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Polyhedral Risk Measures Vadym Omelchenko, Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic. The presentationโs 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 โข ฮณ = ๐ฟ๐ (๐น, โ) be the usual Banach space of real random variables on some probability space (ฮฉ, F, P) for some p โ [1,โ). Polyhedral Probability Functionals โข Definition. A probability functional ๐ : ๐พ โ โ is called ๐๐๐๐ฆโ๐๐๐๐๐ if there exist ๐0 , ๐1 โ โ, ๐ค๐ , ๐๐ โ โ๐๐ , ๐ โ 0,1 , and non-empty polyhedral sets ๐๐ โ โ๐๐ , ๐ โ 0,1 , such that ๏ฌ ๏ผ v1 ๏ L p F , R k1 , ๏ฏ๏ฏ ๏ฏ๏ฏ R (Y ) ๏ฝ sup ๏ญ c0 , v0 ๏ซ E ๏จ c1 , v1 ๏ฉ vi ๏ Vi , i ๏ 0, 1 , ๏ฝ ๏ฏ ๏ฏ w , v ๏ซ w , v ๏ฝ Y ๏ฏ๏ฎ ๏ฏ๏พ 0 0 1 1 ๏จ ๏ป ๏ฉ ๏ฝ โข for every Y โ ๐พ . Here โ ,โ denote scalar products on โ๐0 and โ๐1 . โข ๐1 โ โ๐1 has to be understood in the sense a.s. Linear Reformulation โข Definition. A probability functional ๐ : ๐พ โ โ is called ๐๐๐๐ฆโ๐๐๐๐๐ if there exist ๐0 , ๐1 โ โ, ๐๐ , ๐ค๐ โ โ๐๐ , matrices ๐ด๐ , and vectors ๐๐ โ โ๐๐ , ๐ โ 0,1 , such that ๏ฌ ๏ฏ ๏ฏ R (Y ) ๏ฝ sup ๏ญ E cT0 ๏ v0 ๏ซ c1T ๏ v1 ๏ฏ ๏ฏ ๏ฎ ๏จ ๏จ ๏ฉ ๏ฉ v1 ๏ L p F , R k1 , ๏ผ ๏ฏ ๏ฏ A1v1 ๏ฃ b1 , a.s., ๏ฝ A0 v0 ๏ฃ b0 , ๏ฏ w0 ๏ v0 ๏ซ w1 ๏ v1 ๏ฝ Y ๏ฏ๏พ Example โข We consider the functional R Y = E u Y โข on ฮณ where u: โ โ โ is of the form ๐ โข ๐ข ๐ฅ = sup ๐, ๐ฃ : ๐ฃ โ โ ๐+๐ + , ๐=1 ๐ฃ๐ = 1, ๐ค, ๐ฃ = ๐ฅ with some ๐, ๐ฃ โ โ ๐+๐ , ๐, ๐ โ โ and hence it is concave and polyhedral in kinks ๐ค๐ , ๐๐ โ โ 2, ๐ = 1, โฆ , ๐. โข According to Rockafellar and Wets (1998), Theorem 14.60, we can reverse the order of sup and E. Example โข We consider the functional R Y = E u Y โข on ฮณ where u: โ โ โ is of the form ๐ โข ๐ข ๐ฅ = sup ๐, ๐ฃ : ๐ฃ โ โ ๐+๐ , + ๐=1 ๐ฃ๐ = 1, ๐ค, ๐ฃ = ๐ฅ with some ๐, ๐ฃ โ โ ๐+๐ , ๐, ๐ โ โ and hence ๐(โ ) is concave and polyhedral in kinks ๐ค๐ , ๐๐ โ โ 2, ๐ = 1, โฆ , ๐. โข See Rockafellar and Wets (1998), Theorem 14.60. Theorem Rockafellar and Wets Popular examples โข CV@R is a polyhedral risk measure. โข Every linear combination of CV@Rs are polyhedral risk measures โข V@R is not polyhedral. Properties of Polyhedral Functionals โข Let R be a functional of the form: ๏ฌ ๏ฏ๏ฏ R(Y ) ๏ฝ sup ๏ญ c0 , v0 ๏ซ E ๏จ c1 , v1 ๏ฏ ๏ฏ๏ฎ ๏ฉ ๏จ ๏ฉ ๏ผ ๏ฏ๏ฏ vi ๏ Vi , i ๏ ๏ป 0, 1 ๏ฝ, ๏ฝ ๏ฏ w0 , v0 ๏ซ w1 , v1 ๏ฝ Y ๏ฏ๏พ v1 ๏ L p F , R k1 , โข Let ๐๐ โ โ๐๐ , ๐ โ 0,1 be polyhedral cones and assume: 1. 