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Abstract

We analyze mean-variance-optimal dynamic hedging strategies in oil futures for oil producers and consumers. In a model for the oil spot and futures market with Gaussian convenience yield curves and a stochastic market price of risk, we find analytical solutions for the optimal trading strategies. An implementation of our strategies in an out-of-sample test on market data shows that the hedging strategies improve long-term return-risk profiles of both the producer and the consumer.

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Acknowledgements

We should like to thank the two anonymous referees and the AE for their constructive criticism and many suggestions that improved the presentation of the paper. We should also like to thank Wolfgang Runggaldier for his valuable feedback on the estimation techniques.

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Correspondence to Johannes Wissel.

Appendix

Appendix

All proofs are given in this appendix. Sections A.1 and A.2 review some key results from the literature on affine models and on quadratic hedging that we use in this paper. Section A.3 contains the proofs of our main results.

1.1 A.1 Proof of Theorem 2.4

Assumption (2.7) implies that

$$\epsilon(t,T) = A(T-t) + B(T-t) X_t $$

with A(τ)=a′(τ) and B(τ)=b′(τ), and thus by (2.8)

for all T>t. Comparing with (2.4) we obtain σ ϵ (t,T)=B(Tt)η, and then (2.5) and (2.11) yield

By separating this into a sum of deterministic terms and linear terms in X t , we obtain

Since B(0)=b′(0)=1, we obtain B(Tt)=e β(Tt) with β=κ+γ 1 η and

with ϵ 0=A(0) and ϵ 1=η(σρβ 1). Integrating A and B then yields (2.12) and (2.13). Finally (2.14) follows by applying Itô’s formula to (2.2).

1.2 A.2 General results on the mean-variance hedging problem

In this appendix, we collect some technical background on the material in Sect. 3.1 and deduce the proof of Theorem 3.3 from the literature on the general mean-variance hedging problem. For a continuous semimartingale F on [0,T] with canonical decomposition F=F(0)+M+A, we write \(\mathbf {F}\in\mathcal {S}^{2}(P)\) if

$$\| \mathbf{F}\|^2 := E \bigg[ \mathbf{F}(0)^2 + \langle M,M \rangle_T + \bigg( \int_0^T |dA_s| \bigg)^2 \bigg] < \infty. $$

Recall the set \(\mathcal{A}\) of admissible strategies in Definition 3.1, the set \(\mathcal{M}\) of measures, and the notation G(π)=∫πd F. Various authors have worked with a different set Θ of admissible strategies. Define

$$\varTheta= \big\{ \pi\text{ predictable and } \mathbf{F} \text{-integrable} \,\big|\, G(\pi) \in\mathcal{S}^2(P) \big\}, $$

and note that the set \(\mathcal{A}\) corresponds to the set \(\bar{\varTheta}\) in Černý and Kallsen [5].

Theorem A.1

  1. (a)

    We have

  2. (b)

    \(\varTheta\subset\mathcal{A}\), and \(G_{T}(\mathcal{A})\) is the closure of G T (Θ) in L 2(P).

Part (a) is Theorem 2.8 in Černý and Kallsen [6], where the inclusion ‘‘⊇’’ follows from Theorems 1.2 and 2.2 in Delbaen and Schachermayer [8], and part (b) is Corollary 2.9 part 1 in Černý and Kallsen [5]. In particular, (a) says that the set of admissible strategies \(\mathcal{A}\) coincides with the set of strategies used in Gouriéroux et al. [13]. Part (b) in particular says that \(G_{T}(\mathcal{A})\) is closed in L 2(P).

