Abstract
In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer’s utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.
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1 Introduction
In the last 15 years, since the start of the energy market deregulation and privatization in Europe and in the US, the study of energy markets became a challenging topic from both a practical and a theoretical perspective. Especially important is the problem of pricing and hedging of energy contracts. This is far from being trivial because of the peculiarity of the models and since these contracts typically have a very complex structure, incorporating optionality features which can be exercised by the buyer at multiple times. The two main examples of products used in energy markets for primary supply are swing contracts and forward contracts. While the structure of the forward contracts is rather simple, swing contracts are much more involved as they give the buyer some degrees of freedom about the quantity of energy to buy for each sub-period, usually with daily or monthly scale, subject to a cumulated constraint over the contract period. This flexibility is much welcomed by the contract buyers, as energy markets are affected by many unexpected events such as peaks in consumption related to sudden weather changes, breakdowns of power plants, financial turmoils and so on. Many other kinds of contract are traded in the energy market, they are often negotiated over-the-counter, and some of them, e.g. virtual storage contracts, also include optionality components similar to the ones of swing contracts.
The pricing of these products has a consolidated tradition in discrete time models (see, e.g. Edoli et al. 2013 or Henaff et al. 2013 and references therein), which is manly based on dynamic programming. The article (Jaillet et al. 2004) use a multilevel lattice method to study the pricing of a swing option on natural gas. The two papers (Pagès et al. 2009, 2010) propose a different method based on optimal quantization theory. In continuous time models, the first approaches were based on optimal switching techniques (e.g., Carmona and Ludkovski 2010) or multiple stopping (e.g., Carmona and Touzi 2008). In all these articles the optionality features of the structured product can be exercised over a discrete set of stopping times, that can be chosen by the buyer. A very detailed comparison of the literature on storage and swing evaluation can be found in Aïd (2015), Table 4.1.
A different approach consists in approximating the contract payoff with its continuous time counterpart. This idea has been proposed in Benth et al. (2012) (and further exploited in Basei et al. (2014)) for swing contracts and in Chen and Forsyth (2007), Felix (2012), Thompson et al. (2009) for virtual storage contracts. Other examples of structured contracts can be treated with the same methodology, see e.g. Benth and Eriksson (2013) for flexible load contracts and tolling agreements. The main advantage of this approach is that it makes the pricing problem more tractable, since it allows using the stochastic control theory in continuous time, based on PDE methods. In those papers, the price of a structured contract is defined—in analogy with American options—as the supremum, over all the strategies available to the buyer, of the expected payoff, where the expectation is taken under a given risk-neutral measure. When the model for the underlying of the structured contract is Markovian, as it happens in most of the models used in practice, the pricing problem reduces to solving the corresponding Hamilton–Jacobi–Bellman (HJB) equation. Notice that the choice of the risk-neutral measure is not at all obvious since energy market models are typically incomplete, because of the presence of assets bearing non tradable risks. Moreover, with the exception of Warin (2012) which focuses on gas storage contracts and uses a delta-hedging approach, the problem of hedging the risk coming from a long position in those structured products is not considered in those papers. Hedging such contracts can be quite a delicate task in energy markets, since the underlying of the contracts is often not tradable, hence the buyer has to trade in some other asset with a good correlation with the underlying. For an extensive review of the existing literature with descriptions of the most traded contracts and a detailed comparison between the main articles we refer once more to the recent book (Aïd 2015, Chapter 4).
Our contribution to the literature consists in building on the idea of continuously approximating the payoff as in Basei et al. (2014), Benth et al. (2012), in order to provide a general framework, where both problems of pricing and hedging of structured contracts can be solved in a consistent fashion. The main novelty of this paper is that the buyer of the given structured contract is allowed to (at least partially) hedge his position by trading in forward contracts, written on the underlying of the structured contract itself or on some asset correlated with the underlying. We model the forward market as a general incomplete multivariate market model with finitely many forward contracts (with different maturities), evolving over time as diffusions whose coefficients depend on a certain number of exogenous stochastic factors with Markovian dynamics. The underlying of the structured contract is defined as a function of such factors. This setting includes many models that have been previously proposed and studied in the literature, e.g. Aïd et al. (2014), Carmona and Ludkovski (2006), Cartea and Villaplana (2008), Schwartz and Smith (2000) to cite a few.
The market being incomplete, we adopt the utility indifference price (henceforth UIP) approach, which is one of the most appealing ways of pricing in incomplete markets, since it naturally incorporates the buyer preferences in the price of the contract. We assume that the preferences of the buyer can be encoded in an exponential utility function with a risk aversion parameter \(\gamma >0\). The UIP approach has been extensively used for pricing European and American options in a wide range of financial market models. We refer to Henderson and Hobson (2009) for an excellent survey on this approach. This approach was already used in Porchet et al. (2009), Ludkovski (2008) for the evaluation of industrial assets (see Remark 2.10) and in Fiorenzani (2006) and Benedetti and Campi (2016) for energy derivatives.
We apply this method for evaluating a rather general structured derivative. Its buying UIP will be characterized as the difference between the two log-value functions of the agent (with and without the contract), that can be obtained as the unique viscosity solutions of a suitable HJB equation. Our results are consistent with the ones in Basei et al. (2014), Benth et al. (2012), Chen and Forsyth (2007), Felix (2012), Thompson et al. (2009), in the case of complete market models. Moreover, the shape of such HJB equation gives reasonable candidates for the optimal withdrawal strategy of the structured product, as well as for the related hedging strategy.
Finally, we push our general results further in two specific examples. One of them is a class of models with two risky assets, one traded and one nontraded, and constant correlation. This includes models with a nontraded asset and basis risk, which have been studied by many authors (see, e.g., the papers Davis 2006, Henderson 2002, Monoyios 2004 to cite only a few). For these models, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk-neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Such a perturbation is due to the dependence of drift and volatility of the traded asset on the nontraded one and depends on the value function without the contract. It seems that such a measure change is new to the incomplete market literature. The second example is based on a slight generalization of the two-factor model developed in Cartea and Villaplana (2008) for energy markets, where the factors can be correlated.
The paper is organized as follows. In Sect. 2 we formulate the problem of pricing, by introducing the general payoff of the structured contracts, the market model and the (exponential) utility indifference price. In Sect. 3, we characterize the UIP in terms of viscosity solutions of suitable HJB equations. In Sect. 4 we consider the two examples described above while Sect. 5 presents some numerical applications of our results. Finally, Sect. 6 concludes.
Notation In what follows, unless explicitly stated, vectors will be column vectors, the symbol “*” will denote transposition and the trace of a square matrix A will be denoted by \(\text {tr}(A)\). Furthermore, \(\langle a, b \rangle := a^* b\) will stand for the Euclidean scalar product. We choose as matricial norm \(| A | = \sqrt{\mathrm{{tr}}(AA^*)}\). On the set \(\mathcal S_n\) of all symmetric squared matrices of order n, we define the order \(A\le B\) if and only if \(B-A \in \mathcal S_n ^+\), the subspace of nonnegative definite matrices in \(\mathcal S_n\). We will denote by \(I_n\) the identity matrix of dimension n.
2 Formulation of the problem
Let \(T>0\) be a finite time horizon. All the processes introduced below will be defined on the canonical probability space \((\Omega , \mathcal F, {\mathbb {P}})\), where \(\Omega := C([0,T];{\mathbb {R}}^d)\) is the space of all continuous functions from [0, T] into \(\mathbb R^d\). For \(\omega \in \Omega \), we set \(W_t (\omega ) = \omega (t)\) and define \((\mathcal F_t)_{t\in [0,T]}\) as the smallest right-continuous filtration such that W is optional. Moreover, \(\mathcal F := \mathcal F_T\). We let \({\mathbb {P}}\) be the Wiener measure on \((\Omega , {\mathbb {F}}_T)\). We can assume without loss of generality that such a filtration is complete.
2.1 Structured products
In this section we give a short description of the main structured products that are traded in energy markets. The typical payoff is given by a family of random variables
indexed by a control u, which typically represents the marginal quantity of commodity purchased and it belongs to a suitable set of admissible controls \(\mathcal U\) that we will specify later. In particular, the admissible controls take values in some bounded interval \([0,{\bar{u}}]\) for a given threshold \({\bar{u}} >0\). The variable P in the Eq. (2.1) above denotes the spot price of the commodity (e.g., gas) and \(Z^u _t := z_0 + \int _0^t u_s ds\) for all \(t \in [0,T]\), for some initial value \(z_0 \ge 0\). For technical reasons, that will become clear in the proofs of our results, we will need the following assumption on the structured products.
Assumption 2.1
The functions \(L: {\mathbb {R}} \times [0,{\bar{u}} T] \times [0,{\bar{u}}] \rightarrow {\mathbb {R}}\) and \(\Phi : {\mathbb {R}} \times [0,{\bar{u}} T] \rightarrow {\mathbb {R}}\) in (2.1) are continuous and bounded.
The most common structured products in energy markets are swing and virtual storage contracts. More details are given just below. See also the subsequent Remark 2.4 explaining how one can safely modify these contracts in order to satisfy Assumption 2.1.
