Stochastic modelling of electricity marketsPricing Forwards and Swaps

Jhonny Gonzalez

School of MathematicsThe University of Manchester

Magical books project

August 23, 2012

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Pricing of forwards and Swaps

Risk-neutral price modelling

Constructing Pricing measures

Pricing Forwards and Swaps

Pricing of forwards and Swaps

Risk-neutral price modelling

Constructing Pricing measures

Pricing Forwards and Swaps

DefinitionsForward Contracts

An electricity forward is a financial contract to purchase or

sell some specified volume of power at a certain future timefor a certain price.

Swap Contracts An electricity swap (futures) is a contract to purchase or sell

some specified volume of power for a certain price with

delivery over a period of time.

There exists physical and financial settlement.

We want to give to these contracts a “fair” price that does not

create arbitrage opportunities.

Motivation. Asset pricing

The price of an asset depends on the risk involved in investing

in the asset.

The riskier the asset the more we ask in return for investing in

it. Investors require a greater incentive when they put their

money on more risky investments.

But if want to calculate the price of a particular asset and its

return, it will vary according to the risk preferences of each

investor. And, we would need to calculate each investor’s risk

preferences.

We need a common a set of risk preferences under which we

can price assets. It should include all investor’s preferences.

It is possible to construct these set of preferences or probability,

and it is commonly known as the risk-neutral probability (ormeasure).

Pricing of forwards and Swaps

Risk-neutral price modelling

Constructing Pricing measures

Pricing Forwards and Swaps

Change of measureClip for this slide

Let (Ω,F ,P) be a probability space. Assume we have a standardnormal r.v. X, whose distribution is

P(X ≤ b) = b

−∞ϕ(x)dx , for all b.

Then obviously E[X ] = 0 and Var [X ] = 1. Now take Y = X + θ,

θ > 0. It is normal but “shifted” (non-standard now).

But what if we want Y to be a std. normal r.v.? We do not want

to subtract θ from Y and change Y .

We want to change the distribution of Y without changingY . We need to change P then to Q, say, such that EQ [Y ] = 0

and VarQ [Y ] = 1.

!10 !8 !6 !4 !2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

X Y=X+2

Less probability to outcomes for which Y (ω) > 0.

More probability when Y (ω) < 0.

This can be formally done in practice and is standard in

Probability. Radon-Nikodym theorem.

We want to get a bit more complicated and do the same not just

with random variables but with stochastic processes.

1. We want to change the distribution of prices without changing

the prices themselves.

2. Change the mean of the whole price process.

Girsanov’s Theorem

Change of measure for continuous processes.Let Wt be a Brownian motion on (Ω,F ,P) and Θ(t) a process

that we know at time t (with the information Ft . Define

Wt = Wt + t

0Θ(s)ds.

Then a probability measure Q can be constructed such that on

(Ω,F ,Q) Wt is a Brownian motion.

Q is equivalent to P in the sense that they agree on what is

possible and what is not. (P(A) = 0 iff Q(A) = 0).

Example. GBM

Take under the actual/true probability or preferences P we have

two possible investments:

Risky dSt = µStdt + σdWtSafe Ct = 1

Dt= ert , or dDt = −rDtdt.

µ mean rate of return of stock.

r interest rate.

σ volatility of stock.

In the safe investment

If today have £1

tomorrow make 1ert.

Back today 1ert(times)Dt .

Consider what we call the discounted stock price process

DtSt"the money to put in the bank account today to get St at t”

It satisfies something like

d(DtSt) = σDtStdt [Θ + dWt ] .

Apply Girsanov’s theorem to get the dynamics under Q.

Put the same money back in the risky investment today but

with the new risk preferences.

What would the price dynamics be under this change of

measure?

dSt = rStdt + σStdWt .

(Ω,F ,P) dSt = µStdt + σdWt(Ω,F ,Q) dSt = rStdt + σdWt

The change from P to Q changes the mean rate of return of

the stock to be the (risk-free) interest rate but not the

volatility.

The volatility says which prices paths are possible.

After the change we still have the same set of stock price

paths (unchanged volatility),

but if µ > r , this change puts more probability on thepaths with lower return, r , so that the overall mean is

reduced from µ to r .

Esscher Transform. Jumps

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Change of measure for processes with jumps.