Proof of Theorem 3.3

(a) This result is obtained as a special case of Theorem 4.10 in Černý and Kallsen [5] as follows. Let L denote the opportunity process, \(\tilde{a}\) the adjustment process, and Q the variance-optimal signed martingale measure in the sense of Definitions 3.3, 3.8 and 3.12 in [5]. Since F is continuous, Q is equal to the variance-optimal martingale measure \(\tilde{P}\) by Theorem 1.3 in [7]. By (3.16) in [5], we have

$$\frac{d\tilde{P}}{dP} = \frac{1}{E[L_0]} \mathcal{E}\biggl( - \int \tilde{a}\cdot d\mathbf{F} \biggr)_T. $$

By Theorem 3.25, equation (3.33) in [5], we have \(\tilde {a}\mathcal{E}( - \int\tilde{a}\cdot d\mathbf{F}) \in\mathcal{A}\), so the process \(\mathcal{E}( - \int\tilde{a}\cdot d\mathbf{F})\) is a \(\tilde{P}\)-martingale by Theorem A.1(a). So the process \(\tilde{Z}\) defined in (3.8) satisfies

Applying Itô’s formula in the last equation and comparing with (3.9), we obtain \(\tilde{a}_{t} = - \frac{\tilde{\zeta }_{t}}{\tilde{Z}_{t}}\). Furthermore, V(λ) is the mean-value process in (4.2) of [5]. Since F and V(λ) are continuous, the predictable covariation does not depend on the probability measure, and therefore \(\tilde{\xi}\) is the pure hedge coefficient in Definition 4.6 of [5] by equation (4.8) of [5]. So (3.11) is equivalent to equation (4.14) in [5] and thus the assertion follows from Theorem 4.10 in [5].

(b)–(d) The minimal value in (3.6) is given by

$$ A(\lambda) = \frac{\ ( \lambda- \mathcal{X}_0 - \tilde{E}[H_T] )^2}{\tilde {Z}_0} + R $$
(A.1)

with \(R = E [ ( \tilde{Z}_{T} \int_{0}^{T} \frac {1}{\tilde{Z}_{s}} \,dL_{s} )^{2} ]\). This follows from rewriting (5.3) in [13] under \(\tilde{P}\) and P, and using the relation (4.13) in [20] between the Galtchouk–Kunita–Watanabe decompositions of the discounted cashflows under the measures \(\tilde{P}\) and \(\tilde{R}\), respectively. Next for each m∈ℝ define

$$ B(m) = \inf_{\pi\in\mathcal{A}} \big\{ \mathop{\mathrm{Var}}[ \mathcal{X}_T(\pi) ] \, \big|\, E [ \mathcal{X}_T(\pi) ] = m \big\}. $$
(A.2)

The same proof as for Proposition 6.6.5 in [17] shows that

$$ B(m) = \sup_{\lambda\in\mathbb{R}} \big( A(\lambda) - (m - \lambda)^2 \big), $$
(A.3)

and if λ m is a maximizer in (A.3), the process π(λ m ) in (3.11) is an optimal control for B(m) in (A.2). Using (A.1), straightforward calculations yield the maximizer \(\lambda_{m} = \frac{\tilde{Z}_{0} m - \mathcal{X}_{0} - \tilde {E}[H_{T}]}{\tilde{Z}_{0} - 1} \) and the value

$$ B(m) = \frac{ ( \mathcal{X}_0 + \tilde{E}[H_T] - m )^2}{\tilde{Z}_0 - 1} + R. $$
(A.4)

This yields (c). For given vR, (d) follows from setting B(m)=v and solving the quadratic equation for its largest root. Finally, by definition of U(a) and B(m) we have

$$ U(a) = \sup_{m \in\mathbb{R}} \big( m - a B(m) \big), $$
(A.5)

and by (A.4) the maximum in (A.5) is attained at \(m^{*} = \frac{\tilde{Z}_{0} - 1}{2a} + \mathcal{X}_{0} + \tilde {E}[H_{T}]\). Moreover the optimal solution to (3.3) is now given by the optimal control to B(m ), which by the above observation is \(\pi (\lambda_{m^{*}})\). Combining the formulas for λ m and m yields (b). □

1.3 A.3 Proofs of Theorems 3.5–3.8

We resume the setup and definitions in Theorem 3.5. We start with

Lemma A.2

Let x(⋅),y(⋅),z(⋅) be differentiable functions which satisfy x(0)=y(0)=z(0)=0. The process

$$ Y_t := e^{x(T-t) + y(T-t)X_t + z(T-t)X_t^2} Z_t^2 $$
(A.6)

for t∈[0,T] is a P-local martingale if and only if x(⋅),y(⋅),z(⋅) are a finite solution to the ODE system

