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

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)


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


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:


Deterministic Search Algorithms


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.


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


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


Randomized Search Algorithms


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


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:


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


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


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 …


Application To Option Pricing


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?


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“)


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


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


Thank you very much for your attention!

Questions?

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

Documents

maxsearch slide

search problems

maxsearch rpp

randomized search algorithms

maxsearch remarks

maxsearch algorithm

maxsearch competitive

option pricing julian