A Two-Factor Model for the Electricity Forward Market Ruediger Kiesel1 1 Department

Gero Schindlmayr

2

Reik H. Boerger

of Financial Mathematics, University of Ulm

Department of Statistics, London School of Economics 2 EnBW 3 Department

Trading GmbH

of Financial Mathematics, University of Ulm

March 29, 2006

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Characteristics of the (German) Futures Market

The Brownian Two-Factor Model

Pricing

Calibration

Discussion

Characteristics

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Electricity Futures – Obligation to buy/sell a specified amount of electricity during a delivery period, typically a month, quarter or year.

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Futures show a decreasing volatility term structure

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Level of volatility depends on length of delivery period

Prices of Futures

Implied Prices of Futures

Volatilities

Model Objectives

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Capture the decreasing volatility term structure

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Model volatility dependent on length of delivery period

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Computational tractable

The Model Framework

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Use observable products, e.g. month futures as building blocks,

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

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Assume the dynamics dF (t, T ) = σ(t, T )F (t, T )dW (t), where σ(t, T ) is an adapted d-dimensional deterministic function and W (t) a d-dimensional Brownian motion.

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Initial value of this SDE is the initial forward curve observed at the market.

Options on Building Blocks

A European call option on F (t, T ) with maturity T0 and strike K can be easily evaluated by the Black-formula V option (0) = e −rT0 (F (0, T )N (d1 ) − K N (d2 )) , where N denotes R T the normal distribution, Σ(T0 , T ) = 0 0 ||σ(s, T )||2 ds and d1 d2

) + 12 Σ(T0 , T ) log F (0,T K p = Σ(T0 , T ) p = d1 − Σ(T0 , T )

(1)

The Model Framework – n-period futures

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Use observable products, e.g. month futures as building blocks,

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Express an n-period delivery futures as Pn e −r (Ti −t) F (t, Ti ) Pn YT1 ,...,Tn (t) = i=1 . −r (Ti −t) i=1 e (Compare modelling of forward swap rates in terms of forward LIBOR rates)

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In case of 1-year-futures, the swap rate is the forward price of the 1-year-futures, which can be also observed in the market.

Options on n-period futures I

We need to compute   e −rT0 E (Y (T0 ) − K )+ ,

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where the distribution of Y as a sum of lognormals is unknown. We approximate Y by a random variable Yˆ , which is lognormal and matches Y in mean and variance. Then, log Yˆ ∼ N (m, s) with s 2 depending on the choice of the volatility functions σ(t, Ti ).

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An analysis of the goodness of the approximation can be found in Brigo-Mercurio (2003).

Options on n-period futures Using this approximation, it is possible to apply a Black-Option formula again to obtain the option value as   V option = e −rT0 E (Y (T0 ) − K )+  +  −rT0 ˆ ≈ e E Y (T0 ) − K = e −rT0 (Y (0)N (d1 ) − K N (d2 )) with log YK(0) + 12 s 2 s = d1 − s

d1 = d2

(2)

A Two-Factor Model

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For a fixed delivery start T and delivery period 1 month, let the dynamics of a Future Ft,T be given by the two factor model:   Z t (1) (2) F (t, T ) = F (0, T ) exp µ(t, T ) + σˆ1 (s, T )dWs + σ2 Wt 0

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W (1) and W (2) independent Brownian motions

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σˆ1 (s, T ) = σ1 e −κ(T −s)

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σ1 , σ2 , κ > 0 constants

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µ(t, T ) being the risk-neutral martingale drift

Model Parameters σ1 affects the level at the short end of the volatility curve

Model Parameters κ affects the slope of the volatility curve at the short end

Model Parameters σ2 affects the level at the long end of the volatility curve

Pricing of Futures

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In this model, all products are expressed using Month-Futures

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Prices of quarterly and yearly Futures are given as an average of the n corresponding monthly Futures.

