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MINI CASE STUDY SERIES:

TWO FACTORS MODEL CALIBRATIONEmmanuel Gincberg

CREDIT AGRICOLE CIB. Head of Commodity Quantitative Analytics.

The views expressed in this document are those of the author and do not necessarily

represent those of his employer.

Marcus Evans. Practical Quantitative Analysis in Com-

modities. 17th June 2010


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MODELLING THE FUTURE CURVE 2

MODELLING THE FUTURE CURVE


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3MODELLING THE FUTURE CURVE


The most activated traded instruments in commodity financial markets

are future and forward contracts.

NYMEX and ICE exchange markets enable to trade future con-tracts up to 72 consecutive months for many energy assets.

Agricultural futures can be traded on the CBOT exchange. Forward agreements are OTC contracts which generally settle on

monthly average of future closing prices.

Commodity Swaps are transactions where the floating price isindexed on future prices.


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Many commodity derivatives depend on different future contracts(Swaptions, time spreads,).

It is therefore important to not only correctly model each futurecontract but also to correctly represent the joint distribution of theseassets.

To do so it is convenient to consider the future curve as a wholerather than a collection of individual points


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BACKWARDED FUTURE CURVE


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CONTANGO FUTURE CURVE


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BACKWARDATION TO CONTANGO


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FUTURE CURVE VARIATIONS

This particular observation illustrates general behaviours of commodityfuture curves:

Commodity prices are governed by supply and demand.

Future curves short ends are generally much more volatile than longends (mean reversion).

Short dated future are used to cover unanticipated demand, theirprices are impacted by:

Weather conditions

Pipeline failures

Political events

Etc...


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SHORT TERM / LONG TERM MODEL

Schwartz-Smith proposed a model to describe the variations of the fu-ture curve where the spot price logarithm is driven by two factors:

ln= +

is the short term process:

= + is the long term equilibrium process:= +

=


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Under these assumptions the future prices are lognormal variables:

lnFt, T= + +

lnFt, T =

+ + +

VarlnFt, T = 1 + + 2 1 is a deterministic function:= + 1

21

2+ + 21


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Although, as shown in the Schwartz-Smith paper, this approach is

equivalent to a stochastic convenience model, it is probably more intui-

tive due to the break down between short and long term drivers.

is a mean reverting process which has more impact on shortdated maturities.

creates parallel shifts of the curve Short dated futures tend to revert to the long run equilibrium

regime which is driven by

These variables are however not observable


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FUTURE CURVE MODEL

The previous approach consists in defining the spot price driving vari-

ables and to deduce the corresponding future prices dynamic. Alterna-

tively we can directly model the future price:dFt, T= Ft, T t, T dW

The short term / long term model corresponds to a specific formula-tion of the volatility functions t, Twhere the number of factors isset to 2.

In a more general setting the number of Brownian processes Wcanbe reduced by PCA techniques.


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MODELLING THE FUTURE CURVE 13

FUTURE CURVE MODEL

A more generic formulation of the short term long term future curve

model can therefore be expressed as follows:

Ft, T= F0, T exp 12 it, T2+ + Where

F0, Tis the current price of the future contract expiring at T.=

=

=

t, T=

+

+


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14CALIBRATION

CALIBRATION


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15CALIBRATION

KALMAN FILTER

In their original papers Gibson-Schwartz and Schwartz-Smith propose toestimate the model parameters with a Kalman filtering technique.

The kalman filter is a recursive method for estimating non observablevariables from observable variables.

Kalman results can be statically unstable. There is no guarantee that this procedure would produce a set of pa-

rameters consistent with implied volatilities.

The model should be in line with the term structure of volatilities if wewant to use the corresponding options for hedging.


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16CALIBRATION

LISTED OPTIONS

Similarly to the future market there exists a listed commodity market

for options: For each future contract there generally exist Call and Put

options expiring shortly before the corresponding future maturity .

These options are the basic instruments of choice for thehegdge of Vega risks linked to more complex derivatives

They imply a term structure of implied volatilities but contraryto other markets the underlying is different for each maturity:

In Equity markets a term strucure of volatility defines a

strip of option prices on the same asset observed at

different times.

In most commodity markets each volatility point

corresponds to a unique future underlying expiry.


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CALIBRATION 17

TERM STRUCTURE OF VOLATILITY

A term structure of volatility represents the variation of volatility withmaturity.

It is common to represent the at-the-money volatilities to illustratethe term structure.

Commodity term structures tend to be downward sloping. It can beexplained by the mean reverting nature of these markets.

Each volatility point of the term structure corresponds to a uniquefuture contract point


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18CALIBRATION



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19CALIBRATION


Although volatility term structure varies with time, the downward slop-

ing profile is a persistent characteristic of volatility term structures.

By time homogeneity we can deduce that the volatility term struc-ture of a fixed future contract should be upward sloping

(Samuelson effect).

This is consistent with the short term/long term model:VarlnFt, T= 22 1 2

2+2 1 +2

The first two terms are bounded as t & T tends to infinityi.e. the corresponding volatility is decreasing towards 0.

When T is fixed the first two terms are increasing with t.


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20CALIBRATION

VOLATILITY TERM STRUCTURE CALIBRATION

But calibration to ATM volatilities will not provide a unique solution.