2. ๐ค1 , ๐1 = โ (complete recourse), ๐ท โ ๐ข โ โ: ๐0 โ ๐ข๐ค0 โ ๐ 0โ, ๐1 โ ๐ข๐ค1 โ ๐ 1โ โ โ (dual feasibility. Then R is finite, concave, and continuous on ฮณ. Properties of Polyhedral Functionals โข Let R be a functional of the form: ๏ฌ ๏ฏ๏ฏ R(Y ) ๏ฝ sup ๏ญ c0 , v0 ๏ซ E ๏จ c1 , v1 ๏ฏ ๏ฏ๏ฎ ๏ฉ ๏จ ๏ฉ ๏ผ ๏ฏ๏ฏ vi ๏ Vi , i ๏ ๏ป 0, 1 ๏ฝ, ๏ฝ ๏ฏ w0 , v0 ๏ซ w1 , v1 ๏ฝ Y ๏ฏ๏พ v1 ๏ L p F , R k1 , โข Let ๐๐ โ โ๐๐ , ๐ โ 0,1 be polyhedral cones and assume: 1. ๐ค1 , ๐1 = โ (complete recourse), 2. ๐ท โ ๐ข โ โ: ๐0 โ ๐ข๐ค0 โ ๐ 0โ, ๐1 โ ๐ข๐ค1 โ ๐ 1โ โ โ (dual feasibility. 3. 1 1 ๐ โ 1, โ given by ๐ + ๐ = 1 Then R admits the dual representation ๐น ๐ = ๐๐๐ ๐ฌ ๐ ๐ : ๐ โ โค โ Where โค โ is a subset of โค = ๐ฟ๐ (๐น, โ) given by โค โ= ๐ โ โค: ๐0 โ ๐ค0 ๐ธ ๐ โ ๐ 0โ, ๐1 โ ๐ค1 ๐ โ ๐ 1โ . Definition of Polyhedral Risk Measures (Multi-Stage) Polyhedral Multi-Period Acceptability Functionals โข Let us denote ๐พ =×๐๐ก=1 ๐ฟ๐ ๐น๐ก , ๐ โ [1, โ). โข Definition. A probability functional ๐ : ๐พ โ โ is called ๐๐๐๐ฆโ๐๐๐๐๐ if there are ๐๐ก โ โ, ๐๐ก โ โ๐๐ก , and nonempty polyhedral sets ๐๐ก โ โ๐๐ก , ๐ก = 0, . . ๐, ๐ค๐ก,๐ โ โ๐๐กโ๐ , ๐ = 0, . . , ๐ก, ๐ก = 0, . . ๐ such that ๏ฌ ๏ฏT R (Y ) ๏ฝ sup ๏ญ๏ฅ E ๏จ ct , vt ๏ฏ t ๏ฝ0 ๏ฎ ๏ฉ ๏จ ๏ฉ ๏ฉ vt ๏ L p F , R k1 , vt ๏ Vt , t ๏ฝ 0,.., T ๏ผ ๏ฏ t ๏ฝ E wt ,๏ด , vt ๏ญ๏ด ๏ฝ Yt , t ๏ฝ 1,.., T ๏ฏ ๏ฅ ๏ด ๏ฝ0 ๏พ ๏จ โข holds for every Y โ ๐พ. Here โ ,โ denotes scalar products on every โ๐๐ก . Conditions for Supremal Values 1. 2. โข ๐๐ก is a polyhedral cone for ๐ก = 0, . . , ๐ and ๐ค๐ก,0 , ๐๐ก = โ holds for every ๐ก = 1, . . , ๐ (complete recourse). There exists ๐ข โ โ๐ such that ๐0 โ ๐๐=1 ๐ค๐,๐โ๐ก ๐ข๐ โ ๐0โ , ๐๐ก โ ๐ โ โ ๐=๐ก ๐ค๐,๐โ๐ก ๐ข๐ โ ๐๐ก , ๐ก = 1, . . , ๐, hold, where sets ๐๐ก are the polar cones to ๐๐ก . (dual feasibility) If 1. and 2. and the polyhedral function is defined by: ๏ฌ ๏ฏT R(Y ) ๏ฝ sup ๏ญ๏ฅ E ๏จ ct , vt ๏ฏ t ๏ฝ0 ๏ฎ โข ๏ฉ ๏จ ๏ฉ ๏ฉ vt ๏ L p F , R k1 , vt ๏Vt , t ๏ฝ 0,.., T ๏ผ ๏ฏ t ๏ฝ E wt ,๏ด , vt ๏ญ๏ด ๏ฝ Yt , t ๏ฝ 1,.., T ๏ฏ ๏ฅ ๏ด ๏ฝ0 ๏พ ๏จ R is finite, positively homogeneous, concave, and continuous on ๐พ Note on Multi-Stage ๏จ ๏ฉ ๏ฉ k1 ๏ฌ v ๏ L F , R , vt ๏ Vt , t ๏ฝ 0,.., T ๏ผ t p T ๏ฏ ๏ฏ R (Y ) ๏ฝ sup ๏ญ๏ฅ E ๏จ ct , vt ๏ฉ t ๏ฝ E wt ,๏ด , vt ๏ญ๏ด ๏ฝ Yt , t ๏ฝ 1,.., T ๏ฏ ๏ฅ ๏ฏ t ๏ฝ0 ๏ด ๏ฝ0 ๏ฎ ๏พ โข The dual solutions that correspond to the constraint is the slope of the R. ๏จ E๏จ w ๏ด , v ๏ด ๏ฉ ๏ฝ Y ๏ฅ ๏ด t ๏ฝ0 t, t๏ญ t โข This problem is solved by means of cost-to-go functions and bellmanโs equation. Note on Multi-Stage ๏จ ๏ฉ ๏ฉ k1 ๏ฌ v ๏ L F , R , vt ๏ Vt , t ๏ฝ 0,.., T ๏ผ t p T ๏ฏ ๏ฏ R (Y ) ๏ฝ sup ๏ญ๏ฅ E ๏จ ct , vt ๏ฉ t ๏ฝ E wt ,๏ด , vt ๏ญ๏ด ๏ฝ Yt , t ๏ฝ 1,.., T ๏ฏ ๏ฅ ๏ฏ t ๏ฝ0 ๏ด ๏ฝ0 ๏ฎ ๏พ โข The dual solutions that correspond to the