lecture iv: option pricing in energy markets · introduction spread options asian options quanto...
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Introduction Spread options Asian options Quanto options Conclusions
Lecture IV: Option pricing in energy markets
Fred Espen Benth
Centre of Mathematics for Applications (CMA)University of Oslo, Norway
Fields Institute, 19-23 August, 2013
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Introduction Spread options Asian options Quanto options Conclusions
Introduction
• OTC market huge for energy derivatives
• Highly exotic products:
• Asian options on power spot• Various (cross-commodity) spread options• Demand/volume triggered derivatives• Swing options
• Payoff depending on spot, indices and/or forwards/futures
• In this lecture: Pricing and hedging of (some) of these exotics
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Introduction Spread options Asian options Quanto options Conclusions
Example of swing options
• Simple operation of a gas-fired power plant: income is∫ τ
te−r(s−t)u(s) (P(s)− G (s)) ds
• P and G power and gas price resp, in Euro/MWh.• Heating rate is included in G ...
• 0 ≤ u(s) ≤ 1 production rate in MWh• Decided by the operator
• Value of power plant
V (t) = sup0≤u≤1
E[∫ τ
te−r(s−t)u(s) (P(s)− G (s)) ds | Ft
]• More fun if there are constraints on production volume....
• Maximal and/or minimal total production• Flexible load contracts, user-time contracts
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Introduction Spread options Asian options Quanto options Conclusions
• Tolling agreement: virtual power plant contract• Strip of European call on spread between power spot and fuel• Fuel being gas or coal
V (t) =
∫ τ
te−r(s−t)E [max (P(s)− G (s), 0) | Ft ] ds
• Spark spread, the value of exchanging gas with power• Dark spread, crack spread, clean spread....
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Introduction Spread options Asian options Quanto options Conclusions
German (EEX) spark spread in 2011
• Green: EEX power (Euro/MWh)
• Blue: Natural gas (Euro/MWH)
• Red: Spark spread, with efficiency factor (heat rate) of49.13%
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Introduction Spread options Asian options Quanto options Conclusions
Example: Asian options
• European call option on the average power spot price
max
(1
τ2 − τ1
∫ τ2
τ1
S(u) du − K , 0
)
• Traded at NordPool around 2000• ”Delivery period” a given month• Options traded until τ1, beginning of ”delivery”
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Introduction Spread options Asian options Quanto options Conclusions
Example: Energy quanto options
• Extending Asian options to include volume trigger
• Sample payoff
max
(1
τ2 − τ1
∫ τ2
τ1
S(u) du − KP , 0
)×max
(1
τ2 − τ1
∫ τ2
τ1
T (u) du − KT , 0
)
• T (u) is the temperature at time u• in a location of interest, or average over some area (country)
• Energy quantos on:• gas and temperature (demand)• or power and wind (supply)• Dependency between energy price and temperature crucial
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Introduction Spread options Asian options Quanto options Conclusions
Spread options (tolling agreements)
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Introduction Spread options Asian options Quanto options Conclusions
• Spread payoff with exercise time τ
max (P(τ)− G (τ), 0)
• P,G bivariate geometric Brownian motion −→ Margrabe’sFormula
• Introducing a strike K 6= 0, no known analytic pricing formula
• Our concern: valuation for exponential non-Gaussianstationary processes
• Exponential Levy semistationary (LSS) models
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Introduction Spread options Asian options Quanto options Conclusions
• Recall definition of LSS process from Lecture III
Y (t) =
∫ t
−∞g(t − s)σ(s) dL(s)
• L a (two-sided) Levy process (with finite variance)
• σ a stochastic volatility process
• g kernel function defined on R+
• Integration in semimartingale (Ito) sense• g(t − · · · )× σ(·) square-integrable
• Y is stationary whenever σ is• Prime example: g(x) = exp(−αx), Ornstein-Uhlenbeck
process
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Introduction Spread options Asian options Quanto options Conclusions
• Bivariate spot price dynamics
lnP(t) = ΛP(t) + YP(t)
lnG (t) = ΛG (t) + YG (t)
• Λi (t) seasonality function, Yi (t) LSS process with kernel giand stochastic volatility σi , i = P,G
• The stochastic volatilites are assumed independent of UP ,UG
• L = (UP ,UG ) bivariate (square integrable) Levy process• Denote cumulant (log-characteristic function) by ψ(x , y).
