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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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 price of risk and ambiguity in an intertemporal general equilibrium model of asset prices

We consider a version of the intertemporal general equilibrium model of Cox et?al. (Econometrica 53:363–384, 1985) with a single production process and two correlated state variables. It is assumed that only one of them, Y ₂, has shocks correlated with those of the economy’s output rate and, simultaneously, that the representative agent is ambiguous about its stochastic process. This implies that changes in Y ₂ should be hedged and its uncertainty priced, with this price containing risk and ambiguity components. Ambiguity impacts asset pricing through two channels: the price of uncertainty associated with the ambiguous state variable, Y ₂, and the interest rate. With ambiguity, the equilibrium price of uncertainty associated with Y ₂ and the equilibrium interest rate can increase or decrease, depending on: (i) the correlations between the shocks in Y ₂ and those in the output rate and in the other state variable; (ii) the diffusion functions of the stochastic processes for Y ₂ and for the output rate; and (iii) the gradient of the value function with respect to Y ₂. As applications of our generic setting, we deduct the model of Longstaff and Schwartz (J Financ 47:1259–1282, 1992) for interest-rate-sensitive contingent claim pricing and the variance-risk price specification in the option pricing model of Heston (Rev Financ Stud 6:327–343, 1993). Additionally, it is obtained a variance-uncertainty price specification that can be used to obtain a closed-form solution for option pricing with ambiguity about stochastic variance.     

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.     

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.     

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).     

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 \(\mathrm{TVaR}\).     

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.     

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.