A change of measure in the BNS model for commodities

# A change of measure preserving the affine structure in the BNS model for commodity markets

Fred Espen Benth
Centre of Mathematics for Applications
University of Oslo
P.O. Box 1053, Blindern
N–0316 Oslo, Norway
http://folk.uio.no/fredb/
Centre of Mathematics for Applications
University of Oslo
P.O. Box 1053, Blindern
N–0316 Oslo, Norway
July 5, 2019
###### Abstract.

For a commodity spot price dynamics given by an Ornstein-Uhlenbeck process with Barndorff-Nielsen and Shephard stochastic volatility, we price forwards using a class of pricing measures that simultaneously allow for change of level and speed in the mean reversion of both the price and the volatility. The risk premium is derived in the case of arithmetic and geometric spot price processes, and it is demonstrated that we can provide flexible shapes that is typically observed in energy markets. In particular, our pricing measure preserves the affine model structure and decomposes into a price and volatility risk premium, and in the geometric spot price model we need to resort to a detailed analysis of a system of Riccati equations, for which we show existence and uniqueness of solution and asymptotic properties that explains the possible risk premium profiles. Among the typical shapes, the risk premium allows for a stochastic change of sign, and can attain positive values in the short end of the forward market and negative in the long end.

We are grateful for the financial support from the project ”Energy Markets: Modeling, Optimization and Simulation (EMMOS)”, funded by the Norwegian Research Council under grant Evita/205328.

## 1. Introduction

Benth and Ortiz-Latorre [9] analysed a structure preserving class of pricing measures for Ornstein-Uhlenbeck (OU) processes with applications to forward pricing in commodity markets. In particular, they considered multi-factor OU models driven by Lévy processes having positive jumps (so-called subordinators) or Brownian motions for the spot price dynamics and analysed the risk premium when the level and speed of mean reversion in these factor processes were changed.

In this paper we continue this study for OU processes driven by Brownian motion, but with a stochastic volatility perturbing the driving noise. The stochastic volatility process is modelled again as an OU process, but driven by a subordinator. This class of stochastic volatility models were first introduced by Barndorff-Nielsen and Shephard [1] for equity prices, and later analysed by Benth [2] in commodity markets. Indeed, the present paper is considering a class of pricing measures preserving the affine structure of the spot price model analysed in Benth [2].

Our spot price dynamics is a generalization of the Schwartz model (see Schwartz [23]) to account for stochastic volatility. The Schwartz model have been applied to many different commodity markets, including oil (see Schwartz [23]), power (see Lucia and Schwartz [20]), weather (see Benth and Šaltytė Benth [4]) and freight (see Benth, Koekebakker and Taib [8]). Like Lucia and Schwartz [20], we analyse both geometric and arithmetic models for the spot price evolution. There exists many extensions of the model, typically allowing for more factors in the spot price dynamics, as well as modelling the convenience yield and interest rates (see Eydeland and Wolyniec [11] and Geman [13] for more on such models). In Benth [2], the Schwartz model with stochastic volatility has been applied to model empirically UK gas prices. Also other stochastic volatility models like the Heston have been suggested in the context of commodity markets (see Eydeland and Wolyniec [11] and Geman [13] for a discussion and further references).

The class of pricing measures we study here allows for a simultaneous change of speed and level of mean reversion for both the (logarithmic) spot price and the stochastic volatility process. The mean reversion level can be flexibly shifted up or down, while the speed of mean reversion can be slowed down. It significantly extends the Esscher transform, which only allows for changes in the level of mean reversion. Indeed, it decomposes the risk premium into a price and volatility premium. It has been studied empirically in some commodity markets for multi-factor models in Benth, Cartea and Pedraz [7]. As we show, the class of pricing measures preserves the affine structure of the model, but leads to a rather complex stochastic driver for the stochastic volatility. For the arithmetic spot model we can derive analytic forward prices and risk premium curves. On the other hand, the geometric model is far more complex, but the affine structure can be exploited to reduce the forward pricing to solving a system of Riccati equations by resorting to the theory of Kallsen and Muhle-Karbe [18]. The forward price becomes a function of both the spot and the volatility, and has a deterministic asymptotic dynamics when we are far from maturity.