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t,Ti Yt,T1 ,...Tn = Y = is the forward price of a n-month e −rTi forward quoted in the market (cp. swap rate)

Pe P

−rTi F

Pricing of Options on Month-Futures I

At time t = 0, the price of a Call-Option with strike K and maturity T0 on a Month-Future Ft,T is given by   e −rT0 E (FT0 ,T − K )+

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Within the model, FT0 ,T is log-normally distributed with known variance Σ(T0 , T ) =

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σ12 −2κ(T −T0 ) (e − e −2κT ) + σ22 T0 2κ

Thus the option’s value is given by the formula (Black 76):   e −rT0 E (FT0 ,T − K )+ = e −rT0 (F0,T N (d1 ) − K N (d2 )) with d1,2 depending on the parameters σ1 , σ2 , κ.

Pricing of Options on quarterly and yearly Futures

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At time t = 0, the price of a Call-Option with strike K and maturity T0 on a n-Month-Future Y is given by " P + # −rTi F   e P −rTt,Ti − K e −rT0 E (Y − K )+ = e −rT0 E e i

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The distribution of the sum is not known within the model. There is no explicit solution to this integral.

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Approximate the random variable Y by a log-normal random variable Yˆ with same mean and variance (depending on the model parameters)

Matching the Variance

Using the moment-generating function of a normal random variable, we get exp(s 2 ) =

Var(Y ) (E (Y ))2

+1=

E (Y 2 ) E (Y )2

From the martingale property E (FT0 ,Ti ) = F0,Ti and P −r (T −T0 ) i e F0,T E (YT1 ,...Tn (T0 )) = P −r (T −T ) i . 0 i e

Matching the Variance

So P 2

E (YT1 ,...Tn (T0 ) ) =

i,j

e −r (Ti +Tj −2T0 ) F0,Ti F0,Tj · exp Covij P −r (T −T )
2 0 i e

with Covij = Cov(log F (T0 , Ti ), log F (T0 , Tj )). The covariance can be computed directly from the explicit solution of the SDE Cov(log F (T0 , Ti ), log F (T0 , Tj )) σ2

= e −κ(Ti +Tj −2T0 ) 2κ1 (1 − e −2κT0 ) + σ22 T0

Pricing of Options on quarterly and yearly Futures

The option value can be computed by Black’s formula  +    e −rT0 E (Y − K )+ ≈ e −rT0 E Yˆ − K = e −rT0 (Y (0)N (d1 ) − K N (d2 )) with d1,2 depending on the parameters σ1 , σ2 , κ.

Parameter Estimation I

Use the approximating Black-formula Option value = e −rT0 (Y (0)N (d1 ) − K N (d2 ))   d1,2 = d1,2 Y (0), K , Var (log Yˆ (T0 ))

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Only the variance Var (log Yˆ (T0 )) depends on the unknown parameters Compute the variances Var (log Yˆ (T0 )) for products observable in the market Choose parameter σ1 , σ2 and κ to minimize the distance of the model variances to the market variances in a given metric (in the least-square sense)

Data

Product Month Month Month Quarter Quarter Quarter Quarter Quarter Year Year Year

Delivery Start October 05 November 05 December 05 October 05 January 06 April 06 July 06 October 06 January 06 January 07 January 08

Strike 48 49 49 48 47 40 42 43 44 43 42

Forward 48.90 50.00 49.45 49.44 48.59 40.71 41.80 43.71 43.68 42.62 42.70

Market Price 2.023 3.064 3.244 2.086 3.637 3.421 3.758 4.566 1.521 3.228 4.286

Table: ATM calls and implied Black-volatility, Sep 14

Implied Vola 43.80% 37.66% 34.72% 35.15% 28.43% 26.84% 27.19% 25.35% 20.19% 19.14% 17.46%

Parameter Estimates

Method Function calls and numerical gradient Least Square Algorithm

Constraints yes

σ1 0.37

σ2 0.15

κ 1.40

Time <1min

no

0.37

0.15

1.41

<1min

Table: Parameter estimates with different optimizers, market data as of Sep 14

Options, which are far away from maturity, will have a volatility of about 15%, which can add up to more than 50%, when time to maturity decreases. A κ value of 1.40 indicates, that disturbances in the futures market halve in κ1 · log 2 ≈ 0.69 years.

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

Discussion Parameter estimates are relatively stable over time.

The model does

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

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The model takes delivery periods into account.

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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 motivates to do research on

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The model implies a spot process without jumps.

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

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The model does not reflect volatility smiles.

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Extension of the model: add jumps; use L´evy-OU process.