A spot price model feasible for electricity forward pricing Part II

A spot price model feasible for electricity forward pricing Part II Fred Espen Benth Centre of Mathematics for Applications (CMA) University of Oslo, Norway Wolfgang Pauli Institute, Wien January 17-18 2012

Plan of this lecture 1. Some preliminaries on Lévy processes 2. A multi-factor arithmetic model 3. Electricity forward pricing and option pricing

Some preliminaries on Lévy processes

L(t) is a Lévy process if Lévy processes 1. The increments L(t) L(s) are stationary 2. The increments L(t) L(s) are independent 3. RCLL paths Example 1: Brownian motion Increments are normally distributed, B(t) B(s) N (0, t s) Continuous paths Example 2: Poisson process Increments are Poisson distributed, N(t) N(s) Poiss(λ(t s)) Pure-jump process Inhomogeneous Lévy process: Remove stationarity condition Also called independent increment (II) process

Poisson random measure associated to a Lévy process N(A (0, t]) = 1( L(s) A) 0<s t L(s) = L(s) L(s ) Borel set A R\{0} N(A (0, t]) counts number of jumps of size A up to time t The compensator measure is defined as l(a (0, t]) = E [N(A (0, t])]

For a Lévy process L(t) l(a (0, t]) = t l(a) l(a) is called the Lévy measure The process t N(A (0, t]) tl(a) is a (local) martingale

Examples of Lévy processes Compound Poisson process N(t) X n L(t) = n=1 N Poisson process with intensity λ {X n } iid random variables Lévy measure is l(dz) = λf X (dz) Paths of finite variation Process of finite activity (finitely many jumps in a neighborhood around zero)

Subordinators (nondecreasing Lévy processes) Only positive jumps Finite variation paths Process of infinite activity (infinitely many jumps in a neighborhood around zero) Condition on the Lévy measure for subordinators 0 max(1, z) l(dz) <

The Lévy-Kintchine formula ln E [exp(ixl(t))] = ψ(x) t ψ(x) = iγx 1 2 x 2 σ 2 + {e ixz 1 ixz1( z < 1)} l(dz) R 0 The characteristic function of L(t) γ is the drift, σ the volatility of the Brownian motion part Brownian motion: l(dz) = 0, γ = 0 Pure-jump Lévy process: σ = 0

A Lévy process consists of three parts a drift a Brownian motion a pure-jump process General condition on the Lévy measure R 0 max(1, z 2 ) l(dz) <

The Ito Formula for jump processes X (t) a semimartingale and f (t, x) C 1,2 dx (t) = U(t) dt + V (t) dl(t) f (t, X (t)) f (0, X (0)) = t 0 f t (s, X (s)) ds + t 0 f x (s, X (s )) dx (s) + 1 t 2 σ2 f xx (s, X (s))v 2 (s) ds 0 + f (s, X (s)) f (s, X (s )) f x (s, X (s )) X (s) 0<s t Jumps X (s) = X (s) X (s ), no initial jump

Example: Solution of the SDE dx (t) = αx (t) dt + dl(t) L(t) a Lévy process Use f (t, x) = x exp(αt) in Ito

t t e αt X (t) X (0) = α e αs X (s) ds + e αs 1 dx (s) + 1 t 0 0 2 σ2 0 ds 0 + e αs X (s) e αs X (s ) e α 1 X (s) = α = 0<s t t 0 t 0 e αs X (s) ds α e αs dl(s) t t X (t) = e αt X (0) + e α(t s) dl(s) 0 0 t e αs X (s) ds + e αs dl(s) 0

A multi-factor arithmetic model

The model and properties The spot price as a sum of non-gaussian OU-processes B., Kallsen and Meyer-Brandis (2007) S(t) = Λ(t) n Y i (t) i=1 dy i (t) = α i Y i (t) dt + dl i (t) Λ(t) deterministic seasonality function L i (t) are independent subordinators Possibly time-inhomogeneous processes (II processes)