By careful analysis of the associated system of Riccati equations, we can study the implied risk premium of our class of measure change as a function of its parameters. The risk premium is defined as the difference between the forward price and the predicted spot price at maturity, and is a notion of great importance in commodity markets since it measures the price for entering a forward hedge position in the commodity (see e.g. Geman [13] for more on this). In particular, under rather mild assumptions on the parameters, we can show that the risk premium may change sign stochastically, and may be positive for short times to maturity and negative when maturity is farther out in time. This is a profile of the risk premium that one may expect in power markets based on both economical and empirical findings. Geman and Vasicek [14] argue that retailers in the power market may induce a hedging pressure by entering long positions in forwards to protect themselves against sudden price increases (spikes). This may lead to positive risk premia, whereas producers induce a negative premium in the long end of the forward curve since they hedge by selling their production. This economic argument for a positive premium in the short end is backed up by empirical evidence from the German power market found in Benth, Cartea and Kiesel [6]. In the geometric model, we show that the sign of the risk premium depends explicitly on the current level of the logarithmic spot price.

We recover the Esscher transform in a special case of our pricing measure. The Esscher transform is a popular tool for introducing a pricing measure in commodity markets, or, equivalently, to model the risk premium. For constant market prices of risk, which are defined as the shift in level of mean reversion, we preserve the affine structure of the model as well as the Lévy property of the driving noises of the two OU processes that we consider (indeed, the spot price dynamics is driven by a Brownian motion). We find such pricing measures in for example Lucia and Schwartz [20], Kolos and Ronn [19] and Schwartz and Smith [24]. We refer the reader to Benth, Šaltytė Benth and Koekebakker [5] for a thorough discussion and references to the application of Esscher transform in power and related markets. We note that the Esscher transform was first introduced and applied to insurance as a tool to model the premium charged for covering a given risk exposure and later adopted in pricing in incomplete financial markets (see Gerber and Shiu [15]). In many ways, in markets where the underlying commodity is not storable (that is, cannot be traded in a portfolio), the pricing of forwards and futures can be viewed as an exercise in determining an insurance premium. Our more general change of measure is still structure preserving, however, risk is priced also in the sense that one slows down the speed of mean reversion. Such a reduction allow the random fluctuations of the spot and the stochastic volatility last longer under the pricing measure than under the objective probability, and thus spreads out the risk.

Although our analysis has a clear focus on the stylized facts of the risk premium in power markets, the proposed class of pricing measures is clearly also relevant in other commodity markets. As already mentioned, markets like weather and freight share some similarities with power in that the underlying ”spot” is not storable. Also in more classical commodity markets like oil and gas there are evidences of stochastic volatility and spot prices following a mean-reversion dynamics, at least as a component of the spot. Moreover, in the arithmetic case our analysis relates to the concept of unspanned volatility in commodity markets, extensively studied by Trolle and Schwartz [25]. The forward price will not depend on the stochastic volatility factor, and hence one cannot hedge options by forwards alone. Interestingly, the corresponding geometric model will in fact span the stochastic volatility.

We present our results as follows. In the next Section we present the spot model, and follow up in Section 3 with introducing our pricing measure validating that this is indeed an equivalent probability. In Section 4 we derive forward prices under the arithmetic spot price model, and analyse the implied risk premium. Section 5 considers the corresponding forward prices and the implied risk premium for the geometric spot price model. Here we exploit the affine structure of the model to analyse the associated Riccati equation, and provide insight into the potential risk premium profiles that our set-up can generate. Both Section 4 and 5 have numerous empirical examples.

## 2. Mathematical model

Suppose that is a complete filtered probability space, where is a fixed finite time horizon. On this probability space there are defined , a standard Wiener process, and a pure jump Lévy subordinator with finite expectation, that is a Lévy process with the following Lévy-Itô representation where is a Poisson random measure with Lévy measure satisfying We shall suppose that and are independent of each other.

As we are going to consider an Esscher change of measure and geometric spot price models, we introduce the following assumption on the existence of exponential moments of .

###### Assumption 1.

Suppose that

 ΘL≜sup{θ∈R+:E[eθL(1)]<∞}, (2.1)

is a constant strictly greater than one, which may be .

Some remarks are in order.

###### Remark 2.1.

In the cumulant (or log moment generating) function is well defined and analytic. As , has moments of all orders. Also, is convex, which yields that and, hence, that is non decreasing. Finally, as a consequence of , we have that is non negative.

