Pricing Swaps on Discrete Realized Higher Moments Under the Lévy Process

Wenli Zhu; Xinfeng Ruan

doi:10.1007/s10614-017-9753-x

Computational Economics

February 2019, Volume 53, Issue 2, pp 507–532 | Cite as

Pricing Swaps on Discrete Realized Higher Moments Under the Lévy Process

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Wenli Zhu
Xinfeng Ruan

Article

First Online: 21 September 2017

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Abstract

This paper designs and prices the swaps on discrete realized higher moments under the Lévy process in order to hedge the higher-moment risks, e.g., skewness and kurtosis risks. A comparison with Monte-Carlo simulations provides a verification of the correctness of our pricing formula. This paper is a further extension of Zhu and Lian’s (Math Finance 21:233–256, 2011; Appl Math Comput 219:1654–1669, 2012), which are under the Heston model and only price the variance swaps.

Keywords

Lévy process Stochastic volatility Skewness swaps Kurtosis swaps

We are grateful to the Editor, two anonymous referees, and seminar participants at 2014 International Symposium on Differential Equations and Stochastic Analysis in Mathematical Finance, Sanya, China for helpful comments and suggestions. Xinfeng Ruan acknowledges the financial support from the University of Otago Doctoral Scholarship. Wenli Zhu has been supported by the Fundamental Research Funds for the Central Universities (JBK170942). We declare that we have no relevant or material financial interests that relate to the research described in this paper. All remaining errors are ours. Two earlier versions of this paper circulated under the title “Compound option approach for moment swaps pricing under a general Lévy model with stochastic volatility” and the title “A closed-form solution for moment swaps pricing under a general Lévy model with stochastic volatility”.

JEL Classification

G12 G13

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Appendix A

A.1 Proof of Proposition 1

We denote $X(\tau ):=\ln S(\tau )$ with $X(T):=X_T=\ln ({{S}_{T}})$ and $u(\tau ,X,V):=U(T-\tau ,{{e}^{X}},V)=U(t,S,V)$, then the PIDE (25) can be rewritten as

$$\begin{aligned} \left\{ {\begin{array}{l} \frac{\partial u}{\partial \tau }=\frac{1}{2}V\frac{{{\partial }^{2}}u}{\partial {{X}^{2}}}+\rho \varepsilon V\frac{{{\partial }^{2}}u}{\partial X\partial V}+\frac{1}{2}{{\varepsilon }^{2}}V\frac{{{\partial }^{2}}u}{\partial {{V}^{2}}}+[ \kappa (\theta -V)]\frac{\partial u}{\partial V}\\ \qquad \quad +\,[r-\frac{1}{2}V-\int _{R}{(e^x-1) \nu (dx)}]\frac{\partial u }{\partial X} -ru+\int _{R}{[u(t,X+x,V)-u] \nu (dx)}, \\ u(0,X,V)=f({{e}^{{{X}_{T}}}}). \\ \end{array}} \right. \end{aligned}$$

(56)

Based on the generalized Fourier transform in Poularikas (2009), we can perform the transformation as

$$\begin{aligned} \mathcal {F}[{{e}^{jh t}}]=2\pi {{\delta }_{h }}(\omega ), \end{aligned}$$

(57)

where $j=\sqrt{-1}$ and ${{\delta }_{h }}(\omega )$ is the generalized delta function satisfying

$$\begin{aligned} \int _{-\infty }^{+\infty }{{{\delta }_{h}}(t)\Phi (t)dt}=\Phi (h ), \end{aligned}$$

