15. 05. 2007 optimal algorithms for k-search with application in option pricing julian lorenz,...
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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
209.10.2007 Julian Lorenz, [email protected]
• Competitive analysis:(MIN cost)
Online Problem k-Search (1/2)
k-max-search:k-min-search:
• Prices =(p1,…,pn) presented sequentially
• Must decide immediately whether or not to buy/sell for pi
Player wants to sell k units for MAX profit
Player wants to buy kunits for MIN cost
5$9$4$
1$
(MAX profit)
309.10.2007 Julian Lorenz, [email protected]
Online Problem k-Search (2/2)
Model for price sequences:
pi [m,Marbitrary in that trading range
M = m fluctuation ratio > 1
Can buy/sell only one unit for each pi
Length of known in advance
m
M
i
509.10.2007 Julian Lorenz, [email protected]
Related LiteratureEl-Yaniv, Fiat, Karp, Turpin (2001):
(=1-max-search)
One-Way-Trading: Can trade arbitrary fractions for each pi
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:
709.10.2007 Julian Lorenz, [email protected]
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.
809.10.2007 Julian Lorenz, [email protected]
Theorem: Deterministic K-Max-Search
RPP withsolution ofwhere
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
0 5 10 15
5
10
15
20
25
30
35
40
45
50
i
pi
909.10.2007 Julian Lorenz, [email protected]
Theorem: Deterministic K-Min-Search
RPP with
solution ofwhere
i) Optimal RPP with competitive ratio
ii) Optimal deterministic online algorithm for k-min-search
Remarks:
Asymptotics:
0 5 10 15
5
10
15
20
25
30
35
40
45
50
i
pi
1109.10.2007 Julian Lorenz, [email protected]
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
1209.10.2007 Julian Lorenz, [email protected]
Theorem: Randomized K-Max-Search
For any randomized k-max-search algorithm RALG, the competitive ratio satisfies
1) Independent of k
Remarks:
2) Algorithm EXPOk achieves
3) Small k significant improvement! ( )
Set all k reservation prices to .
EXPOk:
1309.10.2007 Julian Lorenz, [email protected]
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
1409.10.2007 Julian Lorenz, [email protected]
Yao‘s 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
1509.10.2007 Julian Lorenz, [email protected]
ALG1 buys at
ALG2 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 …
1709.10.2007 Julian Lorenz, [email protected]
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?
1809.10.2007 Julian Lorenz, [email protected]
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“)
1909.10.2007 Julian Lorenz, [email protected]
Drawback of Classical Option Pricing
What if Black Scholes model assumptions no good?
price geometric Brownian motion trading not continuous …
DeMarzo, Kremer, Mansour (STOC’06):
Bounds for European options using competitive trading algorithms
In fact, in reality
Weaker model assumptions
„Robust“ bounds for option price
2009.10.2007 Julian Lorenz, [email protected]
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
