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

Julian Lorenz, 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)

Julian Lorenz, 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

Julian Lorenz, 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

Julian Lorenz, 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:

Julian Lorenz, 6 Deterministic Search Algorithms

Julian Lorenz, 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.

Julian Lorenz, 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

Julian Lorenz, 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:

Julian Lorenz, 10 Randomized Search Algorithms

Julian Lorenz, 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

Julian Lorenz, 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 :

Julian Lorenz, 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

Julian Lorenz, 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

Julian Lorenz, 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 …

Julian Lorenz, 16 Application To Option Pricing

Julian Lorenz, 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?

Julian Lorenz, 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)

Julian Lorenz, 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

Julian Lorenz, 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

Julian Lorenz, 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

Julian Lorenz, 22 Thank you very much for your attention! Questions?