(A.7)
(A.8)
(A.9)

on [0,T]. In this case the process \(\hat{Z}:= \frac{Y }{Z }\) satisfies \(\hat{Z}_{T} = Z_{T}\) and

$$ \hat{Z}_t = \hat{Z}_0 - \int_0^t \hat{Z}_s \psi_s \cdot d\hat{W}_s, \quad t \in [0,T], $$
(A.10)

where \(\hat{W}_{t} = W_{t} + \int_{0}^{t} \hat{\theta}_{u} du\) and \(\psi_{t} = ( \hat{\theta}_{t}^{0} , \, \hat{\theta}_{t}^{1} - ( \eta y(T-t) + 2 \eta z(T-t) X_{t} ), \, 0 )\).

Proof

Recall from (2.11) that \(\hat{\theta}_{t} = ( \beta _{0} + \gamma_{0} X_{t}, \beta_{1} + \gamma_{1} X_{t}, 0 )\). From (2.8) we have

Applying Itô’s formula to (A.6),

(A.11)

Using κ=βγ 1 η and writing the drift in (A.11) as a quadratic function in X t with deterministic coefficients, we find that the drift vanishes (that is, Y is a P-local martingale) if and only if (A.7), (A.9) hold true. Finally, we note that Itô’s formula, (A.11) and \(dZ_{t} = - Z_{t} \hat{\theta}_{t} \cdot dW_{t}\) imply

$$d\hat{Z}_t = \hat{Z}_t \Bigl( - \hat{\theta}_t^0 \, d\hat{W}_t^0 + \big( \eta y(T - t) + 2 \eta z(T - t) X_t - \hat{\theta}_t^1 \big)\, d\hat{W}_t^1 \Bigr) $$

which gives (A.10). □

The solution of the Ricatti equation system (A.7)–(A.9) can be expressed in closed form.

Lemma A.3

For constant coefficients a,b,c,f,h,k∈ℝ with a,c>0, define \(d=\sqrt{b^{2} - 4ac}\) and \(g=\frac{d+b}{d-b}\). Then the ODE system

$$\everymath{\displaystyle} \begin{array}{l@{\quad}l} z'(\tau) = a + b z(\tau) + c z(\tau)^2, & z(0) = 0, \\[9pt] y'(\tau) = f + \frac{b}{2} y(\tau) + h z(\tau) + c y(\tau) z(\tau), & y(0) = 0, \\[9pt] x'(\tau) = k + \frac{h}{2} y(\tau) + \frac{1}{4} c y(\tau)^2 + \frac{1}{2} c z(\tau), & x(0) = 0 \end{array} $$

has the solution

with

The above formulas are to be understood as their analytic continuation if d=0. The solution exists on the open interval [0,T max) with

$$T_{\max} = \left\{ \begin{array}{l@{\quad}l} \infty &\text{\textit{if} } b \leq- \sqrt{4ac}, \\[5pt] \frac{1}{\sqrt{b^2 - 4ac}} \log\frac{b+\sqrt{b^2 - 4ac}}{b-\sqrt{b^2 - 4ac}} & \text{\textit{if} } b > - \sqrt{4ac}. \end{array} \right. $$

For \(b \in(-\sqrt{4ac},\sqrt{4ac}]\), the function \(b \mapsto\frac {1}{\sqrt{b^{2} - 4ac}} \log\frac{b+\sqrt{b^{2} - 4ac}}{b-\sqrt{b^{2} - 4ac}}\) is to be understood as its analytic continuation out of the domain \(( \sqrt{4ac}, \infty)\).

Proof

The solution formulas are verified by lengthy but straightforward computations. The time horizon \(T_{\rm max}\) is determined by the smallest positive zero of the function τ↦1+ge . □