###### Remark 2.2.

Thanks to the Lévy-Kintchine representation of we can express and its derivatives in terms of the Lévy measure We have that for

 κL(θ) =∫∞0(eθz−1)ℓ(dz)<∞, κ(n)L(θ) =∫∞0zneθzℓ(dz)<∞,n∈N,

showing, in fact, that

Consider the OU processes

 X(t) =X(0)−α∫t0X(s)ds+∫t0σ(s)dW(s)t∈[0,T], (2.2) σ2(t) =σ2(0)−ρ∫t0σ2(s)ds+L(t),t∈[0,T], (2.3)

with Note that, in equation 2.2, is written as a sum of a finite variation process and a martingale. We may also rewrite equation 2.3 as a sum of a finite variation part and pure jump martingale

 σ2(t)=σ2(0)+∫t0(κ′L(0)−ρσ2(s))ds+∫t0∫∞0z~NL(ds,dz),t∈[0,T],

where is the compensated version of . In the notation of Shiryaev [22], page 669, the predictable characteristic triplets (with respect to the pseudo truncation function ) of and are given by

 (BX(t),CX(t),νX(dt,dz))=(−α∫t0X(s))ds,∫t0σ2(s)ds,0),t∈[0,T],

and

 (Bσ2(t),Cσ2(t),νσ2(dt,dz))=(∫t0(κ′L(0)−ρσ2(s))ds,0,ℓ(dz)dt),t∈[0,T],

respectively. In addition, applying Itô’s Formula to and one can find the following explicit expressions for and

 X(t) =X(s)e−α(t−s)+∫tsσ(u)e−α(t−u)dW(u), (2.4) σ2(t) =σ2(s)e−ρ(t−s)+κ′L(0)ρ(1−e−ρ(t−s))+∫ts∫∞0e−ρ(t−u)z~NL(du,dz), (2.5)

where

Using the notation in Kallsen and Muhle-Karbe [18], we have that the process has affine differential characteristics given by

 β0 =(κ′L(0)0),γ0=(0000),φ0(A)=∫∞01A(z,0)ℓ(dz),∀A∈B(R2) β1 =(−ρ0),γ1=(0001),φ1(A)≡0,∀A∈B(R2), β2 =(0−α),γ2=(0000),φ2(A)≡0,∀A∈B(R2).

These characteristics are admissible and correspond to an affine process in

## 3. The change of measure

We will consider a parametrized family of measure changes which will allow us to simultaneously modify the speed and the level of mean reversion in equations (2.2) and (2.3). The density processes of these measure changes will be determined by the stochastic exponential of certain martingales. To this end, consider the following family of kernels

 Gθ1,β1(t) ≜σ−1(t)(θ1+αβ1X(t)),t∈[0,T], (3.1) Hθ2,β2(t,z) (3.2)

The parameters and will take values on the following sets where and is given by equation (2.1). By Assumption 1 and Remarks 2.1 and 2.2, these kernels are well defined.

###### Example 3.1.

Typical examples of and are the following:

1. Bounded support: has a jump of size i.e. In this case and

2. Finite activity: is a compound Poisson process with exponential jumps, i.e., for some and In this case and

3. Infinite activity: is a tempered stable subordinator, i.e., for some and In this case also and

Next, for define the following family of Wiener and Poisson integrals

 ~Gθ1,β1(t) ≜∫t0Gθ1,β1(s)dW(s),t∈[0,T], ~Hθ2,β2(t) ≜∫t0∫∞0(Hθ2,β2(s,z)−1)~NL(ds,dz),t∈[0,T],

associated to the kernels and respectively.

We propose a family of measure changes given by with

 dQ¯θ,¯βdP∣∣∣Ft≜E(~Gθ1,β1+~Hθ2,β2)(t),t∈[0,T]. (3.3)

Let us assume for a moment that is a strictly positive true martingale (this will be proven in Theorem 3.4 below): Then, by Girsanov’s theorem for semimartingales (Theorems 1 and 3, pages 702 and 703 in Shiryaev [22]), the process and become

