Swing Options Valuation With Jumps2

8/2/2019 Swing Options Valuation With Jumps2

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Risk-neutral valuation of Swing options

in presence of jumps

Jess Prez Colino

[email protected]

January 2011

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Goal of this presentation

The main goal of this presentation is to show the basic framework for therisk-neutral pricing of swing options valuation where the logarithm of the

spot price is the sum of a deterministic seasonal trend and theOrnstein-Uhlenbeck process driven by a Lvy process (jump-diusion).

This presentation has been designed for quants analyst, and the content ofthis presentation can hurt your feelings if you do not used to work withstochastic dierential equations, PDE-PIDE and some measure theory.

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Questions addressed in this presentation:

Which is the stochastic dierential equation that better ts the "mostcorrect" dynamics of the commodities involved in a swing option?

Can we build a risk-neutral model that allow us to model the forwardprices?

Could we build an analytical pricing framework that allows us to estimatenot only the prices but also the Greeks for the swing option hedging?

How can we obtain the most robust parameters that we have to introducein the model such that guarantee the most stable possible pricing andhedging?

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Agenda

1. Introduction

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Introduction

DenitionA Swing option is a nancial contract with the following payo characteristics:

1 Maturity contract: runs over [0, T]

2 Strike: xed price K Eur/MWh

3

Swing action times: nite set of dates fTng

N

n=1 with0 T1 < T2 < ... < TN< T

4 Swing action: At each swing action date Tn the holder decides on theamount of energy Bdn MWh to be bought at xed price K Eur/MWh overeach of the D periods

Tdn, T

d+1n

, 1 d D.

5 Total and partial boundaries: assume that Bdn 2 O [0,) where O is

either a closed interval O = [B

, B] or a discrete set. Additionally, the holdermust buy at least M

MWh, and the most M MWh in total

6 Settlement: All swing options are nancially settled.



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Introduction

The outline of this presentation (excluding this introduction) is the following:

2. Section 1: Deng (2000) spot price model

- This section introduce the model developed by Deng (2000) where thelogarithm of the spot price is the sum of a seasonality term and aOrnstein-Uhlenbeck process driven by a jump diusion.

3. Section 2: Model calibration- Section 2 is devoted to the calibration or estimation of the parameters of the

model, using a combination between a least-squares method (Lucia andSchwartz (2002)) and the Fourier transform based in the maximum likelihoodapproach by Singleton (2001).

4. Section 3: Swing option price estimation and numerical algorithm- In the last section, we proceed to develop the swing option pricing algorithm

based in the discretization of a PIDE using a nite dierence method, similaras Cont and Tankov (2003).



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Agenda

1. Introduction

2. Section 1:Deng (2000) spot price model

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Agenda

1. Introduction

2. Section 1:Deng (2000) spot price model



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Section 1: Deng (2000) spot price model

We consider that power spot prices S = fSt : 0 t T?g lives in a

continuous-time trading economy in [0, T

?

] driven by a compensated Lvyprocess L on a suciently rich stochastic basis (, L,L,P) .

DenitionAssume that the power-spot prices follows the P-dynamics:

St = exp (f(t) + Xt)

dXt = Xtdt + dLt(1)

where > 0 is xed, f(t) is a deterministic seasonal trend and Lt is acompensated Lvy process that accept the canonical Lvy-Ito decomposition

with triplet (, , 0) such that

L(t) =Zt

0W(ds) +

Zt0

Zjxj>

x (JX ) (ds, dx)

= Wt + Uxt EP [x] t (2)

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The solution St to the SDE (1) started in t0 is given by

St = St0e(tt0) EP [x]

1 e(tt0)

+Ztt0

e(tt0)dWs +Nt

i=Nt0

e(tti)Xi (3)

The density ofSt is generally not known explicitly, but using the FT for anyt 2 R+

t (z) = e(tt0)(z) with z 2 Rd (4)

where j (z) is the Lvy-Khitchine exponent with the following

representation

(z) = izxe(tt0) 2z2

4

1 e2(tt0)

+ ZRd e

izx 1 izxe(tt0) (dx) (5)Jess Prez Colino ([email protected]) Swinging between jumps January 2011 9 / 18
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FactForward prices, under a diusive risk-neutral measure, has the following

expression:

F(t, T) = exp

f(T) + (log St f(t)) e

(Tt)

exp

1 e(Tt)

+ 2

4

1 e2(Tt)

exp

ZRd

ex 1 xe(Tt)

(dx)

(6)

Fseason Fdiffusion Fjump

Proof.

