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.     

A note on the condition of no unbounded profit with bounded risk

As a corollary to Delbaen and Schachermayer’s fundamental theorem of asset pricing (Delbaen in Math. Ann. 300:463–520, 1994; Stoch. Stoch. Rep. 53:213–226, 1995; Math. Ann. 312:215–250, 1998), we prove, in a general finite-dimensional semimartingale setting, that the no unbounded profit with bounded risk (NUPBR) condition is equivalent to the existence of a strict sigma-martingale density. This generalizes the continuous-path result of Choulli and Stricker (Séminaire de Probabilités XXX, pp. 12–23, 1996) to the càdlàg case and extends the recent one-dimensional result of Kardaras (Finance and Stochastics 16:651–667, 2012) to the multidimensional case. It also refines partially the second main result of Karatzas and Kardaras (Finance Stoch. 11:447–493, 2007) concerning the existence of an equivalent supermartingale deflator. The proof uses the technique of numéraire change.     

An empirical investigation of asset pricing models under divergent lending and borrowing rates

Asset pricing theory implies that the estimate of the zero-beta rate should fall between divergent lending and borrowing rates. This paper proposes a formal test of this restriction using the difference between the prime loan rate and the 1-month Treasury bill rate as a proxy for the difference between borrowing and lending rates. Based on simulations, this paper shows that in the ordinary least squares case, the Fama and MacBeth (J Pol Econ 81:607–636, 1973) t-statistic has high power against a general alternative, which is not true of the Shanken (Rev Financ Stud 5:1–33, 1992) and Kan et al. (J Financ doi:10.1111/jofi.12035, 2013) t-statistics. In the generalized least squares case, all three t-statistics have high power. The empirical investigation highlights that only the intertemporal capital asset pricing model reasonably prices the zero-beta portfolio. Other models, such as the Fama and French (J Financ Econ 33:3–56, 1993) model, do not assign the correct value to the zero-beta rate.     

Option hedging for small investors under liquidity costs

Following the framework of Çetin et al. (Finance Stoch. 8:311–341, 2004), we study the problem of super-replication in the presence of liquidity costs under additional restrictions on the gamma of the hedging strategies in a generalized Black–Scholes economy. We find that the minimal super-replication price is different from the one suggested by the Black–Scholes formula and is the unique viscosity solution of the associated dynamic programming equation. This is in contrast with the results of Çetin et al. (Finance Stoch. 8:311–341, 2004), who find that the arbitrage-free price of a contingent claim coincides with the Black–Scholes price. However, in Çetin et al. (Finance Stoch. 8:311–341, 2004) a larger class of admissible portfolio processes is used, and the replication is achieved in the L ² approximating sense.     

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).     

Option pricing under non-normality: a comparative analysis

This paper carries out a comparative analysis of the calibration and performance of a variety of options pricing models. These include Black and Scholes (J Polit Econ 81:637–659, 1973), the Gram–Charlier (GC) approach of Backus et al. (1997), the stochastic volatility (HS) model of Heston (Rev Financ Stud 6:327–343, 1993), the closed-form GARCH process of Heston and Nandi (Rev Financ Stud 13:585–625, 2000) and a variety of Lévy processes including the Variance Gamma (VG), Normal Inverse Gaussian (NIG), and, CGMY and Kou (Manag Sci 48:1086–1101, 2002) jump-diffusion models. Unlike most studies of option pricing, we compare these models using a common point-in-time data which reflects the perspective of a new investor who wishes to choose between models using only the most minimal recent data set. For each of these models, we also examine the accuracy of delta and delta-gamma approximations to the valuation of both individual options and an illustrative option portfolio.     

Large Deviations for the Extended Heston Model: The Large-Time Case

We study here the large-time behaviour of all continuous affine stochastic volatility models [in the sense of Keller-Ressel (Math Finan 21(1):73–98, 2011)] and deduce a closed-form formula for the large-maturity implied volatility smile. We concentrate on (rescaled) strikes around the money, which are the most common in practice, and extend the results in Forde and Jacquier (Finan Stoch 15(4):755–780, 2011) and Gatheral and Jacquier (Quant Finan 11(8):1129–1132, 2011).     

Equilibrium model with default and dynamic insider information

We consider an equilibrium model à la Kyle–Back for a defaultable claim issued by a given firm. In such a market the insider observes continuously in time the value of the firm, which is unobservable by the market makers. Using the construction in Campi et al. (http://hal.archives-ouvertes.fr/hal-00534273/en/, 2011) of a dynamic three-dimensional Bessel bridge, we provide the equilibrium price and the insider’s optimal strategy. As in Campi and Çetin (Finance Stoch. 11:591–602, 2007), the information released by the insider while trading optimally makes the default time predictable in the market’s view at the equilibrium. We conclude the paper by comparing the insider’s expected profits in the static and dynamic private information case. We also compute explicitly the value of the insider’s information in the special cases of a defaultable stock and a bond.     