Example 2.2
(Swing contract) For a swing contract one has (see, e.g., Basei et al. 2014; Benth et al. 2012)
where K is the purchase price or strike price, and u is any admissible control. These products usually include some additional features, such as inter-temporal constraints on u or on the cumulated control \(Z^u\) or some penalty function appearing in the payoff. More precisely, constraints on u and \(Z^u\) are typically of the form \(Z_T^u \in [m,M]\), with \(0 \le m < M\), with possibly further intermediate constraints on \(Z_{t_i}^u\), \(t_i < T\), \(i = 1,\ldots ,k\). In the absence of such additional constraints, a penalty is usually present which can be expressed as a function \(\Phi \) of the terminal spot price \(P_T\) and cumulated consumption \(Z^u_T\). A typical form of \(\Phi \) is
for constants \(C>0\) and \(0 \le m < M\) (see Basei et al. 2014; Benth et al. 2012 and references therein). We will focus on the latter case, i.e., a non-zero penalty function \(\Phi (P_T ,Z_T^u)\) without any other contraints on the admissible controls.
Example 2.3
(Virtual storage contract) These products replicate a physical gas storage position, while being handled as pure trading contracts. In this case one has
with \(C,M > 0\) suitable constants, \(a(z,u):= \bar{a} \mathbbm {1}_{u<0}\) and where the control u represents the gas injected into the reservoir and is such that
where \(u_{\mathrm {in}},u_{\mathrm {out}}\) are suitable deterministic functions given by the physics of fluids: their typical shapes are
with \(Z_b,K_i > 0\), \(i = 1,2,3\), given constants (Chen and Forsyth 2007; Felix 2012; Thompson et al. 2009).
Remark 2.4
The boundedness of L as in Assumption 2.1 is not verified in the two Examples 2.2 and 2.3, where L is linear in p, which can in principle take any real value. In practice, one can artificially bound L, for example by introducing
so that \(|{\tilde{L}}(p,z,u)| \le \kappa \) for all (p, z, u), for a suitably chosen and large enough threshold \(\kappa >0\) such that the instantaneous profit should not be larger than \(\kappa \) in absolute value with high probability. The same truncation argument can be applied to the penalty function \(\Phi (p,z)\). Alternatively, one could truncate the unbounded variable p appearing in both payoffs L and \(\Phi \).
2.2 The market model
The spot price of the commodity, P, underlying the structured products, is modelled as \(P_t:= p(t,X_t)\), where \(p:[0,T] \times {\mathbb {R}}^m \rightarrow {\mathbb {R}}\) is a measurable function and X represents the factors driving the market. We assume that the process X has Markovian dynamics
with drift \(b:[0,T] \times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^m\) and volatility matrix \(\Sigma :[0,T] \times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^{d \times m}\).
We also assume that \(n\le d\) forward contracts on the commodity P are traded in the market, with maturities \(T_1< \cdots < T_n\), with \(T_1 \ge T\). Letting \(F^i\) to denote the price of the forward contract with maturity \(T_i\), \(i = 1,\ldots ,n\), we assume that the dynamics of \(F:= (F^1,\ldots ,F^n)\) is given by
for some functions \(\mu _F: [0,T]\times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^n\) and \(\sigma _F: [0,T]\times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^{d\times n}\).
Assumptions on the coefficients of X and F are given below. We will always assume throughout the paper that the interest rate is zero.
Notice that the forward contracts are not necessarily written on the commodity with spot price P, as they could also be written on some correlated commodity. For instance, P could be the spot price of gasoline, while the F’s are written on oil, as in Carmona and Ludkovski (2006), Fiorenzani (2006). This can be also due to illiquidity or to the fact that forward contracts relative to the commodity do not exist: for a detailed discussion of this phenomenon, see Carmona and Ludkovski (2006), Sect. 2.3.
We will always work under the following standing assumptions on the coefficients of the model:
Assumption 2.5
-
(i)
The function \(p{:}\, [0,T] \times {\mathbb {R}}^m \rightarrow \mathbb R\) is continuous.
-
(ii)
The coefficients \(b{:}\,[0,T] \times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^m\) and \(\Sigma :[0,T] \times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^{d \times m}\) of the factor process X are continuous functions, Lipschitz in x uniformly in t and with linear growth in x uniformly in t.
-
(iii)
The drift \(\mu _F{:}\, [0,T]\times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^n\) and the volatility \(\sigma _F: [0,T]\times {\mathbb {R}}^m \rightarrow {\mathbb {R}}^{d\times n}\) are continuous functions, Lipschitz in x uniformly in t and with linear growth in x uniformly in t.
Under such assumptions, the SDEs (2.3) and (2.4) are well-known to admit a unique strong solution (X, F) such that \(X_0 =x\) and \(F_0 = f_0\) [see, e.g., Theorem 13.1 in Rogers and Williams (2000), Chapter V].
2.3 Admissible strategies and utility indifference price
We consider an agent whose preferences are modelled by an exponential utility function \(U(x)=-\frac{1}{\gamma }e^{-\gamma x}\) , \(x \in {\mathbb {R}}\), with risk aversion parameter \(\gamma > 0\). We assume that (s)he has a long position in \(q \ge 0\) units of a given structured product with payoff \(C_T = (C_T ^u)_{u\in \mathcal U}\) with \(C_T ^u \) as in (2.1). Moreover, in order to hedge away the risk attached to such a contract, (s)he trades in the financial market of forward contracts described in the previous section. At any time \(s \in [0,T]\), the agent invests the amount of wealth \(\pi ^i_s\) in the forward contracts \(F^i\) with \(i = 1,\ldots ,n\). Hence the evolution of the agent’s portfolio is
where we recall that \(\langle \cdot , \cdot \rangle \) denotes the Euclidean scalar product in \(\mathbb {R}^n\) and we use the notation
At this point, we need to specify the set \(\mathcal A\) of admissible strategies.
Definition 2.6
Let \({\bar{u}} >0\) be a given threshold. The set of admissible controls \(\mathcal A\) is the set of all couples \((u,\pi )\), where u is any adapted process such that \(u_t \in [0,{\bar{u}}]\) for all \(t\in [0,T]\), and \(\pi \) is any progressively measurable \({\mathbb {R}}^n\)-valued process such that
for some \(\varepsilon >0\). We will denote by \(\mathcal U\) the set of all admissible controls u. Moreover, \(\mathcal A_t\) (resp. \(\mathcal U_t\)) will be the set of admissible controls \((u,\pi )\) (resp. admissible controls u) starting from t.
Now, we are ready to introduce the utility indifference (buying) price of q units of the structured product \(C_T\). We will use the notation \(C_{t,T}^u\) for the payoff of the structured contract \(C_T^u\) starting at time t, i.e.,
Moreover, we set \(C_T^u = C_{0,T}^u\).
Definition 2.7
The utility indifference (buying) price at time t for a position \(q \ge 0\) in the structured product, when starting from the initial portfolio value \(y_t\), is defined as the unique \(\mathcal F_t\)-measurable random variable \(v_t \) solution (whenever it exists) to
where
and \({\mathbb {E}}_t\) stands for the conditional expectation given \(\mathcal F_t\).
Clearly, \(V(y_0,q)\) gives the maximal expected utility from terminal wealth, computed at time 0, that an agent with an exponential utility can obtain starting from an initial wealth \(y_0\) and having a position \(q \ge 0\) in the structured product. Therefore, the (buying) UIP defined above represents the highest price the buyer is willing to pay for q units of the structured contract.
The maximization problem (2.7) can be easily translated into a standard Markovian control problem by suitably redefining the set of state variables as follows. Let \(t \in [0,T]\). First, using Equation (2.1), we can rewrite the terminal wealth as follows
Using the fact that \(P_t = p(t,X_t)\) is a function of the factor process X, we obtain that the value function in (2.7) equals
where the reward function G is given by
and the state variables \((X,Y^{u,\pi },Z^u)\) evolve as
with initial conditions \((X_t, Y^{u,\pi }_t,Z^u _t) = (x,y,z)\). Notice that the linear growth properties set in Assumption 2.5 combined with the boundedness of L in Assumption 2.1 give the following estimate for the controlled state process \((X,Y^{u,\pi },Z^u)\):
for some constant \(C_{u,\pi }>0\), which depends possibly on the control \((\pi , u)\) and is uniform in t.
Remark 2.8
Observe that the linear growth condition on b and \(\Sigma \) [cf. Assumption 2.5(ii)] imply, through an application of Gronwall’s lemma, that
for some \(\eta >0\).
Within this formulation, the UIP of \(q \ge 0\) units of the structured product is the unique solution \(v_t = v(t,x,y,z;q)\) (whenever it exists) to
Remark 2.9
In principle, the controls associated to the virtual storage contract described in Example 2.3 do not satisfy Definition 2.6, where the control \(u_t\) belongs to \([0,{\bar{u}}]\) with \({\bar{u}}\) constant. However, this example can be reduced to our setting by simply reparameterizing the control. In fact, one could define a new control c with values in [0, 2] such that the old control u satisfies \(u_t = f(c_t,Z_t)\) for a suitable function f(c, z) given by
Z solves
and \(L(p,z,c) = - p (f(c,z) - a(z,f(c,z)))\).
