Optimal Algorithms for k-Search with
Application in Option Pricing
Julian Lorenz, Konstantinos Panagiotou, Angelika Steger
Institute of Theoretical Computer Science, ETH Zürich
15. 05. 2007
Online Problem k-Search (1/2)
4$ 9$
5$
1$
k-max-search:
k-min-search:
Player wants to sell k units for MAX profit
Player wants to buy kunits for MIN cost
• Prices  =(p1,…,pn) presented sequentially
• Must decide immediately whether or not to buy/sell for pi
• Competitive analysis:
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(MAX
profit)
(MIN cost)
Julian Lorenz, jlorenz@inf.ethz.ch
2
Online Problem k-Search (2/2)
Model for price sequences:
M
 pi  [m,M],arbitrary in that
trading range
 M = m j,fluctuation ratio j > 1
 Can buy/sell only one unit for
each pi
 Length of  known in advance
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Julian Lorenz, jlorenz@inf.ethz.ch
m
i
3
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
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Julian Lorenz, jlorenz@inf.ethz.ch
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Deterministic Search Algorithms
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Julian Lorenz, jlorenz@inf.ethz.ch
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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.
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Julian Lorenz, jlorenz@inf.ethz.ch
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Theorem: Deterministic K-Max-Search
50
RPP with
where
45
40
solution of
35
30
p
i
25
20
15
i) Optimal RPP with competitive ratio
10
5
0
5
10
ii) Optimal deterministic online algorithm for k-max-search
i
Remarks:
1) Asymptotics:
2) “Bridging“ Timeseries-Search and One-Way-Trading
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Julian Lorenz, jlorenz@inf.ethz.ch
8
15
Theorem: Deterministic K-Min-Search
50
RPP with
where
45
40
solution of
35
30
pi
25
20
15
i) Optimal RPP with competitive ratio
10
5
0
5
10
ii) Optimal deterministic online algorithm for k-min-search
i
Remarks:
Asymptotics:
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Julian Lorenz, jlorenz@inf.ethz.ch
9
15
Randomized Search Algorithms
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Julian Lorenz, jlorenz@inf.ethz.ch
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Randomized k-Max-Search
Consider k=1: Optimal deterministic RPP has
.
Randomized algorithm EXPO:
Fix base
. Choose
random, set RP to
.
Competitive ratio
uniformly at
(El-Yaniv et. al., 2001).
We can prove: In fact, asymptotically optimal.
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Julian Lorenz, jlorenz@inf.ethz.ch
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Theorem: Randomized K-Max-Search
For any randomized k-max-search algorithm RALG,
the competitive ratio satisfies
Remarks:
1) Independent of k
2) Algorithm EXPOk achieves
EXPOk:
Set all k reservation prices to
3) Small k
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significant improvement! (
Julian Lorenz, jlorenz@inf.ethz.ch
.
)
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
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Julian Lorenz, jlorenz@inf.ethz.ch
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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() !
Best deterministic algorithm
for fixed input distribution
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Julian Lorenz, jlorenz@inf.ethz.ch
Lower bound for best
randomized algorithm
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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 …
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Julian Lorenz, jlorenz@inf.ethz.ch
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Application To Option Pricing
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Julian Lorenz, jlorenz@inf.ethz.ch
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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?
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Julian Lorenz, jlorenz@inf.ethz.ch
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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“).
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Julian Lorenz, jlorenz@inf.ethz.ch
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Drawback of Classical Option Pricing
What if Black Scholes model assumptions no good?
In fact, in reality
 price  geometric Brownian motion
 trading not continuous
 …
DeMarzo, Kremer, Mansour (STOC’06):
Bounds for European options using competitive
trading algorithms
Weaker model assumptions
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Julian Lorenz, jlorenz@inf.ethz.ch
„Robust“ bounds for
option price
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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
Instead of GBM assumption: • Trading range
• Discrete-time trading
Robust bound for option price, qualitatively and
quantitatively similar to Black Scholes price
V = price of lookback call on k shares
Under no-arbitrage assumption
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Julian Lorenz, jlorenz@inf.ethz.ch
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Thank you very much
for your attention!
Questions?
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Julian Lorenz, jlorenz@inf.ethz.ch
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