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The Jacobi stochastic volatility model

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Abstract

We introduce a novel stochastic volatility model where the squared volatility of the asset return follows a Jacobi process. It contains the Heston model as a limit case. We show that the joint density of any finite sequence of log-returns admits a Gram–Charlier A expansion with closed-form coefficients. We derive closed-form series representations for option prices whose discounted payoffs are functions of the asset price trajectory at finitely many time points. This includes European call, put and digital options, forward start options, and can be applied to discretely monitored Asian options. In a numerical study, we show that option prices can be accurately and efficiently approximated by truncating their series representations.

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Fig. 5

Notes

  1. We conjecture that (3.3) holds for any \(k\ge0\) also when \(v_{\min}=0\) (and \(\kappa\theta>0\)) or \(\rho^{2}=1\). For the Heston model (2.5) with \(Q(v)=v\) and \(\rho^{2}<1\), the conjecture follows from Dufresne [24, Theorem 4.1].

  2. The Gram–Charlier A series expansion of a density function \(g(x)\) on the real line is defined as the formal infinite series \(\sum_{n\ge0} \ell_{n} H_{n}(x)w(x)\), where \(\ell_{n}= \int_{\mathbb{R}} H_{n}(x)g(x)\,dx\) for all \(n\ge0\).

  3. Similar recursive relations of the Fourier coefficients for the physicist Hermite polynomial basis can be found in Drimus et al. [22]. The physicist Hermite polynomial basis is the orthogonal polynomial basis of the \(L^{2}_{w}\)-space equipped with the weight function \(w(x)=\mathrm{e} ^{-x^{2}}\) so that \((H_{n},H_{n})_{w}=\sqrt{2\pi}2^{n}n!\).

  4. In practice, depending on the model parameters, this may be not always feasible, in which case the truncation order \(N\) should be increased.

  5. The derivation of the tightness of \((V_{t}^{(n)},X_{t}^{(n)})\) from (B.5) is also stated without proof in Rogers and Williams [53, Theorem II.85.5]. For the sake of completeness, we give a short self-contained argument here.

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Acknowledgements

We thank the participants at the 2014 Stochastic Analysis in Finance and Insurance Conference in Oberwolfach, the 2015 AMaMeF and Swissquote Conference in Lausanne, the 2016 ICMS Workshop in Edinburgh, and the seminar at Mannheim Mathematics Department, as well as Stefano De Marco, Julien Hugonnier, Wahid Khosrawi-Sardroudi, Martin Larsson and Peter Tankov for their comments. We thank an anonymous referee, an anonymous Associate Editor, and Chris Rogers (Co-Editor) for their careful reading of the manuscript and suggestions.

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Correspondence to Damien Ackerer.

Additional information

The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) ERC Grant Agreement n. 307465-POLYTE. The research of Sergio Pulido benefited from the support of the Chair Markets in Transition (Fédération Bancaire Française) and the project ANR 11-LABX-0019.

Appendices

Appendix A: Hermite moments

To describe more explicitly how the Hermite moments \(\ell_{0},\dots, \ell_{N}\) in (3.12) can be efficiently computed for a fixed truncation order \(N\ge1\), we apply Theorem 2.4. We let \(M=\dim\mathrm{Pol}_{N}\) and \(\pi:{\mathcal{E}}\rightarrow\{1, \ldots, M\}\) be an enumeration of the set of exponents

$$ {\mathcal{E}}=\{(m,n): m,n\ge0,\,m+n\le N\}. $$

The polynomials

$$ h_{\pi(m,n)}(v,x) = v^{m} H_{n}(x), \qquad (m,n)\in{\mathcal{E}}, $$
(A.1)

then form a basis of \(\mathrm{Pol}_{N}\). In view of the elementary property

$$ H_{n}'(x)=\frac{\sqrt{n}}{\sigma_{w}}H_{n-1}(x), \qquad n\ge1, $$

we obtain that the \(M\times M\)–matrix \(G\) representing \({\mathcal{G}}\) on \(\mathrm{Pol}_{N}\) has at most 7 nonzero elements in column \(\pi(m,n)\) with \((m,n)\in{\mathcal{E}}\); they are given by

$$ \begin{aligned} G_{\pi(m-2,n),\pi(m,n)} &=-\frac{\sigma^{2} m(m-1) v_{\max}v_{ \min}}{2(\sqrt{v_{\max}}-\sqrt{v_{\min}})^{2}}, \qquad m\ge2,\\ G_{\pi(m-1,n-1),\pi(m,n)}&=-\frac{\sigma\rho m\sqrt{n} v_{\max}v_{\min}}{\sigma_{w}(\sqrt{v_{\max}}-\sqrt{v_{\min}})^{2}}, \qquad m,n\ge1,\\ G_{\pi(m-1,n),\pi(m,n)}&=\kappa\theta m+\frac{\sigma ^{2}m(m-1) (v_{\max}+v_{\min})}{2(\sqrt{v_{\max}}-\sqrt{v_{ \min}})^{2}}, \qquad m\ge1,\\ G_{\pi(m,n-1),\pi(m,n)}&=\frac{(r-\delta)\sqrt{n}}{ \sigma_{w}}+\frac{\sigma\rho m\sqrt{n} (v_{\max}+v_{\min})}{ \sigma_{w}(\sqrt{v_{\max}}-\sqrt{v_{\min}})^{2}}, \qquad n\ge1,\\ G_{\pi(m+1,n-2),\pi(m,n)}&=\frac{\sqrt{n(n-1)}}{2\sigma _{w}^{2}}, \qquad n\ge2,\\ G_{\pi(m,n),\pi(m,n)}&=-\kappa m-\frac{\sigma^{2} m(m-1)}{2(\sqrt{v _{\max}}-\sqrt{v_{\min}})^{2}} \qquad \text{for all $m,n$},\\ G_{\pi(m+1,n-1),\pi(m,n)}&=-\frac{ \sqrt{n}}{2\sigma_{w}}-\frac{\sigma\rho m\sqrt{n}}{\sigma _{w}(\sqrt{v _{\max}}-\sqrt{v_{\min}})^{2}}, \qquad n\ge1. \end{aligned} $$

Theorem 2.4 now implies the following result.

