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.     

A BSDE approach to fair bilateral pricing under endogenous collateralization

Nie and Rutkowski (Int. J. Theor. Appl. Finance 18:1550048, 2015; Math. Finance, 2016, to appear) examined fair bilateral pricing in models with funding costs and an exogenously given collateral. The main goal of this work is to extend results from Nie and Rutkowski (Int. J. Theor. Appl. Finance 18:1550048, 2015; Math. Finance, 2016, to appear) to the case of an endogenous margin account depending on the contract’s value for the hedger and/or the counterparty. Comparison theorems for BSDEs from Nie and Rutkowski (Theory Probab. Appl., 2016, forthcoming) are used to derive bounds for unilateral prices and to study the range for fair bilateral prices in a general semimartingale model. The backward stochastic viability property, introduced by Buckdahn et al. (Probab. Theory Relat. Fields 116:485–504, 2000), is employed to examine the bounds for fair bilateral prices for European claims with a negotiated collateral in a diffusion-type model. We also generalize in several respects the option pricing results from Bergman (Rev. Financ. Stud. 8:475–500, 1995), Mercurio (Actuarial Sciences and Quantitative Finance, pp. 65–95, 2015) and Piterbarg (Risk 23(2):97–102, 2010) by considering contracts with cash-flow streams and allowing for idiosyncratic funding costs for risky assets.     

An explicit martingale version of the one-dimensional Brenier theorem

By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge–Kantorovich mass transport problem was introduced in (Beiglböck et al. in Finance Stoch. 17:477–501, 2013; Galichon et al. in Ann. Appl. Probab. 24:312–336, 2014). Further, by suitable adaptation of the notion of cyclical monotonicity, Beiglböck and Juillet (Ann. Probab. 44:42–106, 2016) obtained an extension of the one-dimensional Brenier theorem to the present martingale version. In this paper, we complement the previous work by extending the so-called Spence–Mirrlees condition to the case of martingale optimal transport. Under some technical conditions on the starting and the target measures, we provide an explicit characterization of the corresponding optimal martingale transference plans both for the lower and upper bounds. These explicit extremal probability measures coincide with the unique left- and right-monotone martingale transference plans introduced in (Beiglböck and Juillet in Ann. Probab. 44:42–106, 2016). Our approach relies on the (weak) duality result stated in (Beiglböck et al. in Finance Stoch. 17:477–501, 2013), and provides as a by-product an explicit expression for the corresponding optimal semi-static hedging strategies. We finally provide an extension to the multiple marginals case.     

Change of numeraire in the two-marginals martingale transport problem

In this paper, we apply change of numeraire techniques to the optimal transport approach for computing model-free prices of derivatives in a two-period setting. In particular, we consider the optimal transport plan constructed in Hobson and Klimmek (Finance Stoch. 19:189–214, 2015) as well as the one introduced in Beiglböck and Juillet (Ann. Probab. 44:42–106, 2016) and further studied in Henry-Labordère and Touzi (Finance Stoch. 20:635–668, 2016). We show that in the case of positive martingales, a suitable change of numeraire applied to Hobson and Klimmek (Finance Stoch. 19:189–214, 2015) exchanges forward start straddles of type I and type II, so that the optimal transport plan in the subhedging problems is the same for both types of options. Moreover, for Henry-Labordère and Touzi’s (Finance Stoch. 20:635–668, 2016) construction, the right-monotone transference plan can be viewed as a mirror coupling of its left counterpart under the change of numeraire.     

A four-factor stochastic volatility model of commodity prices

The number of factors driving the uncertain dynamics of commodity prices has been a central consideration in financial literature. While the majority of empirical studies relies on the assumption that up to three factors are sufficient to explain all relevant uncertainty inherent in commodity spot, futures, and option prices, evidence from Trolle and Schwartz (Rev Financ Stud 22(11):4423–4461, 2009b) and Hughen (J Futures Mark 30(2):101–133, 2010) indicates a need for additional risk factors. In this article, we propose a four-factor maximal affine stochastic volatility model that allows for three independent sources of risk in the futures term structure and an additional, potentially unspanned stochastic volatility process. The model principally integrates the insights from Hughen (2010) and Tang (Quant Finance 12(5):781–790, 2012) and nests many well-known models in the literature. It can account for several stylized facts associated with commodity dynamics such as mean reversion to a stochastic level, stochastic volatility in the convenience yield, a time-varying correlation structure, and time-varying risk-premia. In-sample and out-of-sample tests indicate a superior model fit to futures and options data as well as lower hedging errors compared to three-factor benchmark models. The results also indicate that three factors are not sufficient to model the joint dynamics of futures and option prices accurately.     

An Empirical Comparison of Asset-Pricing Models in the Shanghai A-Share Exchange Market

This paper evaluates and compares the performance of three-asset pricing models—the capital asset pricing model of Sharpe (J Finance 19:425–442, 1964), the three-factor model of Fama and French (J Financ Econ 33:3–56, 1993), and the five-factor model (Fama and French in J Financ Econ 123:1–22, 2015)—in the Shanghai A-share exchange market. Our results do not support the superiority of the five-factor model and show that the three-factor model outperforms the other models. We also verify the redundancy of the book-to-market factor and confirm the findings of Fama and French (2015).     