(58)

with h being any complex number. Applying the transform to the PIDE (56) with respect to the variable X, we obtain the following problem for $\tilde{u}(\tau ,\omega ,V):=\mathcal {F}[u(\tau ,X,V)]$,

$$\begin{aligned} \left\{ {\begin{array}{l} \frac{\partial \tilde{u}}{\partial \tau }=\frac{1}{2}{{\varepsilon }^{2}}V\frac{{{\partial }^{2}}\tilde{u}}{\partial {{V}^{2}}}+[ \kappa \theta +(j\rho \varepsilon \omega - \kappa )V]\frac{\partial \tilde{u}}{\partial V} \\ \quad +\,[rj\omega -r-\frac{1}{2}(j\omega +{{\omega }^{2}})V+\int _{R}{[{e^{j\omega x }}-1] \nu (dx)}-j\omega \int _{R}{(e^x-1) \nu (dx)}]\tilde{u}, \\ \tilde{u}(0,\omega ,V)=\mathcal {F}[u(0,X,V)]. \\ \end{array}} \right. \end{aligned}$$

(59)

We guess a solution of (59) as following form,

$$\begin{aligned} \tilde{u}(\tau ,\omega ,V)={{\text {e}}^{A(\omega ,\tau )+B(\omega ,\tau )V}}\tilde{u}(0,\omega ,V). \end{aligned}$$

(60)

Substituting (60) into (59), then the PDE (59) can be reduced to two ordinary differential equations,

$$\begin{aligned} \left\{ {\begin{array}{l} \frac{\partial A(\omega ,\tau )}{\partial \tau }= \kappa \theta B(\omega ,\tau )+[r(j\omega -1)+\int _{R}{({e^{j\omega x}}-1-j\omega (e^x-1) ) \nu (dx)}], \\ \frac{\partial B(\omega ,\tau )}{\partial \tau }=\frac{1}{2}{{\varepsilon }^{2}}{{B}^{2}}(\omega ,\tau )+(j\rho \varepsilon \omega - \kappa )B-\frac{1}{2}(j\omega +{{\omega }^{2}}), \\ \end{array}} \right. \end{aligned}$$

(61)

with the initial conditions $A(\omega ,0)=0;B(\omega ,0)=0$.

The solutions of these equations can be easily found as

$$\begin{aligned} \left\{ {\begin{array}{l} A(\omega ,\tau )=[r(j\omega -1)+J]\tau -\frac{ \kappa \theta }{{{\varepsilon }^{2}}}[{{\xi }_+}\tau +2\ln \frac{{{\xi }_{-}}+{{\xi }_+}{{e}^{-\varsigma \tau }}}{2\varsigma }], \\ B(\omega ,\tau )=-(j\omega +{{w}^{2}})\frac{1-{{e}^{-\varsigma \tau }}}{{{\xi }_{-}}+{{\xi }_+}{{e}^{-\varsigma \tau }}}, \\ \end{array}} \right. \end{aligned}$$

(62)

where

$$\begin{aligned} {{\xi }_{\pm }}= & {} \varsigma \mp ( \kappa -\rho \varepsilon \omega j), \quad \varsigma =\sqrt{{{( \kappa -\rho \varepsilon \omega j)}^{2}}+{{\varepsilon }^{2}}({{\omega }^{2}}+\omega j)}, \end{aligned}$$

(63)

$$\begin{aligned} J= & {} \int _R{({e^{j\omega x}}-1-j\omega (e^x-1))v^Q(dx)}. \end{aligned}$$

(64)

Therefore, the solution of (56) can be obtained after the inverse Fourier transform in the form as

$$\begin{aligned} u(\tau ,X,V)={{\mathcal {F}}^{-1}}[\tilde{u}(\tau ,\omega ,V)]={{\mathcal {F}}^{-1}}[{{\text {e}}^{A(\omega ,\tau )+B(\omega ,\tau )V}}\mathcal {F}[f({{e}^{{{X}_{T}}}})]]. \end{aligned}$$

(65)