Theorem A.1

The coefficients \(\ell_{n}\) are given by

$$ \ell_{n}= \begin{pmatrix} h_{1}(V_{0},X_{0}) & \cdots& h_{M}(V_{0},X_{0}) \end{pmatrix} \,\mathrm{e}^{TG}\, \mathbf{e}_{\pi(0,n)}, \qquad 0\le n\le N, $$
(A.2)

where \(\boldsymbol{e}_{i}\) is the \(i\)th standard basis vector in \({\mathbb{R}} ^{M}\).

Remark A.2

The choice of the basis polynomials \(h_{\pi(m,n)}\) in (A.1) is convenient for our purposes because (i) each column of the \(M\times M\)-matrix \(G\) has at most seven nonzero entries, and (ii) the coefficients \(\ell_{n}\) in the expansion of prices (3.10) can be obtained directly from the action of \(\mathrm{e}^{G T}\) on \({\boldsymbol{e}}_{\pi_{(0,n)}}\) as specified in (A.2). In practice, it is more efficient to compute directly this action, rather than computing the matrix exponential \(\mathrm{e}^{G T}\) and then selecting the \(\pi_{(0,n)}\)-column.

We now extend Theorem A.1 to a multidimensional setting. The following theorem provides an efficient way to compute the multidimensional Hermite moments defined in (4.3). Before stating the theorem, we fix some notation. Set \(N=\sum_{i=1}^{d} n _{i}\) and \(M=\dim\mathrm{Pol}_{N}\). Let \(G^{(i)}\) be the matrix representation of the linear map \({\mathcal{G}}\) restricted to \(\mathrm{Pol}_{N}\) with respect to the basis, in row vector form,

$$ h^{(i)}(v,x)=\big(h^{(i)}_{1}(v,x) \cdots h^{(i)}_{M}(v,x)\big), $$

with \(h^{(i)}_{\pi(m,n)}(v,x)=v^{m}H^{(i)}_{n}(x)\) as in (A.1), where \(H_{n}^{(i)}\) is the generalized Hermite polynomial of degree \(n\) associated to the parameters \(\mu_{w_{i}}\) and \(\sigma_{w_{i}}\); see (3.8). Define the \(M\times M\)-matrix \(A^{(k,\ell)}\) by

$$ A^{(k,\ell)}_{i,j} = \textstyle\begin{cases} H_{n}^{(\ell)}(0), \quad & \text{if $i=\pi(m,k)$ and $j=\pi(m,n)$ for some $m,n\in{\mathbb{N}}$,} \\ 0, \quad & \text{otherwise.} \end{cases} $$

Theorem A.3

For any \(n_{1},\ldots,n_{d}\in{\mathbb{N}}_{0}\), the multidimensional Hermite moment in (4.3) can be computed through

$$ \ell_{n_{1},\ldots,n_{d}}=h^{(1)}(V_{0},0) \bigg(\prod_{i=1}^{d-1} \mathrm{e}^{G^{(i)} \Delta t_{i}}A^{(n_{i},i+1)} \bigg)\mathrm{e}^{G ^{(d)}\Delta t_{d}}{\boldsymbol{e}}_{\pi(0,n_{d})}, $$

where \(\Delta t_{i}=t_{i}-t_{i-1}\).

Proof

By an inductive argument, it is sufficient to illustrate the case \(n=2\). Applying the law of iterated expectations, we obtain

$$ \ell_{n_{1},n_{2}}= {\mathbb{E}}[H^{(1)}_{n_{1}}(Y_{t_{1}})H^{(2)} _{n_{2}}(Y_{t_{2}})] = {\mathbb{E}}\big[H^{(1)}_{n_{1}}(X_{t_{1}}-X _{0}){\mathbb{E}}_{t_{1}}[H^{(2)}_{n_{2}}(X_{t_{2}}-X_{t_{1}})]\big]. $$

Since the increment \(X_{t_{2}}-X_{t_{1}}\) does not depend on \(X_{t_{1}}\), we can rewrite, using Theorem 2.4,

$$ {\mathbb{E}}_{t_{1}}[H^{(2)}_{n_{2}}(X_{t_{2}}-X_{t_{1}})] = {\mathbb{E}}[H ^{(2)}_{n_{2}}(X_{\Delta t_{2}})| X_{0}=0, V_{0}=V_{t_{1}}] = h^{(2)}(V _{t_{1}},0) v^{(n_{2},2)}, $$

where \(v^{(n_{2},2)}=e^{G^{(2)} \Delta t_{2}}{\boldsymbol{e}}_{\pi(0,n_{2})}\). Note that this last expression is a polynomial solely in \(V_{t_{1}}\), i.e.,

$$ h^{(2)}(V_{t_{1}},0) v^{(n_{2},2)} = \sum_{n=0}^{n_{2}} a_{n} \, V _{t_{1}}^{n} \qquad \text{with } a_{n} = \sum_{n+j\le n_{2}} \, H_{j}^{(2)}(0) \, v^{(n _{2},2)}_{\pi(n,j)}. $$

Theorem 2.4 now implies that the Hermite coefficient is given by

$$ \ell_{n_{1},n_{2}}= {\mathbb{E}}[p(V_{t_{1}},X_{t_{1}})| X_{0}=0] = h ^{(1)}(V_{0},0) \mathrm{e}^{G^{(1)} \Delta t_{1}} \vec{p}, $$

where \(\vec{p}\) is the vector representation in the basis \(h^{(1)}(v,x)\) of the polynomial

$$ p(v,x) = \sum_{n=0}^{n_{2}} a_{n} \, v^{n} \, H_{n_{1}}(x) = h^{(1)}(v,x) \vec{p} . $$

We conclude by observing that the coordinates of the vector \(\vec{p}\) are given by \(\mathrm{e}_{i}^{\top}\, \vec{p} = a_{n}\), if \(i=\pi(n, n_{1})\) for some integer \(n\le n_{2}\), and equal to zero otherwise, which in turn shows that \(\vec{p} = A^{(n_{1},2)} \, v^{(n _{2},2)} \). □

Appendix B: Proofs

This appendix contains the proofs of all theorems and propositions in the main text.