= 116%,

= 110%,

= 23%, = 88%


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21CALIBRATION


Another satisfactory calibration:

= 223%, = 90%, = 18%, = 20%


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CALIBRATION 22


The previous calibrations are obtained by global minimization of thedifferences between listed options implied volatilities and the corre-

sponding volatilities computed with the two factors model parame-

ters (with constant instantaneous volatilities and ). Another approach consists in extrapolating from the term struc-

ture of volatility as it corresponds to the implied volatility limit

(, ) when t, Ttend toward infinity:Each point of the ATM volatility term structure can then be

exactly matched by introducing piecewise-constant instanta-

neous volatilities.However there is no unique solution: each couple , willlead to a different profile.


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CALIBRATION 23


In order to illustrate the previous calibration method let us denote byt, the market option maturities and corresponding ATM impliedvolatilities (to simplify the calculations we assume

= 0).

If we suppose that , are known it remains to define = for , such as

+

= = 1

It implies the following result:

=

1 2

It can only be solved if

1

0

CALIBRATION 24


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CALIBRATION 24


We have seen that calibrating , to the term structure of volatility cangenerate different results:

Two acceptable calibrations can create different correlation profiles

of the future curve.

The volatility term structure profile,of a given future contractimplied by the model is mainly driven by the mean reversion

level:

, = 1

2+ 2 1

+

CALIBRATION 25


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CALIBRATION 25


Volatility term structure of a fixed contract

CALIBRATION 26


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CALIBRATION 26

HISTORICAL FUTURE PRICES

When no additional type of option quotes (swaptions, time spread

options) relevant for the calibration of the mean reversion/correlation

are available, an alternative solution is to use historical future prices.

Let us consider historical future pricesFt, t + , where trepresent past observation dates, and are constant expiries (theyare in fact approximated by the average time to maturity of the k-th

nearest future).

We focus on the stochastic term itof the corresponding loga-rithmic returns

Ft+t,t+t+iFt,t+i :

t= +

CALIBRATION 27


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CALIBRATION 27


With a local expansion around t~0, we can writet = + t + t

Up to the first order of t, the stochastic terms of the logarithmic re-turns are independent and equi-distributed:t~0, + t Similarly we can compute the covariances ,, correspondingto two different contracts:

,, =

+

t

The calibration method consists in minimizing the differences betweenthe model and the historical covariances.

CALIBRATION 28


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= 84%, = 17%, = 20%, = 19.59%

29CALIBRATION


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VOLATILITY SEASONALITY

In some markets, for example Henry Hub natural Gas, the term struc-ture of volatility is strongly seasonal.

It can be explained by higher demand, and therefore volatility, duringheating periods in winter.

The short term/long term breakdown cannot explain these term struc-tures, alternative approaches have to be considered.

Each month of the year can be modelled by a distinct underlying creat-ing twelve different term structure of implied volatilities.

30CALIBRATION


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A typical Natural Gas term structure of ATM volatilities

31CONCLUDING REMARKS


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Natural Gas ATM volatilities with theAugust and November months highlighted.

CALIBRATION 32


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Each season can be calibrated separately and it remains to define the

correlations between summer and winter:

lnFt, T=

+ +

lnFt, T = + + The summer-winter correlations can be estimated by historical

analysis or implied from Calendar spread options, the most liquid

pairs being March-April and October-January.

=

,

=

= , =

CALIBRATION 33


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HEDGING CONSIDERATIONS

Different calibration methods will imply different hedging strategies.

Let us assume two different approaches where the mean reversion iscalibrated to historical covariances. We want to price a calendar spread

option:

, , Forward model:

dFt, T= Ft, T t, T dWEach volatility function

t, T

is calibrated separately to the appropri-

ate ATM volatility point.

CALIBRATION 34


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Short term/long term model ( = 0):lnFt, T= + + is calibrated to the implied volatility curve asymptote:

VarlnFt, Tt , =

is a piecewise constant function calibrated to the ATM volatilitycurve.

+ = i

= 1 2

CALIBRATION 35


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With the Forward model the value of the Calendar spread option de-pends on both volatilities iand i.

With the short term/long term model the value of the Calendar spreadoption depends on both volatilities iand .

Alternatively a short term/long term model can be calibrated to a fewfixed points of the ATM volatility term structure (the more liquid) and

the Vega profile of any options will be depend on the calibrated points

only.

CALIBRATION 36


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CONCLUDING REMARKS

Multiple factors are necessary to correctly represent the variationsof the future curve.

To obtain satisfactory calibration - stable and in line with vanilla op-tions - one has to combine different approaches (minimization, sta-

tistical estimation).

The calibration procedure impacts directly the risk profile and there-fore should be decided accordingly to a hedging strategy.

37REFERENCES


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REFERENCES

Rajna Gibson; Eduardo S. Schwartz. Stochastic Convenience Yield and the Pricing of Oil

Contingent Claims. The Journal of Finance, Vol. 45, No. 3, Papers and Proceedings, Forty-

ninth Annual Meeting, American Finance Association, Atlanta, Georgia, December 28-30,

1989. (Jul., 1990), pp. 959-976.

Eduardo Schwartz; James E. Smith. Short-Term Variations and Long-Term Dynamics in

Commodity Prices. Management Science Vol. 46, No. 7, July 2000 pp. 893911

Helyette Geman. Commodities and Commodity Derivatives: Pricing and Modeling Agricul-

tural, Metals and Energy, January 2005, Wiley Finance

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