constraint is the slope of the R. ๏จ E๏จ w ๏ด , v ๏ด ๏ฉ ๏ฝ Y ๏ฅ ๏ด t ๏ฝ0 t, t๏ญ t โข This problem is solved by means of cost-to-go functions and bellmanโs equation. Vt (Y ) ๏ฝ max๏ปCt (Y , x) ๏ซ E ๏จVt ๏ซ1 (Yt ๏ซ1 (Yt , x)) | Y ๏ฉ๏ฝ x๏๏ Note on V@R โข 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 โข 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. โข More requirements on Risk management. Liberalization/Deregulation of the Energy Markets โข Mathematical modeling of such optimization problems with uncertain input data results in mixedinteger large-scale stochastic programming models with a risk measure in the objective. โข Often Multi-Stage problems are solved in the framework of either dynamic or stochastic programming. โข Simultaneous optimization of profits and risks. Applications of polyhedral Risk Measures The problem of finding a strategy that yields the optimal (or near optimal) profit under taking into account technical constraint and risks. min โ 1 โ ๐พ โ ๐ ๐๐ ๐ โ ๐พ โ ๐๐๐๐๐๐ก Specification of the Problem โข 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. โข 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) โข Prices depend on demands and vice versa โข Tri-variate ARMA models (demand for heat&electricity and spot prices). โข Spot prices AR-GARCH. โข The futures prices are calculated aposteriori from the spot prices in the scenario tree. (month average) Parameters Decision Variables Objective Objective โ Cash Values โข Cash values are what we earn from producing heat and electricity. We of course take into account technical constraints. Objective Simulation Results โข The best strategy is to not use any contracts. โข Minimizing without a risk measure causes high spread for the distribution of the overall revenue. โข 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 โข Polyhedral risk measures enable us to incorporate more realistic features of the problem and to preserve its convexity and linearity. โข Hence, they enable the tractability of many problems. โข V@R is a less sophisticated risk measure, but many problems cannot be solved by using V@R unlike CV@R. Bibliography โข 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. โข A. Shapiro, W. Tekaya, J.P. da Costa, and M.P. Soares. Risk neutral and risk averse Stochastic Dual Dynamic Programming method. 2013. โข G. C Pflug, W. Roemisch. Modeling, Measuring and Managing Risk. 2010. โข A. Eichhorn, W. Römisch, Mean-risk optimization of electricity portfolios using multiperiod polyhedral risk measures. 2005