Proof

of Theorem 2.1 For strong existence and uniqueness of (2.1), it is enough to show strong existence and uniqueness for the SDE for \((V_{t})\) which is

$$ dV_{t} = \kappa(\theta- V_{t})\,dt + \sigma\sqrt{Q(V_{t})}\,dW _{1t}. $$
(B.1)

Since the interval \([0,1]\) is an affine transformation of the unit ball in ℝ, weak existence of a \([v_{\min},v_{\max}]\)-valued solution can be deduced from Larsson and Pulido [44, Theorem 2.1]. Pathwise uniqueness of solutions follows from Yamada and Watanabe [55, Theorem 1]. Strong existence of solutions for the SDE (B.1) is a consequence of pathwise uniqueness and weak existence of solutions; see, for instance, Yamada and Watanabe [55, Corollary 1].

Now let \(v\in[v_{\min},v_{\max})\). The occupation times formula from Revuz and Yor [51, Corollary VI.1.6] implies

$$ \int_{0}^{\infty}\mathbf{1}_{\{V_{t}=v\}} \sigma^{2} Q(v)\,dt =0, \qquad v>v_{\min}. $$

Since \(\sigma^{2} Q(v)>0\), this proves (2.2) for \(v>v_{\min}\). We can show that the local time at \(v_{\min}\) of \((V_{t})\) is zero as in Filipović and Larsson [30, Theorem 5.3], which in turn proves (2.2) for \(v=v_{\min}\) by applying [30, Lemma A.1].

To conclude, Proposition 2.2 in Larsson and Pulido [44] shows that we have \(V_{t}\in(v_{\min},v_{\max})\) if and only if \(V_{0}\in(v_{ \min},v_{\max})\) and condition (2.3) holds. □

2.1 B.1 Proof of Theorem 2.3

The proof of Theorem 2.3 builds on the following four lemmas.

Lemma B.1

Suppose that \(Y\) and \(Y^{(n)}\), \(n\ge1\), are random variables in \({\mathbb{R}}^{d}\) for which all moments exist. Assume further that

$$ \lim_{n\to\infty}{\mathbb{E}}[p(Y^{(n)})]={\mathbb{E}}[p(Y)] $$
(B.2)

for any polynomial \(p(y)\) and that the distribution of \(Y\) is determined by its moments. Then the sequence \((Y^{(n)})\) converges weakly to \(Y\) as \(n\to\infty\).

Proof

Theorem 30.2 in Billingsley [11] proves this result for the case \(d=1\). Inspection shows that the proof is still valid for the general case. □

Lemma B.2

The moments of the finite-dimensional distributions of the diffusions \((V_{t}^{(n)},X_{t}^{(n)})\) converge to the respective moments of the finite-dimensional distributions of \((V_{t},X_{t})\). That is, for any \(0\le t_{1}<\cdots<t_{d}<\infty\) and for any polynomials \(p_{1}(v,x), \ldots,p_{d}(v,x)\), we have

$$ \lim_{n\to\infty}{\mathbb{E}}\bigg[\prod_{i=1}^{d} p_{i}(V^{(n)} _{t_{i}},X^{(n)}_{t_{i}})\bigg]={\mathbb{E}}\bigg[\prod_{i=1}^{d} p _{i}(V_{t_{i}},X_{t_{i}})\bigg]. $$
(B.3)

Proof

Let \(N=\sum_{i=1}^{d} \deg p_{i}\). Throughout the proof, we fix a basis \(h_{j}(v,x)\) of \(\mathrm{Pol}_{N}\), where \(1\le j\le M=\dim \mathrm{Pol}_{N}\), and for any polynomial \(p(v,x)\), we denote by \(\vec{p}\) its coordinates with respect to this basis. We denote by \(G\) and \(G^{(n)}\) the respective \(M\times M\)-matrix representations of the generators restricted to \(\mathrm{Pol}_{N}\) of \((V_{t},X_{t})\) and \((V^{(n)}_{t},X^{(n)}_{t})\). We then define recursively the polynomials \(q_{i}(v,x)\) and \(q^{(n)}_{i}(v,x)\) for \(1\le i\le d\) by

$$ \begin{aligned} q_{d}(v,x) &=q^{(n)}_{d}(v,x)=p_{d}(v,x),\\ q_{i}(v,x)&= p_{i}(v,x) \begin{pmatrix} h_{1}(v,x) & \cdots& h_{M}(v,x) \end{pmatrix} \mathrm{e}^{(t_{i+1}-t_{i})G}\vec{q}_{i+1} , \qquad 1\le i< d,\\ q^{(n)}_{i}(v,x)&= p_{i}(v,x) \begin{pmatrix} h_{1}(v,x) & \cdots& h_{M}(v,x) \end{pmatrix} \mathrm{e}^{(t_{i+1}-t_{i})G^{(n)}}\vec{q}^{(n)}_{i+1}, \qquad 1\le i< d. \end{aligned} $$

As in the proof of Theorem A.3, a successive application of Theorem 2.4 and the law of iterated expectations implies that

$$\begin{aligned} {\mathbb{E}}\bigg[\prod_{i=1}^{d} p_{i}(V_{t_{i}},X_{t_{i}})\bigg] &= {\mathbb{E}}\bigg[\prod_{i=1}^{d-1} p_{i}(V_{t_{i}},X_{t_{i}}) {\mathbb{E}}[ p_{d}(V_{t_{d}},X_{t_{d}}) | {\mathcal{F}}_{t_{d-1}}] \bigg] \\ &=\cdots= \begin{pmatrix} h_{1}(V_{0},X_{0}) & \cdots& h_{M}(V_{0},X_{0}) \end{pmatrix} \mathrm{e}^{t_{1}G}\vec{q}_{1}, \end{aligned}$$

and similarly,

$$ {\mathbb{E}}\bigg[\prod_{i=1}^{d} p_{i}(V^{(n)}_{t_{i}},X^{(n)}_{t _{i}})\bigg]=\big(h_{1}(V_{0}^{(n)},X_{0}^{(n)}) \cdots h_{M}(V_{0} ^{(n)},X_{0}^{(n)})\big)\mathrm{e}^{t_{1}G^{(n)}}\vec{q}_{1}^{(n)}. $$

We deduce from (2.6) that

$$ \lim_{n\to\infty} G^{(n)}=G. $$
(B.4)

This is valid also for the limit case \(v_{\max}=\infty\), that is, \({Q(v)=v-v_{\min}}\). This fact together with an inductive argument shows that \(\lim_{n\to\infty}\vec{q}_{1}^{(n)}=\vec{q}_{1}\). This combined with (B.4) proves (B.3). □

Lemma B.3

The finite-dimensional distributions of \((V_{t},X_{t})\) are determined by their moments.

