Recent Advances in Financial Engineering 2012

The following sections are included:

• Preface
• International Workshop on Finance 2012
• Program
• Contents

We analyse the forward price dynamics for contracts written on a spot following a continuous-time autoregressive dynamics. Prime examples of such spots could be power or freight rates, or weather variables like temperature and wind speed. It is shown that the forward price evolves according to template term structure functions, which are scaled by the deseasonalized spot and its derivatives. These template term structure functions can be expressed as a series of exponentially decaying functions with rates given by the (real parts) of the eigenvalues of the autoregressive dynamics. Moreover, the continuous-time autoregressive spot dynamics is differentiable up to an order less than the autoregressive order, and this is precisely the derivatives needed in the representation. The template term structures may produce humps in the forward curve. We consider several empirical examples for illustration based on a model relevant for the temperature market. A particular result of our analysis is that the paths of the forward price are non-differentiable, although the underlying spot is smooth. Our results offer insight into the dynamics of forward and futures prices for contracts in the markets for weather, shipping and power.

We consider a bottom-up Markovian copula model of portfolio credit risk where instantaneous contagion is possible in the form of simultaneous defaults. Due to the Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a two-steps procedure, much like in a standard static copula set-up. In this sense this model solves the bottom-up top-down puzzle which the CDO industry had been trying to do for a long time. It can be applied to any dynamic credit issue like consistent valuation and hedging of CDSs, CDOs and counterparty risk on credit portfolios.

In this paper, we prove that the conditional dependence structure of default times in the Markov model of [4] belongs to the class of Marshall-Olkin copulas. This allows us to derive a factor representation in terms of “common-shocks”, the latter beeing able to trigger simultaneous defaults in some pre-specified groups of obligors. This representation depends on the current default state of the credit portfolio so that fast convolution pricing schemes can be exploited for pricing and hedging credit portfolio derivatives. As emphasized in [4], the innovative breakthrough of this dynamic bottom-up model is a suitable decoupling property between the dependence structure and the default marginals as in [9] (like in static copula models but here in a full-flesh dynamic “Markov copula” model). Given the fast deterministic pricing schemes of the present paper, the model can then be jointly calibrated to single-name and portfolio data in two steps, as opposed to a global joint optimization procedures involving all the model parameters at the same time which would be untractable numerically. We illustrate this numerically by results of calibration against market data from CDO tranches as well as individual CDS spreads. We also discuss hedging sensitivities computed in the models thus calibrated.

In this note we link the Kusuoka limit theorem on super replication of European contingent claim under transaction costs with the multi-asset mainstream and we study the asymptotic behavior of the hedging sets in the context of topological convergence of subsets of ℝ².

In this paper, we study an optimal execution problem in the case of uncertainty in market impact to derive a more realistic market model. Our model is a generalized version of that in [6], where a model of optimal execution with deterministic market impact was formulated. First, we construct a discrete-time model as a value function of an optimal execution problem. We express the market impact function as a product of a deterministic part (an increasing function with respect to the trader's execution volume) and a noise part (a positive random variable). Then, we derive a continuous-time model as a limit of a discrete-time value function. We find that the continuous-time value function is characterized by an optimal control problem with a Lévy process and investigate some of its properties, which are mathematical generalizations of the results in [6]. We also consider a typical example of the execution problem for a risk-neutral trader under log-linear/quadratic market impact with Gamma-distributed noise.

This paper examines the optimal investment threshold (timing) and volume strategies under debt borrowing constraints on the condition that an increase in investment volume increases cash inflow. As debt borrowing limits increase, the firm is more likely to prefer market debt to bank debt. Although debt borrowing constraints distort the investment threshold, they have no influence on the optimal investment volume. Debt borrowing constraints have the possibility of distorting the interactions between equity values and the investment threshold.

Fractional Brownian motion (fBm) was first introduced within a Hilbert space framework by Kolmogorov [1], and further studied and coined the name ‘fractional Brownian motion’ in the 1968 paper by Mandelbrotand Van Ness [2]…

In this article, we provide an alternative approach, based on mean-field theory, on establishing the pre-commitment optimal solution to the continuous time mean-variance portfolio selection problem. By considering the probability distribution of the underlying wealth process and introducing a suitable adjoint function, we show that the optimal control (pre-commitment solution) satisfies a HJB equation that also involves the probability distribution of the corresponding optimal wealth process, which naturally satisfies Kolmogorov's Forward Equation. For if the probability density function exists, it will also satisfy a Fokker-Planck (Kolmogorov's Backward Equation), and hence together with the mentioned HJB, the whole problem becomes a special problem in the theory of mean-field games. Solving by Ansatz, we further obtain the explicit optimal asset allocation, which coincides with that first obtained in the work of Li and Zhou (2000) [18].