Proof of Theorem 3.5

Define Y as in (A.6) with (A.7), (A.9). Since z(Tt)>0 for all t<T, there exists a constant c>0 such that

$$e^{x(T-t) + y(T-t)X_t + z(T-t)X_t^2} \geq e^{x(T-t) - \frac {y(T-t)^2}{4z(T-t)}} \geq c $$

for t<T and thus \(Y_{t} \geq c Z_{t}^{2}\) for all t∈[0,T]. Since Y is a continuous process, the stopping times

$$\tau_n = \inf\big\{ t \in[0,T] \,\big|\, Y_t \geq n \big\} \wedge T $$

satisfy τ n T for n→∞, and the processes \(Y^{\tau_{n}}\) and \(Z^{\tau_{n}}\) are bounded P-martingales by Lemma A.2 and \(Z_{t} \leq\sqrt{\frac{1}{c} Y_{t}}\). Applying Doob’s inequality to \(Z^{\tau_{n}}\), we find

$$E \Big[ \sup_{0 \leq t \leq T} Z_{t \wedge\tau_n}^2 \Big] \leq c_2 E \big[ Z_{\tau_n}^2 \big] \leq\frac{c_2}{c} E \big[ Y_{\tau_n} \big] = \frac{c_2}{c} Y_0 $$

for some constant c 2>0. Letting n→∞ and applying monotone convergence in the last inequality, we obtain \(E [ \sup_{0 \leq t \leq T} Z_{t}^{2} ] \leq\frac{c_{2}}{c} Y_{0} < \infty\). So \(\hat{P}\in\mathcal {M}\). □

For the proof of Theorem 3.6 we need the following result.

Lemma A.4

Let W be a d-dimensional Brownian motion on some filtered probability space \((\varOmega,\mathcal{F},\mathbb{F},P)\), and a and-valued and b,σ,νd×d-valued deterministic functions. Let V be and-valued and S an ℝ-valued adapted process satisfying S 0>0 and

Then S is a martingale.

Proof

The proof follows the ideas in Sin [24]. S is a positive local martingale and hence a supermartingale, so it suffices to show that E[S T ]=S 0 for each T>0. Define the stopping times

$$\tau_n = \inf\biggl\{ t \geq0 \,\bigg|\, \int_0^t \| \nu_u \cdot V_u \|^2 \,du \geq n \bigg\}. $$

Since νV is a locally bounded process, we have τ n ↗∞ P-a.s. for n→∞. Moreover, the stopped process \(S^{\tau_{n}} \) is a martingale by Novikov’s condition. Hence we can define a probability measure P nP by \(\frac{dP^{n}}{dP} = \frac {S_{T}^{\tau_{n}}}{S_{0}^{\tau_{n}}}\). Then the process

$$W^n = W - \int\nu_u \cdot V_u \, I_{ \{ u \leq\tau_n \wedge T \} }\,du $$

is a d-dimensional P n-Brownian motion by Girsanov’s theorem, and V satisfies

$$dV_t =\big(a_t + ( \sigma_t \cdot\nu_t I_{ \{ t \leq\tau_n \wedge T \} } + b_t ) \cdot V_t \big) \,dt + \sigma_t \cdot dW_t^n. $$

Now define a process \(\hat{V} \) by \(\hat{V}_{0} = V_{0}\) and

$$d\hat{V}_t = \big(a_t + ( \sigma_t \cdot\nu_t I_{ \{ t \leq T \} } + b_t ) \cdot\hat{V}_t \big) \,dt + \sigma_t \cdot dW_t $$

and a sequence of stopping times \(\hat{\tau}_{n}\) by

$$\hat{\tau}_n = \inf\bigg\{ t \geq0 \,\bigg|\, \int_0^t \| \nu _u \cdot\hat{V}_u \|^2 \,du \geq n \bigg\}. $$

Then the distribution of τ n under P n is the same as the distribution of \(\hat{\tau}_{n}\) under P. Moreover, \(\hat{\tau}_{n} \nearrow\infty\) P-a.s. for n→∞ since \(\nu_{t} \cdot\hat{V}_{t}\) is locally bounded. Monotone convergence therefore yields

This finishes the proof. □

Applying Lemma A.4 to V=X immediately yields

Corollary A.5

Let x(⋅),y(⋅),z(⋅) be a solution to (A.7), (A.9) which satisfies x(0)=y(0)=z(0)=0. Then the process \(\hat{Z}= \frac{Y }{Z }\) in Lemma A.2 is a \(\hat{P}\)-martingale.

