We propose a macroeconomic model in which adverse selection in investment amplifies macroeconomic fluctuations, in line with the prominent role played by the credit crunch during the financial crisis. Endogenous lending standards emerge due to an informational asymmetry between borrowers and lenders about the riskiness of borrowers. By using loan approval probability as a screening device, banks ration credit following increases in lending risk, generating large endogenous movements in TFP, explaining why productivity often falls during crises. Furthermore, the mechanism implies that financial instability is heightened when interest rates are low.

In this paper, we study carbon emission trading whose market is gaining in popularity as a policy instrument for global climate change. The mathematical model is presented for pricing options on CO2 emission allowance futures with jump diffusion processes, and a so-called fitted finite volume method is proposed to solve the pricing model for the spatial discretization, in which the Crank-Nicolson is employed for time stepping. In addition, the stability and the convergence of the fully discrete scheme are given, and some numerical results, which are compared with the closed form solution and the Monte Carlo simulation solution, are provided to demonstrate the rates of convergence and the robustness of the numerical method.

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

We consider an investment problem where observing and trading are only possible at random times. In addition, we introduce drawdown constraints which require that the investor's wealth does not fall under a prior fixed percentage of its running maximum. The financial market consists of a riskless bond and a stock which is driven by a Lévy process. Moreover, a general utility function is assumed. In this setting we solve the investment problem using a related limsup Markov decision process. We show that the value function can be characterized as the unique fixed point of the Bellman equation and verify the existence of an optimal stationary policy. Under some mild assumptions the value function can be approximated by the value function of a contracting Markov decision process. We are able to use Howard's policy improvement algorithm for computing the value function as well as an optimal policy. These results are illustrated in a numerical example.

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

In this article we investigate the minimal entropy martingale measure for continuous-time Markov chains. The conditions for absence of arbitrage and existence of the minimal entropy martingale measure are discussed. Under this measure, expressions for the transition intensities are obtained. Differential equations for the arbitrage-free price are derived.

Consider a general bivariate Lévy-driven risk model. The surplus process Y, starting with Y0=x > 0, evolves according to dYt= Yt- dRt -dPt for t > 0, where P and R are two independent Lévy processes respectively representing a loss process in a world without economic factors and a process describing the return on investments in real terms. Motivated by a conjecture of Paulsen, we study the finite-time and infinite-time ruin probabilities for the case in which the loss process P has a Lévy measure of extended regular variation and the stochastic exponential of R fulfills a moment condition. We obtain a simple and unified asymptotic formula as x→∞, which confirms Paulsen's conjecture.

In this paper we propose a class of financial market models which are based on telegraph processes with alternating tendencies and jumps. It is assumed that the jumps have random sizes and that they occur when the tendencies are switching. These models are typically incomplete, but the set of equivalent martingale measures can be described in detail. We provide additional suggestions which permit arbitrage-free option prices as well as hedging strategies to be obtained.

A Hawkes process is also known under the name of a self-exciting point process and has numerous applications throughout science and engineering. We derive the statistical estimation (maximum likelihood estimation) and goodness-of-fit (mainly graphical) for multivariate Hawkes processes with possibly dependent marks. As an application, we analyze two data sets from finance.

For a bivariate Lévy process (ξt,ηt)t≥ 0 and initial value V0 define the generalised Ornstein–Uhlenbeck (GOU) process Vt:=eξt (V0+∫t0 e-ξs-dηs), t≥0, and the associated stochastic integral process Zt:=∫0t e-ξs-dηs, t≥0. Let Tz:=inf{t>0: Vt<0|V0=z} and ψ(z):=P(Tz<∞) for z≥0 be the ruin time and infinite horizon ruin probability of the GOU process. Our results extend previous work of Nyrhinen (2001) and others to give asymptotic estimates for ψ(z) and the distribution of Tz as z→∞, under very general, easily checkable, assumptions, when ξ satisfies a Cramér condition.

In this paper we study the excursion time of a Brownian motion with drift outside a corridor by using a four-state semi-Markov model. In mathematical finance, these results have an important application in the valuation of double-barrier Parisian options. We subsequently obtain an explicit expression for the Laplace transform of its price.

Within an anticipative stochastic calculus framework, we study a market game with asymmetric information and feedback effects. We derive necessary and sufficient criteria for the existence of Nash equilibria and study how general welfare is affected by the level of information. In particular, we show that, under certain conditions in a competitive environment, an increased level of information may in fact lower the level of general welfare, leading to the so-called Hirshleifer effect (see Hirshleifer (1971)). Finally, we determine equilibrium prices for particular pieces of information, by extending our market game with a pre-stage, in which information is traded.

