Showing papers in "Finance and Stochastics in 2015"

Aggregation-robustness and model uncertainty of regulatory risk measures

Abstract: Research related to aggregation, robustness and model uncertainty of regulatory risk measures, for instance, value-at-risk (VaR) and expected shortfall (ES), is of fundamental importance within quantitative risk management. In risk aggregation, marginal risks and their dependence structure are often modelled separately, leading to uncertainty arising at the level of a joint model. In this paper, we introduce a notion of qualitative robustness for risk measures, concerning the sensitivity of a risk measure to the uncertainty of dependence in risk aggregation. It turns out that coherent risk measures, such as ES, are more robust than VaR according to the new notion of robustness. We also give approximations and inequalities for aggregation and diversification of VaR under dependence uncertainty, and derive an asymptotic equivalence for worst-case VaR and ES under general conditions. We obtain that for a portfolio of a large number of risks, VaR generally has a larger uncertainty spread compared to ES. The results warn that unjustified diversification arguments for VaR used in risk management need to be taken with much care, and they potentially support the use of ES in risk aggregation. This in particular reflects on the discussions in the recent consultative documents by the Basel Committee on Banking Supervision.

Robust price bounds for the forward starting straddle

Abstract: We consider the problem of giving a robust, model-independent, lower bound on the price of a forward starting straddle with payoff $|F_{T_{1}} - F_{T_{0}}|$ , where 0

Spot volatility estimation using delta sequences

Abstract: We introduce a unifying class of nonparametric spot volatility estimators based on delta sequences and conceived to include many of the existing estimators in the field as special cases. The full limit theory is first derived when unevenly sampled observations under infill asymptotics and fixed time horizon are considered, and the state variable is assumed to follow a Brownian semimartingale. We then extend our class of estimators to include Poisson jumps or financial microstructure noise in the observed price process. This work makes different approaches (kernels, wavelets, Fourier) comparable. For example, we explicitly illustrate some drawbacks of the Fourier estimator. Specific delta sequences are applied to data from the S&P 500 stock index futures market.

Necessary and sufficient conditions in the problem of optimal investment with intermediate consumption

Abstract: We consider the problem of optimal investment with intermediate consumption in the framework of an incomplete semimartingale model of a financial market. We show that a necessary and sufficient condition for the validity of key assertions of the theory is that the value functions of the primal and dual problems are finite.

How non-arbitrage, viability and numéraire portfolio are related

Abstract: This paper proposes two approaches that quantify the exact relationship among viability, absence of arbitrage, and/or existence of the numeraire portfolio under minimal assumptions and for general continuous-time market models. Precisely, our first and principal contribution proves the equivalence between the no-unbounded-profit-with-bounded-risk condition (NUPBR hereafter), the existence of the numeraire portfolio, and the existence of the optimal portfolio under an equivalent probability measure for any “nice” utility and positive initial capital. Herein, a “nice” utility is any smooth von Neumann–Morgenstern utility satisfying Inada’s conditions and the elasticity assumptions of Kramkov and Schachermayer. Furthermore, the equivalent probability measure—under which the utility maximization problems have solutions—can be chosen as close to the real-world probability measure as we want (but might not be equal). Without changing the underlying probability measure and under mild assumptions, our second contribution proves that NUPBR is equivalent to the “local” existence of the optimal portfolio. This constitutes an alternative to the first contribution, if one insists on working under the real-world probability. These two contributions lead naturally to new types of viability that we call weak and local viabilities.

Multi-portfolio time consistency for set-valued convex and coherent risk measures

Abstract: Equivalent characterizations of multi-portfolio time consistency are deduced for closed convex and coherent set-valued risk measures on $L^{p}({\varOmega,\mathcal{F},\mathbb{P}; \mathbb{R}^{d}})$ with image space in the power set of $L^{p}({\varOmega,\mathcal{F}_{t},\mathbb{P}; \mathbb{R}^{d}})$ . In the convex case, multi-portfolio time consistency is equivalent to a cocycle condition on the sum of minimal penalty functions. In the coherent case, multi-portfolio time consistency is equivalent to a generalized version of stability of the dual variables. As examples, the set-valued entropic risk measure with constant risk aversion coefficient is shown to satisfy the cocycle condition for its minimal penalty functions; the set of superhedging portfolios is shown to have in markets with proportional transaction costs the stability property and to satisfy in markets with convex transaction costs the composed cocycle condition; and a multi-portfolio time-consistent version of the set-valued average value at risk, the composed AV@R, is given, and its dual representation deduced.