• We suppose that spot model is under Q• Pricing measure
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Introduction Spread options Asian options Quanto options Conclusions
Fourier approach to pricing
• To compute the expected value under Q for the spread:
• Factorize out the gas component
ΛP(τ)E
[eYG (τ)
(eYP(τ)−YG (τ) − h
ΛG (τ)
ΛP(τ)
)+
| Ft
]
• Apply the tower property of conditional expectation,conditioning on σi ,
• Recall being independent of L• Gt = Ft ∨ {σi (·) , i = P,G}
ΛP(τ)E
[E
[eYG (τ)
(eYP(τ)−YG (τ) − h
ΛG (τ)
ΛP(τ)
)+
| Gt
]| Ft
]
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Introduction Spread options Asian options Quanto options Conclusions
• For inner expectation, use that Z is the density of an Esschertransform for t ≤ τ
Z (t) = e∫ t−∞ gG (τ−s)σG (s) dUG (s)−
∫ t−∞ ψG (−igG (τ−s)σG (s)) ds
• ψG (y) = ψ(0, y), the cumulant of UG .• The characteristics of L is known under this transform
• This ”removes” the multiplicative term exp(YG (τ)) frominner expectation
• Finally, apply Fourier method
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Introduction Spread options Asian options Quanto options Conclusions
• Define , for c > 1,
fc(x) = e−cx(
ex − hΛG
ΛP
)+
• fc ∈ L1(R), and its Fourier transform fc ∈ L1(R)
• Representation of fc :
fc(x) =1
2π
∫Rfc(y)eixy dy
• Gives general representation for a random variable X
E[f (X )] =1
2π
∫fc(y)E
[ei(y−ic)X
]dy
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Introduction Spread options Asian options Quanto options Conclusions
TheoremSuppose exponential integrability of L. Then the spread option hasthe price at t ≤ τ
C (t, τ) =e−r(τ−t) ΛP(τ)
2π
∫Rfc(y)Φc
(YP(t, τ),YG (t, τ)
)Ψc,t,τ (y) dy
where, for i = P,G,
Yi (t, τ) =
∫ t
−∞gi (τ − s)σi (s) dUi (s)
Φc(u, v) = exp ((y − ic)u + (1− (iy + c))v)
and
Ψc,t,τ (y) = E[e∫ τt ψ((y−ic)gP(τ−s)σP ,((c−1)i−y)gG (τ−s)σG (s)) ds | Ft
]
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Introduction Spread options Asian options Quanto options Conclusions
• Note: spread price not a function of the current power andgas spot, but on Yi (t, τ), i = P,G
• Recalling theory from Lecture III: Yi (t, τ) isl given by thelogarithmic forward price....
ln fi (t, τ) = Xi (t, τ, σi (t)) + Yi (t, τ)
• No stochastic volatility, σi = 1: Xi is a deterministic function
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Introduction Spread options Asian options Quanto options Conclusions
Some remarks on hedging
• Power spot not tradeable, gas requires storage facilities
• Alternatively, hedge spread option using forwards!
• But incomplete model, so only partial hedging possible• Quadratic hedging, for example• May also depend on stochastic volatility, making model ”more
incomplete”
• In real markets: forwards on power and gas deliver over agiven time period
• Further complication, as we cannot easily express spread insuch forwards
• Further approximation of partial hedging strategy
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Introduction Spread options Asian options Quanto options Conclusions
Asian options
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Introduction Spread options Asian options Quanto options Conclusions
Asian options
• Options on the average spot price over a period• Traded at NordPool up to around 2000 for ”monthly periods”
• Recall payoff function
max
(1
τ2 − τ1
∫ τ2
τ1
S(u) du − K , 0
)
• Geometric LSS spot model:
lnS(t) = Λ(t) + Y (t)
• Y an LSS process with kernel g and stochastic volatility σ
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Introduction Spread options Asian options Quanto options Conclusions
• Pricing requires simulation• Propose an efficient Monte Carlo simulation of the path of an
LSS process
• Suppose that gλ(u) := exp(λu)g(u) ∈ L1(R) and its Fouriertransform is in L1(R)
Y (t) =1
2π
∫Rgλ(y)Yλ,y (t) dy
• Yλ,y (t) complex-valued Ornstein-Uhlenbeck process
Yλ,y (t) =
∫ t
−∞e(iy−λ)(t−s)σ(s) dL(s)
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Introduction Spread options Asian options Quanto options Conclusions
• Paths of Ornstein-Uhlenbeck processes can be simulatediteratively
Yλ,y (t+δ) = e(iy−λ)δYλ,y (t)+e(iy−λ)δ
∫ t+δ
te(iy−λ)(t−s)σ(s) dL(s)
• Numerical integration (fast Fourier) to obtain paths of Y• Extend g to R if g(0) > 0• Let g(u) = 0 for u < 0 if g(0) = 0.• Smooth out g at u = 0 if singular in origo
• Error estimates in L2-norm of the paths in terms oftime-stepping size δ
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Introduction Spread options Asian options Quanto options Conclusions
• Asian call option on Y over [0, 1], with strike K = 5
• Y BSS-process, with σ = 1, Y (0) = 10, and kernel (modifiedBjerksund model)
g(u) =1
u + 1exp(−u)
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Introduction Spread options Asian options Quanto options Conclusions
Issues of hedging
• Let F (t, τ1, τ2) be forward price for contract delivering powerspot S over τ1 to τ2: At t = τ2,
F (τ2, τ1, τ2) =1
τ2 − τ1
∫ τ2
τ1
S(u) du
• Asian option: call option on forward with exercise time τ2
• In power and gas, forwards are traded with delivery period• Hence, can price, but also hedge using these• Analyse based on forward price model rather than spot!