A simulation of S(t) fitted to EEX electricity data Calibration in B., Kiesel and Nazarova (2010) Top: simulated, bottom: EEX prices

Dynamics of S(t) ds(t) = { ( ) } X (t) α n Λ (t) S(t) dt + Λ(t) d L(t) Λ(t) AR(1)-process, with stochastic mean and seasonality Mean-reversion to stochastic base level n 1 X (t) = Λ(t) (α n α i )Y i (t) Seasonal speed of mean-reversion α n Λ (t)/λ(t) Seasonal jumps, where d L(t) = n i=1 dl i(t), dependent on the stochastic mean i=1

L i (t) jumps only upwards Jump size is a positive random variable Y i will mean-revert to zero However, Y i is always positive Ensures that S(t) is positive No Brownian motion component in the factors would give a probability for S(t) becoming negative EEX have several occurences of negative prices Hence, reasonable with a Brownian component in this case Calibration becomes simpler

Simulation of two processes Y i Spike process Y 1 with fast speed of reversion Normal variation Y 2 with slow mean-reversion Y 2 can be thought of as the stochastic mean Y 2 visually like a Brownian motion driven AR(1)-process

Y i (t) is stationary Recall from Ito s Formula Y i (u) = Y i (t)e α i (u t) + u t e α i (u s) dl i (s) The log-characteristic function of Y i (t) is, when t [ u ] ln E [exp(ixy i (u)] = iy i (0)e α i u + ln E exp(ix e α i (u s) dl i (s)) = iy i (0)e α i u + 0 u 0 ψ i (xe α i s ) ds Y (t) has a stationary distribution. 0 ψ i (xe α i (u s) ) ds

Autocorrelation function for S(t) := S(t)/Λ(t) We calculate, Cov( S(t + τ), S(t)) n n = Cov( Y i (t + τ), Y i (t)) = = i=1 i=1 n Cov(Y i (t + τ), Y i (t)) i=1 n Var(Y i (t)) e α i τ i=1

In conclusion ρ(t, τ) = corr[ S(t), S(t + τ)] = n ω i (t, τ)e α i τ i=1 If Y i are stationary, ω i (t, τ) = ω i The weights ω i sum to 1 The theoretical ACF can be used in practice as follows: 1. Find the number of factors n required 2. Find the speeds of mean-reversion by calibration to empirical ACF

Forward pricing

Definition of the electricity forward price with constant interest rate Delivery over [T 1, T 2 ]. Assuming financial settlement at maturity T 2 F (t, T 1, T 2 ) = E Q [ 1 T 2 T 1 T2 T 1 S(u) du F t Any Q P a risk-neutral probability/pricing measure ]

By commuting expectation and integration F (t, T 1, T 2 ) = 1 T 2 T 1 T2 T 1 f (t, u) du Here, f (t, u) is the price of a forward with fixed-delivery time at u, f (t, u) = E Q [S(u) F t ]

Restrict to a subclass of measures Q: the Esscher transform Common choice in energy when models have jumps Idea: structure preserving measures In the further analysis, let us for simplicity assume n = 1...i.e. only one factor Y 1 (t), denoted Y (t) no seasonlity Λ(u) = 1

The Esscher transform Define a martingale Z as θ is a constant Z(t) = exp (θl(t) φ(θ) t) φ(x) ψ( ix) is the log-mgf of L(1) Radon-Nikodym derivative for measure change: dq = Z(t) dp Ft

Effect of measure change on L: ψ θ (x) ln E θ [exp (ixl(1))] = ln E [exp ((ix + θ)l(1))] exp ( ψ( iθ)) = ψ(x iθ) ψ( iθ) = i(x iθ) + = ixγ + 0 i( iθ)γ {e i(x iθ)z 1} l(dz) L Lévy process under Q, with drift γ and Lévy measure exp(θz)l(dz) 0 0 {e i( iθ)z 1} l(dz) {e ixz 1} e θz l(dz)

