Lecture 3 - Spread Options p. 1/19 Commodity and Energy Markets (Princeton RTG summer school in financial mathematics) Lecture 3 - Spread Option Pricing Michael Coulon and Glen Swindle June 17th - 28th, 2013 mcoulon@princeton.edu
Lecture 3 - Spread Options p. 2/19 Commodity Spread Options A general spread option payoff (at time T ) has the form: (ax T by T K) + where X T and Y T are different commodity prices (spot or forward): Input / Output (e.g., dark if X T is electricity, Y T is coal) Input / Output (e.g., crack if X T is refined product, Y T is crude) Calendar (e.g., X T is Dec13 forward,y T is Jun13 forward) Locational (e.g.,x T is Henry Hub gas, Y T is NorthEast gas) Spread options are of utmost importance, due to their strong link with physical assets (hence hedging and valuation tools). Examples above: Coal power plant, Oil refinery, Gas storage facility, Pipeline Optimal (unconstrained) operation mimics a string of spread options.
Lecture 3 - Spread Options p. 3/19 Classical Spread Option Pricing Margrabe (1978) derived a well-known closed-form formula for spread options when K = 0 (ie, exchange options ) and assets follow GBMs: ds (1) t = rs (1) t dt+σ 1 S (1) t dw (1) t ds (2) t = rs (2) t dt+σ 2 S (2) t dw (2) t dw (1) t dw (2) t = ρdt Then via a clever use of change of numeraire: V t = e r(t t) E Q t [ ( ) ] + S (1) T S(2) T = S (2) t E Q t ( S (1) T S (2) T 1 ) + we obtain (where σ 2 = σ 2 1 +σ 2 2 2ρσ 1 σ 2 ): V t = S (1) t Φ(d + ) S (2) t Φ(d ), and d ± = ( ) log S (1) t /S (2) t ± 1 2 σ2 (T t) σ. T t
Lecture 3 - Spread Options p. 4/19 Power Plants / Tolling Deals Goal: Power plant value approximated as string of spread options Plant Value exp( rt j )E Q[ ( PTj h g G Tj e g A Tj K ) ] + j J Main challenge: Capturing multi-commodity dependence structure (and link with demand), while retaining mathematical tractability. How can we attempt to model these relationships? Reduced-Form: Correlated Lognormals, etc. (including Margrabe and its extensions; see e.g. Carmona & Durrleman ( 03) ) Full Fundamental: Via production cost optimization problem. Structurally: Embedded into a model for spot power: Power = f(gas, Coal, Carbon,...). What are some of the problems with using Margrabe in this case?
Lecture 3 - Spread Options p. 5/19 Electricity Markets Many differences when compared with other commodity markets: Non-storability of electricity Hourly matching of supply & demand required for market clearing Wide price variation across different locations within a grid Clear links to costs of production and demand patterns Dependence on local conditions and market structure Recently deregulated, but rules can still matter (eg, bidding). These lead to many effects which we would like to capture: Highly complicated periodicities / seasonality Sudden price spikes! (jumps followed by rapid recoveries) High volatility, skew and kurtosis Mean reversion - at multiple time scales Complex forward curve movements (many factors needed) Correlation / cointegration with fuel prices
Lecture 3 - Spread Options p. 6/19 Electricity Price Spikes! Dramatic spikes in peak hours early Aug 2011 during Texas heatwave: 9:;.<"=<>?."3@,5;78" %!!!" $'!!" $!!!" #'!!" #!!!" '!!".A.?B<>?>BC"D=:B"=<>?.".A.?B<>?>BC"2./012","A:02" #!!" +!" *!" )!" (!" '!" &!" %!" $!" #!" -./012"34!!!"5678"!" *,#,##" *,$,##" *,%,##" *,&,##" *,',##" *,(,##" *,),##" *,*,##" *,+,##" *,#!,##"!"