The Esscher transform is a generalisation of the Girsanov’s

theorem for Brownian motion to jump processes.

It has a similar formulation to the G.T., but in terms of jumps.

For this work it provides a risk-neutral measures that are

tractable for pricing in the presence of jumps.

Introduced by Esscher in 1932, and used for pricing in

financial markets starting with Gerber and Shiu (1994),

Option pricing with Esscher transforms.

Change of measure in the presence of jumps

Some consequences. Change of measure affects1. Intensity for a Poisson process

2. Intensity and jump size for a Compound Poisson Process

3. Mean of BM, and intensity and jump size of jump process.

Example. Geometric Poisson Process

(Ω,F ,P) dSt = µStdt + σSt−dMt .

(Ω,F ,Q) dSt = rStdt + σSt−dMt .

Mt = Nt − λt.

Changing from P to Q changes the intensity of the CPP

Mt = Nt − λt, with

λ = λ − µ−rσ .

Hence Mt = Mt +µ−r

σ = Mt + Θ.

There is a smaller intensity.

Remarks

1. Under the Q the return is always r, that is why we call it the

risk-neutral measure.

2. It is not the real probability, it is different but depends on the

real one.

3. It does not assume that we live in a risk-free world. It is a

probability, it still makes the future uncertain.

4. It does not assume investors (market players) do not care

about risk. They do care about risk. But they can use it to

price assets as it contains all risk preferences.

5. Solutions under a risk-neutral measure or pricing measure are

valid in the real world where real risk preferences apply.

Remarks

The risk-neutral measure is the only measure that gives

arbitrage-free prices (in complete markets).

This method is only a very useful computational tool, but it

is artificial.

In mathematical finance they allow to solve PDEs more easily.

Pricing measures and pricing formula.

"Initial capital" = "discounted expected payoff"

A Common set of preferences

Ω,F , Ftt≥0 ,P

(outcomes, relative information, preferences)

Ω,F , Ftt≥0 ,Q

(outcomes, relative information, pricing preferences)

Pricing of forwards and Swaps

Risk-neutral price modelling

Constructing Pricing measures

Pricing Forwards and Swaps

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Let us price forwards and swaps with general spot price S(t).We assume there exists a pricing measure Q equivalent to the

actual measure.

ForwardsAssume we buy a forward contract at time t promising future

delivery at τ, 0 ≤ t ≤ τ. The agreed price to pay upon delivery is

f (t, τ). The underlying product has price dynamics S(t). Include

a risk-free asset yielding a continuously compounded rate of return

r > 0, and initial price equal to one.

At τ the payoff is

S(τ)− f (t, τ).

Since it is costless to enter in a forward contract, risk-neutral

valuation gives (integrability conditions apply)

e−r(τ−t)EQ [S(τ)− f (t, τ) | Ft ] = 0.

Assuming f (t, τ) is Ft measurable (set the price with the

information available up to time t), we get

f (t, τ) = EQ [S(τ) | Ft ] .

ForwardsAssume we buy a forward contract at time t promising future

delivery at τ, 0 ≤ t ≤ τ. The agreed price to pay upon delivery is

f (t, τ). The underlying product has price dynamics S(t). Include

a risk-free asset yielding a continuously compounded rate of return

r > 0, and initial price equal to one.

At τ the payoff is

S(τ)− f (t, τ).

Since it is costless to enter in a forward contract, risk-neutral

valuation gives (integrability conditions apply)

e−r(τ−t)EQ [S(τ)− f (t, τ) | Ft ] = 0.

Assuming f (t, τ) is Ft measurable (set the price with the

information available up to time t), we get

f (t, τ) = EQ [S(τ) | Ft ] .

ForwardsAssume we buy a forward contract at time t promising future

delivery at τ, 0 ≤ t ≤ τ. The agreed price to pay upon delivery is

f (t, τ). The underlying product has price dynamics S(t). Include

a risk-free asset yielding a continuously compounded rate of return

r > 0, and initial price equal to one.

At τ the payoff is

S(τ)− f (t, τ).

Since it is costless to enter in a forward contract, risk-neutral

valuation gives (integrability conditions apply)

e−r(τ−t)EQ [S(τ)− f (t, τ) | Ft ] = 0.