That can be obtained following Lucia and Schwartz(2002)

Jess Prez Colino ([email protected]) Swinging between jumps J an uar y 2 01 1 1 0 / 18


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Agenda

1. Introduction

2. Section 1:

Deng (2000) spot price model

3. Section 2:Model Calibration



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Agenda

1. Introduction

2. Section 1:





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Agenda

1. Introduction

2. Section 1:





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Section 2: Model Calibration

1 Let us dene a periodic seasonal trend f( tj) with

= (A0,An, Bn)Nn=1such that

f( tj) = A0 +N

n=1

An cos (2fnt + Bn) (7)

where the parameter vector is unknown and is to be estimated from data.2 Notice that the stationary covariance function of the ln St is given by

Cov (ln St+, ln St) = eVar (ln St) so is estimated from sample

covariances and variances.3 Next, we subtract the seasonal component f( tj) from the ln St, and we

estimate the remaining parameters with the maximum likelihood method.Notice that if we unknown the density function we may form the log

likelihood function using the inverse Fourier transformation of thecharacteristic function, following Deng(2000) or Singleton(2001).4 Finally, to estimate the market price of risk , we minimized the distance

between market and model prices, such that

= arg minK

k=1

F (t, T) F (t, T)2 (8)Jess Prez Colino ([email protected]) Swinging between jumps J an uar y 2 01 1 1 2 / 1 8
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Agenda

1. Introduction

2. Section 1:Deng(2000) spot price model


4. Section 3:Swing option price estimation


A d
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Agenda

1. Introduction





A d
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Agenda

1. Introduction





A d
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Agenda

1. Introduction





S ti 3 S i ti i ti ti
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Section 3: Swing option price estimation

Let us remind that at time Tn, the swing option holder decide to buy Bdn

MWh at K EUR/MWh.

A swing action is described as a D-vectorB1n...BDn

, and dened as

n = Dd=1 B

dn such that n 2 S = [0, DB] . A swing action means choosing

n without violating the contract constraints, and receiving the amountg (Tn, s,n) .

Finally, let us dene Zt = jn=1 n for t 2 Tj, Tj+1 .

DenitionWe can dene the value of a swing option at moment t, when s = St andz = Zt, as V(t, s, z) given by

V(t, s, z) =

8>>>>>:

supNn=j

er(TnTj)Ehg (Tn, STn ,n)j FTj

it = Tj

er(TnTj)EhVTj, STj , z

FTj

i Tj < t < Tj, j > 1t < T1

(9)


S ti 3 S i ti i ti ti
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Theorem

There exists at least one optimal swing action strategy fngNn=j such that thesupremum is attainable.

According with the Feynman-Kac theorem, the swing option price V(t, s, z) isthe unique solution to the following parabolic partial integro-dierential equation

(PIDE) V

t+ LxV rV= 0 (10)

where the operator Lx ca be splitted into two parts (integral and dierential) suchthat Lx = Dx + Ix with

DxV(t, x, z) = 2

22Vx2

( + x) VxIxV(t, x, z) =

RR

hV(t, xl + y, z) V(t, xl, z) y

Vx

if(y)dy

(11)

that can be solved numerically splitting the diusion or continuous part, from theintegral or discontinuous part.




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The space derivatives are discretized using nite dierences:

Vx

n

(

Vn+1Vnx if + x < 0VnVn1

x if + x 02V

x2

n

Vn+1 2Vn Vn1

(x)2

In order to approximate the integral terms one can use the trapezoidalquadrature rule with the same grid resolution x. More specically, if wedene:

Vn = Zx(n+ 12 )

x(n 12 )f(y)dy

then IxV(t, x, z) may approximated as:

IxV(tk+1, xl, z) N

n=N

hVl+nk+1 V

lk+1

n

2

Vl+1k+1 V

l1k+1

in

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Answers:

1. We have introduced a stochastic process Lt, that has "some good"properties for the commodities price modelling.

2. We have built the risk-neutral dynamics for forward prices where we haveincluded seasonality, mean-reversion, Brownian motion and jumps.

3. We have proposed a calibration methodology, that include the marketprice of risk for the diusion part.

4. We have dened a pricing framework for swing option based in a PIDE.

The solution of the PIDE can be found with a numerical pricing algorithmbased in nite dierences (using a combination of explicit-implicitschemes).


References
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References

R. Cont and P. Tankov. Financial Modelling with Jump Processes. Chapman& Hall, London (2003)

S.Deng. Stochastic models of energy commodity prices and their applications:mean reversion with jumps and spikes, UC Energy Institute (1998)

P.Jaillet, E.Ronn and S.Tompaidis. Valuation of commodity-based swingoptions. Management Science, 50(7) (2004)

J.Jacod and A.N.Shiryaev. Limit Theorems for Stochastic Processes. Springer(1987)

J.Lucia and E.Schwartz. Electricity prices and power derivatives: evidencefrom the nordic power exchange. Review of Derivatives Research (2002)

M.Kjaer. Pricing of swing options in a mean reverting model with jumps.Goteborg University. (2007)

T. Kluge. Pricing Swing options and other Electricity Derivatives. Oxford(2006)

... among others.

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Swing Options Valuation With Jumps2

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