Proof

The proof of this result is contained in the proof of Filipović and Larsson [30, Lemma 4.1]. □

Lemma B.4

The family of diffusions \((V_{t}^{(n)},X_{t}^{(n)})\), \(n \in{\mathbb{N}}\), is tight.

Proof

Fix a time horizon \(N\in{\mathbb{N}}\). We first observe that by Karatzas and Shreve [41, Problem V.3.15], there is a constant \(K\) independent of \(n\) such that

$$ {\mathbb{E}}[\vert(V_{t}^{(n)},X_{t}^{(n)})-(V_{s}^{(n)},X_{s}^{(n)}) \vert^{4}]\leq K|t-s|^{2}, \qquad 0\le s< t\le N. $$
(B.5)

Now fix any positive \(\alpha<1/4\). Kolmogorov’s continuity theorem (see Revuz and Yor [51, Theorem I.2.1]) implies that

$$ {\mathbb{E}}\bigg[\bigg(\sup_{0\le s< t\le N} \frac{\vert(V_{t}^{(n)},X _{t}^{(n)})-(V_{s}^{(n)},X_{s}^{(n)})\vert}{|t-s|^{\alpha}}\bigg)^{4} \bigg]\le J $$

for a finite constant \(J\) that is independent of \(n\). The modulus of continuity

$$ \Delta(\delta,n)=\sup\{ \vert(V_{t}^{(n)},X_{t}^{(n)})-(V_{s}^{(n)},X _{s}^{(n)})\vert: 0\le s< t\le N,\, |t-s|< \delta\} $$

thus satisfies

$$ {\mathbb{E}}[ \Delta(\delta,n)^{4}] \le\delta^{\alpha}J . $$

Using Chebyshev’s inequality, we conclude that for every \(\epsilon>0\),

$$ {\mathbb{Q}}\left[ \Delta(\delta,n)>\epsilon\right] \le\frac{ {\mathbb{E}}[ \Delta(\delta,n)^{4}]}{\epsilon^{4}}\le\frac{ \delta^{\alpha}J}{\epsilon^{4}}, $$

and thus \(\sup_{n \in{\mathbb{N}}} {\mathbb{Q}}[ \Delta(\delta,n)> \epsilon]\to0\) as \(\delta\to0\). This together with the property that the initial states \((V_{0}^{(n)},X_{0}^{(n)})\) converge to \((V_{0},X _{0})\) as \(n\to\infty\) proves the lemma; seeFootnote 5 Rogers and Williams [53, Theorem II.85.3]. □

Remark B.5

Kolmogorov’s continuity theorem (see Revuz and Yor [51, Theorem I.2.1]) and (B.5) imply that the paths of \((V_{t},X_{t})\) are \(\alpha\)-Hölder continuous for any \(\alpha<1/4\).

Proof

of Theorem 2.3 Lemmas B.1B.3 imply that the finite-dimensional distributions of the diffusions \((V_{t}^{(n)},X _{t}^{(n)})\) converge weakly to those of \((V_{t},X_{t})\) as \(n\to\infty\). Theorem 2.3 thus follows from Lemma B.4 and Rogers and Williams [53, Lemma II.87.3]. □

Proof of Theorem 3.7

We claim that the solution of the recursion (3.18) is given by

$$ I_{n}(\mu;\nu)=\int_{\mu}^{\infty}{\mathcal{H}}_{n}(x)\mathrm{e} ^{\nu x}\phi(x)\,dx, \qquad n\ge0. $$
(B.6)

Indeed, for \(n=0\), the right-hand side of (B.6) equals

$$ \int_{\mu}^{\infty}{\mathcal{H}}_{0}(x)\mathrm{e}^{\nu x}\phi(x)\,dx= \mathrm{e}^{\frac{\nu^{2}}{2}}\int_{\mu-\nu}^{\infty}\phi(x)\,dx, $$

which is \(I_{0}(\mu;\nu)\). For \(n\ge1\), we recall that the standard Hermite polynomials \({\mathcal{H}}_{n}(x)\) satisfy

$$ {\mathcal{H}}_{n}(x)= x{\mathcal{H}}_{n-1}(x)-{\mathcal{H}}_{n-1}'(x). $$
(B.7)

Integration by parts and (B.7) then show that

$$\begin{aligned} \int_{\mu}^{\infty}{\mathcal{H}}_{n}(x)\mathrm{e}^{\nu x}\phi(x)\,dx &= \int_{\mu}^{\infty}{\mathcal{H}}_{n-1}(x)\mathrm{e}^{\nu x}x\phi(x) \,dx-\int_{\mu}^{\infty}{\mathcal{H}}_{n-1}'(x)\mathrm{e}^{\nu x} \phi(x)\,dx \\ &=- {\mathcal{H}}_{n-1}(x)\mathrm{e}^{\nu x}\phi(x)\big|_{\mu}^{ \infty}+ \int_{\mu}^{\infty}{\mathcal{H}}_{n-1}(x)\nu\mathrm{e} ^{\nu x}\phi(x)\,dx \\ &= {\mathcal{H}}_{n-1}(\mu)\mathrm{e}^{\nu\mu}\phi(\mu)+\nu \int_{\mu}^{\infty}{\mathcal{H}}_{n-1}(x)\mathrm{e}^{\nu x}\phi(x) \,dx , \end{aligned}$$

which proves (B.6).