Assuming f (t, τ) is Ft measurable (set the price with the

information available up to time t), we get

f (t, τ) = EQ [S(τ) | Ft ] .

SwapsAssume now the buyer of an electricity futures receives power

during the period [τ1, τ2], physically or financially against paying a

fixed price F (t, τ1, τ2), t ≤ τ1.

At time t the value of the payoff is

τ2

τ1e−r(u−t) (S(u)− F (t, τ1, τ2)) du.

Since it is costless to enter an electricity futures

EQ

τ2

τ1e−r(u−t) (S(u)− F (t, τ1, τ2)) du | Ft

= 0.

Assuming F (t, τ1, τ2) is Ft measurable, we get

F (t, τ1, τ2) = EQ

τ2

τ1

re−ru

e−rτ1 − e−rτ2S(u)du | Ft

.

SwapsAssume now the buyer of an electricity futures receives power

during the period [τ1, τ2], physically or financially against paying a

fixed price F (t, τ1, τ2), t ≤ τ1.

At time t the value of the payoff is

τ2

τ1e−r(u−t) (S(u)− F (t, τ1, τ2)) du.

Since it is costless to enter an electricity futures

EQ

τ2

τ1e−r(u−t) (S(u)− F (t, τ1, τ2)) du | Ft

= 0.

Assuming F (t, τ1, τ2) is Ft measurable, we get

F (t, τ1, τ2) = EQ

τ2

τ1

re−ru

e−rτ1 − e−rτ2S(u)du | Ft

.

SwapsAssume now the buyer of an electricity futures receives power

during the period [τ1, τ2], physically or financially against paying a

fixed price F (t, τ1, τ2), t ≤ τ1.

At time t the value of the payoff is

τ2

τ1e−r(u−t) (S(u)− F (t, τ1, τ2)) du.

Since it is costless to enter an electricity futures

EQ

τ2

τ1e−r(u−t) (S(u)− F (t, τ1, τ2)) du | Ft

= 0.

Assuming F (t, τ1, τ2) is Ft measurable, we get

F (t, τ1, τ2) = EQ

τ2

τ1

re−ru

e−rτ1 − e−rτ2S(u)du | Ft

.

If the settlement takes place financially at τ2, then

F (t, τ1, τ2) = EQ

τ2

τ1

1

τ2 − τ1S(u)du | Ft

.

With the function

ω(u) =

1 settlement at τ2e−ru settlement over [τ1, τ2]

define the weight function ω(u, s, t) = ω(u) ts ω(v)dv . Hence, in

general we have

F (t, τ1, τ2) = EQ

τ2

τ1ω(u, τ1, τ2)S(u)du | Ft

.

FactSuppose EQ

τ2τ1

|ω(u, τ1, τ2)S(u)|du< ∞. Then

F (t, τ1, τ2) = τ2

τ1ω(u, τ1, τ2)f (t, u)du.

Intuitively, holding a swap can be considered as holding a

continuous stream of forwards.

FactSuppose EQ [|S(τ)|] < ∞. Then

limt↑τ

f (t, τ) = S(τ).

At delivery there is no difference between entering the forward or

buying the commodity in the spot market.

FactSuppose EQ

τ2τ1

|ω(u, τ1, τ2)S(u)|du< ∞. Then

F (t, τ1, τ2) = τ2

τ1ω(u, τ1, τ2)f (t, u)du.

Intuitively, holding a swap can be considered as holding a

continuous stream of forwards.

FactSuppose EQ [|S(τ)|] < ∞. Then

limt↑τ

f (t, τ) = S(τ).

At delivery there is no difference between entering the forward or

buying the commodity in the spot market.

FactSuppose EQ

τ2τ1

|ω(u, τ1, τ2)S(u)|du< ∞. Hence, a.s.,

limt↑τ1

F (t, τ1, τ2) = τ2

τ1ω(u, τ1, τ2)f (τ1, u)du.

If delivery takes place over a period of time, swap prices do not

converge to the spot price at delivery. If S(t) is a Q martingale

the convergence of the swap to the spot holds. A swap contract

delivering the commodity at a single point in time is a forward.

FactSuppose EQ

τ2τ1

|ω(u, τ1, τ2)S(u)|du< ∞. Then

limτ2→τ1

F (t, τ1, τ2) = f (t, τ1).