To study Q-dynamics of Y, define Lθ (t) L(t) E θ [L(t)] = L(t) φ (θ)t Lθ is a Q-martingale dy (t) = ( φ (θ) αy (t) ) dt + d L θ (t) When θ 0, we have a change in the level of Y (t) φ (0) φ (θ) θ often called the market price of risk

Derivation of the forward price Calculate first f (t, u) using the Q-dynamics of Y Y (u) = Y (t)e α(u t) + φ (θ) u α (1 e α(u t) )+ e α(t s) d L θ (s) t Independent increment property of L θ yields f (t, u) = E θ [Y (u) F t ] = Y (t)e α(u t) + φ (θ) α (1 e α(u t) )

Integrating over the delivery period [T 1, T 2 ] yields the electricity forward price F (t, T 1, T 2 ) = Y (t)α(t, T 1, T 2 ) + φ (θ) α (1 α(t, T 1, T 2 )) α(t, T 1, T 2 ) = 1 (e ) α(t1 t) e α(t 2 t) α(t 2 T 1 ) Note that when T 1, T 2 are large, F (t, T 1, T 2 ) φ (θ)/α In the long end, forward prices vary very little

Dynamics of the forward price F (t, T 1, T 2 ) is a martingale (under Q) Q-dynamics of Y then yields, df (t, T 1, T 2 ) = α(t, T 1, T 2 ) d L θ (t)

α is the average of exp( α(u t)) over u [T 1, T 2 ] α i (t, T 1, T 2 ) = 1 T 2 T 1 T2 T 1 e α i (u t) du Hence, we have an average Samuelson effect exp( α(u t)) increasing when time to maturity u t goes to zero Volatility goes up as we approach delivery at time u Delivery over a period, so we average

Pricing of options on forwards Let g be the payoff of an option E.g, a put option g(x) = max(k x, 0) Call options require a damping factor in what follows (or one can use the put-call parity) Option price is p(t, T ; T 1, T 2 ) = e r(t t) E θ [max (K F (T, T 1, T 2 ), 0) F t ] Calculate this using Fourier transformation

The Fourier transform ĝ(y) = g(x) exp( ixy) dx The inverse Fourier transform: g(x) = 1 ĝ(y) exp(ixy) dy 2π From the dynamics of F we have F (T, T 1, T 2 ) = F (t, T 1, T 2 ) φ (θ) T T + α(s, T 1, T 2 ) dl(s) t t α(s, T 1, T 2 ) ds

By the independent increment property E θ [g(f (T, T 1, T 2 )) F t ] = 1 2π = 1 2π = 1 2π ĝ(y)e θ [ e iyf (T,T 1,T 2 ) F t ] dy ĝ(y)e iy(f (t,t 1,T 2 ) φ (θ) R [ T t α(s,t 1,T 2 ) ds) E θ e iy R ] T t α(s,t 1,T 2 ) L(s) dy ĝ(y)e iy(f (t,t 1,T 2 ) φ (θ) R T t α(s,t 1,T 2 ) ds) R T e t ψ θ (yα(s,t 1,T 2 )) ds dy

Fourier expression for option price ( the convolution product) p(t, T ; T 1, T 2 ) = e r(t t) (g Φ t,t ) (F (t, T 1, T 2 ) φ (θ) where T ( T ) Φ t,t (y) = exp ψ θ (y α(s, T 1, T 2 )) ds t t α(s, T 1, T 2 ) ds) Implementable using FFT techniques

Conclusions so far... A small introduction to the basics for Lévy processes Defined a multi-factor spot price model additive positive prices ensured possible to calibrate to data Pricing of electricity forwards on the spot model Explicit price and dynamics Possible to price options by transform methods

References Barndorff-Nielsen and Shephard (2001). Non-Gaussian OU based models and some of their uses in financial economics. J. Royal Statist. Soc. B, 63. Benth, Kallsen and Meyer-Brandis (2007). A non-gaussian OU process for electricity spot price modelling and derivatives pricing. Appl. Math. Finance, 14. Benth, Kiesel and Nazarova (2012). A critical empirical study of two electricity spot price models. To appear in Energy Economics