Proof of Theorem 3.6

(a) We start by noting that \(\hat{P}\) is a signed Θ-martingale measure in the sense of Sect. 1 of [22]. Indeed, since \(\hat{P}\in \mathcal{M}\), the process G(π) is a \(\hat{P}\)-martingale for each πΘ by Theorem A.1; hence ZG(π) is a P-martingale, and thus \(E \big[ \frac{d\hat{P}}{dP} G_{T}(\pi) \big] = 0\). To show that \(\hat{P}\) is the variance-optimal measure, by Lemma 1 (c) in [22] it now suffices to show that

$$ Z_{T} = M_0 + J_{T} $$
(A.12)

where M 0∈[1,∞) and J T is in the L 2(P)-closure of G T (Θ), that is, in \(G_{T}(\mathcal{A})\) by Theorem A.1 b). To prove this we proceed in three steps.

Step 1. Let \(\mathbb{G}= (\mathcal{G}_{t})_{t \in[0,T]}\) be the filtration \(\mathcal{G}_{t} := \sigma( (\hat{W}_{s}^{0},\hat{W}_{s}^{1}) \, |\, s \leq t )\) generated by the 2-dimensional \(\hat{P}\)-Brownian motion \((\hat{W}^{0},\hat{W}^{1})\) and define the \(\mathbb{G}\)-stopping times

$$\tau_k = \inf\big\{ t \geq0 \,\big|\, |X_t| \geq k \big\} \wedge T \quad\text{for } k\in\mathbb{N}. $$

Since X is continuous, we have τ k T a.s. for k→∞. Next define the processes

Since Z T and \(Z_{\tau_{k}}\) are \(\mathcal{G}_{T}\)-measurable and \(\mathcal{F} _{t} = \mathcal{G}_{t} \vee\sigma(W_{s}^{2} \, |\, s \leq t )\) with \(W^{2} = \hat{W}^{2}\) independent of \(\mathbb{G}\), we obtain \(M_{t} = \hat{E}[ Z_{T} \, |\, \mathcal{G}_{t} ]\) and \(M_{t}^{(k)} = \hat{E}[ Z_{\tau_{k}} \, |\, \mathcal{G}_{t} ]\), and hence by Itô’s representation theorem

for some predictable processes h=(h 0,h 1,0) and h (k)=g(h (k,0),h (k,1),0). Setting \(J_{T} = \int _{0}^{T} h_{s} \cdot d \hat{W}_{s}\), we obtain (A.12) with \(M_{0} = E [ Z_{T}^{2} ] \geq E [ Z_{T} ]^{2} = 1\).

Step 2. It remains to show that J T is in \(G_{T }(\mathcal{A})\). To this end recall that Z is a square-integrable P-martingale by Theorem 3.5, so dominated convergence and Doob’s inequality imply that

for k→∞, and therefore

$$\int_0^{T} h_s^{(k)} \cdot d \hat{W}_s\longrightarrow\int_0^{T} h_s \cdot d \hat{W}_s = J_{T} \quad\text{in } L^2(P). $$

Since \(G_{T}(\mathcal{A})\) is closed in L 2(P), it thus suffices to show that \(\int_{0}^{T} h_{s}^{(k)} \cdot d \hat{W}_{s} \in G_{T }(\mathcal {A})\) for each k. To verify this, first note that the nonsingularity of the volatility matrix of F allows us to write

$$\int_0^{t} h_s^{(k)} \cdot d \hat{W}_s = \int_0^{t} \zeta_s^{(k)} \cdot d\mathbf{F} (s) = G_{t} \big( \zeta^{(k)} \big), \quad t \in[0,T] $$

for a suitable predictable and F-integrable process ζ (k). By Theorem A.1(a), the assertion now follows once we show that G(ζ (k)) is a Q-martingale for each \(Q \in \mathcal{M}\).

