Pricing of commodity derivatives on processes with memory
Spot option prices, forwards and options on forwards relevant for the commodity markets are computed when the underlying process is modelled as an exponential of a process with memory as e.g. a Lévy semi-stationary process. Moreover a risk premium representing storage costs, illiquidity, convenience yield or insurance costs is explicitly modelled as an Ornstein-Uhlenbeck type of dynamics with a mean level that depends on the same memory term as the commodity. Also the interest rate is assumed to be stochastic. To show the existence of an equivalent pricing measure for we relate the stochastic differential equation for to the generalised Langevin equation. When the interest rate is deterministic the process has an affine structure under the pricing measure and an explicit expression for the option price is derived in terms of the Fourier transform of the payoff function.
Keywords: Equivalent measures, derivatives pricing, commodity markets, Langevin equation, affine processes, Fourier transform
In financial markets the arbitrage-free price of a derivative is derived by a risk neutral probability. In complete markets, the risk-neutral probability is unique, leading to a single arbitrage-free price dynamics. Most financial markets are, however, incomplete, with commodity markets as a typical case. For example, the power spot market is only accessible for physical players that can produce or transmit electricity, whereas the forwards market on power is financial. See [7, 12, 16] for a discussion on pricing in energy and commodity markets, and  for a general treatment of the arbitrage pricing theory in financial markets. Power serves as the extreme example of an incomplete market, as the spot is considered not financially tradeable in addition to a price dynamics with highly non-Gaussian features such as price spikes.
There is no unique risk-neutral probability in an incomplete market. In this paper, we focus on a class of probabilities that can be represented as a “deviation” from the risk-neutral one, in the sense that the price dynamics of the underlying asset will have a mean rate of return which can be represented as the sum of a risk-free interest rate and an additional yield under this probability . The probability will not be risk-neutral, but only equivalent to the market probability . It is referred to as a pricing measure. With this class of probabilities, we model the risk premium by the additional yield. In commodity markets, this yield can be interpreted as storage costs, transportation, insurance, convenience yield and other illiquidity costs. Thus, the market price of risk is viewed as the compensation for financial risk and illiquidity risk.
Mean reversion and stationarity play an important role in the price dynamics of commodity prices (see [12, 16]). We consider a spot price model where the logarithmic price dynamics follows a generalised Langevin equation. In our framework, we allow for dependency on the past in the current price, as well as jumps. The dependency on the past in the dynamics comes in as a memory term in the drift of the Langevin equation, being modelled as a weighted average of the historical logarithmic prices. Our model includes the class of Lévy semi-stationary processes and continuous-time autoregressive moving average processes, popular modelling tools for power, gas and oil prices (see [2, 8, 25]) and weather variables like temperature and wind (see ), as well as volatility and turbulence (see ). We prove existence and uniqueness of a solution of our proposed general Langevin equation model.
Our main result is that the class of pricing measures that we propose are indeed probabilities. This entails in proving that the density process in the Girsanov theorem is a true martingale. We appeal to the criteria in the extended Beneš method, developed in . In our analysis, we allow for stochastic interest rates and a stochastic dynamics for the yield in the risk premium. Both processes are modelled by jump-diffusions of Ornstein-Uhlenbeck type, with an explicit dependency on the memory part of the Langevin dynamics in the price dynamics.
We perform an in-depth study of pricing of options and forwards using our pricing measure in the special cases when the logarithmic spot price dynamics is of Lévy semi-stationary type or a continuous-time autoregressive moving average process. Due to the affine structure, we obtain reasonably explicit expressions for call and put option prices in the former case using Fourier methods. For Lévy semi-stationary processes, the forward price is also available explicitly as a function of the spot and the risk premium. Furthermore, we express the price dynamics for put and call options on forwards for this model class. Plain vanilla European options are typically traded on forwards in many power markets. Further, we consider Wiener-driven continuous-time autoregressive moving average dynamics and extend the results in  to introduce a class of pricing measures . For these models, we derive the forward price dynamics under our pricing measure. We remark that the explicit price expressions in all cases are derived under the assumption of deterministic interest rates and hence future and forward prices coincide.
Our analysis and results are presented as follows. Section 2 provides some motivation from commodity markets on pricing and risk-neutral probabilities, and presents the stochastic dynamics of the market that we will analyse in this paper. As background material, we also include some results on affine processes that will be needed later in the paper. The generalised Langevin equation modelling the logarithmic spot prices is analysed in Section 3, and in Section 4 we prove the validity of our proposed measure change. Finally, Section 5 derives prices for various derivatives like options and forwards in the case of Lévy semi-stationary processes. An extensive analysis for the case of continuous-time autoregressive moving average processes is also contained in this section.
2 Set up and preliminaries
Suppose that is a given probability space satisfying the usual conditions, see e.g. , and denotes an -Wiener process. Furthemore, we let be an -Lévy process with characteristic triplet . Assume that the Lévy measure of the process satisfies , e.g., is a square-integrable Lévy process. From the characteristic triplet of the process , we know that the latter admits the following decomposition
where and is a compensated Poisson random measure. That is and is the Poisson random measure such that .