 X(t) =X(0)+BXQ¯θ,¯β(t)+σ(t)WQ¯θ,¯β(t),t∈[0,T], σ2(t) =σ2(0)+Bσ2Q¯θ,¯β(t)+∫t0∫∞0z~NLQ¯θ,¯β(ds,dz),t∈[0,T],

with

 BXQ¯θ,¯β(t) =∫t0(θ1−α(1−β1)X(s))ds,t∈[0,T], (3.4) Bσ2Q¯θ,¯β(t) =∫t0(κ′L(0)−ρσ2(s))ds+∫t0∫∞0z(Hθ2,β2(s,z)−1)ℓ(dz)ds (3.5) =∫t0{(κ′L(0)−ρσ2(s))+∫∞0z(eθ2z−1)ℓ(dz) +ρβ2κ′′L(θ2)∫∞0z2eθ2zℓ(dz)σ2(s−)}ds =∫t0(κ′L(θ2)−ρ(1−β2)σ2(s))ds,t∈[0,T],

where is a -standard Wiener process and the -compensator measure of (and ) is

 vσ2Q¯θ,¯β(dt,dz)=vLQ¯θ,¯β(dt,dz)=Hθ2,β2(t,z)ℓ(dz)dt.

In conclusion, the semimartingale triplet for and under are given by and respectively.

###### Remark 3.2.

Under the process is affine with differential characteristics given by

 β0 =(κ′L(θ2)θ1),γ0=(0000),φ0(A)=∫∞01A(z,0)eθ2zℓ(dz),∀A∈B(R2), β1 =(−ρ(1−β2)0),γ1=(0001),φ1(A)=∫∞01A(z,0)ρβ2κ′′L(θ2)zeθ2zℓ(dz),∀A∈B(R2), β2 =(0−α(1−β1)),γ2=(0000),φ2(A)≡0,∀A∈B(R2).

These characteristics are admissible and correspond to an affine process in

###### Remark 3.3.

Under still satisfies the Langevin equation with different parameters, that is, the measure change preserves the structure of the equations for . However, the process is not a Lévy process under , but it remains a semimartingale. The equation for is the same under the new measure but with different parameters. Therefore, one can use Itô’s Formula again to obtain the following explicit expressions for and

 X(t) =X(s)e−α(1−β1)(t−s)+θ1α(1−β1)(1−e−α(1−β1)(t−s)) (3.6) +∫tsσ(u)e−α(1−β1)(t−u)dWQ¯θ,¯β(u), σ2(t) =σ2(s)e−ρ(1−β2)(t−s)+κ′L(θ2)ρ(1−β2)(1−e−ρ(1−β2)(t−s)) (3.7) +∫ts∫∞0e−ρ(1−β2)(t−u)z~NLQθ,β(du,dz),

where

We prove that is a true probability measure, that is, is a strictly positive true martingale under for . We have the following theorem.

###### Theorem 3.4.

Let . Then is a strictly positive true martingale under .

###### Proof.

That is strictly positive follows easily from the fact that the Lévy process is a subordinator as this yields strictly positive jumps of . It holds that the quadratic co-variation between and is identically zero, by Yor’s formula in Protter [21, Theorem 38]. Hence, we can write

 E(~Gθ1,β1+~Hθ2,β2)(t)=E(~Gθ1,β1)(t)E(~Hθ2,β2)(t),t∈[0,T]. (3.8)

By classical martingale theory, we know that is a true martingale if and only if

 EP[E(~Gθ1,β1+~Hθ2,β2)(T)]=1,

which, using Yor’s formula, is equivalent to showing that

 EP[E(~Gθ1,β1)(T)E(~Hθ2,β2)(T)]=1.

Let, be the -algebra generated by up to time then we have that

 EP[E(~Gθ1,β1)(T)E(~Hθ2,β2)(T)]=EP[EP[E(~Gθ1,β1)(T)|FTL]E(~Hθ2,β2)(T)].

If we show that then we will have finished, because by Theorem 3.10 in Benth and Ortiz-Latorre [9], we have that is a true martingale and, hence, The idea of the proof is based on the fact that is the expectation of assuming that is a deterministic function that, in addition, is bounded below by Using this information one can show that, conditionally on knowing is a true martingale and, hence, Let us sketch the proof that is basically the same as in Section 3.1 in [9] but, now, with being a function. First, we show that, conditionally on is a square integrable -martingale because

 EP[(~Gθ1,β1)2|FLT] =EP[∫T0σ−2(t)(θ1+αβ1X(t))2dt|FLT] ≤2σ(0)−2e2ρT(θ21T+α2EP[∫T0X2(t)dt])<∞,

(see Proposition 3.6. in [9]). To show that is a -martingale on we consider a reducing sequence of stopping times for and, proceeding as in Theorem 3.7 in [9], we define a sequence of probability measure with Radon-Nykodim densities given by Doing the same reasonings as in Theorem 3.7 in [9], we reduce the problem to prove that

 supn≥1EQnθ1,β1[∫T01[0,τn](Gθ1,β1(t))2dt]<∞.