A change of variables, using (3.8) and (B.6), shows that

$$\begin{aligned} f_{n} &= \mathrm{e}^{-rT} \int_{k}^{\infty}( \mathrm{e}^{x} - \mathrm{e}^{k}) H_{n}(x) w(x)\,dx \\ &= \mathrm{e}^{-rT} \int_{\frac{k-\mu_{w}}{\sigma_{w}}}^{\infty}( \mathrm{e}^{\mu_{w}+\sigma_{w} z} -\mathrm{e}^{k}) H_{n}(\mu_{w}+ \sigma_{w} z) w(\mu_{w}+\sigma_{w} z)\sigma_{w}\,dz \\ &= \mathrm{e}^{-rT} \frac{1}{\sqrt{n!}} \int_{\frac{k-\mu_{w}}{\sigma_{w}}}^{\infty}( \mathrm{e}^{\mu_{w}+ \sigma_{w} z} -\mathrm{e}^{k}) {\mathcal{H}}_{n}(z) \phi(z)\,dz \\ &=\mathrm{e}^{-rT+\mu_{w}}\frac{1}{\sqrt{n!}}I_{n}\bigg(\frac {k-\mu _{w}}{\sigma_{w}};\sigma_{w}\bigg)-\mathrm{e}^{-rT+k}\frac{1}{ \sqrt{n!}}I_{n}\bigg(\frac{k-\mu_{w}}{\sigma_{w}};0\bigg). \end{aligned}$$

Formula (3.17) follows from the recursion formula (3.18). □

Proof

of Theorem  3.8 As before, a change of variables, using (3.8) and (B.6), gives

$$\begin{aligned} f_{n} &=\mathrm{e}^{-rT} \int_{k}^{\infty}H_{n}(x) w(x)\,dx \\ &= \frac{\mathrm{e}^{-rT}}{\sqrt{n!}} \int_{\frac{k-\mu_{w}}{\sigma_{w}}}^{\infty}{\mathcal{H}}_{n}(z) \phi(z)\,dz =\frac{\mathrm{e}^{-rT}}{\sqrt{n!}} I_{n}\bigg(\frac{k- \mu_{w}}{\sigma_{w}};0\bigg). \end{aligned}$$

Formula (3.19) follows directly from (3.18). □

Proof

of Lemma  3.9 We use similar notation as in the proof of Theorem 4.1. In particular, with \(C_{T}\) as in (3.1) and \(M_{T}\) as in (3.21), we denote by

$$ G_{T}(x)=(2\pi C_{T})^{-\frac{1}{2}}\exp\left( -\frac{(x-M_{T})^{2}}{2C _{T}}\right) $$
(B.8)

the conditional density of \(X_{T}\) given \(\{V_{t},t\in[0,T]\}\), so that \(g_{T}(x)={\mathbb{E}}[G_{T}(x)]\) is the unconditional density of \(X_{T}\). Lemma 3.9 now follows from observing that we have \(G_{T}(x)=\phi(x,M_{T},C_{T})\) and \(w(x) = \phi(x,\mu_{w},\sigma _{w}^{2})\). □

Proof

of Lemma  3.10 We first recall that by Cramér’s inequality (see, for instance, Erdélyi et al. [25, Sect. 10.18]), there exists a constant \(K>0\) such that for all \(n\ge0\),

$$ \mathrm{e}^{-(x-\mu_{w})^{2}/4\sigma_{w}^{2}}|H_{n}(x)|=(n!)^{-1/2} \mathrm{e}^{-(x-\mu_{w})^{2}/4\sigma_{w}^{2}}\left| {\mathcal{H}} _{n}\left( \frac{x-\mu_{w}}{\sigma_{w}}\right) \right| \leq K. $$
(B.9)

Additionally, as in the proof Theorem 4.1, since \(1/4\sigma^{2}_{w}<1/(2v_{\max}T)\),

$$ {\mathbb{E}}\left[ \int_{{\mathbb{R}}}\mathrm{e}^{(x-\mu_{w})^{2}/4 \sigma_{w}^{2}}G_{T}(x)\,dx\right] < \infty, $$

where \(G_{T}(x)\) is given in (B.8). This implies

$$\begin{aligned} {\mathbb{E}}\bigg[\int_{\mathbb{R}}|H_{n}(x)|G_{T}(x)\,dx\bigg] &= {\mathbb{E}}\bigg[\int_{\mathbb{R}}|H_{n}(x)|\mathrm{e}^{-(x-\mu _{w})^{2}/4 \sigma_{w}^{2}}\mathrm{e}^{(x-\mu_{w})^{2}/4\sigma _{w}^{2}}G_{T}(x)\,dx \bigg] \\ &\leq K {\mathbb{E}}\bigg[ \int_{\mathbb{R}}\mathrm{e}^{(x-\mu_{w})^{2}/4 \sigma_{w}^{2}}G_{T}(x)\, dx\bigg]< \infty. \end{aligned}$$

We can therefore use Fubini’s theorem to deduce that

$$ \ell_{n}=\int_{{\mathbb{R}}}H_{n}(x)g_{T}(x)\,dx ={\mathbb{E}}\left[ \int_{{\mathbb{R}}}H_{n}(x)G_{T}(x)\,dx\right] ={\mathbb{E}}[Y_{n}]. $$
(B.10)

We now analyze the term inside the expectation in (B.10). A change of variables shows that

$$ Y_{n}=\int_{\mathbb{R}}H_{n}(x)G_{T}(x)\,dx=(2\pi n!)^{-1/2} \int_{{\mathbb{R}}}{\mathcal{H}}_{n}(\alpha y+\beta)\mathrm{e}^{-y ^{2}/2}\,dy, $$

where we define \(\alpha=\frac{\sqrt{C_{T}}}{\sigma_{w}}\) and \(\beta=\frac{M_{T}-\mu_{w}}{\sigma_{w}}\). We recall that

$$ 0< (1-\rho^{2})v_{\min}T\leq C_{T}\leq v_{\max}T< \sigma_{w}. $$
(B.11)

The inequalities in (B.11) together with the fact that \((V_{t})\) is a bounded process yield for \(\alpha\), \(\beta\) the uniform bounds

$$ 1-q=\frac{(1-\rho^{2})v_{\min}T}{\sigma_{w}^{2}}\leq\alpha^{2} \leq v_{\max}T/\sigma_{w}^{2}< 1, \qquad |\beta|\leq R, $$
(B.12)

with constants \(0< q<1\) and \(R>0\). Define

$$ x_{n}=(2\pi)^{-1/2}\int_{{\mathbb{R}}}{\mathcal{H}}_{n}(\alpha y+ \beta)\mathrm{e}^{-y^{2}/2}\,dy $$

so that

$$ Y_{n}=\int_{\mathbb{R}}H_{n}(x)G_{T}(x)\,dx=(n!)^{-1/2}x_{n}. $$

An integration by parts argument using (B.7) and the identity

$$ {\mathcal{H}}'_{n}(x)=n{\mathcal{H}}_{n-1}(x) $$

shows the recursion formula

$$ x_{n}=\beta x_{n-1}-(n-1)(1-\alpha^{2})x_{n-2}, $$

with \(x_{0}=1\) and \(x_{1}=\beta\). This recursion formula is closely related to the recursion formula of the Hermite polynomials which helps us deduce the explicit expression

$$ x_{n}=n!\sum_{m=0}^{\lfloor n/2\rfloor} \frac{(\alpha^{2}-1)^{m}}{m!(n-2m)!}\frac{\beta^{n-2m}}{2^{m}}. $$
(B.13)

