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 Sm´ ıd 2 V´ aclav Kozm´ ık 2

1Czech Academy of Sciences, Institute of Information Theory and Automation

March 29th, 2019

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 1 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Contents

1

Motivation

2

General Model

3

Solution via HMC Approximation

4

Computation

5

Discussion Work in progress...

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 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

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 3 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Example - Optimal Emission Management

The problem: Optimally cover exogenous emissions by buying allowances (spots), futures and options.

200 300 400 500 600 700 800 900 1000 1100 1200 2004 2006 2008 2010 2012 2014 2016 demand

Emissions

• Model. 4-stage Multistage linear problem with

nested or multiperiod Mean-CVaR risk criterion

3 4 5 6 7 8 9 10 Apr Jul Oct 2016 Apr Jul Oct EUA_SPOT EUA_F17 EUA_F18 EUA_F19 EUA_F20

Spots and Futures (17, 18, 19, 20)

Random parameters. Emissions and income from production: AR(1) Spots (Pt): Geometrically Brownian Future prices (Qt): Noised cost-of-carry model Qt = eaτ+ηtPt, ηt ∼ N(0, τ 2b2), τ = tmaturity − t

• 0.4
• 0.2

0.2 0.4 0.6 0.8 2016 y17

• 0.4
• 0.2

0.2 0.4 0.6 0.8 1 2016 y18

• 0.4
• 0.2

0.2 0.4 0.6 0.8 2016 y19

• 0.1
• 0.05

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 2016 y20

Spot-future spreads Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 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′

0x0, . . . , c′ TxT)

xt ∈ Rdt

+,

xt ∈ Ft, At xt−1 xt

• = b,

1 ≤ t ≤ T, ρ(c0, . . . , cT) = c0 + ρ1(ιc1 + ρ2(ι2c2 · · · + ρT(ιTcT) . . . )) At = αt(ξt, ηt), ct = γt(ξt, ηt), bt =

t

• τ=1

βt,τ(ξτ, ητ), Ft = σ((Aτ, bτ, cτ)τ≤T) , ρt - risk mapping, ι - discount factor. αt, βt, γt - mappings, ξt ∈ Rpt - Markov, ηt ∈ Rqt - i.i.d. W.l.o.g. bt = βt(ξt, ηt) (via “artificial” decision variables) ⇒ some problems with time-dependent bt (e.g. VAR) may be reformulated as stage-wise (sometimes true for liabilities, never for prices)

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 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 skeleton + smoothing Roughly keeps dependence properties Continuous distribution (good for risk management)

t

ξ

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 6 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Approximation of The Markov Parameter - detail

Covering and representatives: C = (C1, . . . , CT), Ct = {C i

t : 1 ≤ i ≤ kt}

E = (E1, . . . , ET), Et = {ei

t : 1 ≤ i ≤ kt}

s.t. C 1

t , . . . , C kt t

are disjoint, covering support(ξt), and ei

t ∈ C i t ∩ support(ξt), 1 ≤ i ≤ kt, 1 ≤ t ≤ T, with e1 0 = ξ0.

ςt - approximating process, ιt - index process, P[ιt = j|ςt−1, ιt−1] = pιt−1,j

t

, pi,j

t

= P[ξt ∈ C j

t |ξt−1 = ei t−1],

P[ςt = •|ςt−1, ιt] = ωιt

t (•)

– ω1

t , . . . , ωkt t

are probabilities s. t. support(ωj

t) ⊆ C j t , 1 ≤ j ≤ kt.

– xt the its history of x up to t Remark: ς1, . . . , ςT are conditionally independent given ι1, . . . , ιT.

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 7 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Two Choices of Smoothing probability

Atomic approximation (only skeleton): ωi

t(•) = δei

t(•),

(i.e. νt = eιt

t )

Continuous “unconditional” approximation ωi

t(•) = P[ξt ∈ • ∩ C i t]

P[ξt ∈ C i

t]

with 0 0 = 0 (1)

t

ξ

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 8 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Assumptions

• Fact. There exist mappings gt, 1 ≤ t ≤ T and i.i.d. random variables Ut

such that ξt = gt(ξt−1, Ut), 1 ≤ t ≤ T Assumptions

1

P[ξt ∈ •|ξt−1] and ω•

t are tight, 1 ≤ t ≤ T.

2

gt(x, u) − gt(y, u) ≤ ht(u)x − y where ht is integrable w.r.t. L(Ut).

• Remark. (i) If ξt = Aξt−1 + ǫt then gt(x, u) − gt(y, u) ≤ Ax − y

(ii) If ξt = ξ′

t−1Aǫt then gt(x, u) − gt(y, u) ≤ uAx − y.

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 9 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Multistage Distance

• Definition. A probability measure π on (Ω × Υ, FT ⊗ GT) is a nested

transportation from P to Q if, for any 0 ≤ t < T, π(A × Υ|Ft ⊗ Gt) = P(A|Ft) (2) π(Ω × B|Ft ⊗ Gt) = Q(B|Gt) (3) for each A ∈ FT, B ∈ GT.