A.2 Proof of Proposition 2

Given $U(T,S,V)=f({{V}_{T}})$, the PIDE (25) can be degenerated into

$$\begin{aligned} \left\{ {\begin{array}{l} 0=-rU+\frac{\partial U}{\partial t}+\frac{\partial U}{\partial S}Sr+\frac{\partial U}{\partial V}[ \kappa ( \theta -V)]+\frac{1}{2}\frac{{{\partial }^{2}}U}{\partial {{S}^{2}}}{{S}^{2}}V+\frac{{{\partial }^{2}}U}{\partial S\partial V}\rho \varepsilon SV \\ \qquad +\frac{1}{2}\frac{{{\partial }^{2}}U}{\partial {{V}^{2}}}{{\varepsilon }^{2}}V+\int _{R}{[U({{t}^{-}},Se^x,V)-U-\frac{\partial U}{\partial S}S(e^x-1)] \nu (dx)}, \\ U(T,S,V)=f({{V}_{T}}). \\ \end{array}} \right. \end{aligned}$$

(66)

However, according to Zhu and Lian (2011, 2012), the proof is trivial, because it is actually implied by the Feynman-Kac formula, which states that the solution of (66) can be derived from the conditional expectation of the payoff function under the risk-neutral probability measure. Therefore, the solution can be expressed in the form of

$$\begin{aligned} U(t,S,V)=E_{\text {t}}^{Q}[{{e}^{-r(T-t)}}f({{V}_{T}})]. \end{aligned}$$

(67)

We find that the expectation is actually not related to the process S, since the payoff function $f({{V}_{T}})$ is independent of S. Hence, if we know the distribution of the stochastic volatility process V(t) given by (7), then the solution of (66) can be easily solved. Fortunately, the needed transition probability density function $p(\left. {{V}_{T}} \right| {{V}_{t}})$ in (30) has been given in Cox et al. (1985). Therefore,

$$\begin{aligned} U(t,S,V)=\int _{0}^{+\infty }{{{e}^{-r(T-t)}}f({{V}_{T}})p(\left. {{V}_{T}} \right| {{V}_{t}})d{{V}_{T}}}, \end{aligned}$$

(68)

and the proof naturally follows. In Appendix B.1, we provide an explicit solution of (68).

A.3 Proof of Proposition 3

From Proposition 1, the solution of (23) is

$$\begin{aligned} {{U}_{i}}(t,S(t),V(t),I(t),m)= & {} {{\mathcal {F}}^{-1}}\left[ {{\text {e}}^{A(\omega ,{{t}_{i}}-t)+B(\omega ,{{t}_{i}}-t)V}}\mathcal {F}\left[ {{\left( \ln \frac{S}{I} \right) }^{m}} \right] \right] ,\nonumber \\&\quad {{t}_{i-1}}\le t\le {{t}_{i}}, \end{aligned}$$

(69)

where

$$\begin{aligned} A(\omega ,{{t}_{i-1}}-t)= & {} [r(j\omega -1)+J]({{t}_{i-1}}-t)-\frac{ \kappa \theta }{{{\varepsilon }^{2}}}[{{\xi }_+}({{t}_{i-1}}-t)\nonumber \\&+\,2\ln \frac{{{\xi }_{-}}+{{\xi }_+}{{e}^{-\varsigma ({{t}_{i-1}}-t)}}}{2\varsigma }], \end{aligned}$$

(70)

$$\begin{aligned} B(\omega ,{{t}_{i-1}}-t)= & {} -(j\omega +{{w}^{2}})\frac{1-{{e}^{-\varsigma ({{t}_{i-1}}-t)}}}{{{\xi }_{-}}+{{\xi }_+}{{e}^{-\varsigma ({{t}_{i-1}}-t)}}},\end{aligned}$$

(71)

$$\begin{aligned} {{\xi }_{\pm }}= & {} \varsigma \mp ( \kappa -\rho \varepsilon \omega j),\quad \varsigma =\sqrt{{{( \kappa -\rho \varepsilon \omega j)}^{2}}+{{\varepsilon }^{2}}({{\omega }^{2}}+\omega j)},\qquad \end{aligned}$$

(72)

$$\begin{aligned} J= & {} \int _{R}{({e^{j\omega x}}-1-j\omega (e^x-1)) \nu (dx)}. \end{aligned}$$

