Polyhedral Risk Measures

Vadym Omelchenko,Institute of Information Theory and Automation,

Academy of Sciences of the Czech Republic.

The presentation’s structure

1. Definition of polyhedral risk measures (Two-stage)

2. Definition of polyhedral risk measures (Multi-stage)

3. Applications in the energy sector (CHP)

Definition of Polyhedral Risk Measures (Two-

Stage)

Polyhedral Risk Measures

• be the usual Banach space of real random variables on some probability space for some ,.

Polyhedral Probability Functionals

• Definition. A probability functional is called if there exist , and non-empty polyhedral sets , such that

• for every Y . Here denote scalar products on and .

• has to be understood in the sense a.s.

Yvwvw

iVv

RFLv

vcEvcYR ii

kp

1100

1

1100

,,

,1,0,

,,

,,sup)(

1

Linear Reformulation• Definition. A probability functional is called if there

exist , matrices , and vectorssuch that

Yvwvw

bvA

sabvA

RFLv

vcvcEYR

kp

TT

1100

000

111

1

110,

.,.,

,,

sup)(

1

0

Example• We consider the functional

• on where is of the form

• with some and hence it is concave and polyhedral in kinks

• According to Rockafellar and Wets (1998), Theorem 14.60, we can reverse the order of sup and E.

Example• We consider the functional

• on where is of the form

• with some and hence () is concave and polyhedral in kinks

• See Rockafellar and Wets (1998), Theorem 14.60.

Theorem Rockafellar and Wets

Popular examples• CV@R is a polyhedral risk measure.

• Every linear combination of CV@Rs are polyhedral risk measures

• V@R is not polyhedral.

Properties of Polyhedral Functionals• Let R be a functional of the form:

• Let , be polyhedral cones and assume: 1. (complete recourse),2. (dual feasibility.

Then R is finite, concave, and continuous on .

Yvwvw

iVv

RFLv

vcEvcYR ii

kp

1100

1

1100

,,

,1,0,

,,

,,sup)(

1

Properties of Polyhedral Functionals• Let R be a functional of the form:

• Let , be polyhedral cones and assume: 1. (complete recourse),2. (dual feasibility.3. given by

Then R admits the dual representation

Where is a subset of given by =.

Yvwvw

iVv

RFLv

vcEvcYR ii

kp

1100

1

1100

,,

,1,0,

,,

,,sup)(

1

Definition of Polyhedral Risk Measures (Multi-

Stage)

Polyhedral Multi-Period Acceptability

Functionals• Let us denote .• Definition. A probability functional is called if there

are , and non-empty polyhedral sets , such that

• holds for every . Here denotes scalar products on every .

TtYvwE

TtVvRFLv

vcEYRt

t

tt

ttk

ptT

ttt

,..,1,,

,..,0,,,

,sup)(

0,0

1

Conditions for Supremal Values

1. is a polyhedral cone for and holds for every (complete recourse).2. There exists such that hold, where sets are the polar cones to . (dual

feasibility)

• If 1. and 2. and the polyhedral function is defined by:

• R is finite, positively homogeneous, concave, and continuous on

TtYvwE

TtVvRFLv

vcEYRt

t

tt

ttk

ptT

ttt

,..,1,,

,..,0,,,

,sup)(

0,0

1

Note on Multi-Stage

• The dual solutions that correspond to the constraint is the slope of the R.

• This problem is solved by means of cost-to-go functions and bellman’s equation.

TtYvwE

TtVvRFLv

vcEYRt

t

tt

ttk

ptT

ttt

,..,1,,

,..,0,,,

,sup)(

0,0

1

t

t

tt YvwE

0

, ,

Note on Multi-Stage

• The dual solutions that correspond to the constraint is the slope of the R.

• This problem is solved by means of cost-to-go functions and bellman’s equation.

TtYvwE

TtVvRFLv

vcEYRt

t

tt

ttk

ptT

ttt

,..,1,,

,..,0,,,

,sup)(

0,0

1

t

t

tt YvwE

0

, ,

YxYYVExYCYV ttttx

t |)),((),(max)( 11

Note on V@R

• If we use V@R, many problems will cease to be linear and convex. However, replacing V@R with CV@R enables us to preserve the convexity of the underlying problem because this measure is polyhedral.

Applications in the Energy Sector (CHP)

Liberalization/Deregulation of the

Energy Markets

• The deregulation of energy markets has lead to an increased awareness of the need for profit maximization with simultaneous consideration of financial risk, adapted to individual risk aversion policies of market participants.

• More requirements on Risk management.

Liberalization/Deregulation of the

Energy Markets• Mathematical modeling of such optimization

problems with uncertain input data results in mixed-integer large-scale stochastic programming models with a risk measure in the objective.

• Often Multi-Stage problems are solved in the framework of either dynamic or stochastic programming.

• Simultaneous optimization of profits and risks.

Applications of polyhedral Risk

MeasuresThe problem of finding a strategy that yields the optimal (or near optimal) profit under taking into account technical constraint and risks.

Specification of the Problem

• The multi-stage stochastic optimization models are tailored to the requirements of a typical German municipal power utility, which has to serve an electricity demand and a heat demand of customers in a city and its vicinity.

• The power utility owns a combined heat and power (CHP) facility that can serve the heat demand completely and the electricity demand partly.

Stochasticity of the Model

Sources:

1. Electricity spot prices

2. Electricity forward prices

3. Electricity demand (load)

4. Heat demand.

Stochasticity of the Model

Multiple layers of seasonality

1. Electricity spot prices (daily, weekly, monthly)2. Electricity demand (daily, weekly, monthly)3. Heat demand (daily, weekly, monthly)

The seasonality is captured by the deterministic part.

Interdependency between the Data (prices&demands)• Prices depend on demands and vice versa

• Tri-variate ARMA models (demand for heat&electricity and spot prices).

• Spot prices AR-GARCH.

• The futures prices are calculated aposteriori from the spot prices in the scenario tree. (month average)

Parameters

Decision Variables

Objective

Objective – Cash Values

• Cash values are what we earn from producing heat and electricity. We of course take into account technical constraints.

Objective

Simulation Results• The best strategy is to not use any contracts.

• Minimizing without a risk measure causes high spread for the distribution of the overall revenue.

• The incorporation of the (one-period) CV@R applied to z(T) reduces this spread considerably for the price of high spread and very low values for z(t) at time t<T.

Simulation Results

Simulation Results

Simulation Results

Simulation Results

Simulation Results

Simulation Results

Conclusion• Polyhedral risk measures enable us to incorporate

more realistic features of the problem and to preserve its convexity and linearity.

• Hence, they enable the tractability of many problems.

• V@R is a less sophisticated risk measure, but many problems cannot be solved by using V@R unlike CV@R.

Bibliography• A. Philpott, A. Dallagi, E. Gallet. On Cutting Plane Algorithms

and Dynamic Programming for Hydroelectricity Generation. Handbook of Risk Management in Energy Production and Trading International Series in Operations Research & Management Science , Volume 199, 2013, pp 105-127.

• A. Shapiro, W. Tekaya, J.P. da Costa, and M.P. Soares. Risk neutral and risk averse Stochastic Dual Dynamic Programming method. 2013.

• G. C Pflug, W. Roemisch. Modeling, Measuring and Managing Risk. 2010.

• A. Eichhorn, W. Römisch, Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures. 2005