Oxford-Man Institute

Numerical Approximations of Non-linear Stochastic Systems. Abstract:  The explicit solution of stochastic differential equations (SDEs can be found only in a few cases. Therefore, there is a need fo accurate numerical approximations that could, for example, enabl  Monte Carlo Simulations. Convergence and stability of these methods are well understood for SDEs with Lipschit  continuous coefficients. Our research focuses on those situations wher  the coefficients of the underlying SDEs are non-Lipschitzian  It was demonstrated in the literature,  that in this case using the classical methods we may fail t  obtain numerically computed paths that are accurate for small step-sizes, or to obtain qualitative information about the behaviour of numerical methods over long time intervals. Our work addresses both of these issues, giving a customized analysis of the most widely used numerical methods.

We consider parabolic scalar conservation laws perturbed by a (conservative) noise. Large deviations are investigated in the singular limit of jointly vanishing viscosity and noise. The model is supposed to feature the same behavior of "asymmetric" particles systems (e.g. TASEP) under Euler scaling.

A first large deviations principle is obtained in a space of Young measures. A "second order" large deviations principle is then discussed, including connections with the Jensen and Varadhan functional. As time allows, more recent "long correlation" models will be treated.

We discuss a backward stochastic differential equation, (BSDE), approach to a risk-based, optimal investment problem of an insurer. A simplified continuous-time economy with two investment vehicles, namely, a fixed interest security and a share, is considered.

The insurer's risk process is modeled by a diffusion approximation to a compound Poisson risk process. The goal of the insurer is to select an optimal portfolio so as to minimize the risk described by a convex risk measure of his/her terminal wealth. The optimal investment problem is then formulated as a zero-sum stochastic differential game between the insurer and the market. The BSDE approach is used to solve the game problem. This leads to a simple and natural approach for the existence and uniqueness of an optimal strategy of the game problem without Markov assumptions. Closed-form solutions to the optimal strategies of the insurer and the market are obtained in some particular cases.

We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-

point model and we give the conditions for the existence of f-minimal equivalent martingale measure. Using the connection between utility maximisation and f-divergence minimisation, we obtain a general formula for optimal strategy in change-point case for initially enlarged filtration and also for progressively enlarged filtration when the utility is exponential. We illustrate our results considering the Black-Scholes model with change-point.

Key words and phrases: f-divergence, exponential Levy models, change-point, optimal portfolio

MSC 2010 subject classifications: 60G46, 60G48, 60G51, 91B70

We present three examples of credit products whose valuation poses challenging modeling problems related to armageddon scenarios and extreme losses, analyzing their behaviour pre- and in-crisis.

The products are Credit Index Options (CIOs), Collateralized Debt Obligations (CDOs), and Credit Valuation Adjustment (CVA) related products. We show that poor mathematical treatment of possibly vanishing numeraires in CIOs and lack of modes in the tail of the loss distribution in CDOs may lead to inaccurate valuation, both pre- and especially in crisis. We also consider the limits of copula models in trying to represent systemic risk in credit intensity models. We finally enlarge the picture and comment on a number of common biases in the public perception of modeling in relationship with the crisis.

This paper uses data on house transactions in the state of Massachusetts over the last 20 years

to show that houses sold after foreclosure, or close in time to the death or bankruptcy of at least

one seller, are sold at lower prices than other houses. Foreclosure discounts are particularly large on

average at 27% of the value of a house. The pattern of death-related discounts suggests that they may

result from poor home maintenance by older sellers, while foreclosure discounts appear to be related

to the threat of vandalism in low-priced neighborhoods. After aggregating to the zipcode level and

controlling for regional price trends, the prices of forced sales are mean-reverting, while the prices

of unforced sales are close to a random walk. At the zipcode level, this suggests that unforced sales

take place at approximately ecient prices, while forced-sales prices re

ect time-varying illiquidity in

neighborhood housing markets. At a more local level, however, we nd that foreclosures that take

place within a quarter of a mile, and particularly within a tenth of a mile, of a house lower the price

at which it is sold. Our preferred estimate of this eect is that a foreclosure at a distance of 0.05 miles

lowers the price of a house by about 1%.

Convertible bonds are hybrid securities that embody the characteristics of both straight bonds and equities. The conflict of interests between bondholders and shareholders affects the security prices significantly. In this paper, we investigate how to use a non-zero-sum game framework to model the interaction between bondholders and shareholders and to evaluate the bond accordingly. Mathematically, this problem can be reduced to a system of variational inequalities. We explicitly derive a unique Nash equilibrium to the game.

Our model shows that credit risk and tax benefit have considerable impacts on the optimal strategies of both parties. The shareholder may issue a call when the debt is in-the-money or out-of-the-money. This is consistent with the empirical findings of “late and early calls"

(Ingersoll (1977), Mikkelson (1981), Cowan et al. (1993) and Ederington et al. (1997)). In addition, the optimal call policy under our model offers an explanation for certain stylized patterns related to the returns of company assets and stock on calls.