We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on... more We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results.
Pricing of American options can be achieved by solving optimal stopping problems. This in turn ca... more Pricing of American options can be achieved by solving optimal stopping problems. This in turn can be done by computing so-called continuation values, which we represent as regression functions defined by the aid of a cash flow for the next few time periods. We use Monte Carlo to generate data and apply nonparametric least squares regression estimates to estimate the continuation values from these data. The parameters of the regression estimates and of the underlying regression problems are chosen data-dependent. Results concerning consistency and rate of convergence of these estimates are presented, and the resulting pricing of American options is illustrated by the aid of simulated data.
The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [2... more The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [23]. We use the structure of the stable and unstable foliation to equip the geodesic ray boundary of the universal covering with a Hölder structure. Gromov's geodesic rigidity and the Theorem of Dinaburg--Manning on the relation between the topological entropy and the volume entropy are generalized to the case of Finsler manifolds.
We propose a simple and easily implementable model of credit contagion in which we include macro-... more We propose a simple and easily implementable model of credit contagion in which we include macro-and microstructural interdependencies among the debtors within a credit portfolio. We show that even for diversified portfolios, moderate microstructural interdependencies have a significant impact on the tails of the loss distribution. This impact increases dramatically for less diversified microstructures. Since the inclusion of microstructural interdependencies acts on the tails, the choice of an appropriate risk measure for credit risk management is a delicate task.
This paper concerns the pricing of American options with stochastic stopping time constraints exp... more This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff–Schwartz algorithm to solve the stochastic Cauchy–Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.
Journal of Financial and Quantitative Analysis, 2010
This paper performs specification analysis on the term structure of variance swap rates on the S&... more This paper performs specification analysis on the term structure of variance swap rates on the S&P 500 index and studies the optimal investment decision on the variance swaps and the stock index. The analysis identifies 2 stochastic variance risk factors, which govern the short and long end of the variance swap term structure variation, respectively. The highly negative estimate for the market price of variance risk makes it optimal for an investor to take short positions in a short-term variance swap contract, long positions in a long-term variance swap contract, and short positions in the stock index. * Egloff, [email protected], QuantCatalyst,
This paper concerns the pricing of American options with stochastic stopping time constraints exp... more This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We suggest a new Monte Carlo method to numerically calculate the option value also for multidimensional Markov processes. Because of presence of stopping time constraints the classical Longstaff-Schwartz least-square Monte Carlo algorithm or its extension introduced in Egloff (2005) cannot be directly applied. We adapt the Longstaff-Schwartz algorithm to solve the stochastic Cauchy-Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.
We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators a... more We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators are based on weighted samples that are neither independent nor identically distributed. Using a new law of iterated logarithm for martingales, we prove the convergence of the adaptive quantile estimators for general distributions with nonunique quantiles thereby extending the work of Feldman and Tucker [Ann. Math. Statist. 37 (1996) 451--457]. We illustrate the algorithm with an example from credit portfolio risk analysis.
We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on... more We extend the Longstaff-Schwartz algorithm for approximately solving optimal stopping problems on high-dimensional state spaces. We reformulate the optimal stopping problem for Markov processes in discrete time as a generalized statistical learning problem. Within this setup we apply deviation inequalities for suprema of empirical processes to derive consistency criteria, and to estimate the convergence rate and sample complexity. Our results strengthen and extend earlier results.
Pricing of American options can be achieved by solving optimal stopping problems. This in turn ca... more Pricing of American options can be achieved by solving optimal stopping problems. This in turn can be done by computing so-called continuation values, which we represent as regression functions defined by the aid of a cash flow for the next few time periods. We use Monte Carlo to generate data and apply nonparametric least squares regression estimates to estimate the continuation values from these data. The parameters of the regression estimates and of the underlying regression problems are chosen data-dependent. Results concerning consistency and rate of convergence of these estimates are presented, and the resulting pricing of American options is illustrated by the aid of simulated data.
The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [2... more The geodesic flow of a compact Finsler manifold with negative flag curvature is an Anosov flow [23]. We use the structure of the stable and unstable foliation to equip the geodesic ray boundary of the universal covering with a Hölder structure. Gromov's geodesic rigidity and the Theorem of Dinaburg--Manning on the relation between the topological entropy and the volume entropy are generalized to the case of Finsler manifolds.
We propose a simple and easily implementable model of credit contagion in which we include macro-... more We propose a simple and easily implementable model of credit contagion in which we include macro-and microstructural interdependencies among the debtors within a credit portfolio. We show that even for diversified portfolios, moderate microstructural interdependencies have a significant impact on the tails of the loss distribution. This impact increases dramatically for less diversified microstructures. Since the inclusion of microstructural interdependencies acts on the tails, the choice of an appropriate risk measure for credit risk management is a delicate task.
This paper concerns the pricing of American options with stochastic stopping time constraints exp... more This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of Menaldi et al., we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We also provide a Monte Carlo method to numerically calculate the option value for multidimensional Markov processes. We adapt the Longstaff–Schwartz algorithm to solve the stochastic Cauchy–Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.
Journal of Financial and Quantitative Analysis, 2010
This paper performs specification analysis on the term structure of variance swap rates on the S&... more This paper performs specification analysis on the term structure of variance swap rates on the S&P 500 index and studies the optimal investment decision on the variance swaps and the stock index. The analysis identifies 2 stochastic variance risk factors, which govern the short and long end of the variance swap term structure variation, respectively. The highly negative estimate for the market price of variance risk makes it optimal for an investor to take short positions in a short-term variance swap contract, long positions in a long-term variance swap contract, and short positions in the stock index. * Egloff, [email protected], QuantCatalyst,
This paper concerns the pricing of American options with stochastic stopping time constraints exp... more This paper concerns the pricing of American options with stochastic stopping time constraints expressed in terms of the states of a Markov process. Following the ideas of we transform the constrained into an unconstrained optimal stopping problem. The transformation replaces the original payoff by the value of a generalized barrier option. We suggest a new Monte Carlo method to numerically calculate the option value also for multidimensional Markov processes. Because of presence of stopping time constraints the classical Longstaff-Schwartz least-square Monte Carlo algorithm or its extension introduced in Egloff (2005) cannot be directly applied. We adapt the Longstaff-Schwartz algorithm to solve the stochastic Cauchy-Dirichlet problem related to the valuation problem of the barrier option along a set of simulated trajectories of the underlying Markov process.
We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators a... more We introduce new quantile estimators with adaptive importance sampling. The adaptive estimators are based on weighted samples that are neither independent nor identically distributed. Using a new law of iterated logarithm for martingales, we prove the convergence of the adaptive quantile estimators for general distributions with nonunique quantiles thereby extending the work of Feldman and Tucker [Ann. Math. Statist. 37 (1996) 451--457]. We illustrate the algorithm with an example from credit portfolio risk analysis.
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Papers by Daniel Egloff