Tobin meets Oates: solidarity and the optimal fiscal federal structure

We explore the implications for the optimal degree of fiscal decentralization when people’s preferences for goods and services—which classic treatments of fiscal federalism (Oates in Fiscal federalism, 1972) place in the purview of local governments—exhibit specific egalitarianism (Tobin in J. Law Econ. 13(2): 263–277, 1970), or solidarity. We find that a system in which the central government provides a common minimum level of the publicly provided good, and local governments are allowed to use their own resources to provide an even higher local level, performs better from an efficiency perspective relative to all other systems analyzed for a relevant range of preferences over solidarity.     

Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (Math. Finance 3:241–276, 1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem.     

Duality and convergence for binomial markets with friction

We prove limit theorems for the super-replication cost of European options in a binomial model with friction. Examples covered are markets with proportional transaction costs and illiquid markets. A dual representation for the super-replication cost in these models is obtained and used to prove the limit theorems. In particular, the existence of a liquidity premium for the continuous-time limit of the model proposed in Çetin et al. (Finance Stoch. 8:311–341, 2004) is proved. Hence, this paper extends the previous convergence result of Gökay and Soner (Math Finance 22:250–276, 2012) to the general non-Markovian case. Moreover, the special case of small transaction costs yields, in the continuous limit, the G-expectation of Peng as earlier proved by Kusuoka (Ann. Appl. Probab. 5:198–221, 1995).     

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.     

Pricing and hedging volatility smile under multifactor interest rate models

The paper extends Amin and Morton (1994), Zeto (2002), and Kuo and Paxson (2006) by considering jump-diffusion model of Das (1999) with various volatility functions in pricing and hedging Euribor options across strikes and maturities. Adding the jump element into a diffusion model helps capturing volatility smiles in the interest rate options markets, but specifying the mean-reversion volatility function improves the most. A humped volatility function with the additional jump component yields better in-sample and out-of-sample valuation, but level-dependent volatility becomes more crucial for hedging. The specification of volatility function is more crucial than merely adding jumps into any model and the effect of jumps declines as the maturity of options is longer.     

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.     

Forecasting Long-Term Interest Rates with a General-Equilibrium Model of the Euro Area: What Role for Liquidity Services of Bonds?

This paper studies the forecasting performance of a general equilibrium model of bond yields where government bonds provide liquidity services and are, as such, an integral part of the monetary transmission mechanism. The model is estimated with Bayesian methods on Euro area data. I compare the out-of-sample predictive performance of the model against a variety of competing specifications, including that of De Graeve et al. (J Monet Econ 56(4):545–559, 2009). Forecast accuracy is evaluated through both univariate and multivariate measures. I also control the statistical significance of the forecast differences using the tests of Diebold and Mariano (J Bus Econ Stat 13(3):253–263, 1995), Hansen (J Bus Econ Stat 23:365–380, 2005) and White (Econometrica 68(5):1097–1126, 1980). The results indicate that accounting for the liquidity services of bonds contributes to generate superior out-of-sample forecasts for both real variables, such as output, and inflation, and for bond yields.     

Optimal loan-to-value ratio and the efficiency gains of default

We study the optimal loan-to-value (LTV) ratio in a monetary general equilibrium model with heterogeneous agents, collateral default, production and a banking sector. We find that the welfare of the debtor is not monotonically increasing in the LTV ratio, i.e. tighter financing constraints can be welfare-improving for the debtor. Moreover, the optimal LTV ratio for both the debtor and the creditor allows for the possibility of ex post default. Collateral default enhances efficiency by allowing for better consumption smoothing and risk hedging. Our result improves the argument in Dubey et al. (Econometrica 73(1):1–37, 2005) and Zame (Am Econ Rev 83(5):1142–1164, 1993), which use default penalties instead of collateral to induce repayment and show the efficiency gains of default.     

Bottleneck options

In the spirit of Kyprianou and Ott (in Acta Appl. Math., to appear, 2013) and Ott (in Ann. Appl. Probab. 23:2327–2356, 2013) we consider an option whose payoff corresponds to a capped American lookback option with floating strike and solve the associated pricing problem (an optimal stopping problem) in a financial market whose price process is modelled by an exponential spectrally negative Lévy process. Despite the simple interpretation of the cap as a moderation of the payoff, it turns out that the optimal strategy to exercise the option looks very different compared to the situation without a cap. In fact, we show that the continuation region has a feature that resembles a bottleneck and hence the name “bottleneck option”.     

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.     

Portfolio allocation using multivariate variance gamma models

In this paper, we investigate empirically the effect of using higher moments in portfolio allocation when parametric and nonparametric models are used. The nonparametric model considered in this paper is the sample approach; the parametric model is constructed assuming multivariate variance gamma (MVG) joint distribution for asset returns.We consider the MVG models proposed by Madan and Seneta (1990), Semeraro (2008) and Wang (2009). We perform an out-of-sample analysis comparing the optimal portfolios obtained using the MVG models and the sample approach. Our portfolio is composed of 18 assets selected from the S&P500 Index and the dataset consists of daily returns observed from 01/04/2000 to 01/09/2011.