Buy low, sell high Vladimir Vovk Wouter M. Koolen GTP 2014 Guanajuato Mexico Saturday 15 th November, 2014
Road map Introduction 1 Intuition 1: Sell high only 2 Intuition 2: Iterated trading strategies 3 Simple counterexample 4 Main result 5 Examples 6 Conclusion 7 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 online 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 online learning style and uncover its surprisingly intricate theory Koolen, Vovk (RHUL) Buy low, sell high 3 / 29
example price Fortune brands 2 share price in € 1.5 1 0.5 1999 2000 2001 2002 2003 time
example price Fortune brands 2 share price in € 1.5 1 start with 1 e 0.5 1999 2000 2001 2002 2003 time
example price Fortune brands 2 share price in € 1.5 1 start with 1 e buy at 0 . 7 e 0.5 1999 2000 2001 2002 2003 time
example price sell at 2 . 1 e Fortune brands 2 share price in € 1.5 1 start with 1 e buy at 0 . 7 e 0.5 1999 2000 2001 2002 2003 time
example price sell at 2 . 1 e Fortune brands 2 share price in € 1.5 payoff: 2 . 1 0 . 7 = 3 e 1 start with 1 e buy at 0 . 7 e 0.5 1999 2000 2001 2002 2003 time
example price Fortune brands 2 Altria group share price in € 1.5 1 0.5 1999 2000 2001 2002 2003 time
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 K 0 := 1 Initial price ω 0 := 1 For day t = 1 , 2 , . . . 1 Investor takes position S t ∈ R 2 Market reveals price ω t ∈ [0 , ∞ ) 3 Capital becomes K t := K t − 1 + S t ( ω t − ω t − 1 ) Koolen, Vovk (RHUL) Buy low, sell high 7 / 29
Protocol Initial capital K 0 := 1 Initial price ω 0 := 1 For day t = 1 , 2 , . . . 1 Investor takes position S t ∈ R 2 Market reveals price ω t ∈ [0 , ∞ ) 3 Capital becomes K t := K t − 1 + S t ( ω t − ω t − 1 ) A position S t < 0 is called short S t > 0 is called long S t > K t − 1 /ω t − 1 is called leveraged Koolen, Vovk (RHUL) Buy low, sell high 7 / 29
Protocol Initial capital K 0 := 1 Initial price ω 0 := 1 For day t = 1 , 2 , . . . 1 Investor takes position S t ∈ R 2 Market reveals price ω t ∈ [0 , ∞ ) 3 Capital becomes K t := K t − 1 + S t ( ω t − ω t − 1 ) A position S t < 0 is called short S t > 0 is called long S t > K t − 1 /ω t − 1 is called leveraged Bankrupt when capital K t < 0 is negative. Koolen, Vovk (RHUL) Buy low, sell high 7 / 29
Protocol Initial capital K 0 := 1 Initial price ω 0 := 1 For day t = 1 , 2 , . . . 1 Investor takes position S t ∈ R 2 Market reveals price ω t ∈ [0 , ∞ ) 3 Capital becomes K t := K t − 1 + S t ( ω t − ω t − 1 ) A position S t < 0 is called short S t > 0 is called long S t > K t − 1 /ω t − 1 is called leveraged Bankrupt when capital K t < 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 Introduction 1 Intuition 1: Sell high only 2 Intuition 2: Iterated trading strategies 3 Simple counterexample 4 Main result 5 Examples 6 Conclusion 7 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 F1(y) F2(y) F3(y) 3 2.5 guaranteed capital 2 1.5 1 0.5 0 1 3 5 7 9 11 13 15 17 19 maximum price reached Koolen, Vovk (RHUL) Buy low, sell high 11 / 29
Adjuster A strategy prescribes position S t based on the past prices ω 0 , . . . , ω t − 1 . Definition A function F : [1 , ∞ ) → [0 , ∞ ) is called an adjuster if there is a strategy that guarantees � � K t ≥ F 0 ≤ s ≤ t ω s max . 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 F u ( y ) := u 1 { y ≥ u } Koolen, Vovk (RHUL) Buy low, sell high 13 / 29
Threshold adjusters Fix a price level u ≥ 1. The threshold adjuster F u ( y ) := u 1 { y ≥ u } is witnessed by the threshold strategy S u 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 � ∞ F ( y ) d y ≤ 1 . y 2 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 � ∞ F ( y ) d y ≤ 1 . y 2 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 ) ≤ F u ( y ) d P ( u ) , again with equality iff F is admissible. Koolen, Vovk (RHUL) Buy low, sell high 14 / 29
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