• Problem: many contracts are not traded in the settlementperiod
• Can hedge up to time τ1
• ..but not all the way up to exercise τ2
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Introduction Spread options Asian options Quanto options Conclusions
Example: quadratic hedging
• Hedge option with payoff X at exercise τ2, using ψ(s) forwards
• Assume Levy (jump) dynamics for the forward price• Martingale dynamics
• Can only trade forward up to time τ1 < τ2
V (t) = V (0) +
∫ t∧τ1
0ψ(s)dF (s)
+ 1{t>τ1}ψ(τ1)(F (t, τ1, τ2)− F (τ1, τ1, τ2))
• Predictable strategies ψ being integrable with respect to F .
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Introduction Spread options Asian options Quanto options Conclusions
• Minimize quadratic hedging error
E[(X − V (τ2))2]
• Solution:• Classical quadratic hedge up to time τ1,• thereafter, use the constant hedge
ψmin =E[X (F (τ2)− F (τ1)) | Fτ1 ]
E[(F (τ2)− F (τ1))2 | Fτ1 ]
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Introduction Spread options Asian options Quanto options Conclusions
Example: geometric Brownian motion
• X call option with strike K at time τ2
• t 7→ F (t, τ1, τ2) geometric Brownian motion with constantvolatility σ
• We suppose the forward is tradeable only up to time τ1 < τ2
• Quadratic hedge• N is the cumulative standard normal distribution function
ψ(t) =
{N(d(t)), t ≤ T1
ψmin, t > T1
where
ψmin =F (τ1)eσ
2(τ2−τ1)N(σ√τ2 − τ1 + d(τ1))− (K + F (τ1))N(d(τ1)) + KN(d(τ1)− σ
√τ2 − τ1)
F (τ1)(eσ2(τ2−τ1) − 1)
d(t) =ln(F (t, τ1, τ2)/K) + 0.5σ2(τ2 − t)
σ√τ2 − t
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Introduction Spread options Asian options Quanto options Conclusions
• Empirical example:• Annual vol of 30%, τ1 = 20, τ2 = 40 days• ATM call with strike 100• Quadratic hedge jumps 1.8% up at τ1 compared to delta hedge
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Introduction Spread options Asian options Quanto options Conclusions
Quanto options
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Introduction Spread options Asian options Quanto options Conclusions
• Recall payoff of an energy quanto option
max
(1
τ2 − τ1
∫ τ2
τ1
S(u) du − KP , 0
)×max (Tindex(τ1, τ2)− KT , 0)
• Tindex(τ1, τ2) temperature index measured over [τ1, τ2]• CAT index, say, or HDD/CDD
• Consider idea of viewing the contract as an option on twoforwards
• Product of two calls,• One on forward energy, and one on temperature (CAT forward)
• Main advantages• Avoid specification of the risk premium in the spot modelling• Can price ”analytically” rather than via simulation
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Introduction Spread options Asian options Quanto options Conclusions
Case study: bivariate GBM
• Consider bivariate GBM model
dFP(t, τ1, τ2) = σP(t, τ1, τ2)FP(t, τ1, τ2) dWP(t)
dFT (t, τ1, τ2) = σT (t, τ1, τ2)FT (t, τ1, τ2) dWT (t)
• σP , σT deterministic volatilities, WP ,WT correlated Brownianmotions
• May express the price of the quanto as a ”Black-76-like”formula
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Introduction Spread options Asian options Quanto options Conclusions
• Price of quanto at time t ≤ τ1 is
C (t) = e−r(τ2−t) {FP(t)FT (t)eρσPσTN(d∗∗∗P , d∗∗∗T )
−FP(t)KTN(d∗∗P , d∗∗T )− FT (t)KPN(d∗P , d∗T )
+KPKTN(dP , dT )}
where
di =ln(Fi (t)/K)− 0.5σ2
i
σi, d∗∗i = di + σi , i = P,T
d∗i = di + ρσj , d∗∗∗i = di + ρσj + σi , i, j = P,T , i 6= j
• N(x , y) bivariate cumulative distribution function withcorrelation ρ, equal to the one between WP , and WT
• σP and σT integrated volatility
σ2i =
∫ τ2
tσ2i (s, τ1, τ2) ds , i = P,T
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Introduction Spread options Asian options Quanto options Conclusions