A note on the valuation of asset management firms

Market capitalization relative to assets under management is often used to value asset management firms. Huberman’s (2004) dividend discount model implies that cross-sectional variations in this metric are explained by cross-sectional differences in operating margins, and yet we find no evidence of this in our data set. We show that a superior model—inspired by the work of Berk and Green (2004)—includes also the level of fees as an explanatory variable. This approach dramatically increases the fit of our valuation model and casts doubt on the relevance of the so-called Huberman puzzle.     

The dispersion anomaly and analyst recommendations

The main objective of this study is to distinguish whether the forecast dispersion anomaly is due to Miller’s (J Finance 32(4):1151–1168, 1977) overpricing hypothesis or idiosyncratic risk, by conditioning the sample on “buy” and “sell” consensus recommendations. Observations on the long and short possibilities provided to the investors by the analyst stock recommendations can help us infer on the impact of short sale constraints even though they are not directly observed. This study provides strong evidence that the impact of analyst forecast dispersion is more pronounced in the group of stocks that receive the least favorable recommendations in a given period, even after controlling for the idiosyncratic risk, Fama–French factors (J Financ Econ 33(1):3–56, 1993; J Financ Econ 116(1):1–22, 2015) and even short-sale constraints. These results are consistent with Miller’s (1977) hypothesis, according to which if short-sale constraints bind, high opinion divergence stocks become overpriced and hence have low subsequent returns.     

Service quality in the private banking business

This paper examines the concept of service quality in private banking theoretically and empirically and identifies factors which contribute to service quality. A multidimensional and hierarchical model is developed based on the work of Rust and Oliver (in Service Quality, pp. 1–20, 1994) and Brady and Cronin (in J. Mark. 65(3):34–49, 2001). The model is then empirically tested among private banking providers with the partial least squares method. Furthermore, the developed model is compared to other approaches, including Grönroos (in Eur. J. Mark. 18(4):36–44, 1984). Another model for comparison excludes the indirect effects of Grönroos (in Eur. J. Mark. 18(4):36–44, 1984) and focuses on the direct effects on service quality. We can conclude that the model based on Rust and Oliver (in Service Quality, pp. 1–20, 1994) and Brady and Cronin (in J. Mark. 65(3):34–49, 2001) produces the best results and can best explain service quality in private banking. Finally, an analysis of various provider groups is conducted in order to identify differences between private banking providers in Germany, Switzerland, Austria and Liechtenstein and between providers with various minimum investment requirements.     

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.     

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.     

Corporate bankruptcy prediction: a high dimensional analysis

Much bankruptcy research has relied on parametric models, such as multiple discriminant analysis and logit, which can only handle a finite number of predictors (Altman in The Journal of Finance 23 (4), 589–609, 1968; Ohlson in Journal of Accounting Research 18 (1), 109–131, 1980). The gradient boosting model is a statistical learning method that overcomes this limitation. The model accommodates very large numbers of predictors which can be rank ordered, from best to worst, based on their overall predictive power (Friedman in The Annals of Statistics 29 (5), 1189–1232, 2001; Hastie et al. 2009). Using a sample of 1115 US bankruptcy filings and 91 predictor variables, the study finds that non-traditional variables, such as ownership structure/concentration and CEO compensation are among the strongest predictors overall. The next best predictors are unscaled market and accounting variables that proxy for size effects. This is followed by market-price measures and financial ratios. The weakest predictors overall included macro-economic variables, analyst recommendations/forecasts and industry variables.     

Beating the DAX,MDAX, and SDAX: investment strategies in Germany

Motivated by two recent papers of Asness et al. (J Portf Manag Fall 40(5):75–92, 2014; J Portf Manag Fall 42(1):34–52, 2015), we investigate whether momentum and value strategies outperformed a buy-and-hold strategy in the three biggest German equity indices, DAX, MDAX, and SDAX from 1988 to 2015. Our findings show that a momentum premium was present only in the SDAX and that value strategies did not work in any of the three indices. Consequently, we conclude that at least the DAX and MDAX are efficient indices and that some supposedly abnormal returns could be illusionary, as limits to arbitrage obstruct any profitable exploitation in practice. Finally, we find a negative correlation between momentum and value in the DAX and show that mixing both strategies can substantially decrease a portfolio’s risk.     