Lecture 3 - Spread Options p. 7/19 What about negative prices? Let s look at West Texas zonal prices, instead of the main North Hub: (Transmission constraints, lots of volatile wind power, and subsidies!) (##$ '##$ "##$ %##$ &##$ #$!&##$!%##$!"##$ )*+!#,$ @.A:$B.C*A$D>+*6$E2F9.A$G%##,!%#&#H$$ -./!#,$ 012!#,$ 3*4!#,$ )56!#,$ 057!#,$ 89:!#,$ ;.9!#,$ )*+!&#$ 3*2!&#$ 012!&#$ )5+!&#$ )56!&#$ <.1!&#$ =>?!&#$
Lecture 3 - Spread Options p. 8/19 What about negative prices? Zooming in on a nine-day period: some very unusual dynamics! ("#$,$-./0$12$3405$647.0$89:;40$<=89$&%$!$>./$)?$&##,@$$ (%#$ (&#$ (##$ '#$ "#$ %#$ &#$ #$!&#$ ($ &$ )$ %$ *$ "$ +$ '$,$!%#$!"#$
Energy Price Correlations Example of power to gas relationship from ERCOT (Texas): #('!!"!!## /"'01$23!$*+,-.$%&'()$3#$4)5&1$467$5"86&"0$!"#$ '!"!!# #(&$!"!!##./012#345# &$"!!# #(&!!"!!## 6789:7;#57<# &!"!!# #(%$!"!!## %$"!!# #(%!!"!!## %!"!!# #($!"!!## $"!!# #()####!"!!# %*%*!+# '*%*!+# $*%*!+# +*%*!+#,*%*!+# %%*%*!+# %*%*!-# '*%*!-# $*%*!-# +*%*!-#,*%*!-# %%*%*!-# %*%*!,# '*%*!,# $*%*!,# +*%*!,#,*%*!,# %%*%*!,# %*%*%!# '*%*%!# $*%*%!# +*%*%!#,*%*%!# %%*%*%!# *+,-.$%&'()$$!"#$%&'()$ Lecture 3 - Spread Options p. 9/19
Observed Demand (Load) Demand is easily observable and follows fairly predictable seasonal patterns with short-term weather-driven fluctuations. (ERCOT daily avg data below) (!" '!" &!" %!" $!" #!"!" )*+,!'" )-.,!'" )*+,!(" )-.,!(" )*+,!/" )-.,!/" )*+,!0" )-.,!0" )*+,!1" )-.,!1" )*+,#!" )-.,#!" )*+,##" )-.,##" 23456"7*8.9":;<=*><"?@*A"BCDEFG" Lecture 3 - Spread Options p. 10/19
Lecture 3 - Spread Options p. 11/19 Structural Models for Power Hybrid / structural models provide a convenient compromise between reduced-form, and full fundamental: Identify Key Factors - Demand, Fuel Prices, Outages, etc. Choose function P t = B(t,D t,g t,...) to map to spot power. Exploit available forward looking market data. (e.g., fuel forward prices, regulatory changes, renewables) Examples from literature include: Spot price as a function of... DemandD t : Barlow (2002) Capacity ξ t : Burger et al. (2004), Cartea et al. (2007) Fuel prices G t : Pirrong, Jermakyan (2005), Aid et al. (2009,11) Big Challenge: Need for multiple fuels in many cases, and complex dependencies! Daily auction data provides a natural starting point.
Lecture 3 - Spread Options p. 12/19 The bid stack function Day-ahead generator bids arranged by price to form the bid stack Spot price P t is highest bid needed to match inelastic demandd t Merit order (of production costs) drives dynamics of the stack 600 PJM sample bid stacks 500 400 1st Feb 2003 1st Mar 2003 oil price ($) 300 200 gas 100 nuclear coal 0 0 10000 20000 30000 40000 50000 60000 70000 80000 quantity (MW)
Lecture 3 - Spread Options p. 13/19 Historical PJM Bid Dynamics Historically, Lower part of the PJM stack driven by coal (eg, 40% of ξ point). Upper part of the PJM stack driven by gas (eg, 70% of ξ point). However, recent evidence for a significant merit order change occurring (due to shale gas discoveries, dropping US gas prices to under $2). $(" $!" )(" )!" #(" #!" '(" '!" (" 0123"45"$!6"718-9"1-"592:;" $!6"1-"592:;" :123"7<8:=" '%!" '$!" '#!" '!!" &!" %!" $!" #!" '%!" '$!" '#!" '!!" &!" %!" $!" #!" 012"32".!4"567+8"6+"2819:".!4"6+"2819:" +18";12"5<79=" '&" '%" '$" '#" '!" &" %" $" #"!" '*+,-*!!" '*+,-*!'" '*+,-*!#" '*+,-*!)" '*+,-*!$" '*+,-*!(" '*+,-*!%" '*+,-*!." '*+,-*!&" '*+,-*!/" '*+,-*'!" '*+,-*''"!"!" '()*+(!!" '()*+(!'" '()*+(!#" '()*+(!," '()*+(!$" '()*+(!-" '()*+(!%" '()*+(!." '()*+(!&" '()*+(!/" '()*+('!" '()*+(''"!"