Remark 2.10
Here, we briefly discuss two papers, Ludkovski (2008) and Porchet et al. (2009), that do not fit (strictly speaking) the literature on structured products but that are somewhat related. Indeed, they both deal with the pricing of a physical/industrial asset using a UIP approach with an investment component. However, even though the optimization problems in Ludkovski (2008), Porchet et al. (2009) are mathematically similar to the one considered here, the controls affecting the asset are switching controls with finitely many states. Hence their methods, that are based on optimal switching and BSDEs, are different from ours. Finally, our model is more specific than theirs as it is tailor-made for the pricing and hedging of structured contracts on energy.
3 Characterization of the UIP with viscosity solutions
In this section we characterize, under some further technical assumptions given below, the UIP in terms of viscosity solutions of suitable Cauchy problems. More precisely, we prove that the log-value functions for problem (2.8) with zero initial wealth, defined as
can be characterized as the unique continuous viscosity solutions with quadratic growth to a suitable Cauchy problem. The UIP is obtained from there as the difference between the two log-value functions, corresponding to the problems with and without the structured products. This is done using some techniques developed in Pham (2002) together with recent results on uniqueness for a class of second order Bellman-Isaacs equations, established in Da Lio and Ley (2006).
3.1 Heuristics on the value function PDE
In this section we derive, in a heuristic way, the PDE that the value functions appearing in the definition of UIP are expected to satisfy. It is a classical property in the presence of an exponential utility function (see, e.g., the papers Becherer 2003, 2004; Becherer and Schweizer 2005; Henderson and Hobson 2009; Valdez and Vargiolu 2013) that one can factor out the initial wealth y so that
Hence, by definition of UIP, we deduce
so that the UIP v is given by
where J denotes the log-value function defined in (3.1). From the general theory of stochastic optimal control with Markovian state variables it is clear that the value function V is expected to satisfy the following HJB equation
with terminal condition \(V(T,x,y,z ; q) = G(x,y,z;q)\) and where
is the generator of the state variable (X, Y, Z). Recalling that \(V(t,x,y,z;q) = - e^{- \gamma y - \gamma J(t,x,z;q)}\), we can easily deduce from (3.3) the following PDE for the log-value function \(J:=J(t,x,y,z;q)\):
The Hamiltonian therein is maximised by the control \({\hat{\pi }^q}\), given by
Substituting \({\hat{\pi }}^q\) into the Eq. (3.4) leads to
where
and B is a \(m \times m\) symmetric matrix given by
The terminal condition for V translates into
Remark 3.1
In order to compute the UIP as in Eq. (2.6), we first calculate J(t, x, z; 0), which satisfies Eq. (3.6) with the terminal condition \(J(T,x,z;0) = \frac{\log \gamma }{\gamma }\). It is a classical and intuitive result that, in this situation, J(t, x, z; 0) does not depend on z. Denoting J(t, x, z; 0) by \(J^0 (t,x)\) for simplicity, we have that it fulfills
Thus, subtracting Eq. (3.10) from Eq. (3.6) and using the fact that
we obtain the following PDE for the UIP v:
with the terminal condition
Notice that solving the HJB equation for the UIP v(t, x, z; q) above requires the knowledge of \(J^0\), which is the log-value function of the optimal investment problem with no claim. This phenomenon is due to the presence of the non-tradable factors X in the dynamics of the forward contracts F and it has been observed in a somewhat different model in Becherer (2004), where the non-tradable factors follow a pure jump dynamics.
3.2 Existence and uniqueness results
In this section we show that the log-value function J is the unique continuous viscosity solution with quadratic growth of Eq. (3.6) with the terminal condition (3.9). From there, the UIP v is easily found via the equality (3.2). We will work under the following standing assumption. Recall that the matrix B has been defined in (3.8).
Assumption 3.2
The following properties hold:
-
(i)
b is \(C^1\), B and \(\Sigma ^* \sigma _F ( \sigma _F ^* \sigma _F)^{-1} \mu _F\) are \(C^1\) and Lipschitz in x uniformly in t.
-
(ii)
\(\mu _F\) is bounded and \(\langle (\sigma _F ^* \sigma _F)^{-1} \mu _F , \mu _F\rangle \) is \(C^1\) and Lipschitz in x uniformly in t.
-
(iii)
\(\sigma _F^* \sigma _F\) is bounded and uniformly elliptic, i.e., for some \(\epsilon > 0\),
$$\begin{aligned} (\sigma _F ^* \sigma _F)(t,x) \ge \epsilon I_n, \quad \text{ for } \text{ all } (t,x) \in [0,T] \times \mathbb {R}^m\text{. } \end{aligned}$$(3.13) -
(iv)
The matrix B is positive semidefinite and there exists a constant \(\delta >0\) (uniform in t, x) such that
$$\begin{aligned} \frac{1}{\delta } | \xi |^2 \le \langle \xi , B \xi \rangle \le \delta | \xi |^2 \end{aligned}$$(3.14)for all vectors \(\xi \in {\mathrm{Im}}(B)\), the image of B.
Some comments on these hypotheses are in order. All the assumptions above, with the exception of \(C^1\)-regularities and boundedness of \(\mu _F\) (linear growth is actually sufficient) have to be imposed in order to apply the method and the results established in Da Lio and Ley (2006). In particular, condition (iv) on B is related to the coercivity hypothesis in Assumption A1 in Da Lio and Ley (2006), which has a crucial role in the proof of their comparison theorem. Such a property has to be verified on a case-by-case basis. Some examples where this assumption is verified are provided in Sect. 4.
The additional \(C^1\)-regularity assumptions as well as the boundedness of \(\mu _F\) allow us to adapt results from Pham (2002) to get the quadratic growth condition of the log-value function \(J^0\) of the investment problem with no claim. Furthermore, thanks to Assumption 2.1 on the structured contract, the latter property will be inherited by the log-value function, J, with the claim.
We are now ready to state the main result of the paper.
Theorem 3.3
Let Assumptions 2.1 and 2.5 hold. Under Assumption 3.2, the log-value function J, defined in (3.1), is the unique continuous viscosity solution with quadratic growth of the Cauchy problem (3.6) with terminal condition (3.9).
Before proving the theorem, we give a preliminary result showing that the value function V is a (possibly discontinuous) viscosity solution of a Hamilton–Jacobi–Bellman (HJB) equation in the interior of its domain. Its proof is postponed to the “Appendix”.
Proposition 3.4
Let Assumptions 2.1 and 2.5 hold. Under Assumption 3.2, the value function V in (2.8) is a (possibly discontinuous) viscosity solution of the HJB equation
with terminal condition \(V(T, x,y,z;q) = G(x,y,z ;q )\), where
At this point we are in position to prove Theorem 3.3.
Proof of Theorem 3.3
We consider the existence first. This is an easy consequence of Proposition 3.4 above, which gives that the value function V is a viscosity solution of Eq. (3.15). It then suffices to use the definition of viscosity solution to check that the log-value function J defined in (3.1) is a (possibly discontinuous) viscosity solution of the PDE (3.6).
To complete the proof, it remains to show that J is unique in the class of all continuous viscosity solutions with quadratic growth for the Cauchy problem (3.6) and (3.9). The main idea for uniqueness is to use the comparison theorem in Da Lio and Ley (2006), Th. 2.1. For reader’s convenience, we split the rest of the proof into two steps.
(i) Reduction to Da Lio and Ley (2006) setting. First, we use a Fenchel-Legendre transform to express the quadratic term in our pricing PDE (3.6) into an infimum over the image of B of a suitable function. More precisely, we apply a classical result in convex analysis [e.g. (Rockafellar 1970, Ch.III, Sect. 12)] to get
for all vectors \(w \in {\mathbb {R}}^m\), where \({\tilde{F}}\) is the conjugate of F, which is also given by
when \(\alpha \in {\mathrm{Im}}(B)\) and \(-\infty \) otherwise. Notice that the first infimum in (3.16) is computed over the image of B since the matrix B is not necessarily invertible in our framework. Using (3.16), we can rewrite Eq. (3.6) as
with \({\bar{b}}\) as in (3.7) and with terminal condition \(J(T,x,z;q) = \frac{\log \gamma }{\gamma } + q\Phi (p(T,x),z)\). Notice that, since the function F above can be written as an infimum as in (3.16), we get a PDE with the same form as in Da Lio and Ley (2006), Eq. (1.1) provided we apply the time reversal transformation \({\widehat{J}} (t,x,z;q):= J(T-t,x,z;q)\). Hence the PDE (3.17) turns into the following
with the initial condition
Notice that this Cauchy problem is a particular case of the one studied in Da Lio and Ley (2006) since our Assumptions 2.1, 2.5 and 3.2 imply Assumptions (A1), (A2), (A3) in Da Lio and Ley (2006). In particular, Assumption 3.2 (iv) implies the same property for \(B^{-1}\), giving (A1) (iii) in Da Lio and Ley (2006). Indeed on the image of B, \(B^{1/2}\) as well its inverse \(B^{-1/2}\) are well-defined. Since \(B^{-1/2}: \text {Im}(B) \rightarrow \text {Im}(B)\), we have that, e.g., the LHS in (3.14) implies \(\delta ^{-1}\vert B^{-1/2} y\vert ^2 \le \langle B^{-1/2} y, BB^{-1/2}y\rangle \) for all \(y\in \text {Im}(B)\), leading to \(\langle y , B^{-1} y \rangle \le \delta \vert y \vert ^2\) for all \(y\in \text {Im}(B)\). The other inequality is obtained in a similar way.
