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Return to Risk Limited website: www.RiskLimited.com Valuation and Hedging of Power-Sensitive Contingent Claims for Power with Spikes: a Non-Markovian Approach Valery A. Kholodnyi February 25, 2004 Houston, Texas

Introduction • As the power markets are becoming deregulated worldwide, the modeling of the dynamics of power spot prices is becoming one of the key problems in the risk management, physical assets valuation, and derivative pricing. • One of the main difficulties in this modeling is to combine the following features: • To provide a mechanism that allows for the absence of spikes in the prices of power-sensitive contingent claims while the power spot prices exhibit spikes, and • To keep the dynamics of the prices of power-sensitive contingent claims consistent with the dynamics of the power spot prices.

Models for Power Spot Prices with Spikes • Mean-Reverting Jump Diffusion Process (Ethier and Dorris, 1999; Clewlow, Strickland and Kaminski, 2000) • the same mechanism is responsible for both the decay of spikes and the reversion of power prices to their equilibrium mean • Mixture of Processes (Goldberg and Read, 2000; Ball and Torous, 1985) • spikes and the regular, that is, inter-spike regime do not persist in time • relatively difficult to estimate parameters • Regime Switching Process (Ethier, 1999; Duffie and Gray 1995) • discreet time regime switching • inconsistent short term option values • relatively difficult to estimate parameters

The Non-Markovian Process for Power Spot Prices with Spikes • Motivation • Different mechanisms should be responsible for: • the reversion of power prices to their equilibrium mean in the regular, that is, inter-spike state • the reversion of power prices to their long term mean in the spike state, that is, for the decay of spikes • This is, in our opinion, due to the substantial difference in the scales of the deviations of power prices from their equilibrium mean in the spike and inter-spike states • For example, power prices in the US Midwest in June 1998 rose to $7,500 per megawatt hour (MWh) compared with typical prices of around $30 per MWh

The Non-Markovian Process for Power Spot Prices with Spikes • Main Features • The spikes are modeled directly as self-reversing jumps, either multiplicative or additive, in continuous time • The parameters that characterize spikes are frequency, • duration, and magnitude • The spikes parameters are directly observable from market data as well as admit structural interpretation • The spike state and the regular, that is, inter-spike state do persist in time

The Non-Markovian Process for Power Spot Prices with Spikes • Formal Definition • Define (Kholodnyi, 2000)the non-Markovian process for the power spot prices with spikes by • t>0 is the power spot price at time t , • is the multiplicative magnitude of spikes at time t , • is the inter-spike power spot price at time t. • Assume that the spike process and inter-spike process are independent Markov processes.

The Non-Markovian Process for Power Spot Prices with Spikes • Underlying Two-State Markov Process • Denote by Mt a two-state Markov process with continuous time t  0. • Denote the 22 transition matrix for the two-state Markov process Mt by • Pss(T,t) and Prs(T,t) are transition probabilities from the spike state at time t to the spike and regular states at time T, and • Psr(T,t) and Prr(T,t) are transition probabilities from the regular state at time t to the spike and regular states at time T.

The Non-Markovian Process for Power Spot Prices with Spikes Generators of the Underlying Two-State Markov Process The family of 22 matrices L = {L(t) : t  0} defined by is said to generate the two-state Markov process Mt, and the 22 matrix is called a generator. In terms of the generators, P(T,t) is given by

The Non-Markovian Process for Power Spot Prices with Spikes Decompositions of the Transition Probabilities of the Underlying Two-State Markov Process It can be shown that Moreover where

The Non-Markovian Process for Power Spot Prices with Spikes Underlying Two-State Markov Process in the Time-Homogeneous Case In the special case of a time-homogeneous two-state Markov process Mt the transition matrix P(T-t) and the generator L are given by and

The Non-Markovian Process for Power Spot Prices with Spikes Construction of the Spike Process t (t,) 1 Time Mt Regular State Spike State Time

The Non-Markovian Process for Power Spot Prices with Spikes Formal Definition of the Spike Process The transition probability density function for the spike process t as a Markov process is given by where (x) is the Dirac delta function.