The existence of dominating local martingale measures

Abstract: We prove that for locally bounded processes the absence of arbitrage opportunities of the first kind is equivalent to the existence of a dominating local martingale measure This is related to and motivated by results from the theory of filtration enlargements

Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach

Abstract: This paper develops an eigenfunction expansion approach to solve discretely monitored first passage time problems for a rich class of Markov processes, including diffusions and subordinate diffusions with jumps, whose transition or Feynman–Kac semigroups possess eigenfunction expansions in $L^{2}$ -spaces. Many processes important in finance are in this class, including OU, CIR, (JD)CEV diffusions and their subordinate versions with jumps. The method represents the solution to a discretely monitored first passage problem in the form of an eigenfunction expansion with expansion coefficients satisfying an explicitly given recursion. A range of financial applications is given, drawn from across equity, credit, commodity, and interest rate markets. Numerical examples demonstrate that even in the case of frequent barrier monitoring, such as daily, approximating discrete first passage time problems with continuous solutions may result in unacceptably large errors in financial applications. This highlights the relevance of the method to financial applications.

Taylor approximation of incomplete Radner equilibrium models

Abstract: In the setting of exponential investors and uncertainty governed by Brownian motions, we first prove the existence of an incomplete equilibrium for a general class of models. We then introduce a tractable class of exponential–quadratic models and prove that the corresponding incomplete equilibrium is characterized by a coupled set of Riccati equations. Finally, we prove that these exponential–quadratic models can be used to approximate the incomplete models we studied in the first part.

Fragility of arbitrage and bubbles in local martingale diffusion models

Abstract: For any positive diffusion with minimal regularity, there exists a semimartingale with uniformly close paths that is a martingale under an equivalent probability. As a result, in models of asset prices based on such diffusions, arbitrage and bubbles alike disappear under proportional transaction costs or under small model mis-specifications. Thus, local martingale diffusion models of arbitrage and bubbles are not robust to small trading and monitoring frictions.

Existence of an endogenously complete equilibrium driven by a diffusion

Abstract: The existence of complete Radner equilibria is established in an economy whose parameters are driven by a diffusion process. Our results complement those in the literature. In particular, we work under essentially minimal regularity conditions and treat the time-inhomogeneous case.

A model for a large investor trading at market indifference prices. I: single-period case.

Abstract: We develop a single-period model for a large economic agent who trades with market makers at their utility indifference prices. We compute the sensitivities of these market indifference prices with respect to the size of the investor’s order. It turns out that the price impact of an order is determined both by the market makers’ joint risk tolerance and by the variation of individual risk tolerances. On a technical level, a key role in our analysis is played by a pair of conjugate saddle functions associated with the description of Pareto optimal allocations in terms of the aggregate utility function.

Asymptotics for fixed transaction costs

Abstract: An investor with constant relative risk aversion trades a safe and several risky assets with constant investment opportunities. For a small fixed transaction cost, levied on each trade regardless of its size, we explicitly determine the leading-order corrections to the frictionless value function and optimal policy.

On a Heath–Jarrow–Morton approach for stock options

Abstract: This paper aims at transferring the philosophy behind Heath–Jarrow–Morton to the modelling of call options with all strikes and maturities. Contrary to the approach by Carmona and Nadtochiy (Finance Stoch. 13:1–48, 2009) and related to the recent contribution (Finance Stoch. 16:63–104, 2012) by the same authors, the key parameterisation of our approach involves time-inhomogeneous Levy processes instead of local volatility models. We provide necessary and sufficient conditions for absence of arbitrage. Moreover, we discuss the construction of arbitrage-free models. Specifically, we prove their existence and uniqueness given basic building blocks.

Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation

Abstract: This paper deals with an investment–consumption portfolio problem when the current utility depends also on the wealth process. Such problems arise e.g. in portfolio optimization with random horizon or random trading times. To overcome the difficulties of the problem, a dual approach is employed: a dual control problem is defined and treated by means of dynamic programming, showing that the viscosity solutions of the associated Hamilton–Jacobi–Bellman equation belong to a suitable class of smooth functions. This allows defining a smooth solution of the primal Hamilton–Jacobi–Bellman equation, and proving by verification that such a solution is indeed unique in a suitable class of smooth functions and coincides with the value function of the primal problem. Applications to specific financial problems are given.

An optimal consumption problem in finite time with a constraint on the ruin probability

Abstract: In this paper, we investigate the following problem: For a given upper bound for the ruin probability, maximize the expected discounted consumption of an investor in finite time. The endowment of the agent is modeled by a Brownian motion with positive drift. We give an iterative algorithm for the solution of the problem, where in each step an unconstrained, but penalized problem is solved. For the discontinuous value function $V(t,x)$ of the penalized problem, we show that it is the unique viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. Moreover, we characterize the optimal strategy as a barrier strategy with continuous barrier function.