(73)

Based on the generalized Fourier transform, we can perform the transformation as

$$\begin{aligned} \mathcal {F}[{{X}^{n}}]=2\pi {{j}^{n}}{{\delta }^{(n)}}(\omega ), \end{aligned}$$

(74)

where $j=\sqrt{-1}$; n is any integer; and ${{\delta }^{(n)}}(\omega )$ is the n-th order derivative of the generalized delta function satisfying

$$\begin{aligned} \int _{-\infty }^{+\infty }{{{\delta }^{(n)}}(\omega )\Phi (\omega )d}\omega ={{(-1)}^{(n)}}{{\Phi }^{(n)}}(0). \end{aligned}$$

(75)

Therefore,

$$\begin{aligned} \mathcal {F}\left[ {{\left( \ln \frac{S}{I} \right) }^{m}} \right] =\mathcal {F}\left[ {{\left( X-\ln I \right) }^{m}} \right] =2\pi \sum \limits _{\text {k}=0}^{m}{\frac{m!}{k!(m-k)!}{{j}^{k}}{{\delta }^{(k)}}(\omega ){{(-1)}^{m-k}}{{\ln }^{m-k}}I}. \end{aligned}$$

(76)

Then, the solution of (23) becomes

$$\begin{aligned}&{{U}_{i}}(t,S(t),V(t),I(t),m) \nonumber \\&\quad ={{\mathcal {F}}^{-1}}\left[ {{\text {e}}^{A(\omega ,{{t}_{i}}-t)+B(\omega ,{{t}_{i}}-t)V}}2\pi \sum \limits _{\text {k}=0}^{m}{\frac{m!}{k!(m-k)!}{{j}^{k}}{{\delta }^{(k)}}(\omega ){{(-1)}^{m-k}}{{\ln }^{m-k}}I} \right] \nonumber \\&\quad =\int _{-\infty }^{+\infty }{\left[ {{\text {e}}^{A(\omega ,{{t}_{i}}-t)+B(\omega ,{{t}_{i}}-t)V}}\sum \limits _{\text {k}=0}^{m}{\frac{m!}{k!(m-k)!}{{j}^{k}}{{\delta }^{(k)}}(\omega ){{(-1)}^{m-k}}{{\ln }^{m-k}}I} \right] {{e}^{X\omega j}}d\omega } \nonumber \\&\quad =\int _{-\infty }^{+\infty }{\left[ \sum \limits _{\text {k}=0}^{m}{\frac{m!}{k!(m-k)!}{{\text {e}}^{A(\omega ,{{t}_{i}}-t)+B(\omega ,{{t}_{i}}-t)V+X\omega j}}{{j}^{k}}{{\delta }^{(k)}}(\omega ){{(-1)}^{m-k}}{{\ln }^{m-k}}I} \right] d\omega } \nonumber \\&\quad =\sum \limits _{\text {k}=0}^{m}{\frac{m!}{k!(m-k)!}{{\Lambda }^{(k)}}(0,t){{j}^{k}}{{(-1)}^{m}}{{\ln }^{m-k}}I}, \end{aligned}$$

(77)

where ${{\Lambda }^{(k)}}(0,t)={{\left. \frac{{{\partial }^{k}}\Lambda (\omega ,t)}{\partial {{\omega }^{k}}} \right| }_{\omega =0}}$, $\Lambda (\omega ,t)={{\text {e}}^{A(\omega ,{{t}_{i}}-t)+B(\omega ,{{t}_{i}}-t)V+X\omega j}}$ , $X(t)=\ln S(t)$ and ${{t}_{i-1}}\le t\le {{t}_{i}}$.