(ii) Uniqueness. We proceed by contradiction. Assume that there exists another continuous viscosity solution with quadratic growth \({\tilde{J}}\) of the Cauchy problem (3.18) with terminal condition (3.19). Then, by calling \(J^*\) and \({\tilde{J}}^*\) their u.s.c. envelopes and \(J_*\) and \({\tilde{J}}_*\) their l.s.c. envelopes, we have, by definition of viscosity solution, that \(J^*\), \({\tilde{J}}^*\) are u.s.c. viscosity subsolutions and \(J_*\), \({\tilde{J}}_*\) are l.s.c. viscosity supersolutions of equation (3.18), obviously with \({\tilde{J}}_* \le {\tilde{J}}^*\). We also have \(J_*(T,x,z;q) \le \frac{\log \gamma }{\gamma }+ q \Phi (p(T,x),z) \le J^*(T,x,z;q)\), by definition of upper and lower envelopes. We now want to prove that
for all \(q \ge 0\), \(x \in {\mathbb {R}}^m, z\in [0,{\bar{u}} T]\). To prove the inequalities (3.20) it suffices to apply Theorem 4.3.2 and subsequent Remark 4.3.5 in Pham (2009).Footnote 1
Moreover it can be proved that J(t, x, z; q) has quadratic growth in (x, z), uniformly in t, for all \(q\ge 0\) (ref. Lemma 8.1 in the “Appendix 2”). Then, by the comparison theorem (Da Lio and Ley 2006, Theorem 2.1), we have that
on \([0,T] \times \mathbb {R}^m \times \mathbb {R}\). This implies that \(J_* = J^* = J = {\tilde{J}}\), and that J is continuous. The proof is therefore complete. \(\square \)
As a consequence of the result in Theorem 3.3, we have a good candidate for the optimal hedging strategy, which is given by
where \(v_x\) is the gradient with respect to the factor variables, when it exists, of the UIP (compare Becherer 2003, 2004). Concerning the optimal exercise policy \({\hat{u}}\) of the structured contract, a candidate in feedback form is given by solving the maximization problem
For an explicit formula, consider the case \(L(p,z,u)=u \ell (p,z)\) with \(\ell \) bounded. In this case, it is easy to see that
Even though working with viscosity solutions does not allow to justify rigorously the optimality of such controls, we observe that they are consistent with the optimal policies that have been obtained in the past literature for more specific models (see e.g. Basei et al. 2014; Benth et al. 2012).
Remark 3.5
Note that we have worked on the log-value function’s PDE (3.6) instead of on the PDE for the price v [cf. Eq. (3.11)]. The reason for doing so is that the latter is more delicate to handle due to the fact that it contains the first derivative \(J_x^0\) of the log-value function with no claim. Applying Da Lio and Ley’s results directly to Eq. (3.11) would require a Lipschitz continuity for \(J_x^0\) uniformly in t, which is difficult to have in general. Nonetheless, when this condition is satisfied as in Cartea–Villaplana (see Sect. 4.2) and in the linear dynamics model in Example 4.6, the same arguments go through with fewer assumptions than in Theorem 3.3. Indeed, the boundedness of the payoffs L and \(\Phi \) implies that the UIP v is bounded and so it has quadratic growth. Therefore, Lemma 8.1 is not needed anymore and neither are all the \(C^1\)-regularities and the boundedness of \(\mu _F\) as in Assumption 3.2. Under the remaining assumptions and when \(\mu _F\) has linear growth in x uniformly in t (replacing its boundedness) we can prove that v is the unique continuous viscosity solution with quadratic growth to Eq. (3.11) with terminal condition (3.12). The proof is analogous to that of Theorem 3.3, it is therefore omitted.
Remark 3.6
(Complete market case) When the market is complete, i.e. \(d = n\) and \(\sigma _F\) has full rank, we have \(B=0\) so that Assumption 3.2(iv) is trivially satisfied and \(J^0_x\) does not appear in the PDE for v anymore. In this case, we can work directly with the PDE for v along the same lines as in the previous Remark 3.5. Therefore, under Assumptions 2.1, 2.5 and 3.2 (i)–(ii)–(iii), one can show that v is the unique continuous viscosity solution with quadratic growth of the HJB equation
with terminal condition
Moreover, one can weaken the boundedness of \(\mu _F\) and require only linear growth in x uniformly in t. This result extends to our setting previous ones in Basei et al. (2014), Benth et al. (2012), Chen and Forsyth (2007), Felix (2012), Thompson et al. (2009), which were obtained for particular types of structured contracts, e.g., swings and virtual storages, and without trading in forward contracts.
4 Examples
4.1 A class of models with two assets and constant correlation
In this section we focus on the following incomplete market model:
where \(W=(W^1,W^2)\) is a bidimensional Brownian motion and \(\rho \in (-1,1)\). This is clearly a particular case of the general model in the previous section with \(\sigma ^* _F (t,x) = ({\bar{\sigma }}_F (t,x), 0)\), \(\Sigma ^* (t,x) = \sigma (t,x) (\rho , \sqrt{1-\rho ^2})\) and \(P_t =p(t,X_t)\) for some continuous function p(t, x). This model is a generalization of the usual Black-Scholes model with basis risk (see Davis 2006; Henderson 2002; Monoyios 2004 among many others), with the additional feature that the non traded asset or factor X can appear in the coefficients of the traded asset F.
We suppose that Assumptions 2.1 and 2.5 are in force. Concerning Assumption 3.2, we are going to specialize it to the present setting as follows. Observe first that the quantity \(\Sigma ^* \sigma _F (\sigma _F ^* \sigma _F)^{-1} \mu _F \) appearing in Assumption 3.2(i) reads as
while the scalar product \(\langle (\sigma _F ^* \sigma _F)^{-1} \mu _F , \mu _F\rangle \) in Assumption 3.2(ii) is
and \((\sigma _F^* \sigma _F)(t,x)\) in Assumption 3.2 (iii) corresponds to \((\sigma _F^* \sigma _F)(t,x)= {\bar{\sigma }}_F^2(t,x)\). Finally, we have \(B(t,x)=(1- \rho ^2)\sigma ^2(t,x)\). Hence, Assumption 3.2 is guaranteed by the conditions listed just below and the general results in Theorem 3.3 can be safely applied.
Assumption 4.1
Let the following properties hold:
-
(i)
\(b\in C^1\), \(\sigma \in C^1\);
-
(ii)
\(\mu _F\) is bounded;
-
(iii)
\(\sigma \) and \({\bar{\sigma }}_F\) are bounded and bounded away from zero;
-
(iv)
\(\frac{\mu _F}{{\bar{\sigma }}_F} \in C^1\) and it is Lipschitz in x uniformly in t.
In this more specific setting, we can obtain more information on the structure of the value function of the buyer of q units of the structured product provided we have the following
Assumption 4.2
Let the log-value function \(J^0 _x\) be Lipschitz in x uniformly in t.
Under this assumption, we do not need to suppose that \(\mu _F\) is bounded as in 4.1(ii) above. Indeed the considerations in Remark 3.5 apply, so that in particular \(\mu _F\) can be a linear function of x as in Example 4.6 below.
Let \(C_T^u\) be the payoff of a given structured contract as in (2.1). Inspired by the results in Oberman and Zariphopoulou (2003), which in turn extend Karoui and Rouge (2000) to American options, we obtain a representation of the UIP of the structured product \(C_T^u\) as the value function of an auxiliary optimization problem with respect to the control u only, under a suitable equivalent martingale measure involving the derivative \(J_x ^0\) of the log-value function of the problem with no claim, and where \(\gamma \) is replaced by a modified risk aversion \({\widetilde{\gamma }}=\gamma (1-\rho ^2)\).
Let us consider the measure \({\mathbb {Q}} ^0\) defined as
where \(W=(W^1 , W^2)^*\) and \(\theta \) is given by
Notice that the stochastic exponential is well defined, since X has continuous paths and \(\mu _F / {\bar{\sigma }}_F\) is continuous, so that the stochastic integral \(\int _0 ^t \theta _u ^1 dW_u ^1\) is well-defined for every t. Moreover, the second integral \(\int _0 ^t \theta ^2 _u dW_u ^2\) is also well-defined thanks to the continuity of \(\sigma (t,X_t)\) and the linear growth of \(J_x ^0\) (cf. Lemma 8.1).
Finally, in order for the equation (4.2) to define a probability measure, we need to impose that \({\mathbb {E}}[D_T ^0] =1\). This equality holds true when, for instance, \(J_x^0\) is bounded, so that in particular Novikov’s criterion applies. More generally, one could use the deterministic criteria proposed in Mijatović and Urusov (2012) (e.g. Theorem 2.1 therein).
Remark 4.3
In the case when the coefficients of F do not depend on the state variable X, when, e.g. both follows geometric Brownian motions with constant correlation, we have that \(J^0 _x \equiv 0\), and \({\mathbb {Q}}^0\) coincides with the minimal entropy martingale measure. Therefore the measure \({\mathbb {Q}}^0\) can be viewed as a perturbation of the minimal entropy martingale measure (see Frittelli 2000) where the correction involves the log-value function \(J^0\) of the optimal pure investment problem.
