Stochastic modelling of electricity markets Pricing Forwards and Swaps

Stochastic modelling of electricity markets Pricing Forwards and Swaps Jhonny Gonzalez School of Mathematics The University of Manchester Magical books project August 23, 2012

Clip for this slide 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

Definitions Forward Contracts An electricity forward is a financial contract to purchase or sell some specified volume of power at a certain future time for 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 (or measure).

Pricing of forwards and Swaps Risk-neutral price modelling Constructing Pricing measures Pricing Forwards and Swaps

Change of measure Clip for this slide Let (Ω, F, P) be a probability space. Assume we have a standard normal r.v. X, whose distribution is P(X b) = b ϕ(x)dx, for all b. Then obviously E[X ]=0andVar [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 changing Y. We need to change P then to Q, say,suchthate Q [Y ]=0 and Var Q [Y ]=1.

0.4 0.35 0.3 0.25 0.2 0.15 0.1 X Y=X+2 0.05 0 10 8 6 4 2 0 2 4 6 8 10 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 W t be a Brownian motion on (Ω, F, P) and Θ(t) aprocess that we know at time t (with the information F t.define t W t = W t + Θ(s)ds. 0 Then a probability measure Q can be constructed such that on (Ω, F, Q) W t 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 iffq(a) =0).

Example. GBM Take under the actual/true probability or preferences P we have two possible investments: Risky ds t = µs t dt + σdw t Safe C t = 1 D t = e rt, or dd t = rd t dt. µ mean rate of return of stock. r interest rate. σ volatility of stock. In the safe investment If today have 1 tomorrow make 1e rt. Back today 1e rt (times)d t.

Consider what we call the discounted stock price process D t S t "the money to put in the bank account today to get S t at t It satisfies something like d(d t S t )=σd t S t dt [Θ + dw t ]. 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? ds t = rs t dt + σs t dw t.

(Ω, F, P) ds t = µs t dt + σdw t (Ω, F, Q) ds t = rs t dt + σd W t 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 the paths with lower return, r, so that the overall mean is reduced from µ to r.

Esscher Transform. Jumps Clip for this slide 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 affects 1. 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) ds t = µs t dt + σs t dm t. (Ω, F, Q) ds t = rs t dt + σs t d M t. M t = N t λt. Changing from P to Q changes the intensity of the CPP M t = N t λt, with λ = λ µ r σ. Hence M t = M t + µ r σ = M t + 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"

ACommonsetofpreferences Ω, F, {Ft } t 0, P (outcomes, relative information, preferences) Ω, F, {Ft } t 0, Q (outcomes, relative information, pricing preferences)

Pricing of forwards and Swaps Risk-neutral price modelling Constructing Pricing measures Pricing Forwards and Swaps

Clip for this slide 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.

Forwards Assume 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). TheunderlyingproducthaspricedynamicsS(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) E Q [S(τ) f (t, τ) F t ] = 0. Assuming f (t, τ) is F t measurable (set the price with the information available up to time t), we get f (t, τ) =E Q [S(τ) F t ].

Swaps Assume now the buyer of an electricity futures receives power during the period [τ 1, τ 2 ], physically or financially against paying a fixed price F (t, τ 1, τ 2 ), t τ 1. At time t the value of the payoff is τ 1 e r(u t) (S(u) F (t, τ 1, τ 2 )) du. Since it is costless to enter an electricity futures E Q e r(u t) (S(u) F (t, τ 1, τ 2 )) du F t = 0. τ 1 Assuming F (t, τ 1, τ 2 ) is F t measurable, we get re ru F (t, τ 1, τ 2 )=E Q e rτ S(u)du 1 e rτ 2 F t. τ 1

If the settlement takes place financially at τ 2,then 1 F (t, τ 1, τ 2 )=E Q S(u)du F t. τ 1 τ 2 τ 1 With the function ˆω(u) = 1 settlement at τ 2 e ru settlement over [τ 1, τ 2 ] define the weight function ω(u, s, t) = general we have τ 1 t s ˆω(u) ˆω(v)dv.Hence,in F (t, τ 1, τ 2 )=E Q ω(u, τ 1, τ 2 )S(u)du F t.

Fact Suppose E Q τ 1 ω(u, τ 1, τ 2 )S(u) du <. Then F (t, τ 1, τ 2 )= τ 1 ω(u, τ 1, τ 2 )f (t, u)du. Intuitively, holding a swap can be considered as holding a continuous stream of forwards. Fact Suppose E Q [ S(τ) ] <. Then lim f (t, τ) =S(τ). t τ At delivery there is no difference between entering the forward or buying the commodity in the spot market.

Fact Suppose E Q τ 1 ω(u, τ 1, τ 2 )S(u) du <. Hence, a.s., lim F (t, τ 1, τ 2 )= ω(u, τ 1, τ 2 )f (τ 1, u)du. t τ 1 τ 1 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. Fact Suppose E Q τ 1 ω(u, τ 1, τ 2 )S(u) du <. Then lim F (t, τ 1, τ 2 )=f(t, τ 1 ). τ 2 τ 1

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) =E Q [S(τ) F t ] and F (t, τ 1, τ 2 )= τ 1 ω(u, τ 1, τ 2 )f (t, u)du.

Geometric and arithmetic models ln S(t) =ln Λ(t)+ where, for i = 1,...,m, S(t) =Λ(t)+ m i=1 m i=1 X i (t)+ X i (t)+ dx i (t) =(µ i (t) α i (t)x i (t))dt + and, for j = 1,...,n, n j=1 n j=1 p k=1 Y i (t), (1) Y j (t) (2) σ ik (t)dw k (t), dy j (t) =(δ j (t) β j (t)y j (t))dt + η j (t)di j (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( )) τ exp µ i (u)e τ u α i (v)dv du exp exp m i=1 n j=1 m i=1 t τ t e τ t δ j (u)e τ u β j (v)dv du α i (v)dv X i (t)+ n j=1 e τ t β j (v)dv Y j (t)

where Θ(t, τ; θ( )) is given by ln Θ(t, τ; θ( )) = n ψ j (t, τ; i(η j ( )e τ β j (v)dv + θ j ( ))) j=1 +ψ j (t, τ; i θ j ( )) + 1 2 + p τ k=1 t p τ k=1 t m i=1 m 2 σ ik (u)e τ u α i (v)dv du i=1 σ ik (u) ˆθ k (u)e τ u α i (v)dv du.

Samuelson effect Clip for this slide Under some conditions the dynamics of t f (t, τ) wrt to Q (when there are no jumps) is p df (t, τ) m τ f (t, τ) = σ ik (t) exp α i (u)du d W k (t). k=1 i=1 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 σ ik (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, τ 1, τ 2 ) = τ 1 ω(u, τ 1, τ 2 ) f (t, u) exp m i=1 e u t α i (v)dv X i (t)+ n j=1 e u t β j (v)dv Y j (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 α i and β j 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