FactSuppose EQ

τ2τ1

|ω(u, τ1, τ2)S(u)|du< ∞. Hence, a.s.,

limt↑τ1

F (t, τ1, τ2) = τ2

τ1ω(u, τ1, τ2)f (τ1, u)du.

If delivery takes place over a period of time, swap prices do not

converge to the spot price at delivery. If S(t) is a Q martingale

the convergence of the swap to the spot holds. A swap contract

delivering the commodity at a single point in time is a forward.

FactSuppose EQ

τ2τ1

|ω(u, τ1, τ2)S(u)|du< ∞. Then

limτ2→τ1

F (t, τ1, τ2) = f (t, τ1).

Pricing of forwards and swaps

In order to derive formulas for the forward and swaps prices for the

geometric and arithmetic models, we use the Esscher transform

along with the formulas

f (t, τ) = EQ [S(τ) | Ft ]

and

F (t, τ1, τ2) = τ2

τ1ω(u, τ1, τ2)f (t, u)du.

Geometric and arithmetic models

ln S(t) = ln Λ(t) +m∑i=1

Xi (t) +n∑j=1

Yi (t), (1)

S(t) = Λ(t) +m∑i=1

Xi (t) +n∑j=1

Yj(t) (2)

where, for i = 1, ...,m,

dXi (t) = (µi (t)− αi (t)Xi (t))dt +p∑k=1

σik(t)dWk(t),

and, for j = 1, ..., n,

dYj(t) = (δj(t)− βj(t)Yj(t))dt + ηj(t)dIj(t).

Pricing of forwards. Geometric model

Let 0 ≤ t ≤ τ and assume S(t) is the geometric spot price model

from above. Unser some conditions we have that

f (t, τ) = Λ(τ)Θ(t, τ; θ(·))

× exp

m∑i=1

τ

tµi (u)e−

τu αi (v)dv du

× exp

n∑j=1

τ

tδj(u)e−

τu βj (v)dv du

× exp

m∑i=1

e− τ

t αi (v)dv Xi (t) +n∑j=1

e− τ

t βj (v)dv Yj(t)

where Θ(t, τ; θ(·)) is given by

ln Θ(t, τ; θ(·)) =n∑j=1

ψj(t, τ;−i(ηj(·)e−

τ· βj (v)dv + θj(·)))

+ψj(t, τ;−i θj(·))

+1

2

p∑k=1

τ

t

m∑i=1

σik(u)e− τ

u αi (v)dv2

du

+m∑i=1

p∑k=1

τ

tσik(u)θk(u)e−

τu αi (v)dv du.

Samuelson effectClip for this slide

Under some conditions the dynamics of t → f (t, τ) wrt to Q(when there are no jumps) is

df (t, τ)f (t, τ)

=p∑k=1

m∑i=1

σik(t) exp

−

τ

tαi (u)du

dWk(t).

The volatilities of the forward are decreasing with time to

delivery, being smaller than the spot volatility.

When time to delivery approaches zero, the forward volatility

converges to the volatilities of the spot σik(t). The arrival of information to the market has a much bigger

effect when there is short time left to maturity than for the

long-term contracts (the market has time to adjust, prices are

mean reverting).

Pricing of swaps

For the geometric spot model we have

F (t, τ1, τ2) = τ2

τ1ω(u, τ1, τ2)f (t, u)

exp

m∑i=1

e− u

t αi (v)dv Xi (t) +n∑j=1

e− u

t βj (v)dv Yj(t)

du.

In general, this integral does not have any analytic solution,

and hence numerical integration is required for its evaluation.

If the speed of mean reversion terms αi and βj are both zero,

an analytic solution exists. However, the Samuelson effect is

not observed with this restricted dynamics.

For arithmetic models...

We get similar formulae.

In many cases the integral can be solved analytically.

What could be done as well?

Price options in the presence of jumps or in their absence.

Call, puts, spark spread option, options in weather markets.

Hedge options on forwards and swaps.

Need more advanced maths, Fourier series, etc.

Benth, F. E., Benth, J. Š. and Koekebakker S..

Stochastic modelling of electricity and related markets.

World Scientific, London. 2008