Buy low, sell high Vladimir Vovk Wouter M. Koolen GTP 2014 - - PowerPoint PPT Presentation

Buy low, sell high

Wouter M. Koolen Vladimir Vovk GTP 2014 Guanajuato Mexico Saturday 15th November, 2014

Road map

1

Introduction

2

Intuition 1: Sell high only

3

Intuition 2: Iterated trading strategies

4

Simple counterexample

5

Main result

6

Examples

7

Conclusion

Koolen, Vovk (RHUL) Buy low, sell high 2 / 29

Motivation

“buy low, sell high” as a porcelain-tile pseudo wisdom

Koolen, Vovk (RHUL) Buy low, sell high 3 / 29

Motivation

“buy low, sell high” as a porcelain-tile pseudo wisdom we take it seriously

Koolen, Vovk (RHUL) Buy low, sell high 3 / 29

Motivation

“buy low, sell high” as a porcelain-tile pseudo wisdom we take it seriously

• nline learning style

Koolen, Vovk (RHUL) Buy low, sell high 3 / 29

Motivation

“buy low, sell high” as a porcelain-tile pseudo wisdom we take it seriously

• nline learning style

and uncover its surprisingly intricate theory

Koolen, Vovk (RHUL) Buy low, sell high 3 / 29

share price in € time example price Fortune brands 0.5 1 1.5 2 1999 2000 2001 2002 2003

share price in € time example price Fortune brands 0.5 1 1.5 2 1999 2000 2001 2002 2003

start with 1e

share price in € time example price Fortune brands 0.5 1 1.5 2 1999 2000 2001 2002 2003

start with 1e buy at 0.7e

share price in € time example price Fortune brands 0.5 1 1.5 2 1999 2000 2001 2002 2003

start with 1e buy at 0.7e sell at 2.1e

share price in € time example price Fortune brands 0.5 1 1.5 2 1999 2000 2001 2002 2003

start with 1e buy at 0.7e sell at 2.1e payoff: 2.1

0.7 = 3e

share price in € time example price Fortune brands Altria group 0.5 1 1.5 2 1999 2000 2001 2002 2003

At a glance

“Buy low, sell high” a desirable target Simple: just need low and high trading price Problem: Good trading prices depend on the data.

Koolen, Vovk (RHUL) Buy low, sell high 6 / 29

At a glance

“Buy low, sell high” a desirable target Simple: just need low and high trading price Problem: Good trading prices depend on the data. Idea: Could we learn best low/high prices? I.e. by online trading guarantee payoff as if we knew them?

Koolen, Vovk (RHUL) Buy low, sell high 6 / 29

At a glance

“Buy low, sell high” a desirable target Simple: just need low and high trading price Problem: Good trading prices depend on the data. Idea: Could we learn best low/high prices? I.e. by online trading guarantee payoff as if we knew them? Answer: a firm and crisp almost

Koolen, Vovk (RHUL) Buy low, sell high 6 / 29

At a glance

“Buy low, sell high” a desirable target Simple: just need low and high trading price Problem: Good trading prices depend on the data. Idea: Could we learn best low/high prices? I.e. by online trading guarantee payoff as if we knew them? Answer: a firm and crisp almost Our work: complete characterisation of that “almost”.

Koolen, Vovk (RHUL) Buy low, sell high 6 / 29

Protocol

Initial capital K0 := 1 Initial price ω0 := 1 For day t = 1, 2, . . .

1 Investor takes position St ∈ R 2 Market reveals price ωt ∈ [0, ∞) 3 Capital becomes Kt := Kt−1 + St(ωt − ωt−1) Koolen, Vovk (RHUL) Buy low, sell high 7 / 29

Protocol

Initial capital K0 := 1 Initial price ω0 := 1 For day t = 1, 2, . . .

1 Investor takes position St ∈ R 2 Market reveals price ωt ∈ [0, ∞) 3 Capital becomes Kt := Kt−1 + St(ωt − ωt−1)

A position St < 0 is called short St > 0 is called long St > Kt−1/ωt−1 is called leveraged

Koolen, Vovk (RHUL) Buy low, sell high 7 / 29

Protocol

Initial capital K0 := 1 Initial price ω0 := 1 For day t = 1, 2, . . .