We show that shortfall risks of American options in a sequence of multinomial approximations of the multidimensional Black-Scholes (BS) market converge to the corresponding quantities for similar American options in the multidimensional BS market with path-dependent payoffs. In comparison to previous papers we consider the multiassets case for which we use the weak convergence approach.

We present conditions that imply the conditional full support (CFS) property, introduced in Guasoni, Rásonyi and Schachermayer (2008), for processes Z := H + ∫K dW, where W is a Brownian motion, H is a continuous process, and processes H and K are either progressive or independent of W. Moreover, in the latter case, under an additional assumption that K is of finite variation, we present conditions under which Z has CFS also when W is replaced with a general continuous process with CFS. As applications of these results, we show that several stochastic volatility models and the solutions of certain stochastic differential equations have CFS.

We study the optimal liquidation strategy for a call spread in the case when an investor, who does not hedge, believes in a volatility that differs from the implied volatility. The liquidation problem is formulated as an optimal stopping problem, which we solve explicitly. We also provide a sensitivity analysis with respect to the model parameters.

We consider variance-optimal hedging in general continuous-time affine stochastic volatility models. The optimal hedge and the associated hedging error are determined semiexplicitly in the case that the stock price follows a martingale. The integral representation of the solution opens the door to efficient numerical computation. The setup includes models with jumps in the stock price and in the activity process. It also allows for correlation between volatility and stock price movements. Concrete parametric models will be illustrated in a forthcoming paper.

We introduce a class of stock models that interpolates between exponential Lévy models based on Brownian subordination and certain stochastic volatility models with Lévy-driven volatility, such as the Barndorff-Nielsen–Shephard model. The driving process in our model is a Brownian motion subordinated to a business time which is obtained by convolution of a Lévy subordinator with a deterministic kernel. We motivate several choices of the kernel that lead to volatility clusters while maintaining the sudden extreme movements of the stock. Moreover, we discuss some statistical and path properties of the models, prove absence of arbitrage and incompleteness, and explain how to price vanilla options by simulation and fast Fourier transform methods.

In this paper we first consider the expectation of the total discounted claim costs up to the time of ruin, and then, more generally, we study the expectation of the total discounted operating costs up to the time of default, which is the first passage time of a surplus process downcrossing a given level. These two quantities include the expected discounted penalty function at ruin or the Gerber–Shiu function, the expected total discounted dividends up to ruin, and other interesting quantities as special cases among a class of risk processes. As an illustration, we consider a piecewise-deterministic compound Poisson risk model. This model recovers many risk models appearing in the literature such as the compound Poisson risk models with interest, absolute ruin, dividends, multiple thresholds, and their dual models. We derive and solve the integro-differential equation for the expected present value of the total discounted operating costs up to default. The solutions to the expected present value of the total discounted operating costs up to default can be used as a unified approach to solving many ruin-related quantities. As applications, we derive explicit solutions for the expected accumulated utility up to ruin, the absolute ruin probability with varying borrowing rates, the expected total discounted claim costs up to ruin, the Gerber–Shiu function with two-sided jumps, and the price for a perpetual American put option with two-sided jumps.

By expressing the discounted net loss process as a randomly weighted sum, we investigate the finite-time ruin probabilities for the Poisson risk model with an exponential Lévy process investment return and heavy-tailed claims. It is found that in finite time, however, the extreme of insurance risk dominates the extreme of financial risk, but, for the case of dangerous investment (see Klüppelberg and Kostadinova (2008) for an accurate definition of dangerous investment), the extreme of financial risk has more and more of an effect on the total risk, and as time passes, the extreme of financial risk finally dominates the extreme of insurance risk.

We propose a new approach to utilities in (state) complete markets that is consistent with state-dependent utilities. Full solutions of the optimal consumption and portfolio problem are obtained in a very general setting which includes several functional forms for utilities used in the current literature, and consider general restrictions on allowable wealths. As a secondary result, we obtain a suitable representation for straightforward numerical computations of the optimal consumption and investment strategies. In our model, utilities reflect the level of consumption satisfaction of flows of cash in future times as they are (uniquely) valued by the market when the economic agents are making their consumption and investment decisions. The theoretical framework used for the model is the one proposed in Londoño (2008). We develop the martingale methodology for the solution of the problem of optimal consumption and investment in this setting.