Before defining our stochastic model for the commodity spot market (see Subsection 2.2), we discuss some classical findings of forward pricing.
2.1 Spot and forwards in commodity markets
Let be a stochastic process defining the spot price dynamics of a commodity given by a geometric Brownian motion under
where are constants. If the spot can be liquidly traded in the commodity market, then we can perfectly hedge a short position in a forward contract by a long position in the spot financed by borrowing at the risk-free rate . This hedging strategy is known as the buy-and-hold strategy and uniquely defines the forward prices (see e.g. [12, 16]). I.e., if is the forward price at time of a contract delivering the commodity at time , then .
When running a buy-and-hold strategy in a commodity market, the commodity must be stored. Thus the hedger will be incurred additional costs reflected in the forward price as an increased interest rate to be paid. On the other hand, holding the commodity has a certain advantage over being long a forward contract due to the greater flexibility. The notion of convenience yield is introduced to explain this additional benefit accrued to the owner of the physical commodity (see [12, 16] for more details on convenience yield). Denoting the yield from storage and convenience, one derives a forward price for . In contrast to a classical commodity market, agriculture say, where the insurer can hedge her risk to some extent, the spot power market on the other hand is completely unhedgeable. Speculators cannot hedge in the power spot, where only physical players can take part. In this respect, a forward on power can be seen as a pure insurance instrument, where a speculator can offer insurance to a producer. To describe the added premium that the insurer charges to take on the risk one can also use an increased interest rate as above. We may refer to as the risk premium.
We will introduce a pricing measure in terms of an explicit risk premium that should explain storage costs, illiquidity, and convenience yield. Let us assume that this and an interest rate are constant rates. Moreover, consider the measure change
Under this new probability measure it follows from Girsanov’s theorem that the rate of return of equals and its dynamics is given by
where is a Wiener process under , with
This implies that the discounted spot process of is given under by
Note that the rate of return in the -dynamics of the logreturns of equals , for . Defining , then
The forward price contracted at with time of delivery is defined such that (see e.g. )
When the dynamics of is given by (3) the latter is equivalent to
From the latter we see that the standard forward pricing theory is using a market price of risk , where is the risk-neutral change and is added due to storage costs and convenience yield, or an insurance premium for illiquidity. In this paper we will introduce such pricing measures for much more general models for the spot than a simple geometric Brownian motion.
2.2 A commodity spot market model with memory and jumps
We introduce our spot price dynamics in a commodity market as follows. Let
with being a generalised Langevin equation of the form
where is a deterministic function and is a strictly positive -adapted càdlàg process. The notation means , i.e., the left-limit of the process.
Furthermore, we consider a stochastic interest rate and an explicit risk premium process given by a bivariate Ornstein-Uhlenbeck (OU) type of dynamics with a mean level that depends on . That is
where , , for and , are deterministic functions uniformly bounded in by a constant. This type of dynamics (6)-(8) involving a memory is relevant for many commodity markets including power (see e.g.  and ). The interest rate is driven by the same stochastic factors and as and . This model is more general than a deterministic model and this particular choice is motivated by the aim to prove the existence of a measure change (see Section 4). In the next section we will state conditions ensuring the existence and uniqueness of a solution to the generalised Langevin equation (6).
2.3 Affine processes
As affine processes will play an important role in our considerations, we include in this section with preliminaries some useful results on this class of processes.
Affine processes are continuous-time Markov processes characterised by the fact that their log-characteristic function depends in an affine way on the initial state vector of the process. Recently affine models have gained significant attention in the finance literature mainly due to their analytical tractability (see for example , , and ). In the sequel we introduce an equivalent martingale measure under which the process introduced in (6)-(8) is going to be of time-inhomogeneous affine type. Since we are interested in pricing contingent claims written on a process , we recall in this subsection a result showing that pricing contingent claims in time-inhomogeneous affine models can be reduced to the solution of a set of Riccati-type ordinary differential equations.
Denote by the transpose of a given vector or a matrix. Assume there exists a unique solution to the following stochastic differential equation (SDE)
where is a -dimensional Brownian motion, for , is a random measure of the jumps of , is the compensator of the jump measure which we assume to be deterministic, , is continuous and such that is continuous for and . Moreover, is continuous in for and such that , for all .
Moreover, assume the following affine structure for the time-dependent parameters of the SDE
where and are matrices and and are -vectors. We consider a real-valued process for which we impose the following affine structure
for and .
Let and be -functions satisfying the following Riccati equations
We compute in the following theorem the discounted moment generating function of , conditional on the information at time in terms of the solution to the Riccati equations (2.3). For a proof we refer to [13, Theorem 2.13]. See also [19, Theorem 5.1].
Let . Suppose that (2.3) admits a unique solution . Then
This result allows for the use of Fourier transform techniques to compute derivative prices written on affine models, as we will return to in Section 5.