Recall that

$$ {\mathcal{H}}_{n}(x)=n!\sum_{m=0}^{\lfloor n/2\rfloor} \frac{(-1)^{m}}{m!(n-2m)!}\frac{x^{n-2m}}{2^{m}}. $$
(B.14)

By (B.13) and (B.14), we have

$$\begin{aligned} x_{n} & =n!(1-\alpha^{2})^{\frac{n}{2}}\sum_{m=0}^{\lfloor n/2\rfloor }\frac{(-1)^{m}}{m!(n-2m)!}\frac{((1-\alpha^{2})^{-\frac {1}{2}}\beta )^{n-2m}}{2^{m}} \\ & = (1-\alpha^{2})^{\frac{n}{2}}{\mathcal{H}}_{n}\big((1-\alpha^{2})^{- \frac{1}{2}}\beta\big) \end{aligned}$$

and

$$ \ell_{n}={\mathbb{E}}\big[(1-\alpha^{2})^{\frac{n}{2}}n!^{- \frac{1}{2}}{\mathcal{H}}_{n}\big((1-\alpha^{2})^{-\frac {1}{2}}\beta \big)\big]. $$

The Cauchy–Schwarz inequality and (B.9) yield

$$\begin{aligned} \ell_{n}^{2} & \leq{\mathbb{E}}\Big[\Big(n!^{-\frac{1}{2}}{\mathcal{H}} _{n}\big((1-\alpha^{2})^{-\frac{1}{2}}\beta\big)\Big)^{2}\Big]{\mathbb{E}}[(1- \alpha^{2})^{n}] \\ & \leq K^{2}{\mathbb{E}}\left[ \exp\Big(\beta^{2}/\big(2(1-\alpha ^{2})\big)\Big)\right] {\mathbb{E}}[(1-\alpha^{2})^{n}]. \end{aligned}$$
(B.15)

Inequalities (B.12) and (B.15) imply the existence of constants \(C>0\) and \(0< q<1\) such that \(\ell_{n}^{2} \leq Cq^{n}\). □

Proof of Theorem 4.1

To shorten the notation, we write \(\Delta Z_{t_{i}}=Z_{t_{i}}-Z_{t_{i-1}}\) for any process \((Z_{t})\). Due to (2.1), the log-price is \(X_{t}= M_{t} + \int _{0}^{t} \sqrt{V_{s} -\rho^{2} Q(V_{s})}\,dW_{2s}\), where \(M_{t}\) is defined in (3.21). In particular, the log-returns \(Y_{t_{i}}=\Delta X_{t_{i}}\) have the form

$$ Y_{t_{i}}=\Delta M_{t_{i}}+ \int_{t_{i-1}}^{t_{i}} \sqrt{V_{s} -\rho ^{2} Q(V_{s})}\,dW_{2s}. $$

In view of property (2.2), we infer that \(\Delta C_{t_{i}}>0\) for \(i=1,\ldots,d\). Motivated by Broadie and Kaya [14], we notice that conditionally on \(\{ V_{t},\, t\in[0,T]\}\), the random variable \((Y_{t_{1}},\ldots,Y_{t_{d}})\) is Gaussian with mean vector \((\Delta M_{t_{1}},\ldots,\Delta M_{t_{d}})\) and covariance matrix \(\mathrm{diag}(\Delta C_{t_{1}},\ldots,\Delta C_{t_{d}})\). Its density \(G_{t_{1},\ldots,t_{d}}(y)\) has the form

$$ G_{t_{1},\ldots,t_{d}}(y) =(2\pi)^{-d/2}\prod_{i=1}^{d} (\Delta C _{t_{i}})^{-1/2}\exp\bigg(-\sum_{i=1}^{d} \frac{ (y_{i}-\Delta M_{t _{i}})^{2}}{2\Delta C_{t_{i}}}\bigg). $$

Fubini’s theorem implies that \(g_{t_{1},\ldots,t_{d}}(y) = {\mathbb{E}}[ G_{t_{1},\ldots,t_{d}}(y)]\) is measurable and satisfies, for any bounded measurable function \(f(y)\),

$$ {\mathbb{E}}[ f(Y_{t_{1}},\ldots,Y_{t_{d}})] ={\mathbb{E}}\bigg[ \int_{{\mathbb{R}}^{d}} f(y)G_{t_{1},\ldots,t_{d}}(y) \,dy\bigg] = \int_{{\mathbb{R}}^{d}} f(y)g_{t_{1},\ldots,t_{d}}(y) \,dy. $$

Hence the distribution of \((Y_{t_{1}},\ldots,Y_{t_{d}})\) admits the density \(g_{t_{1},\ldots,t_{d}}(y)\) on \({\mathbb{R}}^{d}\). Dominated convergence implies that \(g_{t_{1},\ldots,t_{d}}(y)\) is uniformly bounded and \(k\) times continuously differentiable on \({\mathbb{R}} ^{d}\) if (4.1) holds. The arguments so far do not depend on \(\epsilon_{i}\) and also apply to the Heston model, which proves Remark 3.3.