The Non-Markovian Process for Power Spot Prices with Spikes Inter-Spike Process For example, can be a diffusion process defined by where: is the drift, is the volatility, and Wt is the Wiener process. In the practically important special case of a geometric-mean reverting process we have where: is the mean-reversion rate, is the equilibrium mean, and is the volatility.

The Non-Markovian Process for Power Spot Prices with Spikes The Expected Time for t to be in the Spike and Inter-Spike States The expected time for t to be in the spike state that starts at time t is: Similarly, the expected time for t to be in the inter-spike state that starts at time t is: In the special case of a time-homogeneous two-state Markov process Mt:

The Non-Markovian Process for Power Spot Prices with Spikes • Interpretation of the Spike State of tas Spikes in Power Prices • If the expected time for the non-Markovian process t to be in the spike state is small relative to the characteristic time of change of the process then the spike state of t can be interpreted as spikes in power spot prices: • t can exhibit sharp upward price movements shortly followed by equally sharp downward prices movements of approximately the same magnitude. • For example, if is a diffusion process then: • and • In this case is the expected lifetime of a spike and is the expected lifetime between two consecutive spikes.

The Non-Markovian Process for Power Spot Prices with Spikes • Estimation of the Spike Parameters • In the special case of a time-homogeneous two-state Markov process the expected life-time of a spike is given by • Similarly, the expected life-time between two consecutive spikes is given by • The estimation of the probability density function (t,) for the spike magnitude can be based on the standard parametric or nonparametric statistical methods • Scaling and asymptotically scaling distributions are of a particular interest in practice

The Non-Markovian Process for Power Spot Prices with Spikes The Non-Markovian Process t as a Markov Process with the Extended State Space The state of the power market at any time t can be fully characterized by a pair of the values of the processes , and at time t. Moreover, although the process t is non-Markovian it can be, in fact, represented as a Markov process that for any time t can be fully characterized by the values of the processes and at time t. Equivalently, the non-Markovian process t can be represented as a Markov process with the extended state space that at any time t consists of all possible pairs with and .

European Contingent Claims on Power in the Absence of Spikes Valuing European Contingent Claims on Power as the Discounted Risk-Neutral expected value of its payoff Denote by the value of the European contingent claim on power with inception time t, expiration time T, and payoff g. The value of this European contingent claim can be found as the discounted risk-neutral expected value of its payoff: where is the risk-neutral transition probability density function.

European Contingent Claims on Power in the Absence of Spikes Example: Geometric Mean-Reverting Process It can be shown (Kholodnyi 1995) that where:

European Contingent Claims on Power in the Absence of Spikes Example: Geometric Mean-Reverting Process For example (Kholodnyi 1995): where: with:

European Contingent Claims on Power in the Presence of Spikes Notation Denote by the value of the European contingent claim on power with inception time t, expiration time T, and payoff The payoff g can explicitly depend, in addition to the power price at time T, on the state, spike or inter-spike state, of the power price and the magnitude of the related spike. If g depends only on the power price at time T we have

European Contingent Claims on Power in the Presence of Spikes General Case The value E(t,T,g) can be found as the discounted risk-neutral expected value of the payoff g where is the the transition probability density function for t represented as a Markov process.

European Contingent Claims on Power in the Presence of Spikes The Case When (t,) is Time-Independent The value E(t,T,g) is given by

European Contingent Claims on Power in the Presence of Spikes The Case of Spikes with Constant Magnitude Consider a special case of spikes with constant magnitude  > 1, that is, when ()is the delta function (- `). The value E(t,T,g) is given by

European Contingent Claims on Power in the Presence of Spikes Linear Evolution Equation for European Contingent Claims on Power with Spikes It can be shown (Kholodnyi 2000) that the value E(t,T,g) of a European contingent claim on power with spikes is the solution of the following linear evolution equation where and are the generators of and as Markov processes.

