1. 15. 05. 2007 Optimal Algorithms for k-Search with Application in Option Pricing Julian Lorenz, Konstantinos Panagiotou, Angelika Steger Institute of Theoretical Computer Science, ETH Zürich

2. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 2 Competitive analysis: (MIN cost) Online Problem k-Search (1/2) k-max-search: k-min-search: Prices = ( p 1,…,p n ) presented sequentially Must decide immediately whether or not to buy/sell for p i Player wants to sell k units for MAX profit Player wants to buy k units for MIN cost 5$ 9$ 4$ 1$ (MAX profit)

3. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 3 Online Problem k-Search (2/2) Model for price sequences: p i [m,M arbitrary in that trading range M = m fluctuation ratio > 1 Can buy/sell only one unit for each p i Length of known in advance m M i

4. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 4 Our Results optimal deterministic algorithms lower bounds for competitive ratio of rand. algorithms Natural application of k-search: optimal randomized algorithms (up to a constant) For both, k-max-search and k-min-search we give Asymmetry between k-max-search and k-min-search! Robust bounds for price of Lookback Options

5. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 5 Related Literature El-Yaniv, Fiat, Karp, Turpin (2001): (=1-max-search) One-Way-Trading: Can trade arbitrary fractions for each p i Other related problems: Search problems with distributional assumption on prices Secretary problems Optimal deterministic One-Way-Trading: Optimal algorithm Optimal randomized & no improvement by randomization Timeseries-Search:

6. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 6 Deterministic Search Algorithms

7. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 7 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.

8. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 8 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

9. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 9 Theorem: Deterministic K-Min-Search RPP with solution of where i) Optimal RPP with competitive ratio ii) Optimal deterministic online algorithm for k-min-search Remarks: Asymptotics:

10. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 10 Randomized Search Algorithms

11. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 11 Randomized k-Max-Search Competitive ratio (El-Yaniv et. al., 2001). random, set RP to. Consider k=1: Optimal deterministic RPP has. Randomized algorithm EXPO: Fix base. We can prove: In fact, asymptotically optimal. Choose uniformly at

12. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 12 Theorem: Randomized K-Max-Search For any randomized k-max-search algorithm RALG, the competitive ratio satisfies 1) Independent of k Remarks: 2) Algorithm EXPO k achieves 3) Small k significant improvement! ( ) Set all k reservation prices to. EXPO k :

13. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 13 Theorem: Randomized K-Min-Search For any randomized k-min-search algorithm RALG, the competitive ratio satisfies 1) Again independent of k Remarks: 2) No improvement over deterministic ALG possible ! Recall CR of RPP for k-minsearch

14. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 14 Yaos Principle (mincost online problems) Finitely many possible inputs Set of deterministic algorithms RALG any randomized algorithm f( ) any fixed probability distribution on With respect to f( ) ! Then: Best deterministic algorithm for fixed input distribution Lower bound for best randomized algorithm

15. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 15 ALG 1 buys at ALG 2 rejects, hoping that next quote is On the Proof of Lower Bound For k-min-search, k=1: f( ) uniform distribution on Essentially only two deterministic algorithms: Similarly for arbitrary k, and for k-max-search …

16. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 16 Application To Option Pricing

17. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 17 Application: Pricing of Lookback Options Two examples of options (there are all kinds of them…): 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) Option price (premium) paid to the option writer at time of issuance. Fair Price of a Lookback Option?

18. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 18 Classical Option Pricing: Black Scholes Model assumption for stock price evolution Geometric Brownian Motion: No-Arbitrage and pricing by replication: Trading algorithm (hedging) for option writer to meet obligation in all possible scenarios. Riskless Replication Hedging cost must be option price. Otherwise: Arbitrage (free lunch). No-Arbitrage Assumption ( efficient markets)

19. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 19 Drawback of Classical Option Pricing What if Black Scholes model assumptions no good? price geometric Brownian motion trading not continuous … DeMarzo, Kremer, Mansour (STOC06): Bounds for European options using competitive trading algorithms In fact, in reality Weaker model assumptions Robust bounds for option price

20. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 20 Bound for Price of Lookback Call Instead of GBM assumption: Trading range Discrete-time trading Use k-min-search algorithm! Robust bound for option price, qualitatively and quantitatively similar to Black Scholes price Under no-arbitrage assumption V = price of lookback call on k shares Hedging lookback call = buying close to min in [0,T] Hedging cost = comp. ratio of k-minsearch = option price

21. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 21 Comparison to Black-Scholes model Expected trading range for Geometric Brownian Motion Option price in BS model and our upper bound Goldman, Sosin, Gatto (79): S 0 = 20, 0.2, k = 100S 0 = 20, 0.2 (Assuming zero interest rate) Stock price volatility

22. 09.10.2007 Julian Lorenz, jlorenz@inf.ethz.ch 22 Thank you very much for your attention! Questions?