American options for energy balancing Jan Palczewski (Leeds) John Moriarty (Manchester) Milton Keynes, 1st June 2015 Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 1 / 18
Real-time market: supply curve and demand process Reproduced from Stephenson and Paun (2001) (highlighting added): Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 2 / 18
UK Electricity Market Reform UK is introducing a capacity market to supplement the real-time market. Reproduced from www.gov.uk : Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 3 / 18
Questions Can battery storage for grid balancing be a viable business? 1 How can such a scheme be organised? 2 Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 4 / 18
Two actors We model a two-agent problem of electricity balancing: The system (network) operator seeks to control deviations in the AC frequency away from 50Hz (i.e., imbalance between generation and demand). The provider of energy balancing services (storage operator) delivers energy to the grid under a ‘call off’ contract. + has battery capable of storing one arbitrary unit of energy + may buy electricity at any time at the prevailing market imbalance price, and without any price impact + the battery charge-discharge cycle incurs round trip costs + the battery charge-discharge cycle gradually decreases capacity Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 5 / 18
Call-off contract Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 6 / 18
The model Ornstein-Uhlenbeck process ( X t ) t ≥ 0 models the imbalance price (time in days) dX t = θ ( D − X t ) dt + σ dW t Baseline level D = 60, θ = 0 . 77, σ = 20 . 81 (07/2011-03/2014, 8am) Exercise occurs when X t ≥ x ∗ = 70 Interest rate is r > 0 (3 % per annum in examples) Aim: To value an American call option on real-time electricity with physical delivery. Approach: Real Options valuation for electricity storage (battery) operator ( ≡ expectation of discounted future cashflows) Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 7 / 18
OU process for APXUK data 100 OU Price 60 20 0 20 40 60 80 100 Days 120 APXUK Price 80 60 40 0 20 40 60 80 100 Days Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 8 / 18
Questions Research questions: 1 The battery operator’s optimal strategy and expected payoff from a) one call option b) a sequence of call options (lifetime contract) Is there mutual benefit? 2 Difficulty: Small price (=large excess supply) means 1 cheaper purchase of power greater expected time until exercise when the strike price K c is received Unclear whether it is optimal to buy energy when the price x is very small. Payoff is non smooth (kink at x ∗ ) 2 Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 9 / 18
Mutual benefit S1. The storage operator has positive expected profit from the offer and exercise of the option. S2. The option cannot lead to a certain financial loss for the system operator. S2 is equivalent to: p c + K c < x ∗ Sufficient conditions: detailed analysis Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 10 / 18
The optimal stopping problem ( X t ) t ≥ 0 the price process Time of exercise by the system operator: τ e = inf { t ≥ 0 : X t ≥ x ∗ } ˆ Expected net present value of the strike price: x ≥ x ∗ , K c , h c ( x ) = E x { e − r ˆ τ e K c } = (1) ψ r ( x ) x < x ∗ . K c ψ r ( x ∗ ) , So, optimally timing the purchase of electricity corresponds to solving the following optimal stopping problem with non-smooth payoff: E x { e − r τ � � V c ( x ) = sup − X τ + p c + h c ( X τ ) } . (2) τ Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 11 / 18
Solution We apply characterisation via concavity of excessive functions for one-dimensional regular diffusions due to Dayanik and Karatzas(2003). Theorem Under p c + K c < x ∗ , an optimal stopping time is given by τ ∗ = inf { t ≥ 0 : X t ≤ x L } , with x L < x ∗ . 60 50 40 present value £ 30 20 10 0 −10 −40 −20 0 20 40 60 stopping boundary: £ per MWh Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 12 / 18
Lifetime contract Lifetime contract = an infinite sequence of call-off contracts back-to-back. Battery deterioration: A = 0 . 9999 remains after cycle 1.0 0.9 0.8 Capacity 0.7 0.6 0.5 0.4 0 2 4 6 8 10 thousands of cycles Value this contract using a fixed point argument. Provides a value of a storage unit on the balancing market. Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 13 / 18
The optimal stopping problem ξ ( x ) - the continuation value Time of exercise by the system operator: τ e = inf { t ≥ 0 : X t ≥ x ∗ } ˆ Expected net present value of the strike price: x ≥ x ∗ , K c + A ξ ( x ) , h ξ ( x ) = E x { e − r ˆ τ e ( K c + A ξ ( X ˆ τ e ) } = (3) [ K c + A ξ ( x ∗ )] ψ r ( x ) x < x ∗ . ψ r ( x ∗ ) , Fixed point argument (non-linear equation for ξ ) E x { e − r τ � � ξ ( x ) = sup − X τ + p c + h ξ ( X τ ) } . (4) τ Theorem Under p c + K c < x ∗ , an optimal stopping time is given by τ ∗ = inf { t ≥ 0 : X t ≤ x L } , with x L < x ∗ . Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 14 / 18
Lifetime value of storage 29280 29270 present value £ 29260 29250 29240 0 20 40 60 80 100 120 140 current £ per MWh x ∗ = 70, p c = 20, K c = 40 = ⇒ x L = 39 . 4 expected time x L → x ∗ is 2.63 days, expected time x ∗ → x L is 4.72 days Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 15 / 18
Is it worth optimising? 30000 25000 20000 present value £ 15000 10000 5000 20 30 40 50 60 stopping boundary: £ per MWh Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 16 / 18
Summary and related work Closed form solutions obtained for (i) single, and (ii) lifetime American-type options with physical delivery to support real-time power system balancing. Model calibration is possible from publicly available data in the UK John Moriarty, Dávid Szabó (i) model imbalance process + price stack function, (ii) put options Jhonny Gonzalez, John Moriarty: estimation (previous talk) Alessandro Balata, John Moriarty: numerical solution with periodic variations Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 17 / 18
Carmona R. and M. Ludkovski (2010): Valuation of energy storage: an optimal switching approach, Quantitative Finance, 10:4, 359-374 Dayanik S. and I. Karatzas (2003). On the optimal stopping problem for one-dimensional diffusions. Stochastic Processes and their Applications. 107:2, 173-212 Moriarty J. and J. Palczewski (2014). American Call Options for Power System Balancing. Available at SSRN: http://ssrn.com/abstract=2508258 Stephenson P . and M. Paun (2001), Electricity Market Trading. Power Engineering Journal , 277-288. Zervos M., T.C. Johnson and F. Alazemi (2013). Buy-Low and Sell-High Investment Strategies. Mathematical Finance, 23(3), 560-578 Jan Palczewski (Leeds) American options for energy balancing Milton Keynes, 1st June 2015 18 / 18
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