1 Investor takes position St ∈ R 2 Market reveals price ωt ∈ [0, ∞) 3 Capital becomes Kt := Kt−1 + St(ωt − ωt−1)

A position St < 0 is called short St > 0 is called long St > Kt−1/ωt−1 is called leveraged Bankrupt when capital Kt < 0 is negative.

Koolen, Vovk (RHUL) Buy low, sell high 7 / 29

Protocol

Initial capital K0 := 1 Initial price ω0 := 1 For day t = 1, 2, . . .

1 Investor takes position St ∈ R 2 Market reveals price ωt ∈ [0, ∞) 3 Capital becomes Kt := Kt−1 + St(ωt − ωt−1)

A position St < 0 is called short St > 0 is called long St > Kt−1/ωt−1 is called leveraged Bankrupt when capital Kt < 0 is negative. No assumptions about price-generating process. Full information

Koolen, Vovk (RHUL) Buy low, sell high 7 / 29

Goal

We want to become rich when the share price exhibits a large upcrossing ([a, b] is upcrossed when the price drops below a before it exceeds b)

Koolen, Vovk (RHUL) Buy low, sell high 8 / 29

Goal

We want to become rich when the share price exhibits a large upcrossing ([a, b] is upcrossed when the price drops below a before it exceeds b) A financial expert claims to have a secret strategy that will accomplish our

• goal. She shows us a function G, and guarantees to

keep our capital above G(a, b) for all upcrossed intervals [a, b]

Koolen, Vovk (RHUL) Buy low, sell high 8 / 29

Goal

We want to become rich when the share price exhibits a large upcrossing ([a, b] is upcrossed when the price drops below a before it exceeds b) A financial expert claims to have a secret strategy that will accomplish our

• goal. She shows us a function G, and guarantees to

keep our capital above G(a, b) for all upcrossed intervals [a, b] Ideally, G(a, b) is close to b/a.

Koolen, Vovk (RHUL) Buy low, sell high 8 / 29

Goal

We want to become rich when the share price exhibits a large upcrossing ([a, b] is upcrossed when the price drops below a before it exceeds b) A financial expert claims to have a secret strategy that will accomplish our

• goal. She shows us a function G, and guarantees to

keep our capital above G(a, b) for all upcrossed intervals [a, b] Ideally, G(a, b) is close to b/a. We would like to find out: Is guaranteeing G possible? Can more than G be guaranteed? Can we reverse engineer a strategy for G?

Koolen, Vovk (RHUL) Buy low, sell high 8 / 29

Road map

1

Introduction

2

Intuition 1: Sell high only

3

Intuition 2: Iterated trading strategies

4

Simple counterexample

5

Main result

6

Examples

7

Conclusion

Koolen, Vovk (RHUL) Buy low, sell high 9 / 29

Sell high only

We want to become rich when the share price is ever high

Koolen, Vovk (RHUL) Buy low, sell high 10 / 29

Sell high only

We want to become rich when the share price is ever high A financial expert claims to have a secret strategy that will accomplish our

• goal. She shows us a function F, and guarantees to

keep our capital above F(y) for all exceeded price levels y

Koolen, Vovk (RHUL) Buy low, sell high 10 / 29

Sell high only

We want to become rich when the share price is ever high A financial expert claims to have a secret strategy that will accomplish our

• goal. She shows us a function F, and guarantees to

keep our capital above F(y) for all exceeded price levels y Ideally, F(y) is close to y.

Koolen, Vovk (RHUL) Buy low, sell high 10 / 29

Sell high only

We want to become rich when the share price is ever high A financial expert claims to have a secret strategy that will accomplish our

• goal. She shows us a function F, and guarantees to

keep our capital above F(y) for all exceeded price levels y Ideally, F(y) is close to y. We would like to find out: Is guaranteeing F possible? Can more than F be guaranteed? Can we reverse engineer a strategy for F?