For the rest of the proof, we assume without loss of generality that \(\epsilon_{i}>0\) for \(i=1,\ldots,d\). Observe that the mean vector and covariance matrix of \(G_{t_{1},\ldots,t_{d}}(y)\) admit the uniform bounds

$$ |\Delta M_{t_{i}}| \le K,\quad{|\Delta C_{t_{i}}|}\le v_{\max}(t _{i}-t_{i-1}), $$

for some finite constant \(K\). Define \(\Delta_{i}=1-2\epsilon_{i} {\Delta C_{t_{i}}}\) and \(\delta_{i}=1-2\epsilon_{i} v_{\max}(t_{i}-t _{i-1})\). Then \(\delta_{i}\in(0,1)\) and \(\Delta_{i}\geq\delta_{i}\). Completing the square implies that

$$\begin{aligned} &\mathrm{e}^{\sum_{i=1}^{d} \epsilon_{i} y_{i}^{2} } G_{t_{1},\ldots ,t_{d}}(y) \\ &=\prod_{i=1}^{d} (2\pi\Delta C_{t_{i}})^{-\frac{1}{2}}\exp\bigg( \epsilon_{i} y_{i}^{2}-\frac{ (y_{i}-\Delta M_{t_{i}})^{2}}{2{\Delta C_{t_{i}}}}\bigg) \\ & =\prod_{i=1}^{d} (2\pi\Delta C_{t_{i}})^{-\frac{1}{2}} \exp \bigg( - \frac{\Delta_{i}}{2{\Delta C_{t_{i}}}}\Big( y_{i}-\frac{ \Delta M_{t_{i}}}{\Delta_{i}}\Big)^{2} + \frac{\Delta M_{t_{i}}^{2}}{2 \Delta C_{t_{i}}}\Big(\frac{1}{\Delta_{i}}-1\Big)\bigg) \\ & =\prod_{i=1}^{d} (2\pi\Delta C_{t_{i}})^{-\frac{1}{2}} \exp \bigg( - \frac{\Delta_{i}}{2\Delta C_{t_{i}}}\Big( y_{i}-\frac {\Delta M_{t_{i}}}{\Delta_{i}}\Big)^{2} + \frac{\epsilon_{i} \Delta M_{t_{i}} ^{2}}{\Delta_{i}}\bigg). \end{aligned}$$
(B.16)

Integration of (B.16) then gives

$$ \int_{{\mathbb{R}}^{d}} \mathrm{e}^{\sum_{i=1}^{d} \epsilon_{i} y_{i} ^{2} } G_{t_{1},\ldots,t_{d}}(y) \,dy = \prod_{i=1}^{d} \frac {1}{\sqrt{ \Delta_{i}}} \exp\bigg(\frac{\epsilon_{i} \Delta M_{t_{i}}^{2}}{ \Delta_{i}}\bigg)\leq\prod_{i=1}^{d} \frac{1}{\sqrt{\delta_{i} }} \exp\bigg(\frac{\epsilon_{i} K^{2}}{\delta_{i} }\bigg) . $$

Hence (3.2) follows by Fubini’s theorem after taking expectation on both sides. We also derive from (B.16) that

$$\begin{aligned} e^{\sum_{i=1}^{d} \epsilon_{i} y_{i}^{2} } g_{t_{1},\ldots,t_{d}}(y) &= {\mathbb{E}}\big[\mathrm{e}^{\sum_{i=1}^{d} \epsilon_{i} y_{i}^{2} } G _{t_{1},\ldots,t_{d}}(y)\big] \\ &\leq{\mathbb{E}}\bigg[ \prod_{i=1}^{d} (2\pi\Delta C_{t_{i}})^{- \frac{1}{2}}\bigg]\prod_{i=1}^{d} \exp\bigg(\frac{\epsilon_{i} K^{2}}{ \delta_{i} }\bigg). \end{aligned}$$

Hence \(\mathrm{e}^{\sum_{i=1}^{d} \epsilon_{i} y_{i}^{2} } g_{t_{1}, \ldots,t_{d}}(y)\) is uniformly bounded and continuous on \({\mathbb{R}} ^{d}\) if (4.1) holds. In fact, for this to hold, it is enough suppose that (4.1) holds with \(k=0\). Moreover, (3.4) implies that \(\Delta C_{t_{i}}\ge(t_{i}-t_{i-1})(1- \rho^{2}) v_{\min}>0\) and (4.1) follows. □

Proof of Theorem 4.4

We assume the Brownian motions \((B_{t})\) and \((W_{1t},W_{2t})\) in (4.5) and (2.1) are independent. We denote by \(\pi_{f,t}\) the time-\(t\) price of the exotic option in the Jacobi model.

For any \(t_{i-1}\le t< t_{i}\) and given a realization \(X_{t_{1}}, \dots,X_{t_{i-1}}\), the time-\(t\) Black–Scholes price of the option is a function \(\pi^{\sigma_{\mathrm{BS}}}_{f}(t,S_{t})\) of \(t\) and the spot price \(S_{t}\), namely

$$\begin{aligned} \mathrm{e}^{-rt} \pi^{\sigma_{\mathrm{BS}}}_{f}(t,s) &={\mathbb{E}}[ f(X _{t_{1}},\dots,X_{t_{i-1}},\log S^{{\mathrm{BS}}}_{t_{i}},\dots,\log S ^{{\mathrm{BS}}}_{t_{d}}) | {\mathcal{F}}_{t},\, S^{{\mathrm{BS}}}_{t}=s] \\ &= {\mathbb{E}}\big[ f\big(X_{t_{1}},\dots,X_{t_{i-1}},\log( s R ^{\mathrm{BS}}_{t,t_{i}}),\dots,\log( s R^{\mathrm{BS}}_{t,t_{d}})\big) \big| {\mathcal{F}}_{t}\big], \end{aligned}$$

where we write

$$ R^{\mathrm{BS}}_{t,t_{i}}=\mathrm{e}^{(r-\delta-\frac{1}{2}\sigma_{\mathrm{BS}} ^{2})(t_{i}-t)+\sigma_{\mathrm{BS}}(B_{t_{i}}-B_{t})}. $$