By noting the fact that $\underset{t\downarrow {{t}_{i-1}}}{\mathop {\lim }}\,\ln S(t)=\ln I$ and due to the definition of I, we have

$$\begin{aligned} G(V({{t}_{i-1}}),m):=\,&\underset{t\downarrow {{t}_{i-1}}}{\mathop {\lim }}\,{{U}_{i}}(t,S(t),V(t),I(t),m) \nonumber \\ =\,&{{U}_{i}}({{t}_{i-1}},S({{t}_{i-1}}),V({{t}_{i-1}}),S({{t}_{i-1}}),m) \nonumber \\ =&\sum \limits _{\text {k}=0}^{m}{\frac{m!}{k!(m-k)!}{{\Lambda }^{(k)}}(0,{{t}_{i-1}}){{j}^{k}}{{(-1)}^{m}}{{X}^{m-k}}}. \end{aligned}$$

(78)

If we denote ${{\Gamma }^{(k)}}(0)={{\left. \frac{{{\partial }^{k}}\Gamma (\omega )}{\partial {{\omega }^{k}}} \right| }_{\omega =0}}$ and $\Gamma (\omega ):=\Lambda (\omega ,{{t}_{i-1}})={{\text {e}}^{A(\omega ,\Delta t)+B(\omega ,\Delta t)V+X\omega j}}$, then the initial value of ${{U}_{i}}(t,S(t),V(t),I(t),m)$, $G(V({{t}_{i-1}}),m)$ can be rewritten as

$$\begin{aligned} G(V({{t}_{i-1}}),m)=\sum \limits _{\text {k}=0}^{m}{\frac{m!}{k!(m-k)!}{{\Gamma }^{(k)}}(0){{j}^{k}}{{(-1)}^{m}}{{X}^{m-k}}}. \end{aligned}$$

(79)

If $i>1$, then the PIDE (24) is valid. Therefore, using Proposition 2, we plug (79) into (30) and then the solution of (24) is

$$\begin{aligned}&{{U}_{i}}(t,S(t),V(t),I(t),m)\nonumber \\&\quad =\int _{0}^{+\infty }{{{e}^{-r({{t}_{i-1}}-t)}}G(V({{t}_{i-1}}),m)p(\left. V({{t}_{i-1}}) \right| V(t))dV({{t}_{i-1}})},\quad 0\le t\le {{t}_{i-1}},\nonumber \\ \end{aligned}$$

(80)

where

$$\begin{aligned} p(\left. {{V}_{{{\text {t}}_{i-1}}}} \right| {{V}_{t}})= & {} h{{e}^{-(u-v)}}{{\left( \frac{v}{u} \right) }^{q/2}}{{I}_{q}}(2\sqrt{uv}), \end{aligned}$$

(81)

$$\begin{aligned} h= & {} \frac{2 \kappa }{{{\varepsilon }^{2}}(1-{{e}^{- \kappa ({{\text {t}}_{i-1}}-t)}})},\end{aligned}$$

(82)

$$\begin{aligned} u= & {} h{{V}_{t}}{{e}^{- \kappa ({{\text {t}}_{i-1}}-t)}},\quad v=h{{V}_{{{\text {t}}_{i-1}}}},\quad q=\frac{2 \kappa \theta }{{{\varepsilon }^{2}}}-1, \end{aligned}$$

(83)

and ${{I}_{q}}$ is the modified Bessel function of the first kind of order q.

Then the initial value of ${{U}_{i}}(t,S(t),V(t),I(t),m)$ becomes

$$\begin{aligned} {{U}_{i}}(0,S(0),V(0),I(0),m)\text {=}\int _{0}^{+\infty }{{{e}^{-r{{t}_{i-1}}}}G(V({{t}_{i-1}}),m)p(\left. V({{t}_{i-1}}) \right| V(0))dV({{t}_{i-1}})}. \end{aligned}$$

(84)