European Contingent Claims on Power in the Presence of Spikes Linear Evolution Equation for European Contingent Claims on Power with Spikes In a practically important special case when is a geometric mean-reverting process the generator is given by The generator is a linear integral operator with the kernel:

European Contingent Claims on Power in the Presence of Spikes Linear Evolution Equation for European Contingent Claims on Power with Spikes In the special case of spikes with constant magnitude the generator (t) can be represented as the 22 matrix L*(t) transposed to the generator L(t) of the Markov process Mt. In turn, v and g can be represented as two-dimensional vector functions Note that (t) represented as L*(t) can also be expressed in terms of the Pauli matrices. This gives rise to an analogy between the linear evolution equation for E(t,T,g) and the Schrodinger equation for a nonrelativistic spin 1/2 particle.

Why Prices of European Claims On Power Do Not Spike Ergodic Transition Probabilities for Mt Assume that the spikes have constant magnitude  and the underlying two-state Markov process Mt is time-homogeneous. The transition probabilities for Mt can be represented as follows: Pss(T,t) = s + O(e-(T - t)a), Psr(T,t) = s + O(e-(T - t)a), Prs(T,t) = r + O(e-(T - t)a), Prr(T,t) = r + O(e-(T - t)a), where: s = b/(a + b) and r = a/(a + b) are the ergodic transition probabilities.

Why Prices of European Claims On Power Do Not Spike Values of European Contingent Claims on Power Far From Expiration The values Et=(t,T,g) and Et=1(t,T,g) of European contingent claims on power coincide up to the terms of order O(e-(T - t)a) and hence can be combined into a single expression as follows (Kholodnyi 2000): When T - t >> , Et=(t,T,g) and Et=1(t,T,g) differ only by an exponentially small term. As a result, prices of European contingent claims on power do not exhibit spikes while the power spot prices do.

Why Prices of European Claims On Power Do Not Spike Values of European Contingent Claims on Power Far From Expiration For example, (Kholodnyi 2000) the values of European call and put options with inception time t, expiration time T, and strike X are given by:

Why Prices of European Claims On Power Do Not Spike Example: Geometric Mean-Reverting Inter-Spike Process It can be shown (Kholodnyi 2000) that the value E(t,T,g) of a European options with inception time t , expiration time T, and payoff g is given by where:

Why Prices of European Claims On Power Do Not Spike Example: Geometric Mean-Reverting Inter-Spike Process For example, (Kholodnyi 2000) the values of European call and put options with inception time t , expiration time T, and strike X are given by where

Why Prices of European Claims On Power Do Not Spike Short-Lived Spikes Consider the case of short-lived spikes, that is . Then for the ergodic transition probabilities we have s = tch + o(tch) and r = 1 - tch + o(tch), where In turn, the value E(t,T,g) can be expressed as a correction to the value Ê(t,T,g):

Why Prices of European Claims On Power Do Not Spike Example: Geometric Mean-Reverting Inter-Spike Process It can be shown (Kholodnyi 2000) that the values of European call and put options with strike X are given by where

Power Forward Prices for Power Spot Prices Without of Spikes Power Forward Prices as Risk-Neutral Expected Power Spot Prices Denote by the power forward price at time t for the forward contract with maturity time T. Power forward price can be found as the risk-neutral expected value of the power spot prices at time T:

Power Forward Prices for Power Spot Prices Without of Spikes Example: Geometric Mean-Reverting Inter-Spike Process It can be shown (Kholodnyi 1995) that power forward prices are given by the following analytical expression: where:

Power Forward Prices for Power Spot Prices Without of Spikes Example: Geometric Brownian Motion (GBM) for Power Forward Prices The risk-neutral dynamics of is described by a geometric Brownian motion: where:

Power Forward Prices for Power Spot Prices With Spikes General Case Denote by the power forward price at time t for the forward contract with maturity time T. Power forward price F(t,T) can be found as the risk-neutral expected value of the power spot prices Tat time T: where is the risk-neutral average magnitudes of spikes

Power Forward Prices for Power Spot Prices With Spikes The Case When (t,) is Time-Independent The risk neutral average magnitude of spikes is given by where is the risk-neutralconditional average magnitude of spikes given by For example, if ()is corresponds to a scaling probability distribution, that is, ()= -1- , then

