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Author: Bender, Christian ×

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    Dual representations for general multiple stopping problems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Bender, Christian; Schoenmakers, John G.M.; Zhang, Jianing
    In this paper, we study the dual representation for generalized multiple stopping problems, hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by integer valued adapted processes and refraction periods modeled by stopping times. As such, this extends the works by Schoenmakers [2010], Bender [2011a], Bender [2011b], Aleksandrov and Hambly [2010] and Meinshausen and Hambly [2004] on multiple exercise options, which either take into consideration a refraction period or volume constraints, but not both simultaneously. We also allow more flexible cashflow structures than the additive structure in the above references. For example some exponential utility problems are covered by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm for constructing confidence intervals for the price of multiple exercise options and exemplify it by a numerical study on the pricing of a swing option in an electricity market.
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    No-arbitrage pricing beyond semimartingales
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Bender, Christian; Sottinen, Tommi; Valkeila, Esko
    We show how no-arbitrage pricing can be extended to some non-semimartingale models by restricting the class of admissible strategies. However, this restricted class is big enough to cover hedges for relevant options. Moreover, we show that the hedging prices depend essentially only on a path property of the stock price process, viz. on the quadratic variation. As a consequence we can incorporate many stylized facts to a pricing model without changing the option prices.
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    Time discretization and Markovian iteration for coupled FBSDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Bender, Christian; Zhang, Jianfeng
    In this paper we lay the foundation for a numerical algorithm to simulate high-dimensional coupled FBSDEs under weak coupling or monotonicity conditions. In particular we prove convergence of a time discretization and a Markovian iteration. The iteration differs from standard Picard iterations for FBSDEs in that the dimension of the underlying Markovian process does not increase with the number of iterations. This feature seems to be indispensable for an efficient iterative scheme from a numerical point of view. We finally suggest a fully explicit numerical algorithm and present some numerical examples with up to 10-dimensional state space.
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    True upper bounds for Bermudan products via non-nested Monte Carlo
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Belomestny, Denis; Bender, Christian; Schoenmakers, John G.M.
    We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic ``true'' stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale may be employed for computing dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Anderson & Broadie (2004)) and nested consumption based (Belomestny & Milstein (2006)) methods . Numerical experiments indicate the efficiency of the non-nested Monte Carlo algorithm and the variance reduced nested one.
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    Solving optimal stopping problems via randomization and empirical dual optimization
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Belomestny, Denis; Bender, Christian; Schoenmakers, John G. M.
    In this paper we consider optimal stopping problems in their dual form. In this way we reformulate the optimal stopping problem as a problem of stochastic average approximation (SAA) which can be solved via linear programming. By randomizing the initial value of the underlying process, we enforce solutions with zero variance while preserving the linear programming structure of the problem. A careful analysis of the randomized SAA algorithm shows that it enjoys favorable properties such as faster convergence rates and reduced complexity as compared to the non randomized procedure. We illustrate the performance of our algorithm on several benchmark examples.