Koolen, Vovk (RHUL) Buy low, sell high 10 / 29

Example guarantees F

guaranteed capital maximum price reached F1(y) F2(y) F3(y) 0.5 1 1.5 2 2.5 3 1 3 5 7 9 11 13 15 17 19

Koolen, Vovk (RHUL) Buy low, sell high 11 / 29

Adjuster

A strategy prescribes position St based on the past prices ω0, . . . , ωt−1.

Definition

A function F : [1, ∞) → [0, ∞) is called an adjuster if there is a strategy that guarantees Kt ≥ F

• max

0≤s≤t ωs

• .

An adjuster F is admissible if it is not strictly dominated.

Koolen, Vovk (RHUL) Buy low, sell high 12 / 29

Threshold adjusters

Fix a price level u ≥ 1. The threshold adjuster Fu(y) := u1{y≥u}

Koolen, Vovk (RHUL) Buy low, sell high 13 / 29

Threshold adjusters

Fix a price level u ≥ 1. The threshold adjuster Fu(y) := u1{y≥u} is witnessed by the threshold strategy Su that takes position 1 until the price first exceeds level u. takes position 0 thereafter

Koolen, Vovk (RHUL) Buy low, sell high 13 / 29

The GUT of Adjusters

Consider a right-continuous and increasing candidate guarantee F.

Theorem (Characterisation)

F is an adjuster iff ∞

1

F(y) y2 dy ≤ 1. Moreover, F is admissible iff this holds with equality.

Koolen, Vovk (RHUL) Buy low, sell high 14 / 29

The GUT of Adjusters

Consider a right-continuous and increasing candidate guarantee F.

Theorem (Characterisation)

F is an adjuster iff ∞

1

F(y) y2 dy ≤ 1. Moreover, F is admissible iff this holds with equality.

Theorem (Representation)

F is an adjuster iff there is a probability measure P on [1, ∞) such that F(y) ≤

• Fu(y) dP(u),

again with equality iff F is admissible.

Koolen, Vovk (RHUL) Buy low, sell high 14 / 29

Road map

1

Introduction

2

Intuition 1: Sell high only

3

Intuition 2: Iterated trading strategies

4

Simple counterexample

5

Main result

6

Examples

7

Conclusion

Koolen, Vovk (RHUL) Buy low, sell high 15 / 29

Adjuster

Definition

A price path ω0, . . . , ωt upcrosses interval [a, b] if there are 0 ≤ ta ≤ tb ≤ t s.t. ωta ≤ a and ωtb ≥ b.

Koolen, Vovk (RHUL) Buy low, sell high 16 / 29

Adjuster

Definition

A price path ω0, . . . , ωt upcrosses interval [a, b] if there are 0 ≤ ta ≤ tb ≤ t s.t. ωta ≤ a and ωtb ≥ b.

Definition

A strategy prescribes position St based on the past prices ω0, . . . , ωt−1.

Koolen, Vovk (RHUL) Buy low, sell high 16 / 29

Adjuster

Definition

A price path ω0, . . . , ωt upcrosses interval [a, b] if there are 0 ≤ ta ≤ tb ≤ t s.t. ωta ≤ a and ωtb ≥ b.

Definition

A strategy prescribes position St based on the past prices ω0, . . . , ωt−1.

Definition

A function G : (0, 1] × [0, ∞) → [0, ∞) is called an adjuster if there is a strategy that guarantees Kt ≥ G (a, b) for each [a, b] upcrossed by ω0, . . . , ωt. An adjuster G is admissible if it is not strictly dominated.

Koolen, Vovk (RHUL) Buy low, sell high 16 / 29

Adjuster

Definition

A price path ω0, . . . , ωt upcrosses interval [a, b] if there are 0 ≤ ta ≤ tb ≤ t s.t. ωta ≤ a and ωtb ≥ b.

Definition

A strategy prescribes position St based on the past prices ω0, . . . , ωt−1.

Definition

A function G : (0, 1] × [0, ∞) → [0, ∞) is called an adjuster if there is a strategy that guarantees Kt ≥ G (a, b) for each [a, b] upcrossed by ω0, . . . , ωt. An adjuster G is admissible if it is not strictly dominated. Sneak peak: Ideal G(a, b) = b/a is not an adjuster. But we can get close.

