportional to the investor’s confidence in her estimate.     

The scaling limit of superreplication prices with small transaction costs in the multivariate case

Kusuoka (Ann. Appl. Probab. 5:198–221, 1995) showed how to obtain non-trivial scaling limits of superreplication prices in discrete-time models of a single risky asset which is traded at properly scaled proportional transaction costs. This article extends the result to a multivariate setup where the investor can trade in several risky assets. The \(G\)-expectation describing the limiting price involves models with a volatility range around the frictionless scaling limit that depends not only on the transaction costs coefficients, but also on the chosen complete discrete-time reference model.     

The space of outcomes of semi-static trading strategies need not be closed

Semi-static trading strategies make frequent appearances in mathematical finance, where dynamic trading in a liquid asset is combined with static buy-and-hold positions in options on that asset. We show that the space of outcomes of such strategies can have very poor closure properties when all European options for a fixed date \(T\) are available for static trading. This causes problems for optimal investment, and stands in sharp contrast to the purely dynamic case classically considered in mathematical finance.     

Stochastic evolution equations in Banach spaces and applications to the Heath–Jarrow–Morton–Musiela equations

The aim of this paper is threefold. Firstly, we study stochastic evolution equations (with the linear part of the drift being a generator of a \(C_{0}\)-semigroup) driven by an infinite-dimensional cylindrical Wiener process. In particular, we prove, under some sufficient conditions on the coefficients, the existence and uniqueness of solutions for these stochastic evolution equations in a class of Banach spaces satisfying the so-called \(H\)-condition. Moreover, we analyse the Markov property of the solutions.Secondly, we apply the abstract results obtained in the first part to prove the existence and uniqueness of solutions to the Heath–Jarrow–Morton–Musiela (HJMM) equations in weighted Lebesgue and Sobolev spaces.Finally, we study the ergodic properties of the solutions to the HJMM equations. In particular, we find a sufficient condition for the existence and uniqueness of invariant measures for the Markov semigroup associated to the HJMM equations (when the coefficients are time-independent) in the weighted Lebesgue spaces.Our paper is a modest contribution to the theory of financial models in which the short rate can be undefined.     

Long-term factorization in Heath–Jarrow–Morton models

The long-term factorization decomposes the stochastic discount factor (SDF) into discounting at the rate of return on the long bond and a martingale that defines a long-term forward measure. We establish sufficient conditions for existence of the long-term factorization in HJM models. A condition on the forward rate volatility ensures existence of the long bond volatility. This yields existence of the long bond and convergence of \(T\)-forward measures to the long forward measure. It contrasts with the familiar risk-neutral factorization that decomposes the SDF into discounting at the short rate and a martingale defining the risk-neutral measure.