Dynamic credit investment in partially observed markets

Abstract: We consider the problem of maximizing the expected utility for a power investor who can allocate his wealth in a stock, a defaultable security, and a money market account. The dynamics of these security prices are governed by geometric Brownian motions modulated by a hidden continuous-time finite-state Markov chain. We reduce the partially observed stochastic control problem to a complete observation risk-sensitive control problem via the filtered regime switching probabilities. We separate the latter into predefault and postdefault dynamic optimization subproblems and obtain two coupled Hamilton–Jacobi–Bellman (HJB) partial differential equations. We prove the existence and uniqueness of a globally bounded classical solution to each HJB equation and give the corresponding verification theorem. We provide a numerical analysis showing that the investor increases his holdings in stock as the filter probability of being in high-growth regimes increases, and decreases his credit risk exposure as the filter probability of being in high default risk regimes gets larger.

Forward equations for option prices in semimartingale models

Abstract: We derive a forward partial integro-differential equation for prices of call options in a model where the dynamics of the underlying asset under the pricing measure is described by a—possibly discontinuous—semimartingale. This result generalizes Dupire’s forward equation to a large class of non-Markovian models with jumps.

Pricing and hedging Asian-style options on energy

Abstract: We solve the problem of pricing and hedging Asian-style options on energy with a quadratic risk criterion when trading in the underlying future is restricted. Liquid trading in the future is only possible up to the start of a so-called delivery period. After the start of the delivery period, the hedge positions cannot be adjusted any more until maturity. This reflects the trading situation at the Nordic energy market Nord Pool, for example. We show that there exists a unique solution to this combined continuous–discrete quadratic hedging problem if the future price process is a special semimartingale with bounded mean–variance tradeoff. Additionally, under the assumption that the future price process is a local martingale, the hedge positions before the averaging period are inherited from the market specification without trading restriction. As an application, we consider three models and derive their quadratic hedge positions in explicit form: a simple Black–Scholes model with time-dependent volatility, the stochastic volatility model of Barndorff-Nielsen and Shephard, and an exponential additive model. Based on an exponential spot price model driven by two NIG Levy processes, we determine an exponential additive model for the future price by moment matching techniques. We calculate hedge positions and determine the quadratic hedging error in a simulation study.

Optimal investment and price dependence in a semi-static market

Abstract: This paper studies the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be traded continuously in time and are modelled as locally bounded semimartingales. Using a general utility function defined on the positive half-line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiability, monotonicity, convexity and limiting properties.

Risk measures for processes and BSDEs

Abstract: The paper analyzes risk assessment for cash flow processes in continuous time. We combine the framework of convex risk measures for processes with a decomposition result for optional and predictable measures to provide a systematic approach to the issues of model ambiguity and uncertainty about the time value of money. We also establish a link between risk measures for processes and BSDEs.

When terminal facelift enforces Delta constraints

Abstract: This paper deals with the superreplication of non-path-dependent European claims under additional convex constraints on the number of shares held in the portfolio. The corresponding superreplication price of a given claim has been widely studied in the literature, and its terminal value, which dominates the claim of interest, is the so-called facelift transform of the claim. We investigate under which conditions the superreplication price and strategy of a large class of claims coincide with the exact replication price and strategy of the facelift transform of this claim. In one dimension, we observe that this property is satisfied for any local volatility model. In any dimension, we exhibit an analytical necessary and sufficient condition for this property, which combines the dynamics of the stock together with the characteristics of the closed convex set of constraints. To obtain this condition, we introduce the notion of first order viability property for linear parabolic PDEs. We investigate in detail several practical cases of interest: multidimensional Black–Scholes model, non-tradable assets, and short-selling restrictions.

Hedge and mutual funds’ fees and the separation of private investments

Abstract: A fund manager invests both the fund’s assets and own private wealth in separate but potentially correlated risky assets, aiming to maximize expected utility from private wealth in the long run. If relative risk aversion and investment opportunities are constant, we find that the fund’s portfolio depends only on the fund’s investment opportunities, and the private portfolio only on private opportunities. This conclusion is valid both for a hedge fund manager, who is paid performance fees with a high-water mark provision, and for a mutual fund manager, who is paid management fees proportional to the fund’s assets. The manager invests earned fees in the safe asset, allocating remaining private wealth in a constant-proportion portfolio, while the fund is managed as another constant-proportion portfolio. The optimal welfare is the maximum between the optimal welfares of each investment opportunity, with no diversification gain. In particular, the manager does not use private investments to hedge future income from fees.