Koolen, Vovk (RHUL) Buy low, sell high 16 / 29

GTP 2010

( t , p ) ^ ^

quick! SELL!

from the future: Incoming message T log price p time t Λ

Koolen, Vovk (RHUL) Buy low, sell high 17 / 29

Sequential Threshold strategies

More of the same Fix price levels α < β. The threshold adjuster Gα,β(a, b) = β α1{a≤α}1{b≥β} is witnessed by the threshold strategy Sα,β that

takes position 0 until the price drops below α takes position 1/α until the price rises above β takes position 0 thereafter

Koolen, Vovk (RHUL) Buy low, sell high 18 / 29

Sequential Threshold strategies

More of the same Fix price levels α < β. The threshold adjuster Gα,β(a, b) = β α1{a≤α}1{b≥β} is witnessed by the threshold strategy Sα,β that

takes position 0 until the price drops below α takes position 1/α until the price rises above β takes position 0 thereafter

Optimal strategies allocate their 1$ to threshold strategies according to some probability measure P(α, β), and hence achieve GP(a, b) =

• Gα,β(a, b) dP(α, β).

Koolen, Vovk (RHUL) Buy low, sell high 18 / 29

Sequential Threshold strategies: Fallacy

More of the same Fix price levels α < β. The threshold adjuster Gα,β(a, b) = β α1{a≤α}1{b≥β} is witnessed by the threshold strategy Sα,β that

takes position 0 until the price drops below α takes position 1/α until the price rises above β takes position 0 thereafter

Optimal strategies allocate their 1$ to threshold strategies according to some probability measure P(α, β), and hence achieve GP(a, b) =

• Gα,β(a, b) dP(α, β).

GP is typically strictly dominated

Koolen, Vovk (RHUL) Buy low, sell high 18 / 29

Road map

1

Introduction

2

Intuition 1: Sell high only

3

Intuition 2: Iterated trading strategies

4

Simple counterexample

5

Main result

6

Examples

7

Conclusion

Koolen, Vovk (RHUL) Buy low, sell high 19 / 29

Mixtures of thresholds are generally dominated

G(a, b) := 1 2G1,2(a, b)+1 2G 1

2 ,1(a, b) = 1{a ≤ 1 and b ≥ 2}+1{a ≤ 1 2 and b ≥ 1}. Koolen, Vovk (RHUL) Buy low, sell high 20 / 29

Mixtures of thresholds are generally dominated

G(a, b) := 1 2G1,2(a, b)+1 2G 1

2 ,1(a, b) = 1{a ≤ 1 and b ≥ 2}+1{a ≤ 1 2 and b ≥ 1}.

1 2 ∞

1 2 1 2

1 2 1 2 1

2, 2

• [1, 2]

1

2, 1

• ⊂

⊂

Koolen, Vovk (RHUL) Buy low, sell high 20 / 29

Mixtures of thresholds are generally dominated

G(a, b) := 1 2G1,2(a, b)+1 2G 1

2 ,1(a, b) = 1{a ≤ 1 and b ≥ 2}+1{a ≤ 1 2 and b ≥ 1}.

1 2

5 4 0

∞

1 1 2

3 4 3 2

1 2

5 4 1 2

1

3 2 1 2

2

2 1 1

1

3 2 1 2

2

2 1

11 12 1 3

1

2, 2

• [1, 2]

1

2, 1

• ⊂

⊂

Koolen, Vovk (RHUL) Buy low, sell high 20 / 29

Road map

1

Introduction

2

Intuition 1: Sell high only

3

Intuition 2: Iterated trading strategies

4

Simple counterexample

5

Main result

6

Examples

7

Conclusion

Koolen, Vovk (RHUL) Buy low, sell high 21 / 29

The GUT of Adjusters

Let G be left/right continuous and de/increasing.

Theorem (Characterisation)

G is an adjuster iff ∞ 1 − exp

• −
• G(a,b)≥h

da db (b − a)2

• dh ≤ 1.