3 Analysis of the generalised Langevin equation
In this section we start from a generalised Langevin equation and its solution found by Laplace transformation to propose a solution to the corresponding SDE introduced in (6).
Consider the following generalised Langevin equation as in 
where stands for the memory kernel, represents Gaussian fluctuations, is a constant and denotes the time-derivative of . When is the Dirac delta function the solution will be Markovian. Using Laplace transforms and the definition
where is defined through its Laplace transform
with, for simplicity, . From (13) it follows that
We include here an example of a memory kernel that defines a corresponding function although it is singular at zero.
Let for . Then we should find a function satisfying (14). This means, after integrating both sides and invoking Fubini,
To solve the Volterra integral equation, we propose that
But the integral on the left hand side is connected to the Beta-distribution:
With we find the recursive relations
Using that , we reach that and
Now, we may ask whether the representation (16) of is a convergent series. By the ratio test we find
By Stirling’s formula, we have an approximation of the Gamma-function for large values being
for some positive constant . But then we have
when , since , and . Hence, is convergent for all .
Motivated by these considerations, we state the following claim.
Consider the generalised Langevin equation (6), which we recall to be
where is a strictly positive -adapted, càdlàg process and satisfying . Assume there exists a unique solution to (14). Further, let and be such that
Then the analogue of the solution (12), namely
is the unique solution to the generalised Langevin equation (6).
Step 1 Computing the differential of in (19), we get
For two solutions and of (20) with , it immediately follows from (20) that for any . Thus (19) is the unique solution to (20). Now, we insert (14) in (20), perform a change of variables by putting , and apply the stochastic Fubini theorem, see e.g. , to arrive at
By observing that the factor between brackets is exactly according to (19), we conclude that (20) and (6) are equivalent. Since (19) is the unique solution to (20) and satisfying (14) is unique, then (19) is also the unique solution to (6).
that also the integral in (20) is well defined.
Let be a real-valued function on and be a differentiable function such that
with a finite value and
Then starting from an SDE of type (20), that is considering
where is an -adapted process, we know from the proof of Proposition 3, that the SDE (23) admits a unique solution given by (19). Furthermore, we know that if there is a unique such that the relation (14) holds and and satisfy the integrability condition (18), then the SDE (23) is equivalent to the SDE (6).
SDEs of the type (23) are common for modelling for example temperature and wind speed in energy markets, see . In the next section, we will exploit the relation of these equations to the Langevin equation (6) to show the existence of equivalent martingale measures for such processes. To conclude this section we state the link between the Langevin equation (6) and Volterra equations, which is of importance for our analysis later.
(A) Notice that the SDE (6) under consideration belongs to the class of Volterra equations driven by a Lévy process. Those are SDEs of the type
where is an -measurable random variable satisfying and . Volterra equations appear naturally in many areas of mathematics such as integral transforms, transport equations, and functional differential equations (we refer to  for an introduction and a general overview of these equations in the deterministic case). They also appear in applications in biology, physics, and finance. For an example in economics (which also applies to population dynamics), we refer to Example 3.4.1 in . In the framework of stochastic delay equations and optimal control theory, we refer to  and . These processes have recently been proposed in the framework of modelling electricity and commodity prices, see e.g.  and .
The existence and uniqueness of the solution to the SDE of type (24) is well studied (see e.g., [22, Theorem 4.6] for Volterra equations driven by Brownian motion and  for Volterra equations driven by semimartingales). In our analysis, we showed the existence and uniqueness of the solution by exploiting the link of the Langevin equation (6) to the SDE (20).
4 Change of measure
As recalled in Section 2, we need the dynamics of under a pricing measure to price contingent claims on the commodity spot . We study a particular class of pricing measures , and start by applying the Itô formula to to derive the dynamics of under the market probability from the dynamics of in (6)
where . Recall the dynamics of in (7). Then, the dynamics of the discounted price process , , is given by
We consider a pricing measure defined by for a density process (see, e.g., the Girsanov Theorem 1.31 in )
with being the risk premium process defined in (8). When the measure exists, then
are respectively a Wiener process and a compensated jump measure under with compensator . The dynamics of the discounted price process of under is
To prove the existence of the measure , we need the following lemma which shows that can be bounded by the maximum of on . To show this crucial bound we apply techniques similar to the proof of the Gronwall Inequality, that works here due to the particular Ornstein-Uhlenbeck-like structure of the dynamics of and .
Let , , and be respectively as in (7), (8), and (6) where we assume , , , , to be uniformly bounded by a constant, and to be functions of bounded variation on , , , -a.s., , for strictly positive constants and , and the initial condition is bounded by a constant -a.s. Further, assume that the kernel function is square integrable over . Define
for some positive constant (depending on ).
Multiplying both sides of (31) with , applying the product rule to , integrating both sides from zero to , and dividing by we get
Applying the triangle inequality, the boundedness of , , and and the assumption on we bound for by