Application of Markov SDDP to Financial Modelling Martin d 2 k 2 - PowerPoint PPT Presentation

Motivation General Model Solution via HMC Approximation Computation Discussion Application of Markov SDDP to Financial Modelling Martin ˇ ıd 2 ık 2 Sm´ V´ aclav Kozm´ 1 Czech Academy of Sciences, Institute of Information Theory and Automation March 29th, 2019 CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 1 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Contents Motivation 1 General Model 2 Solution via HMC Approximation 3 Computation 4 Discussion 5 Work in progress... CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 2 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Motivation Many real-life decision problems are equivalent to problems of Asset-Liability Management: (uncertain) amounts of something is to by delivered at distinct time points the goal is to satisfy these obligations in an optimal way Examples: pension funds, option writing, carbon emission management Problems with solution Deterministic equivalent unusable, as sparse scenarios preclude effective risk management (do not model tails) SDDP assumes stage-wise independence of random parameters, which is not the case with prices ADDP allows for Markov dependence, but only for discrete distributions CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 3 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Example - Optimal Emission Management 1200 1100 Optimally cover exogenous 1000 The problem: 900 800 demand emissions by buying allowances (spots), futures 700 600 500 and options. 400 300 200 2004 2006 2008 2010 2012 2014 2016 Emissions 10 EUA_SPOT EUA_F17 9 EUA_F18 EUA_F19 EUA_F20 8 Model. 4-stage Multistage linear problem with 7 6 nested or multiperiod Mean-CVaR risk criterion 5 4 3 Apr Jul Oct 2016 Apr Jul Oct Spots and Futures (17 , 18 , 19 , 20) Random parameters. y17 y18 0.8 1 Emissions and income from production: AR(1) 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0 0 Spots ( P t ): Geometrically Brownian -0.2 -0.2 -0.4 -0.4 2016 2016 Future prices ( Q t ): Noised cost-of-carry model y19 y20 0.8 0.4 0.35 0.6 0.3 0.25 0.4 0.2 Q t = e a τ + η t P t , 0.2 0.15 0.1 0 0.05 -0.2 0 -0.05 -0.4 -0.1 2016 2016 η t ∼ N (0 , τ 2 b 2 ) , τ = t maturity − t Spot-future spreads CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 4 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion General Model covering ALM models Multi-stage Linear Model with a Nested Risk Measure inf ρ ( c ′ 0 x 0 , . . . , c ′ T x T ) � x t − 1 � x t ∈ R d t x t ∈ F t , 1 ≤ t ≤ T , + , A t = b , x t ρ ( c 0 , . . . , c T ) = c 0 + ρ 1 ( ι c 1 + ρ 2 ( ι 2 c 2 · · · + ρ T ( ι T c T ) . . . )) t � A t = α t ( ξ t , η t ) , c t = γ t ( ξ t , η t ) , b t = β t ,τ ( ξ τ , η τ ) , τ =1 F t = σ (( A τ , b τ , c τ ) τ ≤ T ) , ρ t - risk mapping, ι - discount factor. α t , β t , γ t - mappings, ξ t ∈ R p t - Markov, η t ∈ R q t - i.i.d. W.l.o.g. b t = β t ( ξ t , η t ) (via “artificial” decision variables) ⇒ some problems with time-dependent b t (e.g. VAR) may be reformulated as stage-wise (sometimes true for liabilities, never for prices) CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 5 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Solution of Problems with Markov Parameters Using a version of SDDP requiring only conditional stage wise independence given a finite Markov chain (Philpott at. al. 2013) In other words, it works with Hidden Markov Chains Models (e.g. bull market / bear market) Complexity should be linear in total number of states Minimum possible: have the Markov parameter discrete Same drawback as with Deterministic Equivalent Idea: Approximate ξ by a Markov discrete ξ skeleton + smoothing Roughly keeps dependence properties Continuous distribution (good for risk management) t CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 6 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Approximation of The Markov Parameter - detail Covering and representatives: C t = { C i C = ( C 1 , . . . , C T ) , t : 1 ≤ i ≤ k t } E t = { e i E = ( E 1 , . . . , E T ) , t : 1 ≤ i ≤ k t } s.t. C 1 t , . . . , C k t are disjoint, covering support ( ξ t ), and t e i t ∈ C i t ∩ support ( ξ t ), 1 ≤ i ≤ k t , 1 ≤ t ≤ T , with e 1 0 = ξ 0 . ς t - approximating process, ι t - index process, P [ ι t = j | ς t − 1 , ι t − 1 ] = p ι t − 1 , j p i , j = P [ ξ t ∈ C j t | ξ t − 1 = e i , t − 1 ] , t t P [ ς t = •| ς t − 1 , ι t ] = ω ι t t ( • ) are probabilities s. t. support ( ω j t ) ⊆ C j – ω 1 t , . . . , ω k t t , 1 ≤ j ≤ k t . t – x t the its history of x up to t Remark: ς 1 , . . . , ς T are conditionally independent given ι 1 , . . . , ι T . CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 7 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Two Choices of Smoothing probability Atomic approximation (only skeleton): (i.e. ν t = e ι t ω i t ( • ) = δ e i t ( • ) , t ) Continuous “unconditional” approximation t ( • ) = P [ ξ t ∈ • ∩ C i t ] with 0 ω i 0 = 0 (1) P [ ξ t ∈ C i t ] ξ t CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 8 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Assumptions Fact. There exist mappings g t , 1 ≤ t ≤ T and i.i.d. random variables U t such that 1 ≤ t ≤ T ξ t = g t ( ξ t − 1 , U t ) , Assumptions P [ ξ t ∈ •| ξ t − 1 ] and ω • t are tight, 1 ≤ t ≤ T . 1 � g t ( x , u ) − g t ( y , u ) � ≤ h t ( u ) � x − y � where h t is integrable w.r.t. 2 L ( U t ). Remark. (i) If ξ t = A ξ t − 1 + ǫ t then � g t ( x , u ) − g t ( y , u ) � ≤ � A �� x − y � (ii) If ξ t = ξ ′ t − 1 A ǫ t then � g t ( x , u ) − g t ( y , u ) � ≤ � u �� A �� x − y � . CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 9 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Multistage Distance Definition. A probability measure π on (Ω × Υ , F T ⊗ G T ) is a nested transportation from P to Q if, for any 0 ≤ t < T , π ( A × Υ |F t ⊗ G t ) = P ( A |F t ) (2) π (Ω × B |F t ⊗ G t ) = Q ( B |G t ) (3) for each A ∈ F T , B ∈ G T . Definition. For any pair of probability measures P and Q defined on R n , the multistage distance d ( P , Q ) is defined by � d ( P , Q ) = inf � ω − υ � π ( d ω, d υ ) π s . t π is a nested transportation from P to Q where � x � = � n i =1 | x i | is the l1 norm. For T = 1 the multistage distance coincides with the Wasserstein distance , which we denote by d . Pflug Pichler. Given some Lipschitz assumptions, the difference of true and approximate optimal value is no more than K d ( ξ T , ς T ) for some K . CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 10 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Upper bound of d ( ξ T , ς T ) Denote e t ( ς ) = � k t i =1 1 [ ς ∈ C i t ] e i t and x the history of x . We have d ( ξ t , ς t ) ≤ (1 + L t ) d ( ξ t − 1 , ς t − 1 ) + L t B t − 1 + B t + D t t t t � � � ≤ (1 + L i ) � ξ 0 − ς 0 � (1 + L i )( L τ B τ − 1 + B τ + D τ ) + τ =1 i = τ +1 i =1 where L t = E h t ( U t ) , � B τ = � s − ǫ τ ( s ) � ψ τ ( ds ) , ψ τ is the c.d.f. of ς τ � D τ = � x − ǫ τ ( x ) � φ τ ( dx ) , k τ − 1 � q i τ − 1 P [ ξ τ ∈ •| ξ τ − 1 = e i q i τ = P [ ς τ ∈ C i φ τ ( • ) = τ − 1 ] , τ ] i =1 CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 11 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion Definition. ( C , E ) is rectangular covering of R p if ) × · · · × [ c i p p , c i p + i C = { [ c i 1 1 , c i 1 +1 ) : 0 ≤ i 1 ≤ k 1 , . . . , 0 ≤ i p ≤ k p } p 1 1 , . . . ., e i p E = { ( e i 1 p ) : 0 ≤ i 1 ≤ k 1 , . . . , 0 ≤ i p ≤ k p } 1 < . . . e j , k j < c j , k j = ∞ , for some −∞ = c 0 j < e 1 j < c 1 j ≤ e 2 1 ≤ j ≤ p . Proposition. (Mˇ S (2004.2009)) Let ( C , E ) be a rectangular covering of R p and let µ be a probability measure on R p . Let µ C , E be a probability measure on E fulfilling µ C , E ( e ) = µ C whenever e ∈ C ∈ C . Then p � d ( µ i , µ i C , E ) = E � ξ − e ( ξ ) � d ( µ, µ C , E ) = i =1 where ξ ∼ µ and µ i is the i -th marginal distribution of µ , similarly with µ C , E CMS 2019, Chemnitz, 27 th to 29 th June, 2019 Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık 12 / 17