Optimal Algorithms for k-Search with Application in Option Pricing

Lorenz, Julian; Παναγιώτου, Κωνσταντίνος; Steger, Angelika

doi:10.1007/978-3-540-75520-3_26

Cited by 24 publications

(61 citation statements)

References 14 publications

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“…In this paper, we propose the general k-search problem, which generalizes the k-search problem proposed in [2] by allowing to sell multiple units of the asset at each period and eliminating the assumption of k n. We mainly present an optimal deterministic algorithm (abbr. DET) for the case where k < n, and for the case where k n, we show by numerical computation that the gap between the upper and the lower bound is quite small for many situations.…”

Section: Our Resultsmentioning

confidence: 99%

“…As stated in [2], the k-search problem can be viewed as a natural bridge between the time series search and oneway trading problems considered in [1] such that it is the time series search problem when k = 1 and the one-way trading problem when k → +∞. In [2], the player sells at most one unit of the asset at each period, while in our model, the player is allowed to sell several units of the asset.…”

Section: Problem Descriptionmentioning

confidence: 99%

“…The game ends either one price is accepted or the last period is met. El-Yaniv et al [1] proved that if prices in all periods are Lorenz et al [2] extended the above model to the ksearch problem, that is, a player has totally k 1 units of the asset to be sold within n k periods, and sells at most one unit of the asset at each period. They presented optimal deterministic and asymptotically optimal randomized algorithms for the problem.…”

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Online algorithms for the general k-search problem

Zhang

Zheng

et al. 2011

Information Processing Letters

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Section: Our Resultsmentioning

confidence: 99%

Section: Problem Descriptionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

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Online algorithms for the general k-search problem

Zhang

Zheng

et al. 2011

Information Processing Letters

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“…By assuming that all prices fall in the range [m, M ] and these bounds m and M are known, they gave an algorithm for this problem with competitive ratio O( √ M/m), i.e., the ratio of the highest price and the price accepted by the algorithm is O( √ M/m). In [14], Lorenz et al generalized the 1-max-search problem to the k-max-search problem, in which the objective is to accept the k highest prices. By requiring that the bounds m and M are known, they gave an optimal algorithm for the problem, which has competitive ratio k+1 √ k k (M/m).…”

Section: Previous Resultsmentioning

confidence: 99%

Competitive algorithms for unbounded one-way trading

Chin

Fu²,

Guo

et al. 2015

Theoretical Computer Science

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Abstract. In the one-way trading problem, a seller has some product to be sold to a sequence σ of buyers u1, u2, . . . , uσ arriving online and he needs to decide, for each ui, the amount of product to be sold to ui at the then-prevailing market price pi. The objective is to maximize the seller's revenue. We note that most previous algorithms for the problem need to impose some artificial upper bound M and lower bound m on the market prices, and the seller needs to know either the values of M and m, or their ratio M/m, at the outset. Moreover, the performance guarantees provided by these algorithms depend only on M and m, and are often too loose; for example, given a one-way trading algorithm with competitive ratio Θ(log(M/m)), its actual performance can be significantly better when the actual highest to actual lowest price ratio is significantly smaller than M/m. This paper gives a one-way trading algorithm that does not impose any bounds on market prices and whose performance guarantee depends directly on the input. In particular, we give a class of one-way trading algorithms such that for any positive integer h and any positive number ϵ, we have an algorithm A h,ϵ that has competitive ratio O(log r, the ratio of the highest market price p * = maxi pi and the first price p1, is large and satisfy log (h) r * > 1, where log (i) x denotes the application of the logarithm function i times to x ; otherwise, A h,ϵ has a constant competitive ratio Γ h . We also show that our algorithms are near optimal by showing that given any positive integer h and any one-way trading algorithm A, we can construct a sequence of buyers σ with log (h) r * > 1 such that the ratio between the optimal revenue and the revenue obtained by A is at least Ω(log r * (log (2) r * ) . . . (log (h−1) r * )(log (h) r * )).

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Section: Previous Resultsmentioning

confidence: 99%

Competitive Algorithms for Unbounded One-Way Trading

Chin

Fu²,

Jiang

et al. 2014

Algorithmic Aspects in Information and Management

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In the one-way trading problem, a seller has L units of product to be sold to a sequence σ of buyers u 1 , u 2 , . . . , u σ arriving online and he needs to decide, for each u i , the amount of product to be sold to u i at the then-prevailing market price p i . The objective is to maximize the seller's revenue. We note that all previous algorithms for the problem need to impose some artificial upper bound M and lower bound m on the market prices, and the seller needs to know either the values of M and m, or their ratio M/m, at the outset.This paper gives a one-way trading algorithm that does not impose any bounds on market prices and whose performance guarantee depends directly on the input. In particular, we give a class of one-way trading algorithms such that for any positive integer h and any positive number ϵ, we have an algorithm A h,ϵ that has competitive ratio O(log r * (log (2) r * ) . . . (log (h−1) r * )(log (h) r * ) 1+ϵ ) if the value of r * = p * /p 1 , the ratio of the highest market price p * = max i p i and the first price p 1 , is large and satisfies log (h) r * > 1, where log (i) x denotes the application of the logarithm function i times to x ; otherwise, A h,ϵ has a constant competitive ratio Γ h . We also show that our algorithms are near optimal by showing that given any positive integer h and any one-way trading algorithm A, we can construct a sequence of buyers σ with log (h) r * > 1 such that the ratio between the optimal revenue and the revenue obtained by A is Ω(log r * (log (2) r * ) . . . (log (h−1) r * )(log (h) r * )). A special case of the one-way trading is also studied, in which the L units of product is comprised of L items, each of which must be sold atomically (or equivalently, the amount of product sold to each buyer must be an integer).Furthermore, a complementary problem to the one-way trading problem, say, the oneway buying problem, is studied in this paper. In the one-way buying problem, a buyer wants to purchase one unit of product through a sequence of n sellers v 1 , v 2 , . . . , v n arriving online, and she needs to decide the fraction to purchase from each v i at the then-prevailing market price p i . Her objective is to minimize the cost. The optimal competitive algorithms whose performance guarantees depend only on the lowest market price p * = min i p i , and one of M and φ, the price fluctuation ratio, are presented.

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Optimal Algorithms for k-Search with Application in Option Pricing

Cited by 24 publications

References 14 publications

Online algorithms for the general k-search problem

Online algorithms for the general k-search problem

Competitive algorithms for unbounded one-way trading

Competitive Algorithms for Unbounded One-Way Trading

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