Spot Price ModelsPJM Day Ahead Price

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6/1/

2000

8/1/

2000

10/1

/200

0

12/1

/200

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2/1/

2001

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8/1/

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Date

Pri

ce

Spot Price Dynamics

• An examination of the previous graph shows several items of interest.– Price series exhibit spikes with seasonal

component (more in summer than other times).– Prices show seasonality (higher in summer with

a secondary peak in winter)– Volatility varies over period– Prices revert to a mean

ReturnsLog Returns

-2

-1.5

-1

-0.5

0

0.5

1

1.5

1

37

73

10

9

14

5

18

1

21

7

25

3

28

9

32

5

36

1

39

7

43

3

46

9

Date

Series1

Objective

• Find model / return generating process that describes the preceding graphs

• Several models have been used.

• No clear cut conclusion

• Trade-off between ease of implementation (use) and descriptive power.

• Art rather than science.

Models Commonly Used

• General Brownian Motion (GBM)

• Mean Reversion (Ornstein-Uhlenbeck)

• Mean Reversion with Jumps

• Stochastic Volatility

• GARCH(p,q)

• Markov switching Model

Geometric Brownian Motion

• GBM model in natural log of price

• ( x= ln(S)).

• Drift does not depend upon x as

x dt z

r c1

22

Simulating Brownian Motion

Brownian Motion in Trees

Mean Reversion Models

• Continuous time model in natural log of price.

is the mean the rate of reversion to the mean the standard deviation

dx

( ) x

1

22

dt dz

Mean Reversion: Simulation

• Discrete version

• Issue of step size– Drift term function of log of price

x

( ) x

1

22

dt t

Aside: half-life of MR

T

1

2

( )ln 2

• Average time to return to one-half a deviation from the average price

• The larger , the shorter the time to revert.

MR Parameter Estimation

• Historical Basis: regression of change in log of price on log of price.

• We now have the slope and intercept and can compute the mean reversion parameters.

x t x t

Scatter Plot Day Ahead

0

1

2

3

4

5

6

-2 -1.5 -1 -0.5 0 0.5 1 1.5

Series1

Parameter Estimation

• From the OLS estimation, we can recover the parameter values of the process.

2( )ln 1

1

( )ln 1 1

2 1

0

1

( )ln 1 1

MR Jump Diffusion Process

• Electricity Prices exhibit jumps.

• The jumps do not persist, but are more like spikes, i.e. quickly return to a ‘base’ level.

• The last term represents the jump component.

d S ( ) ( )ln S S dt S dz S dq

Estimation of Jump Parameters

• The MR jump process can be estimated via a number of techniques. These range from the heuristic through formal statistical methods such as maximum likelihood.

• ML tends to overestimate jump intensity hence we will simply use the heuristic approach.

• Estimation via recursive filter• Need the intensity, size and variance of jumps.

Estimation Procedure

• Determine some (arbitrary) level at which a return is a jump.

• Count number of ‘jumps’ and divide by time in years to obtain (frequency).

• Compute jump return and standard deviation.

• Repeat until estimates converge.

Stochastic Volatility

dS S dt S dzd 2 ( )2 2

dt 2dw

• This is an extension to GBM wherein the volatility is no longer constant, but random.

• Two factors: Price and Volatility.

• Mean reverting in volatility.

GARCH

• Generalized Autoregressive Conditional Heteroskedacity

• Bollerslev (1986)

• Process for variance.

t2

0

i u t i2

i t i

2

Markov Switching

• These models are based upon the idea that the returns will be due to multiple regimes.

• Simplest model is two states, one a regular state and the other a spike.

• Inputs:– Transition matrix– two stochastic equations.

Markov Switching

• Transition matrix

• p(t) is the probability of a spike on day t.

• q(t) is the probability that a spike ends at t.

P

1 ( )p t ( )p t

q 1 q

Comparison of Models

• Issues in simulation

• Terminal values more widely dispersed in GBM.

• Mean expressed in terms of ln(S).