SCIENTIFIC PROGRAMS AND ACTIVITIES

Actuarial Science and Mathematical Finance Group Meetings 2010-11
at the Fields Institute

We develop an efficient Monte Carlo method for the valuation of a financial contract with payoff dependent on discretely realized variance. We assume a general model in which asset returns are random shocks modulated by a stochastic volatility process. Realized variance is the sum of squared daily returns, depending on the sequence of shocks to the asset and the realized path of the volatility process. The price of interest is the expected payoff, represented as a high dimensional integral over the fundamental sources of randomness. We compute it through the combination of deterministic integration over a two dimensional manifold defined by the sum of squared shocks to the asset and the path average of the modulating variance process, followed by exact conditional Monte Carlo sampling. The deterministic integration variables capture most of the variability in realized variance therefore the residual variance in our estimator is much smaller than that in standard Monte Carlo. We derive theoretical results that quantify the variance reduction achieved by the method. We test it for the Hull-White, Heston, and Double Exponential models and show that the algorithm performs significantly better than standard Monte Carlo for realistic computational budgets.

(joint work with Leonardo Vicchi, Center of Applied Mathematics, Universidad Nacional de San Martin, San Martin, Argentina )

This is joint work with Georges Dionne and Olfa Maalaoui Chun

Stochastic dynamic programming approaches for the valuation of natural gas storage, and the determination of the optimal continuous-time injection/withdrawal strategy, give rise to HJB P(I)DEs which are typically solved using finite differences [Thompson et. al., 2009]. A semi-Lagrangian discretization was analyzed by [Chen and Forsyth, 2007], who demonstrated first-order convergence to the viscosity solution.

This talk will show how a semi-Lagrangian approach for such problems can be formulated in such a way that it generates a second-order accurate discretization in time. Combined with a hybrid Fourier/finite difference discretization in the remaining dimensions, the resulting method can provide efficiency gains over existing approaches.

Bruno Rémillard (Department of Management Sciences, HEC Montreal)
Optimal hedging in discrete and continuous time

In this talk I will do a quick survey of the literature on mean-variance hedging in discrete and continuous time. Then I will find the optimal solution of the hedging problem in continuous time when the underlying assets are modeled by a regime-switching geometric Lévy process or a stochastic volatility model. It is also shown that the continuous time solution can be approximated by discrete time Markov models processes. In some cases, the optimal prices corresponds to prices under an equivalent martingale measure, making that measure a natural choice for pricing. However, even if the optimal hedging strategy is not the usual delta hedging, it can be easily computed by Monte Carlo methods.

In the wake of the current financial crisis, there is an ongoing debate on the importance of managing systemic risk in the financial sector. Much of the conventional regulation focuses on the safeguarding of the solvency of individual firms. The recent crisis has highlighted the importance of systemic risk and the shortcomings of pure firm specific regulation. This paper proposes the use of the Co Conditional Tail Expectation(CoCTE) to measure systemic risk since adapted from CoVaR by Adrian and Brunnermeier. We explain how CoCTE can be used in constructing a fund to protect the financial sector in times of severe crises. The second goal of this paper is to endogenize the pro-cyclicality of capital requirements. Using a regime switching model we show how to determine counter-cyclical risk charge for systemic insurance fund.

This is joint work with Phelim Boyle

Carole Bernard (Department of Statistics and Actuarial Science, University of Waterloo)
Explicit Representation of Cost-Efficient Strategies: Suboptimality of path-dependent strategies

This paper uses the preference free framework proposed by Dybvig (1988) and Cox and Leland (1982,2000) to analyze dynamic portfolio strategies. In general there will be a set of dynamic strategies that have the same payoff distribution. We are able to characterize a lowest cost strategy (a “cost-efficient” strategy) and to give an explicit representation of it. As an application, for any given path-dependent strategy, we show how to construct a financial derivative that dominates in the sense of first-order stochastic dominance. We provide new cost-efficient strategies with the same payoff distributions as some well-known option contracts and this enables us to compute the relative efficiency of these standard contracts. We illustrate the strong connections between cost-efficiency and stochastic dominance.

Pascale Valery (Department of Finance, HEC Montreal)
Wald-type tests when rank conditions fail: a smooth regularization approach
Abstract