A convergence result for the Emery topology and a variant of the proof of the fundamental theorem of asset pricing

Abstract: We show that no unbounded profit with bounded risk (NUPBR) implies predictable uniform tightness (P-UT), a boundedness property in the Emery topology introduced by Stricker (Seminaire de Probabilites de Strasbourg XIX, pp. 209–217, 1985). Combining this insight with well-known results of Memin and Slominski (Seminaire de Probabilites de Strasbourg XXV, pp. 162–177, 1991) leads to a short variant of the proof of the fundamental theorem of asset pricing initially proved by Delbaen and Schachermayer (Math. Ann. 300:463–520, 1994). The results are formulated in the general setting of admissible portfolio wealth processes as laid down by Kabanov (Statistics and Control of Stochastic Processes, pp. 191–203, World Sci. Publ., River Edge, 1997).

On the forward rate concept in multi-state life insurance

Abstract: Similarly to the notion of modeling credit risk by using forward credit default spread rates, mortality risk in life insurance contracts is nowadays often modeled by using forward mortality (spread) rates. More recently, this concept has also been discussed for more complex life insurances that include multiple lives or intermediate states that correspond to the health status of the insured. For consistency purposes and for technical reasons, most authors assume that the underlying financial and demographic events are stochastically independent. In the present paper, we study sufficient and necessary conditions under which general transition forward rates are indeed consistent with respect to the relevant insurance claims. This shows the theoretical limitations of the forward rate concept in life insurance. Our study is based on a model where the underlying financial and demographical developments are diffusion processes driven by a multivariate Brownian motion. This allows us to investigate independence properties by analyzing the asymptotic behavior of mixed (conditional) moments. In particular, we obtain that for joint life and disability insurance policies, some specific demographic events need to be dependent in order to ensure consistency.

When do creditors with heterogeneous beliefs agree to run

Abstract: This paper explores, in a multiperiod setting, the funding liquidity of a borrower that finances its operations through short-term debt. The short-term debt is provided by a continuum of creditors with heterogeneous beliefs about the prospects of the borrower. In each period, creditors observe the borrower’s fundamentals and decide on the amount they invest in its short-term debt. We formalize this problem as a coordination game, and we show that there exists a unique reasonable Nash equilibrium. We show that the borrower is able to refinance if and only if the liquid net worth is above an illiquidity barrier, and we explicitly find this barrier in terms of the distribution of capital and beliefs across creditors.

Approximate hedging for nonlinear transaction costs on the volume of traded assets

Abstract: This paper is dedicated to the replication of a convex contingent claim h(S 1) in a financial market with frictions, due to deterministic order books or regulatory constraints. The corresponding transaction costs can be rewritten as a nonlinear function G of the volume of traded assets, with G′(0)>0. For a stock with Black–Scholes midprice dynamics, we exhibit an asymptotically convergent replicating portfolio, defined on a regular time grid with n trading dates. Up to a well-chosen regularization h n of the payoff function, we first introduce the frictionless replicating portfolio of $h^{n}(S^{n}_{1})$ , where S n is a fictitious stock with enlarged local volatility dynamics. In the market with frictions, a suitable modification of this portfolio strategy provides a terminal wealth that converges in $\mathbb{L}^{2}$ to the claim h(S 1) as n goes to infinity. In terms of order book shapes, the exhibited replicating strategy only depends on the size 2G′(0) of the bid–ask spread. The main innovation of the paper is the introduction of a “Leland-type” strategy for nonvanishing (nonlinear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, which we overcome by using an innovative approach based on the Malliavin calculus representation of the Greeks.

Portfolio optimization with insider’s initial information and counterparty risk

Abstract: We study the gain of an insider having private information which concerns the default risk of a counterparty. More precisely, the default time τ is modelled as the first time a stochastic process hits a random threshold L. The insider knows this threshold (as it can be the case for the manager of the counterparty) and this information is modelled by using an initial enlargement of filtration. The standard investors only observe the value of the threshold at the default time and estimate the default event by its conditional density process. The financial market consists of a risk-free asset and a risky asset whose price is exposed to a sudden jump at the default time of the counterparty. All investors aim to maximize the expected utility from terminal wealth given their own information at the initial date. We solve the optimization problem under short-selling and buying constraints and we compare through numerical illustrations the optimal processes for the insider and the standard investors.

Addendum to: Multilevel dual approach for pricing American style derivatives

Abstract: In this note, we show how the dual approach in its particular form presented in Andersen and Broadie (Manag. Sci. 50:1222–1234, 2004) can be fitted into the framework of the recent work (Belomestny et al., Finance Stoch. 17:717–742, 2013).

Static hedging under maturity mismatch

Abstract: Can shorter maturity European options be statically hedged with longer maturity plain vanilla options? This problem appears, for example, when analysing options on forwards in relation to liquid options on the spot underlying. Under mild assumptions on the underlying security price process and the option’s payoff function, we show that approximate static hedges exist and we provide a recipe for constructing them. Examples illustrate the power of the hedge and its sensitivity to modelling assumptions. The results can be extended to formulating semi-static hedging strategies for discretely monitored path-dependent contingent claims.