• Definition. For any pair of probability measures P and Q defined on Rn,

the multistage distance d(P, Q) is defined by d(P, Q) = inf

π

• ω − υπ(dω, dυ)

s.t π is a nested transportation from P to Q where x = n

i=1 |xi| 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 Kd(ξT, ςT) for some K.

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 10 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Upper bound of d(ξT, ςT)

Denote et(ς) = kt

i=1 1[ς ∈ C i t]ei t and x the history of x. We have

d(ξt, ςt) ≤ (1 + Lt)d(ξt−1, ςt−1) + LtBt−1 + Bt + Dt ≤

t

• τ=1

t

• i=τ+1

(1 + Li)(LτBτ−1 + Bτ + Dτ) +

t

• i=1

(1 + Li)ξ0 − ς0 where Lt = Eht(Ut), Bτ =

• s − ǫτ(s)ψτ(ds),

ψτ is the c.d.f. of ςτ Dτ =

• x − ǫτ(x)φτ(dx),

φτ(•) =

kτ−1

• i=1

qi

τ−1P[ξτ ∈ •|ξτ−1 = ei τ−1],

qi

τ = P[ςτ ∈ C i τ]

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 11 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

• Definition. (C, E) is rectangular covering of Rp if

C = {[ci1

1 , ci1+1 1

) × · · · × [cip

p , cip+i p

) : 0 ≤ i1 ≤ k1, . . . , 0 ≤ ip ≤ kp} E = {(ei1

1 , . . . ., eip p ) : 0 ≤ i1 ≤ k1, . . . , 0 ≤ ip ≤ kp}

for some −∞ = c0

j < e1 j < c1 j ≤ e2 1 < . . . ej,kj < cj,kj = ∞,

1 ≤ j ≤ p.

• Proposition. (Mˇ

S (2004.2009)) Let (C, E) be a rectangular covering of Rp and let µ be a probability measure on Rp. Let µC,E be a probability measure on E fulfilling µC,E(e) = µC whenever e ∈ C ∈ C. Then d(µ, µC,E) =

p

• i=1

d(µi, µi

C,E) = Eξ − e(ξ)

where ξ ∼ µ and µi is the i-th marginal distribution of µ, similarly with µC,E

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 12 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Corollary.

If (Ct, Et), 1 ≤ t ≤ T are rectangular then, for any 1 ≤ τ ≤ T Bτ =

pτ

• i=1
• |s − ǫτ(s)|ψi

τ(ds),

ψ′

τ is the c.d.f. of ςi τ

Dτ =

pτ

• i=1
• |x − ǫτ(x)|φi

τ(dx),

φi

τ is the i-th marginal of φτ

⇒ numerically computable bounds

• Remark. Ppper bounds exist (Mˇ

S 2004,2009).

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 13 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Towards Convergence ...

Proposition (Mˇ S 2004,2009): Let µ be an absolutely continuous distribution on Rp with marginals having finite first moments. Then there exists a sequence of rectangular coverings such that

• x − en(x)µ(dx) → 0. If the tails of the marginals vanish

exponentially then the convergence is o(n

1 p +δ) for any δ > 0.

Unfortunately, our upper bounds depend on the covering. . . . What is problem: We need to have

• x − et(x)ψt(dx) →
• x − et(x)µt(dx) where µt is the c.d.f.
• f ξt. By induction, we can get d(ψt, µt) → 0, but x − et(x) is

not Lipshitz (may have jumps on the edges of the covering sets)... Similarly φt...

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 14 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Example - continued

T = 3 (four stage), nested mean CVaR a risk measure Emissions (and production) AR(1) - handled by artificial variables Future-spot spreads i.i.d. Prices Geometrical Brownian martingale - approximated Unexpected obstacle - the arbitrage Continuous ς can never be martingale (the conditioning value is always the skeleton one) Neither the atomic approximation is exactly martingale (the approximation is not moment matching) Mean-CVar sometimes tolerates arbitrage (which distorts all the results) ⇒ atomic approximation with correction for martingale used

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 15 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Example - continued - computation experience

spots, futures and four options with maturities in all future stages used (23 − 20 − 14 − 6 variables by stages) Independent random parameters variables: production and emissions residuals and spot future spreads (5 − 4 − 3 − 2 by stage, option price computed by BS) Generated scenario size: 1200 per Markov state Skeleton size: 1 − 15 − 21 − 30 (67 states at all) Average computational time: 8.4 hour on Intel I7core.

• Remark. No options selected.
• Remark. Upper bounds volatile, algorithm stopped due to the

non-improvement of the lower bound.

• Remark. The time seems quadratic in number of Markov states.

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 16 / 17

Motivation General Model Solution via HMC Approximation Computation Discussion

Discussion

Curse of dependence lasts: trade-off between the approximation of marginals and that of the dependence No matter the objectivee function is Lipschitz, arbitrage stemming from the approximation can destroy the results However, the approximation may be close as required ⇒ To be done: formulate the conditions of arbitrage and approximate exactly enough so that they are not fulfilled.

Martin ˇ Sm´ ıd, V´ aclav Kozm´ ık CMS 2019, Chemnitz, 27th to 29th June, 2019 17 / 17