Power Forward Prices for Power Spot Prices With Spikes The Case of Spikes with Constant Magnitude Consider a special case of spikes with constant magnitude  > 1, that is, when ()is the delta function (- `). The risk neutral average magnitude of spikes is given by

Why Power Forward Prices Do Not Spike Ergodic Transition Probabilities for Mt Assume again that the spikes have constant magnitude  and the underlying two-state Markov process Mt is time-homogeneous. The transition probabilities for Mt can be represented as follows: Pss(T,t) = s + O(e-(T - t)a), Psr(T,t) = s + O(e-(T - t)a), Prs(T,t) = r + O(e-(T - t)a), Prr(T,t) = r + O(e-(T - t)a), where: s = b/(a + b) and r = a/(a + b) are the ergodic transition probabilities.

Why Power Forward Prices Do Not Spike Ergodic Average Magnitude of Spikes The risk-neutral average magnitudes of spikes and coincide up to the terms of order O(e-(T - t)a). Therefore, and can be combined into a single expression as follows: where is the risk-neutralergodic average magnitude of spikes given by

Why Power Forward Prices Do Not Spike Power Forward Prices far From Maturity The power forward prices Ft=(t,T) and Ft=1(t,T) coincide up to the terms of order O(e-(T - t)a). Therefore, Ft=(t,T) and Ft=1(t,T) can be combined into a single expression as follows: When T - t >> , Ft=(t,T) and Ft=1(t,T) differ only by an exponentially small term. As a result, power forward prices do not exhibit spikes while the power spot prices do.

Why Power Forward Prices Do Not Spike Short-Lived Spikes Consider the case of short-lived spikes, that is . Then for the ergodic transition probabilities we have s = tch + o(tch) and r = 1 - tch + o(tch), where For the average magnitude of spikes we have In turn, F(t,T) can be expressed as a correction to

Why Power Forward Prices Do Not Spike • Example: GBM for Power Forward Prices • Assume that the power forward prices follow a geometric Brownian motion. • this is, for example, the case when the power spot prices • follow a geometric mean-reverting process. • Then power forward prices F(t,T) far from maturity also follow the same geometric Brownian motion. • This, for example, can be used for: • the estimation of the volatility for the geometric Brownian motion for , • the estimation of the volatility and the mean-reversion rate for the geometric mean-reverting process for , and • dynamic hedging of derivatives on forwards on power.

European Contingent Claims on Forwards on Power with Spikes Geometric Mean-Reverting Inter-Spike Process and Spikes with Constant Magnitude It can be shown (Kholodnyi 2000) that the value of a European contingent claim (on forwards on power for power with spikes) with inception time t, expiration time T, and payoff g is given by:

European Contingent Claims on Forwards on Power with Spikes Geometric Mean-Reverting Inter-Spike Process and Spikes with Constant Magnitude For example, (Kholodnyi 2000) the values of European call and put options (on forwards on power for power with spikes) with inception time t, expiration time T, and strike X are given by:

European Contingent Claims on Forwards on Power with Spikes Geometric Mean-Reverting Inter-Spike Process and Short-Lived Spikes with Constant Magnitude It can be shown (Kholodnyi 2000) that the value of a European contingent claim (on forwards on power for power with spikes) with inception time t, expiration time T, and payoff g can be represented as the following correction:

European Contingent Claims on Forwards on Power with Spikes Geometric Mean-Reverting Inter-Spike Process and Short-Lived Spikes with Constant Magnitude For example, (Kholodnyi 2000) the values of European call and put options (on forwards on power for power with spikes) with inception time t, expiration time T, and strike X can be represented as the following corrections:

Extensions of the Model • Both positive and negative spikes as well as spikes of more complex shapes can be considered • European contingent claims on power with spikes and another commodity that does not exhibit spikes can also be valued. Those include fuel and weather sensitive derivatives such as spark spread options and full requirements contracts • European options on power at two distinct points on the grid with spikes in both power prices can also be valued. Those include transmission options • Contingent claims of a general type such as universal contingent claims on power with spikes can be valued with the help of the semilinear evolution equation for universal contingent claims (Kholodnyi, 1995). Those include Bermudan and American options.

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