Moreover, G is admissible iff this holds with equality and G is saturated.

Koolen, Vovk (RHUL) Buy low, sell high 22 / 29

The GUT of Adjusters

Let G be left/right continuous and de/increasing.

Theorem (Characterisation)

G is an adjuster iff ∞ 1 − exp

• −
• G(a,b)≥h

da db (b − a)2

• dh ≤ 1.

Moreover, G is admissible iff this holds with equality and G is saturated. Lower bound from option pricing Upper bound from explicitly constructed strategy Temporal reasoning evaporated.

Koolen, Vovk (RHUL) Buy low, sell high 22 / 29

Road map

1

Introduction

2

Intuition 1: Sell high only

3

Intuition 2: Iterated trading strategies

4

Simple counterexample

5

Main result

6

Examples

7

Conclusion

Koolen, Vovk (RHUL) Buy low, sell high 23 / 29

Simple adjusters

Corollary (Sell high Dawid, De Rooij, Gr¨

unwald, Koolen, Shafer, Shen, Vereshchagin, Vovk (2011))

Let G(a, b) := F(b ∨ 1). G is an adjuster iff ∞

1

F(y) y2 dy ≤ 1.

Koolen, Vovk (RHUL) Buy low, sell high 24 / 29

Simple adjusters

Corollary (Sell high Dawid, De Rooij, Gr¨

unwald, Koolen, Shafer, Shen, Vereshchagin, Vovk (2011))

Let G(a, b) := F(b ∨ 1). G is an adjuster iff ∞

1

F(y) y2 dy ≤ 1.

Corollary (Length)

Let G(a, b) := F(b − a). G is an adjuster iff ∞ F(y)e−1/y y2 dy ≤ 1.

Koolen, Vovk (RHUL) Buy low, sell high 24 / 29

Simple adjusters

Corollary (Sell high Dawid, De Rooij, Gr¨

unwald, Koolen, Shafer, Shen, Vereshchagin, Vovk (2011))

Let G(a, b) := F(b ∨ 1). G is an adjuster iff ∞

1

F(y) y2 dy ≤ 1.

Corollary (Length)

Let G(a, b) := F(b − a). G is an adjuster iff ∞ F(y)e−1/y y2 dy ≤ 1.

Corollary (Ratio)

Let G(a, b) := F(b/a) for some unbounded F. Then G is not an adjuster.

Koolen, Vovk (RHUL) Buy low, sell high 24 / 29

Our favourite adjuster

Let 0 ≤ q < p < 1. Then G(a, b) := (b − a)p aq

• ≈b/a

( p−q

p )p

Γ(1 − p)

• normalisation

is an admissible adjuster.

Koolen, Vovk (RHUL) Buy low, sell high 25 / 29

Our favourite adjuster

Let 0 ≤ q < p < 1. Then G(a, b) := (b − a)p aq

• ≈b/a

( p−q

p )p

Γ(1 − p)

• normalisation

is an admissible adjuster. Strategy: In situation ω with minimum price m take position S(ω) = (p − q) m1−p+q Φ

• m

p−q p

• XG(ω)Γ(1 − p)

1/p

• where Φ is the CDF of the Gamma distribution.

Koolen, Vovk (RHUL) Buy low, sell high 25 / 29

Road map

1

Introduction

2

Intuition 1: Sell high only

3

Intuition 2: Iterated trading strategies

4

Simple counterexample

5

Main result

6

Examples

7

Conclusion

Koolen, Vovk (RHUL) Buy low, sell high 26 / 29

What just happened

We took “buy low, sell high” as the learning target We consider parametrised payoff guarantees We classified candidate guarantees using a simple formula

(≤ 1) Attainable adjuster (= 1) Admissible adjuster (> 1) Not an adjuster

Looked at some interesting example adjusters

Koolen, Vovk (RHUL) Buy low, sell high 27 / 29

Open problems

Sell high, buy low, then sell high again. . . .

Koolen, Vovk (RHUL) Buy low, sell high 28 / 29

Thank you!

Koolen, Vovk (RHUL) Buy low, sell high 29 / 29