In what follows we will need the following preliminary lemma, stating the dynamics of the spot price under the martingale measure \({\mathbb {Q}}^0\). Its proof is based on a standard application of Girsanov’s theorem, and it is therefore omitted.
Lemma 4.4
Assume \({\mathbb {E}}[D_T ^0] =1\). Then the dynamics of X under \({\mathbb {Q}}^0\) is given by
where
and
defines a \({\mathbb {Q}}^0\)-Brownian motion and \({{\tilde{\gamma }}} = \gamma (1 - \rho ^2)\).
The following proposition extends to our setting the characterisation in Oberman and Zariphopoulou (Oberman and Zariphopoulou 2003 Prop. 10).
Proposition 4.5
Let the standing Assumptions 2.1, 2.5, 4.1 (i)–(iii)–(iv) and 4.2 hold. Then the UIP \(v = v(t,x,z;q)\) satisfies
where \({\mathbb {E}}^0 _{t,x,z}\) denotes the conditional expectation under \({\mathbb {Q}}^0\).
Proof
We prove the result by showing that the candidate function
satisfies Eq. (3.11) with terminal condition (3.12) and we conclude using the comparison theorem in Da Lio and Ley (2006), Th. 2.1. To this end, write \({\tilde{v}}\) as
with
The value function w above solves the following Cauchy problem in a viscosity sense
with
The corresponding Cauchy problem for \({\tilde{v}}\) is immediately obtained:
with
which is a particular case of Eq. (3.11) in this setting.
To identify \({\tilde{v}}\) with the UIP v and conclude, we need a uniqueness result for the Cauchy problem (4.7). Since \(J_x ^0\) is assumed to be Lipschitz in x uniformly in t, we can use Remark 3.5 to get the existence of a unique continuous viscosity solution with quadratic growth to the Cauchy problem (4.7). Finally, the boundedness of the payoff \(C^u _{t,T}\) (cf. Assumption 2.1) clearly implies that the value function \({\tilde{v}} (t,x,z)\) has quadratic growth. Thus the proof is complete. \(\square \)
The previous proposition suggests the following approach to compute the UIP and the corresponding (partial) hedging strategy of a given structured product in this setting:
-
first, solve the pure optimal investment problem V(t, x, y; 0) with no claim;
-
second, compute the x-derivative of the log-value function \(J^0\) giving the new probability measure \({\mathbb {Q}}^0\) as well as the corresponding dynamics of X;
-
finally, solve the maximisation problem in (4.5), which is now computed with respect to the control u only; its value function gives the UIP while its derivative with respect to x gives the hedging strategy via (3.5).
Example 4.6
(Linear dynamics model) This example is a slight generalization of the model studied in Carmona and Ludkovski (2006), Sect. 2.2 and Fiorenzani (2006):
where \(a,k,{\bar{\sigma }}_F , \delta , \theta , \sigma \) are real constants, the correlation \(\rho \) belongs to \((-1,1)\), and \((W^1, W^2)\) is a bidimensional Brownian motion as before. Here F represents the price of a liquid forward contract with maturity T written on a commodity, while instead X is the log-price of another, less liquid, commodity on which the structured product is written (i.e. \(p(t,x)=e^x\) in this case). In practical applications, one searches for a liquidly traded forward F written on a commodity correlated with \(P_t = e^{X_t}\), with a correlation coefficient \(\rho \) as close to 1 as possible [for practical examples, see Carmona and Ludkovski (2006), Sect. 2.3 and Fiorenzani (2006)]. When \(k = 1\) we obtain exactly the model in Carmona and Ludkovski (2006), Sect. 2.2, while for \(k = 0\) we obtain the model in Fiorenzani (2006).
Notice that if \({\bar{\sigma }}_F>0\), \(\sigma >0\) and \(k=0\), then Assumption 4.1 holds true, while in the general case when \(k \ne 0\) Assumptions 4.1 (ii) is not satisfied. Nevertheless, as we are going to see, in this example \(J_x^0\) is Lipschitz, so that Remark 3.5 applies. Hence we can take \(\mu _F\) linear in x as above.
To see that \(J_x ^0\) is Lipschitz, consider Eq. (3.10) which in this setting becomes
Then, in analogy with Benth and Karlsen (2005), one guesses that the solution \(J^0\) has the general form
such that \(J^0(T,x) \equiv \frac{\log \gamma }{\gamma }\). This ansatz gives the system of first order ODEs
with final condition
The system above is solvable in closed form, as the third equation is a Riccati equation in \(\Gamma \), the second one is a linear equation in \(\beta \), which can be solved once that \(\Gamma \) is known, and, finally, the first one can be solved in \(\alpha \) just by integration.
Notice that, if the parameter k appearing in the forward drift is zero then the dynamics of the forward contract does not depend on X, so that \(J^0\) does not depend on x, thus leading to \(\beta \equiv \Gamma \equiv 0\).
Finally, Eq. (3.11) is given in this case by
with terminal condition
4.2 The Cartea–Villaplana model with correlation
Here we consider a slight generalization of the two factor model for the electricity spot price introduced by Cartea and Villaplana (2008). While the two factors are assumed independent in the original paper (Cartea and Villaplana 2008), here we allow for a possibly non zero (constant) correlation between them. We recall briefly the main features of the model. The electricity spot log-price \(P_t\) at time t is decomposed into the sum of two stochastic factors \(X^C\) and \(X^D\), i.e.,
with \(\alpha _C < 0\) and \(\alpha _D > 0\), where \(\eta \) represents a seasonal continuous deterministic component. The factors \(X^i_t\), \(i = C,D\), are Ornstein-Uhlenbeck processes driving, respectively, the capacity of power plants and the demand of electricity. Their dynamics is given by
where \(k^i\) are constant coefficients, \(\sigma _i (t)\) are deterministic measurable functions of time and each \(W^i\), for \(i=C,D\), is a unidimensional Brownian motion such that \( d\langle W^C, W^D \rangle _t = \rho dt \) with a constant correlation \(\rho \in (-1,1)\). Notice that the Cartea–Villaplana model reduces to the Schwarz–Smith model (2000) when \(\alpha _C = \alpha _D = 1\) and \(k^C = 0\) (or \(k^D = 0\)). In this example we work under the following standing assumptions:
Assumption 4.7
Let \(\sigma _C(t)\) and \(\sigma _D (t)\) be continuous, bounded and bounded away from zero over [0, T].
Since the interest rate is zero, the price at time t of a forward contract with maturity T can be computed via the usual formula \(F_t = \mathbb {E}^\mathbb {Q}[P_T|\mathcal{F}_t]\), \(t\in [0,T]\), for a suitable choice of risk-neutral measure \({\mathbb {Q}}\) preserving the Gaussian structure of the model as in Cartea and Villaplana (2008), Sect. 5. Following the approach in Cartea and Villaplana (2008) we can obtain the dynamics of the forward price under the risk-neutral measure \({\mathbb {Q}}\) as
where \(W^{{\mathbb {Q}}, C}\) and \(W^{{\mathbb {Q}}, D}\) are two \(\mathbb Q\)-Brownian motions with correlation \(\rho \). Choosing suitably the market prices of risk as in Cartea and Villaplana (2008) and using Assumption 4.7, we can obtain the following forward dynamics under the objective probability \({\mathbb {P}}\):
where the drift \(\mu _F (t)\) is a bounded function of time.
We deal separately with two different situations: the incomplete market case with one forward contract (recall that we have two stochastic factors) and the complete one with two forward contracts.
4.2.1 The case of one forward contract
In this case the agent is allowed to hedge the structured product by trading only in one forward contract. The Cartea–Villaplana model fits the general setting of Sect. 2.3 with \(X = (X^C,X^D)^*\), whose coefficients are
Notice that \(\Sigma \) has full rank unless \(\rho = \pm 1\), as
Let us consider a forward contract F with maturity T. Here \(\sigma _F(t,X_t)\) depends only on t, so that for simplicity we set \(\sigma _F (t):= \sigma _F(t,X_t)\), and we have
We note that, since the correlation between the spot and forward log-prices is not constant, this model does not fit the setting in Sect. 4.1.
In this model the matrix B has rank equal to one. In fact, by definition [cf. Eq. (3.8)] we have
with
Consider \(x = \Sigma ^{-1} \sigma _F\). Then \(x \ne 0\) and we have
Therefore, working on the image of B in Eq. (3.16) is fully justified here, as \(\mathrm {rank}(B) = 1\). Now, we show that Assumption 3.2(iv) is satisfied in this case. Indeed, a direct computation shows that
where
Hence, the two eigenvalues of B are \(\lambda _1 (t) \equiv 0\) and
By Assumption 4.7 we have that \(\sigma _C(t)\) and \(\sigma _D (t)\) are bounded and bounded away from zero over [0, T], yielding \(\frac{1}{\delta } \le \lambda _2 (t) \le \delta \) for some \(\delta >0\) independent of \(t \in [0,T]\). This implies Assumption 3.2 (iv).