Lecture 3 - Spread Options p. 14/19 An alternative perspective Can look at bid stack as a histogram of bids Merit order is often visible through clusters of bids 12000 10000 nuc coal PJM sample bid histogram natural gas + a few higher bids in tail bid amount (MW) 8000 6000 4000 2000 0 0 16 32 48 64 96 128 160 192 224 256 288 320 480 640 800 bid price ($)
Lecture 3 - Spread Options p. 15/19 Distribution-based Bid Stack Model Coulon / Howison (09) - Stochastic Behaviour of the Electricity Bid Stack... Fuel typesi = 1,...,n with weights (relative capacities) w 1,...,w n Bid distributionsf 1 (x),...,f N (x) (proportion of bids belowx). Require 0 < D t / ξ < 1. (demand D t cannot exceed max capacity ξ) The bid stack function is then the quantile function of the distribution of bids. Then choose two-parameter distributions for bids such as Gaussian, etc. Set parameters (m i,s i ) to be linear in fuel price for each technology. Finally, pick typical processes (eg, exp OU) for factors C t, G t,d t, ξ t. The spot power price P t solves: N w i F i (P t ) = i=1 N ( ) Pt m i w i Φ i=1 s i = D t ξ t Key idea: Clusters of bids of each fuel type moving together (with fuel price).
Lecture 3 - Spread Options p. 16/19 Next challenge: closed-form! Multi-fuel case: no explicit expressions even for spot or forward. Alternative: allow slightly less flexibility in the stack but with the benefit of closed-form expressions for forwards, options and even spark or dark spread options. (e.g., payoff V T = (P T HG T ) + ) (Carmona / Coulon / Schwarz ( 13), Electricity Price Modeling and...) Key assumption: within each fuel type, heat rate differences lead to exponential bid stacks. (multiplicative in fuel price) Assume coal and gas generators only, with capacity ξ c and ξ g. Then aggregation of coal bids produces the sub bid stack : b c (D) = C t e k c+m c D, for 0 D ξ c and similarly for gas: b g (D) = G t e k g+m g D, for 0 D ξ g
Lecture 3 - Spread Options p. 17/19 Case of exponential sub bid stacks The total market bid stack (as a function of demand) is given by: B(x) = (b 1 c +b 1 g ) 1 (D), for 0 D ξ = ξ c + ξ g Hence, the result is piecewise exponential, although the precise form depends on ordering of start and endpoints of coal and gas stacks. Possible Expressions for P t Criteria Marginal Fuel Type b c (D) = C t e k c+m c D b c (D) < b g (0) Coal (no gas used) b g (D) = G t e k g+m g D b g (D) < b c (0) Gas (no coal used) b c (D ξ g ) = C t e k c+m c (D ξ g ) b g (D ξ c ) = G t e k g+m g (D ξ c ) b g ( ξ g ) < b c (D ξ g ) Coal (all gas used) b c ( ξ c ) < b g (D ξ c ) Gas (all coal used) b cg (D) = C α c t G α g t e β+γd otherwise Both (overlapping) α c = m g, α g = 1 α c, β = k cm g +k g m c, γ = m cm g, m c +m g m c +m g m c +m g
Lecture 3 - Spread Options p. 18/19 Exponential Stacks - Power vs Fuel Depicting power price P t as a function ofg t (or similarly C t ) leads to three different demand regimes, with three cases each (note: ξ c > ξ g below): Low Demand Medium Demand High Demand P4 High Demand: D > ξ c (i.e.,d > max( ξ c, ξ g )) Power Price P1 P5 P3 Medium Demand: ξ g < D < ξ c P2 Low Demand: D < ξ g (i.e.,d < min( ξ c, ξ g )) 0 Gas Price P1 to P5 on plot match rows 1 to 5 of previous table. Quadrilateral in middle represents region of coal and gas price overlap (ie, both generators at margin).
Lecture 3 - Spread Options p. 19/19 Exponential Bid Stack Model Topic to be continued in next lecture...