On a Market Model for Electricity Futures and Options

On a Market Model for Electricity Futures andOptions

Reik H. Borger

September 22, 2005

A Market Model for Electricity Futures and Options 1

Outline

• Characteristics of the (German) Futures Market

• Part I

– The Brownian Two-Factor Model– Pricing– Parameter Estimation– Discussion

• Part II

– The Jump-Diffusion Model– Model Properties– Discussion

University of Ulm

Department of Financial Mathematics

Reik H. Borger

September 22, 2005


Goal

Modelling and Pricing of Electricity Futures and Options consideringthe term structure of volatility

University of Ulm


Reik H. Borger

September 22, 2005


Characteristics of the (German) Futures Market

• Electricity Future – Obligation to buy/sell a specified amount of electricityduring a delivery period, typically a month, quarter or year.

• Futures show a decreasing volatility term structure

• Level of volatility depends on length of delivery period

University of Ulm


Reik H. Borger

September 22, 2005



University of Ulm


Reik H. Borger

September 22, 2005



Part IThe Brownian Two-Factor Model

Pricing

Parameter Estimation

Discussion

Part IIThe Jump-Diffusion Model

Model Properties

Discussion

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Reik H. Borger

September 22, 2005


The Brownian Two-Factor Model

• Modelling observable products, e.g. month futures

• Under a risk-neutral measure month forward prices F (t, Ti, Tj) = F (t, T )have to be martingales,

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Reik H. Borger

September 22, 2005


The Brownian Two-Factor Model

• Modelling observable products, e.g. month futures

• Under a risk-neutral measure month forward prices F (t, Ti, Tj) = F (t, T )have to be martingales,

• For a fixed delivery start T and delivery period 1 month, let the dynamics ofa Future Ft,T be given by the two factor model:

F (t, T ) = F (0, T ) exp{

µ(t, T ) +∫ t

0

σ1(s, T )dW (1)s + σ2W

(2)t

}with

– W (1) and W (2) independent Brownian motions– σ1(s, T ) = σ1e

−κ(T−s)

– σ1, σ2, κ > 0 constants– µ(t, T ) being the risk-neutral martingale drift

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Reik H. Borger

September 22, 2005


Pricing of Futures

• In this model, all products besides Month-Futures are derivatives

• Prices of quarterly and yearly Futures are given independent of the model asan average of the n corresponding monthly Futures:

1n

n∑i=1

e−r(Ti−t) (Ft,Ti−D)

• Yt,T1,...Tn = Y =P

e−rTiFt,TiPe−rTi

is the forward price of a n-month forward

quoted in the market (cp. swap rate)

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Reik H. Borger

September 22, 2005


Pricing of Options on Month-Futures

• At time t = 0, the price of a Call-Option with strike K and maturity T0 ona Month-Future Ft,T is given by

e−rT0E[(FT0,T −K)+

]• Within the model, FT0,T is lognormally distributed with known variance,

thus the option’s value is given by the formula (Black 76):

e−rT0E[(FT0,T −K)+

]= e−rT0 (F0,TN (d1)−KN (d2))

with d1,2 depending on the parameters σ1, σ2, κ.

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Reik H. Borger

September 22, 2005


Pricing of Options on quarterly and yearly Futures

• At time t = 0, the price of a Call-Option with strike K and maturity T0 ona n-Month-Future Y is given by

e−rT0E[(Y −K)+

]= e−rT0E

[(∑e−rTiFt,Ti∑

e−rTi−K

)+]

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Reik H. Borger

September 22, 2005




e−rT0E[(Y −K)+

]= e−rT0E


e−rTi−K

)+]

• The distribution of the sum is not known within the model. There is noexplicit solution to this integral.

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Reik H. Borger

September 22, 2005




e−rT0E[(Y −K)+

]= e−rT0E


e−rTi−K

)+]

• The distribution of the sum is not known within the model. There is noexplicit solution to this integral.

• Approximate the random variable Y by a logormal random variable Y withsame mean and variance (depending on the model parameters)

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September 22, 2005



• Approximate the random variable Y by a longormal random variable Y withsame mean and variance (depending on the model parameters)

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Reik H. Borger

September 22, 2005



• Approximate the random variable Y by a longormal random variable Y withsame mean and variance (depending on the model parameters)

• Then the option value can be computed by Black’s formula

e−rT0E[(Y −K)+

]≈ e−rT0E

[(Y −K

)+]

= e−rT0 (Y (0)N (d1)−KN (d2))

with d1,2 depending on the parameters σ1, σ2, κ.

