Abstract
The authors aim to develop numerical schemes of the two representative quadratic hedging strategies: locally risk-minimizing and mean-variance hedging strategies, for models whose asset price process is given by the exponential of a normal inverse Gaussian process, using the results of Arai et al. (Int J Theor Appl Financ 19:1650008, 2016) and Arai and Imai (A closed-form representation of mean-variance hedging for additive processes via Malliavin calculus, preprint. Available at https://arxiv.org/abs/1702.07556). Here normal inverse Gaussian process is a framework of Lévy processes that frequently appeared in financial literature. In addition, some numerical results are also introduced.
Keywords
JEL Classification: G11, G12
Mathematics Subject Classification (2010): 91G20, 91G60, 60G51
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Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers 15K04936 and 17K13764.
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Appendix
Appendix
1.1 Proof of Proposition 3.1
In order to see Condition 1, it suffices to show \(\int _1^\infty (e^x-1)^4\nu (dx)<\infty \) and \(\int ^{-1}_{-\infty }(e^x-1)^4\nu (dx)<\infty \).
Firstly, we see \(\int _1^\infty (e^x-1)^4\nu (dx)<\infty \). Noting that the Sommerfeld integral representation for the function K 1 (see, e.g., Appendix A of [9]):
for z ≥ 0, we have
Remark that the above first inequality is given from \((e^{\frac {z}{\alpha }}-1)^4 \leq e^{\frac {4z}{\alpha }}\) for any z ∈ [α, ∞).
Next, we show \(\int ^{-1}_{-\infty }(e^x-1)^4\nu (dx)<\infty \) by a similar argument to the above. Noting that \((e^{\frac {z}{\alpha }}-1)^4\leq 1\) for any z ∈ (−∞, −α], we have
Thus, Condition 1 holds true.
To confirm Condition 2, we need some preparations. The following lemma is proven later.
Lemma A.1
For any v ∈ [0, ∞) and any \(a\in (\frac {3}{2},2]\) , we have
In addition, (11) still holds for the case where (v, a) = (0, 1) and (v, a + 1).
We have \(\int _{{\mathbb R}_0}(e^x-1)\nu (dx)=W(0,1)={\delta \alpha }(\sqrt {M_2}-\sqrt {M_1(0,1)})\). Assumption 3.1 implies
from which the inequality \(0 \geq \int _{{\mathbb R}_0}(e^x-1)\nu (dx)\) holds true. To see the second inequality, since we have
it suffices to show W(0, 2) − W(0, 1) > 0. Firstly, we have
On the other hand, it holds that
by Assumption 3.1. As a result, the inequality \(\int _{{\mathbb R}_0}(e^x-1)\nu (dx) > -\int _{{\mathbb R}_0}(e^x-1)^2\nu (dx)\) holds under Assumption 3.1. \(\Box \)
1.2 Proof of Lemma A.1
We begin with the following lemma:
Lemma A.2
For any γ ≥ 0 and any M > 0, we have
Proof
Remark that the characteristic function of the Gamma distribution with parameters θ > 0 and k > 0 is given as
for any \(u\in {\mathbb R}\), where Γ(⋅) is the Gamma function. We have then
for any M > 0 and any \(u\in {\mathbb R}\). Thus, we obtain
Putting \(x=\sqrt {M^2+u^2}\), we have
and
This completes the proof of Lemma A.2. \(\Box \)
Now, let us go back to the proof of Lemma A.1. For any v ∈ [0, ∞) and any \(a\in (\frac {3}{2},2]\), the same sort of argument as in the proof of Proposition 3.1 implies that
Since we have
we obtain
On the other hand, we have
by M 1, M 2 > 0. As a result, using Lemma A.2, we obtain
from which (11) follows for any v ∈ [0, ∞) and any \(a\in (\frac {3}{2},2]\).
For v ≥ 0, we see that (11) still holds for a + 1. To this end, it is enough to make sure that M 1(v, a + 1) and b(v, a + 1) remain nonnegative. In fact, we have
and
by Assumption 3.1. Similarly, (11) follows for the case of (v, a) = (0, 1), since
and b(0, 1) = 0. \(\Box \)
1.3 Proof of Proposition 3.2
Noting that 0 ≥ h > −1 by Assumption 3.1 and Proposition 3.1, we have
by (2). This completes the proof of Proposition 3.2. \(\Box \)
1.4 Proof of Proposition 3.3
To show Proposition 3.3, we start with the following lemma:
Lemma A.3
We have
Proof
The Sommerfeld integral representation (10) implies that
\(\Box \)
Note that we do not need Assumption 3.1 in the above proof. Now, we show Proposition 3.3. By Lemma A.3 and Proposition 3.2, we have
Remark that W(v, a;α, 1 + β, −hδ) is well-defined and satisfies (11), since we have M 1(v, a;α, β + 1) = M 1(v, a + 1;α, β) ≥ 0 and b(v, a;α, β + 1) = b(v, a + 1;α, β) ≥ 0. (3) implies that
from which Proposition 3.3 follows. \(\Box \)
1.5 Proof of Proposition 3.5
To see Proposition 3.5, we prepare one proposition and one lemma. In order to emphasize the parameters α, β, and δ, we write M 1(v, a), M 2, and b(v, a) as M 1(v, a;α, β), M 2(α, β), and b(v, a;α, β), respectively.
Proposition A.1
For any v ∈ [0, ∞) and any t ∈ [0, T), we have
where C(t) is given in (8).
Proof
Proposition 3.3 implies that
Note that the last inequality follows from the fact that α 2 − (a + β)2 > 0 and α 2 − (a + 1 + β)2 > 0 hold by Assumption 3.1. \(\Box \)
Lemma A.4
For any v ∈ [0, ∞) and any \(a\in (\frac {3}{2},2]\) ,
holds.
Proof
Denoting \(M_1^{\prime }:=M_1(v,a+1)\), b′ := b(v, a + 1), M 1 := M 1(v, a), and b := b(v, a), for short, we have
Since a + β > 0, we have
and
which imply that
Setting
we have p > 0 and q > 0 for any \(a\in (\frac {3}{2},2]\) by Assumption 3.1 and
This completes the proof of Lemma A.4. \(\Box \)
Proof of Proposition 3.5.
Firstly, Lemma A.4 implies that
where p is defined in the proof of Lemma A.4. Remark that the last inequality in (16) holds since
by Assumption 3.1. Now, note that
Thus, Proposition A.1 implies that
As a result, noting that w > 1, we obtain
This completes the proof of Proposition 3.5. \(\Box \)
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Arai, T., Imai, Y., Nakashima, R. (2018). Numerical Analysis on Quadratic Hedging Strategies for Normal Inverse Gaussian Models. In: Kusuoka, S., Maruyama, T. (eds) Advances in Mathematical Economics. Advances in Mathematical Economics, vol 22. Springer, Singapore. https://doi.org/10.1007/978-981-13-0605-1_1
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