Step 3. Fix k∈ℕ and \(Q \in\mathcal{M}\). Clearly G(ζ (k)) is a Q-local martingale. To show the martingale property under Q, we start by computing

$$Z_{t \wedge\tau_k} = \mathcal{E}\Bigl( - \int\hat{\theta}^{(k)} \cdot dW \Bigr)_{t} = \mathcal{E}\Bigl( - \int\hat{\theta}^{(k)} \cdot d\hat{W}\Bigr)_{t} e^{ \int_0^t \| \hat{\theta}_s^{(k)} \|^2 \,ds } = N_t B_t, $$

where \(\hat{\theta}_{t}^{(k)} = \hat{\theta}_{t} I_{ \{ t < \tau_{k} \} }\) is a process bounded by some constant c k depending on k and the model parameters, \(B_{t} = e^{ \int_{0}^{t} \| \hat{\theta}_{s}^{(k)} \|^{2} \,ds }\), and \(N = \mathcal{E}( - \int\hat{\theta}^{(k)} \cdot d\hat{W}) \) is a \(\hat{P} \)-martingale by Novikov’s condition. Hence

It follows that sup t∈[0,T]|G t (ζ (k))|∈L 2(P) by Theorem 3.5 and Doob’s inequality. Hence

by the Cauchy–Schwarz inequality, and so G(ζ (k)) is a Q-martingale.

(b) By (a) we have \(\tilde{P}= \hat{P}\) and thus \(\frac{d\tilde{P}}{dP} = Z_{T} = \hat{Z}_{T}\). Since \(\hat{Z} \) is a \(\hat{P}\)-martingale by Corollary A.5, it follows that

$$\tilde{Z}_t = \tilde{E}\bigg[ \frac{d\tilde{P}}{dP} \, \big|\, \mathcal{F}_t \bigg ] = \hat{E} \big[ \hat{Z}_{T} \, \big|\, \mathcal{F}_t \big] = \hat{Z}_t. $$

Equation (3.14) now follows from Lemma A.2. For (3.15), note that (2.14) and \(d\hat{W}_{t} = dW_{t} + \hat{\theta}_{t}\, dt\) imply

and

(A.13)

Plugging this into (3.14) yields (3.15). Finally \(( \tilde{\zeta} ^{i}, \zeta^{h} ) \in\mathcal{A}\) follows from using the uniqueness of the VOMM and the representations (3.9) and (3.15). □

Proof of Theorem 3.7

We give the proof under the assumption ϕ>β>0, which is satisfied for the parameter estimates we find in our calibration procedure. The result can be easily extended to general parameter values of ϕ and β.

(a) By definition of the spot-futures spread in (2.1) with T 1=T 1(t), we have

$$e^{-ru} S_u = e^{-T_1(u) r} F \big(u, T_1(u) \big) e^{Y_u}. $$

From (3.16) we then compute

(A.14)

Fix u∈[0,T] and T j . We claim that

$$ \tilde{E}\big[ F (u, T_j ) e^{Y_u} \,\big|\, \mathcal{F}_t \big] = F (t,T_j) e^{m_j(u-t) + n_1(u-t) Y_t + n_2(u-t) X_t}, \quad t \leq u, $$
(A.15)

for deterministic functions m j (τ),n 1(τ),n 2(τ) such that m j (0)=n 2(0)=0 and n 1(0)=1. Indeed, Itô’s formula for \(M_{t}^{j}(u) := F (t,T_{j} ) e^{m_{j}(u-t) + n_{1}(u-t) Y_{t} + n_{2}(u-t) X_{t}}\) and using

(A.16)

from (2.14), (2.8) and (2.15), gives, writing n 1=n 1(ut), n 1=n 1(ut) and m j =m j (ut),

(A.17)

Hence the drift of M j(u) is zero if m j ,n 1,n 2 satisfy the ODE system

with m j (0)=n 2(0)=0 and n 1(0)=1, and lengthy but straightforward calculations show that the solution to this system is given by

(A.18)
(A.19)
(A.20)

where \(\alpha= \frac{\nu(\gamma_{0} c_{0} + \gamma_{1} c_{1})}{\phi- \beta}\) and

In this case, M j(u) is a \(\tilde{P}\)-local martingale, and since the diffusion coefficient is of the form \(M_{t}^{j}(u) c(t)\) with a (deterministic) bounded function c(t), the process M j(u) is a \(\tilde{P} \)-martingale by Novikov’s condition. Now (A.15) follows from \(M_{u}^{j}(u) = F (u, T_{j} ) e^{Y_{u}}\). Together with (A.14) we obtain (3.17).