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Reik H. Borger

September 22, 2005



• Use the approximating Black-formula

Option value = e−rT0 (Y (0)N (d1)−KN (d2))

d1,2 = d1,2

(Y (0),K, V ar(log Y (T0))

)Only the variance V ar(log Y (T0)) depends on the unknown parameters

• Compute the variances V ar(log Y (T0)) for products observable in the market

• Choose parameter σ1, σ2 and κ such that the model variances match themarket variances as good as possible (in the least-square sense)

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Reik H. Borger

September 22, 2005


Discussion

It has been the goal to set up a model that fits the volatility term structure...

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Reik H. Borger

September 22, 2005


Discussion

Parameter estimates are relatively stable over time.

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Reik H. Borger

September 22, 2005


Discussion

+ The model fits the volatility structure of Futures options reasonably well.

+ The model takes delivery periods into account.

+ The model can be used for pricing all the relevant products in the market,calibration time and accuracy is within the usual bounds of the market.

University of Ulm


Reik H. Borger

September 22, 2005


Discussion

+ The model fits the volatility structure of Futures options reasonably well.

+ The model takes delivery periods into account.

+ The model can be used for pricing all the relevant products in the market,calibration time and accuracy is within the usual bounds of the market.

- The model implies a spot process without jumps.

- Only few products observable, which can be used for calibration.

- The model does not reflect volatility smiles.

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Reik H. Borger

September 22, 2005



Part IThe Brownian Two-Factor Model

Pricing


Discussion

Part IIThe Jump-Diffusion Model

Model Properties

Discussion

University of Ulm


Reik H. Borger

September 22, 2005


The Jump-Diffusion-Model

• For a fixed delivery start T and delivery period 1 month, let the dynamics ofa Future Ft,T be given by the two factor model:

F (t, T ) = F (0, T ) exp

{µ(t, T ) +

∫ t

0

σ1(s, T )dW (1)s + σ2W

(2)t +

Nt∑Yi

}

with

– W (1) and W (2) independent Brownian motions– N a Poisson process with intensity λ– Yi independent and normal distributed with mean m and variance δ2

– σ1(s, T ) = σ1e−κ(T−s)

– σ1, σ2, κ > 0 constants– µ(t, T ) being the risk-neutral martingale drift

University of Ulm


Reik H. Borger

September 22, 2005


Model Properties

• Spot process is given by

St = F (t, t) = F (0, t) exp

{µ(t, t) + χt +

Nt∑Yi

}

χ being a mean-reverting process. This spot process also shows jumps andseasonality through F (0, t).

• Option prices on 1-month futures available as a series of Black-formulas.

• Approximate option prices on n-month futures available as a series ofBlack-formulas.

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Reik H. Borger

September 22, 2005


Calibration

• Risk-neutral calibration is possible using option prices as before. More dataavailable since also non-ATM-options can be used.

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Reik H. Borger

September 22, 2005


Calibration

• Risk-neutral calibration is possible using option prices as before. More dataavailable since also non-ATM-options can be used.

• In order to have a reasonable integrated spot-forward-model, jump parame-ters can be estimated historically, term structure parameters risk-neutrally(time-saving).

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Reik H. Borger

September 22, 2005


Discusion

The market term structure...

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Reik H. Borger

September 22, 2005


Discusion

...with a Brownian two-factor-model...

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Reik H. Borger

September 22, 2005


Discusion

...and a jump diffusion model.

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Reik H. Borger

September 22, 2005


Discusion

Volatility smile for the year 2007...

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Reik H. Borger

September 22, 2005


Discusion

...with the Brownian two-factor model implied smiles...

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Reik H. Borger

September 22, 2005


Discusion

...and a jump-diffusion model.

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Reik H. Borger

September 22, 2005


Discusion

Finally, a jump-diffusion fitted explicitly to this option maturity.

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Reik H. Borger

September 22, 2005


Discussion

• Jump-Diffusion-Model superior to Two-Factor-Brownian motion

• Time consuming

• How well are spot characteristics represented?

University of Ulm


Reik H. Borger

September 22, 2005

On a Market Model for Electricity Futures and Options

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