Since in this example the two factors \(X^C\) and \(X^D\) do not enter in the coefficients of the forward contract dynamics, we expect that the derivative \(J^0 _x \) of the log-value function is zero. Indeed, this can be obtained from the PDE (3.10) satisfied by \(J^0\). Since \(\mu _F\) and \(\sigma _F\) do not depend on X, such a PDE simplifies to
which gives
Therefore \(J_x^0 \equiv 0\), and Eq. (3.11) for the UIP becomes
Hence, under Assumption 4.7, the considerations in Remark 3.5 apply and give that the UIP v is the unique viscosity solution with quadratic growth of the PDE above.
Finally, in this case the candidate optimal hedging strategy is given by \({\hat{h}}^q = {\hat{\pi }}^q - {\hat{\pi }}^0 = - (\sigma _F ^* \sigma _F)^{-1} \sigma _F ^* \Sigma v_x\) as in (3.21), where \(\sigma _F ^* \sigma _F\) is as in (4.13) and
4.2.2 The case of two forward contracts
We look now at the much simpler situation where the agent can hedge the structured product by trading in two forward contracts \(F^1\) and \(F^2\) with respective maturities \(T_1\) and \(T_2\), with \(T \le T_1 < T_2\). Then we have
Of course, in this case \(B = 0\), since \(\sigma _F\) is invertible. Hence, the market model is complete and we are in the situation described in Remark 3.6. Analogously to the previous case, it is possible to find an explicit expression for \(J^0\), which is now given by
Here again \(J_x^0 \equiv 0\), so that Remark 3.5 applies and Eq. (3.11) for the UIP becomes
Finally the candidate optimal hedging strategy is given by \({\hat{h}}^q = -(\sigma _F^* \sigma _F)^{-1} \sigma _F^* \Sigma v_x\) as before, where this time
5 Numerical results
In this section we present some numerical applications of our results to swing options (see Example 2.2).Footnote 2 We focus on this type of contract for essentially two reasons: first, swing options are the main type of volumetric contracts in commodity markets and, second, we want to compare our results to those in Benth et al. (2012).
More specifically, in Sect. 5.1 we consider the benchmark case with strike price \(K=0\) and minimal cumulated quantity \(m=0\) in order to compare the prices obtained following the UIP approach to those in Benth et al. (2012); in Sect. 5.2 we consider more general swing options with \(K > 0\) and \(m >0\). In both parts, we compute the solution of the relevant PDEs using finite difference schemes, as suggested in Benth et al. (2012).
5.1 Comparison with the results in Benth et al. (2012)
Here, we compare the UIP, obtained by solving the non-linear PDE (4.11), with the classical linear pricing rule which is used in the energy market literature (e.g. Basei et al. 2014; Benth et al. 2012; Chen and Forsyth 2007; Felix 2012; Thompson et al. 2009). The latter is given in terms of a PDE which is essentially linear, except for the first derivative in z and which has the same form as Eq. (3.23), namely Eq. (4.11) without the quadratic term in \(v_x\). In both cases, the optimal strategy \({\hat{u}}(t,x,z;q)\) is given by Eq. (3.22) with \(\ell (p,z) = p-K\).
We consider, as in Benth et al. (2012), one swing option (i.e., we take \(q=1\)) with parameter values
i.e., the control u belongs to [0, 1] and the holder faces the problem of picking the most favorable price of the commodity, up to a certain total volume M. We set the risk-free interest rate to zero. Moreover, in order be as close as possible to the setting considered in Benth et al. (2012), where \(Z^u\) is constrained to fulfil \(Z^u _T \le M = 0.5\), we use the penalty function
with \(C=1000\). Indeed, the authors in Basei et al. (2014) prove that when \(C \rightarrow \infty \) the price of a contract with penalty \(\Phi \) as in (5.1) converges to the price of a contract with the constraint on \(Z^u\) as above.
Moreover, with a view towards the comparison with Benth et al. (2012), we choose a special case of the linear dynamics model of Example 4.6 with \(k = 0.01\) and where
Finally, the risk-aversion parameter is set to be \(\gamma = 0.02\).
Remark 5.1
Notice that the coefficients \(\delta , \theta \) and \(\sigma \) above correspond, respectively, to \(\kappa , \mu \) and \(\sigma \) in Benth et al. (2012), and they have the same numerical values as in Benth et al. (2012). The remaining coefficients \(\sigma _F\) and a refer to the dynamics of the forward contract F, which is not part of the model in Benth et al. (2012), and \(\rho \) is the correlation between (the logarithms of) the spot price P and F.
“Classical” price (above) of a swing contract and UIP (below). Prices are computed at \(t=0.5\) with \(\gamma = 0.02\)
We compute both kinds of price (the risk-neutral price and the UIP) for such a contract, solving numerically the corresponding PDE via finite difference methodology with a backward time stepping scheme. In all the numerical experiments we use an approximating domain for the logarithm of the spot price which is wider than the one in Benth et al. (2012) (where \(x_{min} = \ln (21.6)\) and \(x_{max} = \ln (73.9)\)) and the domain for Z is obviously \([0,{\bar{u}} T] = [0,1]\), thus leading to a global domain
with \(x_{min} = \ln (0.001)\) and \(x_{max} = \ln (500)\). Notice that \([x_{min}, x_{max}] \) here is wider with respect to the interval used in Benth et al. (2012), so that the probability that X belongs to this interval is higher, thus leading to more stable numerical results. The boundary conditions are the same as in Benth et al. (2012) as well as the numerical approximations of \(v_t\) and \(v_z\): denoting by \(v_{i,j}^n\) the approximation of \(v(t_n, x_i,z_j;1)\) with \(n \in \{ 0, \dots ,N \}, i \in \{ 0, \dots , I \}\) and \(j \in \{ 0, \dots , J \}\) we have
with \(\Delta t:= \frac{T}{N}, \Delta z:= \frac{1}{J}\), while we use a fully explicit scheme also for the derivatives in x
with \(\Delta x:= \frac{ x_{max} - x_{min}}{I}\). We set
in order to have convergence of our numerical solution to the UIP. The proof of the convergence can be found in “Appendix 3”.
5.1.1 Numerical results
We plot in Fig. 1 the prices of the swing contract at time \(t=0.5\), obtained with the two approaches (a similar picture can be provided at any other date). In order to stress the difference between the two prices, we do not plot the surfaces for \(z \in [0.25 , 0.5]\) (remember that \(M=0.5\)). As we can see, the two price surfaces have similar shapes, even though the “classical” procedure slightly overprices the option with respect to the UIP when the log spot price is high. The difference between the two prices is clearly due to the risk aversion \(\gamma \) and, secondarily, to the correlation \(\rho \) between the underlying and the forward market where the buyer can invest.
We conclude this part by illustrating in Tables 1 and 2 below the effect that those two parameters separately have on the UIP. Concerning the dependence of the UIP on \(\gamma \), which are summarized in Table 1, we choose x, z and t so that the difference between the UIP for \(\gamma \) that varies and the UIP for \(\gamma = 0.01\) is as large as possible (on the domain of Fig. 1 with \(x_{min} = \ln (21.6)\) and \(x_{max} = \ln (73.9)\)). Similarly, Table 2 shows how the UIP varies with \(\rho \). As we can see, the UIP is decreasing in \(\gamma \), while it is neither increasing, nor decreasing in \(\rho \). The first effect is very natural, since a higher risk aversion for the buyer is expected to induce a lower price. Concerning \(\rho \), one would expect a higher price as the correlation \(\rho \) with the forward market increases (in absolute value), since this widens the hedging opportunities for the buyer. Nevertheless, in this model also \(J_0\) (i.e., the log-value function without investing in the structured product) depends on \(\rho \) and this seems to produce a more complicated, non necessarily monotonic, dependence on \(\rho \). The combined effect of \(\gamma \) and \(\rho \) is not clear in general.
5.2 A more realistic example
We now focus on computing the UIP of a more realistic swing option contract, with \(q=1\),
Indeed, swing contracts usually have strictly positive strike price and a nonzero minimal cumulated quantity to be purchased. The penalty function we use is the one in Equation (2.2) with \(C = 1000\). We keep working under the linear dynamics model in Example 4.6, with \(k=0.01\) and with parameters as in (5.2). We solve the PDE for v using a backward time stepping finite difference method on the domain \( {\mathcal D} = [0,T] \times [x_{min}, x_{max}] \times [0,1]\), where \(x_{min}= \ln (0.001), x_{max}= \ln (500)\).
The approximating schemes for \(v_t, v_z,v_x\) and \(v_{xx}\) are as in Sect. 5.1, as well as the boundary conditions, except for \(x=x_{min}\): in fact, if \(x=x_{min}\) the optimal operational behavior still consists in waiting as long as possible before exercising (this is because \(x_{min}\) is much smaller than the expectation of X in the long run and the price is thus expected to increase), but now we have to take into account the constraint \(m =0.1\) [recall that \(m=0\) in Benth et al. (2012)]. Hence we set:
With this choice of u, it is possible to explicitly compute the approximating price (recall that in the linear dynamics model in Example 4.6 the spot price is \(P_t = e^{X_t}\) and that \(\bar{u} =1\))
as done in Benth et al. (2012), Appendix A.
In Fig. 2 we plot the price of the swing option at two different dates.