(b) Plugging \(M_{t}^{j}(u) = F (t,T_{j} ) q_{j}(t,u)\) into (A.17) and using (A.16), we obtain

(A.21)

Moreover by (3.17) we have

$$V_t(\lambda) = \lambda- \mathcal{X}_0 + \int_0^t e^{-ru} S_u \,du + \sum _{j=1}^k e^{-T_j r} \int_{T_{j-1}}^{T_j} M_t^j(u) \,du. $$

Applying Itô’s formula and using that V(λ) and M j(u) are \(\tilde{P}\)-martingales gives

$$d V_t(\lambda) = \sum_{j=1}^k e^{-T_j r} \int_{T_{j-1}}^{T_j} dM_t^j(u) \, du. $$

Plugging in (A.21) here, we obtain

with a \(\tilde{P}\)-local martingale L orthogonal to F. Plugging (A.13) into the last equation yields (3.18). □

Proof of Theorem 3.8

The structure of the proof is analogous to the proof of Theorem 3.7(a), so we only give a sketch. As in (A.14) we obtain

$$E[H_T] = - \sum_{j=1}^k e^{-T_j r} \int_{T_{j-1}}^{T_j} E \big[ F (u, T_j ) e^{Y_u} \big] \,du, $$

so (3.21) follows once we show that for all tu,

$$ E \big[ F (u, T_j ) e^{Y_u} \,\big|\, \mathcal{F}_t \big] = F (t,T_j ) e^{\ell _j(u-t) + p(u-t) Y_t + s_j(u-t) X_t} $$
(A.22)

with deterministic functions p, j ,s j satisfying p(0)=1 and s j (0)= j (0)=0. To this end, we apply Itô’s formula to the RHS of (A.22), use (2.14), (2.8) and (2.15), and as in the proof of Theorem 3.7(a), we find that the RHS of (A.22) is a P-local martingale, and then indeed a martingale, if and only if the functions p,s j , j fulfill a system of ODEs. This system can be solved explicitly, and lengthy but straightforward computations yield

(A.23)
(A.24)
(A.25)

where \(\alpha= \frac{\eta\gamma_{1}}{\beta\kappa} - \frac{\sigma }{\kappa} ( \rho_{0} \gamma_{0} + \rho_{1} \gamma_{1} )\) and

To verify (3.22), similarly as above we compute

$$E[H_T^2] = \sum_{i=1}^k \sum_{j=1}^k e^{-(T_i + T_j) r} \int _{T_{i-1}}^{T_i} \Bigg( \int_{T_{j-1}}^{T_j} E \big[ F (u, T_i ) F (v, T_j ) e^{Y_u + Y_v} \big] \,dv \Bigg) \,du. $$

Hence the assertion follows once we show

$$ E \big[ F (u, T_i ) F (v, T_j ) e^{Y_u + Y_v} \big] = F (0,T_i ) F (0,T_j ) q_{ij}(u,v) $$
(A.26)

for all u,v, and it suffices to establish (A.26) for uv by symmetry of the function q ij (u,v) in u and v. So let uv. We note that by (A.22) we have

and thus (A.26) follows once we prove for all t∈[0,v]

(A.27)

with deterministic functions w,w ij ,m ij satisfying the equations m ij (0)=0, w(0)=1+p(uv), and w ij (0)=s i (uv). To this end, we proceed as above. We apply Itô’s formula to the RHS of (A.27), use (2.14), (2.8) and (2.15), and as in the proof of Theorem 3.7(a), we find that the RHS of (A.27) is a P-local martingale, and then indeed a martingale, if and only if the functions w,w ij ,m ij fulfil a system of ODEs. This system can be solved explicitly, and lengthy but straightforward computations yield that

(A.28)
(A.29)
(A.30)

where p(⋅) and s i (⋅) are defined in (A.23), (A.24), \(\alpha= \frac{\eta\gamma_{1}}{\beta\kappa} - \frac {\sigma }{\kappa} ( \rho_{0} \gamma_{0} + \rho_{1} \gamma_{1} )\), and

This finishes the proof. □

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Wang, L., Wissel, J. Mean-variance hedging with oil futures. Finance Stoch 17, 641–683 (2013). https://doi.org/10.1007/s00780-013-0203-x

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