The investment performance,attributes, and investment behavior of ethical equity mutual funds in the US: an empirical investigation

This paper examines the investment performance of US ethical equity mutual funds relative to the market and their traditional counterparts using a survivorship-bias-free database. We detect selectivity and market timing performance of fund managers using two models. First, we use Treynor and Mazuy’s (Harv Bus Rev 44:131–136, 1966) model to determine these performances from a quadratic regression of fund returns on market returns. Second, we use a comprehensive and integrated model derived by Bhattacharya and Pfleiderer (A note on performance evaluation. Technical Report 714, Stanford, California, Stanford University, Graduate School of Business, 1983) and Lee and Rahman (J Bus 63:261–278, 1990) to simultaneously capture stock selection and market timing skill of fund managers. This model extracts timing skill from the relationship between managers’ forecast and realized market return. In addition, the R² approach developed by Amihud and Goyenko (Rev Financ Stud 26:667–694, 2013) for evaluating selectivity is also used in this paper. Our empirical results indicate that ethical funds perform no worse than their traditional counterparts, although ethical and traditional funds do not outperform the market. We find some evidence of superior security selection and/or market timing skill among a very small number of ethical and traditional funds. It appears that matching traditional funds have slightly more abnormal (superior as well as inferior) performance than ethical funds in our sample.     

Pathwise superreplication via Vovk’s outer measure

Since Hobson’s seminal paper (Hobson in Finance Stoch. 2:329–347, 1998), the connection between model-independent pricing and the Skorokhod embedding problem has been a driving force in robust finance. We establish a general pricing–hedging duality for financial derivatives which are susceptible to the Skorokhod approach.Using Vovk’s approach to mathematical finance, we derive a model-independent superreplication theorem in continuous time, given information on finitely many marginals. Our result covers a broad range of exotic derivatives, including lookback options, discretely monitored Asian options, and options on realized variance.     

The SIML Estimation of Integrated Covariance and Hedging Coefficient Under Round-off Errors,Micro-market Price Adjustments and Random Sampling

For estimating the integrated volatility and covariance by using high frequency data, Kunitomo and Sato (Math Comput Simul 81:1272–1289, 2011; N Am J Econ Finance 26:289–309, 2013) have proposed the separating information maximum likelihood (SIML) method when there are micro-market noises. The SIML estimator has reasonable finite sample properties and asymptotic properties when the sample size is large when the hidden efficient price process follows a Brownian semi-martingale. We shall show that the SIML estimation is useful for estimating the integrated covariance and hedging coefficient when we have round-off errors, micro-market price adjustments and noises, and when the high-frequency data are randomly sampled. The SIML estimation is consistent, asymptotically normal in the stable convergence sense under a set of reasonable assumptions and it has reasonable finite sample properties with these effects.     

Pricing currency options under double exponential jump diffusion in a Markov-modulated HJM economy

Extending the framework of Amin and Jarrow (J Int Money Financ 10:310–329, 1991) and Bo et al. (Insur Math Econ 46:461–469, 2010), this study provides a theoretical exploration of currency options pricing under the presence of interest-rate regime shifts and exchange-rate asymmetric jumps. Evidence of interest-rate regime shifts inferred from UK and US zero coupon bond yields provides support for the regime-switching specifications which we reflect upon the domestic and foreign forward rates. Results of statistical tests conducted on JPY/USD and EUR/USD FX rates provide further support the rationale behind using a double exponential jump diffusion process within a Markov modulated Heath–Jarrow–Morton economy. Our numerical results suggest that, the pricing performance of our model is closely comparable to the Bo-Wang-Yang model for at-the-money options, yet yields improvements in percentage root mean errors for in-the-money options.     

Risk measures with the CxLS property

In the present contribution, we characterise law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (Math. Finance 16:419–441, 2006), we show that these risk measures can be identified with a class of generalised shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (Math. Finance, 2014, http://onlinelibrary.wiley.com/doi/10.1111/mafi.12080/abstract) and Bellini and Bignozzi (Quant. Finance 15:725–733, 2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterisation of robustness for convex risk measures in terms of a weak notion of mixture continuity.     

Another look at the integral of exponential Brownian motion and the pricing of Asian options

It is shown that Marc Yor’s formula (Adv. Appl. Probab. 24:509–531, 1992) for the density of the integral of exponential Brownian motion taken over a finite time interval is an extremal member of a family of previously unknown integral formulae for the same density. The derivation is independent from the one by Yor and obtained from a simple time-reversibility feature, in conjunction with a Fokker–Planck type argument. Similar arguments lead to an independent derivation of Dufresne’s result (Scand. Actuar. J. 90:39–79, 1990) for the law of the integral taken over an infinite time interval. The numerical aspects of the new formulae are developed, with concrete applications to Asian options.     

The pricing of common exchange rate factors in the U.S. equity market

We extend Lustig et al. (Rev Financ Stud 24:3731–3777, 2011) and Brusa et al. (The International CAPM Redux, 2014) by examining if the common exchange rate factors, the dollar and carry factors, are priced in the US equity market. Our results suggest that while the carry factor has incremental pricing information relative to the US market factor, the dollar factor (or the trade-weighted exchange rate index) is redundant. Our results have important theoretical as well as practical implications. Theoretically, we suggest that financial economists take an endogenous perspective of exchange rates. Practically, we suggest that practitioners incorporate in the carry factor to measure the exposure of exchange rate risk.