By assumption, we infer that \(\pi^{\sigma_{\mathrm{BS}}}_{f}(t,s)\) is convex in \(s>0\). Moreover, \(\pi_{f}^{\sigma_{\mathrm{BS}}}(t,s)\) satisfies the PDE

$$ r \pi_{f}^{{\sigma_{\mathrm{BS}}}}(t,s) = \frac{\partial\pi _{f}^{{\sigma_{\mathrm{BS}}}}(t,s)}{\partial t} + (r-\delta) s \frac{\partial\pi_{f} ^{{\sigma_{\mathrm{BS}}}}(t,s)}{\partial s} + \frac{1}{2} \sigma_{\mathrm{BS}} ^{2} s^{2}\frac{\partial^{2} \pi_{f}^{{\sigma_{\mathrm{BS}}}}(t,s)}{\partial s^{2}} $$
(B.17)

and has terminal value satisfying \(\pi_{f}^{{\sigma_{\mathrm{BS}}}}(T,S _{T})=\pi_{f,T}\). Write

$$\begin{aligned} & \pi_{f,t}^{\sigma_{\mathrm{BS}}} = \pi^{\sigma_{\mathrm{BS}}}_{f}(t,S_{t}), \qquad \Theta^{\sigma_{\mathrm{BS}}}_{f,t}= - \frac{\partial\pi_{f}^{{\sigma_{ \mathrm{BS}}}}(t,S_{t})}{\partial t}, \\ & \Delta_{f,t}^{\sigma_{\mathrm{BS}}} = \frac{\partial\pi_{f}^{{\sigma _{\mathrm{BS}}}}(t,S _{t})}{\partial s}, \qquad \Gamma^{\sigma_{\mathrm{BS}}}_{f,t}= \frac{\partial^{2} \pi _{f}^{{\sigma_{\mathrm{BS}}}}(t,S _{t})}{\partial s^{2}} \end{aligned}$$

and \(dN_{t} =\rho\, \sqrt{Q(V_{t})}\,dW_{1t} + \sqrt{V_{t} -\rho ^{2} \,Q(V_{t})}\,dW_{2t} \) for the martingale driving the asset return in (2.1) such that, using (B.17),

$$\begin{aligned} d(e^{-rt}\pi_{f,t}^{{\sigma_{\mathrm{BS}}}}) &= \mathrm{e}^{-rt} \bigg( -r \pi_{f,t}^{{\sigma_{\mathrm{BS}}}} -\Theta^{\sigma_{\mathrm{BS}}}_{f,t} +(r- \delta) S_{t} \Delta^{\sigma_{\mathrm{BS}}}_{f,t} +\frac{1}{2} V_{t} S _{t}^{2} \Gamma^{\sigma_{\mathrm{BS}}}_{f,t}\bigg)dt \\ & \phantom{=:} + \mathrm{e}^{-rt} \Delta_{f,t}^{{\sigma_{\mathrm{BS}}}} S_{t} \,dN_{t} \\ &= \frac{1}{2} \mathrm{e}^{-rt} (V_{t}-\sigma_{\mathrm{BS}}^{2}) S_{t} ^{2} \Gamma^{\sigma_{\mathrm{BS}}}_{f,t}\,dt + \mathrm{e}^{-rt}\Delta_{f,t} ^{{\sigma_{\mathrm{BS}}}} S_{t} \,dN_{t}. \end{aligned}$$

Consider the self-financing portfolio with zero initial value, long one unit of the exotic option and short \(\Delta_{f,t}^{{\sigma_{\mathrm{BS}}}} \) units of the underlying asset. Let \(\Pi_{t}\) denote the time-\(t\) value of this portfolio. Its discounted price dynamics then satisfies

$$\begin{aligned} d(e^{-rt}\Pi_{t}) &= d(e^{-rt}\pi_{f,t}) - \Delta_{f,t}^{{\sigma_{ \mathrm {BS}}}} \left( d(e^{-rt}S_{t}) + \mathrm{e}^{-rt}S_{t}\delta\,dt\right) \\ &= d(e^{-rt}\pi_{f,t}) - \Delta_{f,t}^{{\sigma_{\mathrm{BS}}}}e^{-rt} S _{t}\,dN_{t} \\ &= d(e^{-rt}\pi_{f,t}) - d(e^{-rt}\pi_{f,t}^{{\sigma_{\mathrm{BS}}}}) + \frac{1}{2}e^{-rt}(V_{t}-\sigma_{\mathrm{BS}}^{2})S_{t}^{2}\Gamma_{f,t} ^{{\sigma_{\mathrm{BS}}}}\,dt. \end{aligned}$$

Integrating in \(t\) gives

$$ \mathrm{e}^{-rT} \Pi_{T} = -\pi_{f,0} + \pi_{f,0}^{{\sigma_{\mathrm{BS}}}} + \frac{1}{2}\int_{0}^{T}e^{-rt}(V_{t}-\sigma_{\mathrm{BS}}^{2})S_{t}^{2} \Gamma_{f,t}^{{\sigma_{\mathrm{BS}}}}\,dt $$
(B.18)

as \(\pi_{f,T} - \pi_{f,T}^{{\sigma_{\mathrm{BS}}}}=0\).

We now claim that the time-0 option price \(\pi_{f,0}=\pi_{f}\) lies between the Black–Scholes option prices for \(\sigma_{\mathrm{BS}}=\sqrt{v _{\min}}\) and \(\sigma_{\mathrm{BS}}=\sqrt{v_{\max}}\), i.e.,

$$ \pi^{\sqrt{v_{\min}}}_{f,0} \le\pi_{f} \le\pi^{ \sqrt{v_{\max}}}_{f,0}. $$
(B.19)

Indeed, let \(\sigma_{\mathrm{BS}}=\sqrt{v_{\min}}\). Because \(\Gamma_{f,t} ^{{\mathrm{BS}}}\ge0\) by assumption, it follows from (B.18) that \(\mathrm{e}^{-rT} \Pi_{T}\ge- \pi_{f,0}+\pi_{f,0}^{\sqrt{v _{\min}}}\). Absence of arbitrage implies that \(\Pi_{T}\) must not be bounded away from zero, hence \(- \pi_{f,0}+\pi_{f,0}^{\sqrt{v_{ \min}}}\le0\). This proves the left inequality in (B.19). The right inequality follows similarly, whence the claim (B.19) is proved.

A similar argument shows that the Black–Scholes price \(\pi^{\sigma_{\mathrm{BS}}}_{f,0}\) is nondecreasing in \(\sigma_{{\mathrm{BS}}}\), whence \(\sqrt{v_{\min}}\le\sigma_{{\mathrm{IV}}}\le\sqrt{v_{\max}}\), and the theorem is proved. □

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Ackerer, D., Filipović, D. & Pulido, S. The Jacobi stochastic volatility model. Finance Stoch 22, 667–700 (2018). https://doi.org/10.1007/s00780-018-0364-8

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