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Optimal Algorithms for k-Search with Application in Option Pricing

Optimal Algorithms for k-Search with Application in Option Pricing. Julian Loren z , Konstantinos Panagiotou, Angelika Steger Institute of Theoretical Computer Science, ETH Zürich. 1$. 9$. 5$. 4$. (MIN cost). (MAX profit). Online Problem k-Search (1/2). k-max-search:.

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Optimal Algorithms for k-Search with Application in Option Pricing

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  1. Optimal Algorithms for k-Search with Application in Option Pricing Julian Lorenz, Konstantinos Panagiotou, Angelika Steger Institute of Theoretical Computer Science, ETH Zürich

  2. 1$ 9$ 5$ 4$ (MIN cost) (MAX profit) Online Problem k-Search (1/2) k-max-search: Player wants to sellk³1 units for MAX profit k-min-search: Player wants to buyk³1 units for MIN cost • Prices  =(p1,…,pn)presented sequentially • Must decide immediately whether or not to buy/sell for pi • Competitive analysis:

  3. M m i Online Problem k-Search (2/2) Model for price sequences: • piÎ [m,M], arbitrary in that trading range • M = m j, fluctuation ratioj > 1 • Can buy/sell only one unit for each pi • Length of  known in advance

  4. Related Literature El-Yaniv, Fiat, Karp, Turpin (2001): • Timeseries-Search: Optimal deterministic (=1-max-search) Optimal randomized • One-Way-Trading: Optimal algorithm & no improvement by randomization One-Way-Trading: Can trade arbitrary fractions for each pi Other related problems: • Search problems with distributional assumption on prices • Secretary problems

  5. Deterministic Search Algorithms

  6. Deterministic K-Search: RPP Reservation price policy (RPP) for k-max-search: • Choose • Process sequentially • Accept incoming price if exceeds current • Forced sale of remaining units at end of sequence … and analogously for k-min-search.

  7. 50 45 40 35 30 p i 25 20 15 10 5 0 5 10 15 i Theorem: Deterministic K-Max-Search RPP with solution of where i) Optimal RPP with competitive ratio ii) Optimal deterministic online algorithm for k-max-search Remarks: 1) Asymptotics: 2) “Bridging“ Timeseries-Search and One-Way-Trading

  8. 50 45 40 35 30 p i 25 20 15 10 5 0 5 10 15 i Theorem: Deterministic K-Min-Search RPP with where solution of i) Optimal RPP with competitive ratio ii) Optimal deterministic online algorithm for k-min-search Remarks: Asymptotics:

  9. Randomized Search Algorithms

  10. Randomized k-Max-Search Consider k=1: Optimal deterministic RPP has . Randomized algorithm EXPO: Fix base . Choose uniformly at random, set RP to . Competitive ratio (El-Yaniv et. al., 2001). We can prove: In fact, asymptotically optimal.

  11. Theorem: Randomized K-Max-Search For any randomized k-max-search algorithm RALG, the competitive ratio satisfies Remarks: 1) Independent of k 2) Algorithm EXPOkachieves EXPOk: Set all k reservation prices to . 3) Small k significant improvement! ( )

  12. Theorem: Randomized K-Min-Search For any randomized k-min-search algorithm RALG, the competitive ratio satisfies Remarks: 1) Again independent of k 2) No improvement over deterministic ALG possible ! Recall CR of RPP for k-minsearch

  13. Yao‘s Principle (mincost online problems) • Finitely many possible inputs • Set of deterministic algorithms • RALG any randomized algorithm • f()any fixed probability distribution on Then: With respect to f() ! Lower bound for best randomized algorithm Best deterministic algorithm for fixed input distribution

  14. On the Proof of Lower Bound For k-min-search, k=1: f()uniform distribution on Essentially only two deterministic algorithms: • ALG1 buys at • ALG2 rejects , hoping that next quote is Similarly for arbitrary k, and for k-max-search …

  15. Application To Option Pricing

  16. Application: Pricing of Lookback Options • European Call Option: right to buy shares for prespecified price at future time T from option writer • Lookback Call Option: right to buy at time T for minimum price in [0,T] (i.e. between issuance and expiry) Two examples of options (there are all kinds of them…): Option price (“premium“) paid to the option writer at time of issuance. Fair Price of a Lookback Option?

  17. Classical Option Pricing: Black Scholes • Model assumption for stock price evolution Geometric Brownian Motion: • No-Arbitrage and pricing by “replication“: Riskless Replication Trading algorithm (“hedging“) for option writer to meet obligation in all possible scenarios. No-Arbitrage Assumption (“efficient markets“) “Hedging cost“ must be option price. Otherwise: Arbitrage (“free lunch“).

  18. Drawback of Classical Option Pricing What if Black Scholes model assumptions no good? In fact, in reality • price ¹ geometric Brownian motion • trading not continuous • … DeMarzo, Kremer, Mansour (STOC’06): Bounds for European options using competitive trading algorithms „Robust“ bounds for option price Weaker model assumptions

  19. Robust bound for option price, qualitatively and quantitatively similar to Black Scholes price Bound for Price of Lookback Call Hedging lookback call = buying “close to min“ in [0,T] _ Use k-min-search algorithm! _ Hedging cost = comp. ratio of k-minsearch = option price • Trading range • Discrete-time trading Instead of GBM assumption: V = price of lookback call on k shares Under no-arbitrage assumption

  20. Thank you very much for your attention!Questions?

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