Empirical study of US gas and temperature
• Temperature index in quanto is based on Heating-degree days
Tindex(τ1, τ2) =
∫ τ2
τ1
max(c − T (u), 0) du
• FT (t, τ1, τ2) HDD forward
• HDD forward prices for New York• Use prices for 7 first delivery months
• NYMEX gas forwards, monthly delivery• Use prices for coming 12 delivery months
• 3 years of daily data, from 2007 on
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Introduction Spread options Asian options Quanto options Conclusions
• Approach modelling of F (t, τ1, τ2) by F (t, τ), forward withfixed maturity date
• Choose the maturity date τ to be middle of delivery period• Price dynamics only for t ≤ τ !
• Two factor structure (long and short term variations)
dFi (t, τ) = Fi (t, τ){γi dWi + βie
−αi (τ−t) dBi
}i = G ,T
• Estimate using Kalman filtering• W and B strongly negatively correlated for both gas and
temperature• W ’s negatively correlated , B’s positively
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Introduction Spread options Asian options Quanto options Conclusions
• Compute quanto-option prices from our formula• The period τ1 to τ2 is December 2011• Current time t is December 31, 2010• Use market observed prices at this date for FG (t),FT (t)
• Prices benchmarked against independent gas and temperature• Quanto option price is equal to the product of two call options
prices, with interest rate r/2
strikes KG ,KT 1100,3 1200,5 1300,7
dependence 596 231 108independence 470 164 74
• Note: long-dated option, long-term components mostinfluencial
• These are negatively correlated, approx. −0.3
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Introduction Spread options Asian options Quanto options Conclusions
Conclusions
• European-style options can be priced using transform-basedmethods
• Example: spread options
• Path-dependent options require simulation of LSS processes• Suggested a method based on Fourier transform• Paths simulated via a number of OU-processes
• Considered ”new” quanto option• Priced using corresponding forwards• Case study from US gas and temperature market
• Discussed hedging based on minimizing quadratic hedge error• Particular consideration of no-trading constraint in delivery
period
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Introduction Spread options Asian options Quanto options Conclusions
References
• Barndorff-Nielsen, Benth and Veraart (2013). Modelling energy spot prices by volatility modulatedLevy-driven Volterra processes. Bernoulli 19(3), 803-845, 2013
• Barndorff-Nielsen, Benth and Veraart (2010). Modelling electricity forward markets by ambit fields.Preprint, CREATES, University of Aarhus.
• Benth, Saltyte Benth and Koekebakker (2008). Stochastic Modelling of Electricity and Related Markets,World Scientific
• Benth and Detering (2013). Quadratic hedging of average-options in energy. In preparation
• Benth, Lange and Myklebust (2012). Pricing and hedging quanto options in energy markets. Available atssrn.com
• Benth, Eyjolfsson and Veraart (2013). Approximating Lvy Semistationary processes via Fourier methods inthe context of power markets. Submitted.
• Benth and Solanilla (2013). Forward prices as weighted average of the spot for non-storable commodities.In preparation
• Benth and Zdanowicz (2013). Pricing spread options for LSS processes. In preparation
• Eberlein, Glau and Papapantoleon (2010). Analysis of Fourier transform valuation formulas andapplications. Applied Mathematical Finance 17, 211240.
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Introduction Spread options Asian options Quanto options Conclusions
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