Price of one swing contract with minimal annual quantity \(m = 0.1\) and maximal quantity \(M = 0.5\), a price at \(t=0.5\), b price at \(t=0.75\)
Notice that in both Fig. 2a, b we cut the domain in z in order to focus on positive prices: for \(0.5 =M< z < 1\) the penalty function plays a crucial role and the price becomes negative. We see that the UIP is decreasing in z [as in Benth et al. (2012)] and increasing in x. Moreover, from Fig. 2b it is clear that for \(z > 0.25\) the price is strictly decreasing. This might be explained as follows: for a fixed value of the log spot x and for \(t=0.75\), if \(z>0.25\) the value of the contract is lower than when \(z \le 0.25\) and it even becomes lower and lower as z increases, since the time to maturity is equal to 0.25 and so if \(z>0.25\) the buyer has less opportunities to exercise the option, hence less possibilities to take advantage of (possibly) higher prices. This is analogous to what happens with linear prices, see e.g. Basei et al. (2014).
Moreover, as an example, in Fig. 3 we show the optimal exercise strategy \({\hat{u}}\) at time \(t=0.75\) as a function of the (log) spot price x and of the cumulated quantity z. In the grey region \({\hat{u}} = {\bar{u}}\), while in the white region \({\hat{u}} =0\).
Optimal control \({\hat{u}}\) at time \(t=0.75\). In the grey region \({\hat{u}} = {\bar{u}}=1\), in the white region \({\hat{u}} =0\)
From Fig. 3 it is clear that, unless the spot price is very low, if the cumulated quantity \(z < m=0.1\), then it is always optimal to exercise the option, to avoid the penalty. Furthermore, when \( x > 2.5\), equivalently the spot price \(e^x\) is bigger than the strike price \( K= \exp (2.5)\) and so the optimal policy consists in exercising the option (i.e., \({\bar{u}} = 1\)) whenever \(z \in [0,0.25]\). On the other hand, if the spot price is higher than the strike, \(x > 2.5\), and if the cumulated quantity satisfies \(z>0.25\) then it is not optimal to exercise the option: in the current state \(m< z < M\), thus we are not incurring the penalty and the more we have used of our control, the higher the spot price has to be before we are willing to exercise.
We conclude this section by showing in Fig. 4 the candidate optimal hedging strategy \({\hat{h}}^1\) found in Eq. (3.21) as a function of the (log) spot price x and of the cumulated quantity z, at time \(t=0.5\).
Candidate optimal hedging strategy \({\hat{h}}^1\) at time \(t=0.5\)
We notice that, being the UIP increasing in x, \(v_x\) is positive on our domain [recall Eq. (3.21)], so that \({\hat{h}}^1\) is always negative: in order to hedge a buyer position in a swing option it is always “optimal” to sell the forward contract. Moreover, for a fixed z, as the (log) spot price increases, the quantity of forward contracts to sell increases. On the other hand, for a fixed x, \({\hat{\pi }}\) is increasing as a function of z, for \(z \in [0,0.5]\) (meaning that as the cumulated quantity z increases towards \(M=0.5\), selling forward contracts is less and less needed), while \({\hat{h}}^1 =0\) for \(z \ge M=0.5\), as expected.
6 Conclusions
In this paper, we considered the problem of pricing and hedging of structured products in energy markets from a buyer’s perspective using the (exponential) utility indifference pricing approach. The main novelty with respect to the existing literature is that buyer has the possibility to trade in the forward market in order to hedge the risk coming from the structured contract.
We characterized the UIP in terms of continuous viscosity solutions of a suitable nonlinear PDE. As a consequence, we were able to identify a candidate for the optimal exercise strategy of the structured product as well as a portfolio strategy partially hedging the financial position.
Moreover, in a more specific setting with two assets and constant correlation, we showed that the UIP equals the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be interpreted as a perturbation of the minimal entropy martingale measure.
Finally, we provided some numerical applications in the case of swing options. In particular, we computed the UIP price as well as the optimal exercise and hedging strategies for a buyer of one swing option in the linear dynamics model, by solving the corresponding nonlinear PDEs via finite difference schemes. We highlighted the differences with respect to the classical price as in Benth et al. (2012) and discussed some qualitative properties.
Notes
Notice that in our case the function G appearing in the statement of Theorem 4.3.2 in Pham (2009) can be chosen to be any positive number.
All the numerical tests were performed in MATLAB R2015b.
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This work was partly supported by the grant CPDA138873-2013 of the University of Padova “Stochastic models with spatial structure and applications to new challenges in Mathematical Finance, with a focus on the post-2008 financial crisis environment and on energy markets”. Part of this work was done while the first and fourth authors were visiting LSE in November 2013 and while the second author was Visiting Scientist at the University of Padova in June 2014 and July 2015: the financial contribution of the two institutions is kindly acknowledged. Moreover, the authors wish to thank René Aïd, Matt Davison, Enrico Edoli, Paola Mannucci, Mario Putti and the participants to the 2014 ISEFI Conference in Paris and the Conference in Mathematics for Energy Markets in Vienna (2016) for valuable comments.
Appendices
Appendix 1: Proof of Proposition 3.4
The maximisation problem (2.8) fits the setting of Section 5 in the paper Bouchard and Touzi (2011) on weak dynamic programming principle. In particular, their Corollary 5.6 applies. More precisely, the essential ingredients in the proof of Corollary 5.6 are the a-priori estimate (5.2) in Bouchard and Touzi (2011), the local boundedness of the value function and the lower semi-continuity of the objective function in (t, x, y, z) for all admissible controls. First, the a-priori estimate holds due to (2.11). Concerning the local boundedness of the value function, it can be easily checked that in our setting the value function is bounded since it is trivially nonpositive and, being \((u,\pi )=(0,0)\) an admissible strategy, we have
since the functions L and \(\Phi \) are bounded (cf. Assumption 2.1). Let \((u,\pi )\) be an admissible given control. Since the control is now fixed, we drop it from the notation of the state variable at maturity and denote them as \(A_T ^{t,a}:= (X_T ^{t,x},Y_T ^{t,a}, Z_T ^{t,a})\) with \(a =(x,y,z)\), to stress the dependence on the initial data. Now consider the objective function
where G is defined in (2.9). From the continuity of the function G and of the state variables \(A^{t,a} _T\) with respect to the initial data (t, a), we get that \(G(A^{t,a} _T )\) is also continuous in (t, a). Moreover, notice that since L and \(\Phi \) are bounded (ref. Assumption 2.1) we have
for some constant \(C>0\). Therefore, to prove the lower semi-continuity of the objective function it suffices to show that the family of random variables
is uniformly integrable. We prove that they are bounded in \(L^2\) for all admissible controls, i.e.
which will imply the uniform integrability. Let \(\mathcal F_{t,T}\) be the smallest filtration generated by the Brownian increment after t and satisfying the usual conditions. Consider the following change of measure on \(\mathcal F_{t,T}\):
which is well defined. Indeed, the boundedness of \(\sigma _F^* \sigma _F\) (cf. Assumption 3.2 (iii)) and the admissibility property (2.5) imply that \(\sup _{t\le s\le T} {\mathbb {E}} [ \exp (\varepsilon | \pi ^*_s \sigma _F ^*(s,X_s^{t,x}) |^2)] < \infty \) for some \(\varepsilon >0\), hence the criterion in Liptser and Shiryaev (1977), Example 3, Sect. 6.2.3 is fulfilled. Moreover, the change of measure (7.1) satisfies \(\sup _{t\in [0,T]} {\mathbb {E}} [ (d{\mathbb {Q}}_t / d{\mathbb {P}})^2 ] < \infty \). This is a consequence of the admissibility of \(\pi \) as in (2.5). Indeed,
giving that
with \(\delta \) such that \(2\delta = (8\gamma )^2 /2\), since the first exponential in the second inequality above is a true martingale. Moreover, since \(\pi \) is admissible we have \({\mathbb {E}}\left[ e^{2\delta \int _0 ^T | \pi ^*_s \sigma ^*_F(s,X^{t,x}_s) | ^2 ds}\right] < \infty \). As a consequence, we obtain that \(d{\mathbb {Q}}_t / d{\mathbb {P}}\) is square integrable.
Therefore we have
Using the linear growth condition of \(\mu _F\) and the boundedness of \(\sigma _F ^* \sigma _F\) [cf. Assumption 3.2 (ii) and (iii)], we have
for some positive constants \(c_1,c_2,c_3\). To conclude it suffices to prove that the RHS above is integrable for \({\mathbb {P}}\). This follows from the admissibility of \(\pi \) as in (2.5) and the exponential uniform bound (2.12) for X.
Finally, even though the space of admissible controls in our setting is smaller than the one in Bouchard and Touzi (2011), the value functions are the same since any controls in their space \(\mathcal U_0\) can be clearly approximated by admissible controls in \(\mathcal A\) through truncation. The result follows.
Appendix 2: Regularity properties of the log-value function
In order to prove the next lemma we follow closely the approach in Pham (2002), which has also been used in Mnif (2007) in a slightly different model with stochastic volatility with jumps and for an agent with exponential utility. Since the proof mimicks closely the arguments in Pham (2002), we only sketch them pointing out the main differences.