Now, one has

 EQnθ1,β1[∫T01[0,τn](Gθ1,β1(t))2dt]≤2σ(0)−2e2ρT(θ21T+α2EQnθ1,β1[∫T01[0,τn]X2(t)dt]).

To bound the last expectation in the previous expression we use that we know the dynamics of for under , which is obtained from equation (3.6) by setting and Therefore, we can write

 EQnθ1,β1[∫T01[0,τn](t)X(t)2dt] ≤2⎧⎨⎩EQnθ1,β1[∫T01[0,τn](t)(X(0)e−α(1−β1)t+θ1α(1−β1)(1−e−α(1−β1)t))2dt] +EQnθ1,β1[∫T01[0,τn](t)(∫t0σ(s)e−α(1−β1)(t−s)dWQnθ1,β1(s))2dt]} ≤2T{(|X(0)|+(|θ1|)T)2+σ(0)−2e2ρTT2}<∞.

Here, we have used that the function for and that

 EQnθ1,β1[(∫t0σ(s)e−α(1−β1)(t−s)dWQnθ1,β1(s))2] =σ(0)−2e2ρT∫t0e−2α(1−β1)(t−s)ds≤σ(0)−2e2ρTT.

The Theorem follows. ∎

We also have the following result on the independence of the driving noise processes after the change of measure:

###### Lemma 3.5.

Under , the Brownian motion and the random measure are independent.

###### Proof.

To prove the independence of and under it is sufficient to prove that

 EQ¯θ,¯β[exp(ik∑j=1(μjWQ¯θ,¯β(tj)+ξj∫tj0∫∞0z~NLQ¯θ,¯β(ds,dz)))] =EQ¯θ,¯β[exp(ik∑j=1μjWQ¯θ,¯β(tj))]EQ¯θ,¯β[exp(ik∑j=1ξj∫tj0∫∞0z~NQ¯θ,¯βL(ds,dz))],

for any and We will make use of the following notation: given a process we will denote by where by convention. We have that

 EQ¯θ,¯β[exp(ik∑j=1(μjWQ¯θ,¯β(tj)+ξj∫tj0∫∞0z~NLQ¯θ,¯β(ds,dz)))] =EP[E(~Gθ1,β1+~Hθ2,β2)(tk)exp(ik∑j=1(μjWQ¯θ,¯β(tj)+ξj∫tj0∫∞0z~NLQ¯θ,¯β(ds,dz)))] =EP[EP[E(~Gθ1,β1)(tk)exp(ik∑j=1μjWQ¯θ,¯β(tj))|FLtk]E(~Hθ2,β2)(tk)exp(ik∑j=1ξj∫tj0∫∞0z~NLQ¯θ,¯β(ds,dz))],

and

 EP[E(~Gθ1,β1)(tk)exp(ik∑j=1μjWQ¯θ,¯β(tj))|FLtk] =EP[E(~Gθ1,β1)(tk−1)exp(∫tktk−1Gθ1,β1(s)dW(s)−12∫tktk−1G2θ1,β1(s)ds) ×exp(ik−1∑j=1μjWQ¯θ,¯β(tj)+iμkWQ¯θ,¯β(tk−1)+iμkΔWQ¯θ,¯β)|FLtk] =EP[E(~Gθ1,β1)(tk−1)exp(ik−1∑j=1μjWQ¯θ,¯β(tj)+iμkWQ¯θ,¯β(tk−1)) ×EP[exp(∫tktk−1Gθ1,β1(s)dW(s)−12∫tktk−1G2θ1,β1(s)ds +iμk(ΔkW−∫tktk−1Gθ1,β1(s)ds))|FLtk∨FWtk−1]|FLtk].

Moreover, using similar arguments to those used in the proof of Theorem 3.4, we have that

 exp(∫t0(Gθ1,β1(s)+iμk)dW(s)−12∫t0(Gθ1,β1(s)+iμk)2ds),

is a -martingale and, then, we get