Lemma 8.1
Let \(q\ge 0\). Let Assumptions 2.1 and 2.5 hold. Under Assumption 3.2 the log-value function J(t, x, z; q) defined as in (3.1) has quadratic growth in (x, z) uniformly in t.
Proof
Since the claim \(C^u _{t,T}\) is bounded in (x, z) uniformly in the controls u (cf. Assumption 2.1), it suffices to prove that \(J^0 (t,x)\), the log-value function of the pure investment problem, has quadratic growth in x uniformly in t.
First of all, repeating exactly the same arguments as in the proof of Theorem 3.1 in Pham (2002), we get that if the PDE (3.10) with terminal condition \(J^0(T,x) = \frac{\log \gamma }{\gamma }\) admits a unique solution belonging to \(C^{1,2}([0,T) \times {\mathbb {R}}^m)\cap C^0 ([0,T]\times \mathbb R^m)\), whose x-derivative has linear growth, then such a solution coincides with \(J^0 (t,x)\).
To conclude the proof, we need to show that the PDE (3.10) has a unique smooth solution as above, whose x-derivative has linear growth. We adapt to our setting the arguments in the proof of Pham (2002), Th. 4.1 under his Assumptions (H3a). Indeed, notice that our Assumption 3.2(i), together with the Lipschitz continuity of b postulated in Assumption 2.5 (ii), corresponds to \(\mathbf (H3a) (i)\) in Pham (2002). Moreover Assumption 3.2 (ii) implies \(\mathbf (H3a) (ii)\), while Assumption 3.2 (iii) guarantees \(\mathbf (H2) (b)\) [see Remark 2.3 in Pham 2002].
Consider the PDE (3.17) in the case \(q=0\), with F(w) replaced by
where \(\mathcal B_k\) is the centered ball in \({\mathbb {R}}^m\) with radius \(k\ge 1\). Recall that \({\tilde{F}}\) is the convex conjugate of F and that is given by
while it equals \(-\infty \) otherwise. Proceeding as in the proof of Pham (2002), Th. 4.1, we can apply Theorem 6.2 in Fleming and Rishel (1975), giving the existence of a unique solution \(J^{0,k} \in C^{1,2}([0,T) \times {\mathbb {R}}^m)\cap C^0 ([0,T]\times {\mathbb {R}}^m)\) with polynomial growth in x, for the parabolic PDE
with terminal condition \(J^{0,k} (T,x) = \frac{\log \gamma }{\gamma }\). Notice that the convex conjugate \({\tilde{F}}\) of F, appearing in the definition of \(F_k (w)\) in (8.1), can take the value \(-\infty \), which is not a problem here since this value does not contribute to the infimum over \(\alpha \).
The next step consists, as in Pham (2002), in using a stochastic control representation of the solution \(J^{0,k}\) to derive a uniform bound on the derivative, independently of the approximation. Indeed, from standard verification arguments we get that
where
where \({\mathbb {B}}_k\) is the set of \({\mathbb {R}}^m\)-valued adapted processes \(\alpha \) bounded by k, and the controlled dynamics of X under \(\mathcal Q\) is given by
where \(W^{\mathcal Q}\) is a d-dimensional Brownian motion under \(\mathcal Q\) and \({\bar{b}}\) has been defined in (3.7). Notice that, since \(\Lambda \) takes the value \(-\infty \) outside the image of B, then the optimal Markov control evaluated along the optimal path \({\hat{\alpha }} (s,{\hat{X}}_s)\) will lie on \(\mathrm {Im} (B)\) a.s. for every \(s\in [t,T]\). We can use Lemma 11.4 in Fleming and Soner (1993) and the same estimates as in Pham (2002), Lemma 4.1 to obtain
for some positive constant C, which does not depend on k. Now we argue as in the proof of Pham (2002), Th. 4.1, Case (H3a), to deduce that \(|{\hat{\alpha }}_k (t,x)| \le C\) for all \(t\in [0,T]\) and \(|x| \le M\) for some positive constant C (independent of k) and an arbitrarily large \(M>0\). Therefore, we get that, for \(k \le C\), \(F_k (J^{0,k}_x) = F(J^{0,k}_x)\) for all \((t,x) \in [0,T]\times \mathcal B_M\). Letting M tend to \(+\infty \), we finally get that \(J^{0,k}\) is a smooth solution with linear growth on derivative to the PDE (3.17) (with \(q=0\)). To conclude, we have that \(J^0 = J^{0,k}\) for k sufficiently large, giving, in particular, that \(J^0\) has quadratic growth in x uniformly in t. Therefore the proof is complete. \(\square \)
Appendix 3: Convergence of the numerical scheme
In this section we show that the value function obtained from the finite difference scheme converges to v. We will follow an approach originally developed by Kushner (1977) and based on stochastic control theory, which specifically requires that the finite difference scheme has a Markov chain interpretation.
First of all we notice that Eq. (4.11) can be written in the form of a Bellman–Isaacs equation, as done in the proof of Theorem 3.3 part (i):
with
where \(\beta (t)\) and \(\Gamma (t)\) can be computed explicitly as solutions of the system of ODEs (4.10).
We now want to use the results in Barles and Souganidis (1991), or equivalently in Fleming and Soner (1993), Ch. IX (in the spirit of Li and Song (2007)), which work well when the min-max is taken on compact sets. To do this, we approximate \(\inf _{\alpha \in \mathbb {R}}\) by \(\inf _{\alpha \in \mathcal B_R}\), where \(\mathcal B_R:= [-R,R]\), \(R\ge 1\) (we will eventually let R go to \(+ \infty \)), obtaining a finite-difference approximation of the form [see the analogous equation (3.26) in Fleming and Soner (1993), Ch. IX]
where
Notice that above quantities can be interpreted as the one-step transition probabilities of, respectively, going up, nowhere or down in x and up in z, when at time \(t_n\) the processes (X, Z) is in the state \((x_i,z_j)\): more explicitly, e.g.,
and the other ones can be written analogously. Hence, we are dealing with a Markov chain approximation of the state variable. Notice that the sum of the above four probabilities is equal to one.
For this Markov chain approximation to be rigorous, we must impose that \(p^{i,n}_{\alpha ,u;i,j} \in [0,1]\) for every \(i \in \{1,2,3,4 \}\) and for \(n \in \{ 0, \dots , N-1 \}\) and for all possible states i, j and controls \(u,\alpha \).
Taking into account that the domain in (x, z) is bounded and \(\alpha ,u\) are also taken to be valued in a compact domain, the two conditions \(p^{1,n}_{\alpha ,u;i,j} \le 1\) and \(p^{3,n}_{\alpha ,u;i,j} \le 1\) are satisfied as soon as \(\Delta t\) is small enough, since they read, respectively:
while imposing \(p^{1,n}_{\alpha ,u;i,j} \ge 0\) and \( p^{3,n}_{\alpha ,u;i,j} \ge 0\) yields
which has to hold true for every control \(\alpha \), every n and every state \(x_i\), so that we find
Moreover \(p^{2,n}_{\alpha ,u;i,j}\) is always smaller than 1, while asking its non-negativity gives as necessary and sufficient condition
which implies the well-known Courant-Friedrichs-Lewy condition \( {\overline{u}} \Delta t \le \Delta z \) [also present in Benth et al. (2012)] implying in turn \(p^{4,n}_{\alpha ,u;i,j} \le 1\). Finally, \(p^{4,n}_{\alpha ,u;i,j} \) is always positive. We are now ready to verify the conditions of monotonicity, stability and consistency required by the framework in Barles and Souganidis (1991), which correspond to assumptions (4.3)–(4.6) in Fleming and Soner (1993), Ch. IX. We proceed as done in Fleming and Soner (1993), Ch. IX, Example 4.1. The monotonicity is automatically given by the Markov chain interpretation in Eq. (9.2). The stability is implied by the same equation (which also has a unique solution) and by the fact that L and \(\Phi \) are bounded on the bounded domain: in fact, one can easily prove by backward induction on n that
where in the last inequality we have used the fact the \(L_{\alpha ,u}\) does not depend on \(\alpha \) and where the upper bound is uniform in i, j and in the discretization step \(\Delta t\). Finally, the consistency property holds because the finite differences converge uniformly on compact sets to the corresponding derivatives (see, e.g., Fleming and Soner 1993, Theorem 4.2 and remember that we are working under the linear dynamics model and in the case of swing contracts). So we now have the solution \( v_R\) to Equation (9.1) where \(\inf _{\alpha \in \mathbb {R}}\) is replaced by \(\inf _{\alpha \in \mathcal B_R}\). Because of the stochastic game interpretation of this equation [as in Barles and Souganidis (1991)], letting \(R \rightarrow + \infty \) gives that the sequence \({(v_R)}_{R \ge 1}\) decreases pointwise and, by Eq. (9.5), it is bounded uniformly in \(\alpha \) (here \(L_{\alpha ,u}\) does not depend on \(\alpha \)). Thus it admits a finite limit, v, which is the solution to Eq. (9.1).
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Callegaro, G., Campi, L., Giusto, V. et al. Utility indifference pricing and hedging for structured contracts in energy markets. Math Meth Oper Res 85, 265–303 (2017). https://doi.org/10.1007/s00186-016-0569-6
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DOI: https://doi.org/10.1007/s00186-016-0569-6