According to (22) and the fact that ${{t}_{i}}=i\Delta t$, we have

$$\begin{aligned} E_{0}^{Q}\left[ {{\left( \ln \frac{S({{t}_{i}})}{S({{t}_{i-1}})} \right) }^{m}} \right]&={{e}^{r{{t}_{i}}}}{{U}_{i}}(0,S(0),V(0),I(0),m) \nonumber \\&={{e}^{r{{t}_{i}}}}\int _{0}^{+\infty }{{{e}^{-r{{t}_{i-1}}}}G(V({{t}_{i-1}}),m)p(\left. V({{t}_{i-1}}) \right| V(0))dV({{t}_{i-1}})} \nonumber \\&={{e}^{r\Delta t}}\int _{0}^{+\infty }{G(V({{t}_{i-1}}),m)p(\left. V({{t}_{i-1}}) \right| V(0))dV({{t}_{i-1}})} . \end{aligned}$$

(85)

When $i=1$, there is no PIDE (24). In this case, the compound option is degenerated into one standard European option [PIDE (23)], then

$$\begin{aligned} E_{0}^{Q}\left[ {{\left( \ln \frac{S({{t}_{1}})}{S(0)} \right) }^{m}} \right] ={{e}^{r\Delta t}}\underset{t\downarrow 0}{\mathop {\lim }}\,{{U}_{i}}(t,S(t),V(t),I(t),m)={{e}^{r\Delta t}}G(V(0),m). \end{aligned}$$

(86)

Denoting ${{G}_{i}}(V(0),m):=E_{0}^{Q}\left[ {{\left( \ln \frac{S({{t}_{i}})}{S({{t}_{i-1}})} \right) }^{m}} \right] $ and ${{G}_{1}}(V(0),m):=E_{0}^{Q}\left[ {{\left( \ln \frac{S({{t}_{1}})}{S(0)} \right) }^{m}} \right] $, we finally obtain

$$\begin{aligned} E_{0}^{Q}\left[ \sum \limits _{i=1}^{N}{{{\left( \ln \frac{S({{t}_{i}})}{S({{t}_{i-1}})} \right) }^{m}}}\times \textit{NA} \right] =\textit{NA}\left[ {{G}_{1}}(V(0),m)+\sum \limits _{i=2}^{N}{{{G}_{i}}(V(0),m)} \right] . \end{aligned}$$

(87)

Appendix B

B.1 Numerical Expressions

For the CIR stochastic volatility process in (5), expectations of V(t), $V{{(t)}^{2}}$, $V{{(t)}^{3}}$ and $V{{(t)}^{4}}$ in risk-neutral probability measure are given by

$$\begin{aligned} E_{0}^{Q}[V(t)]=\,&{{e}^{-{{\kappa }^{Q}}t}}{{V}_{0}}+{{\theta }^{Q}}\left( 1-{{e}^{-{{\kappa }^{Q}}t}}\right) , \end{aligned}$$

(88)

$$\begin{aligned} E_{0}^{Q}[V{{(t)}^{2}}]=\,&{{e}^{-2{{\kappa }^{Q}}t}}V_{0}^{2}+\frac{2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}}}{{{\kappa }^{Q}}}({{V}_{0}}-{{\theta }^{Q}})\left( {{e}^{-{{\kappa }^{Q}}t}}-{{e}^{-2{{\kappa }^{Q}}t}}\right) \nonumber \\&+\frac{{{\theta }^{Q}}(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{2{{\kappa }^{Q}}}\left( 1-{{e}^{-2{{\kappa }^{Q}}t}}\right) , \end{aligned}$$

(89)

$$\begin{aligned} E_{0}^{Q}[V{{(t)}^{3}}]=\,&{{e}^{-3{{\kappa }^{Q}}t}}V_{0}^{3}+\frac{3{{\kappa }^{Q}}{{\theta }^{Q}}+3{{\varepsilon }^{2}}}{{{\kappa }^{Q}}}\left[ V_{0}^{2}{{e}^{-3{{\kappa }^{Q}}t}}\right. \nonumber \\&-\frac{({{V}_{0}}-\frac{1}{2}{{\theta }^{Q}})(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{{{\kappa }^{Q}}}\left( {{e}^{-2{{\kappa }^{Q}}t}}-{{e}^{-3{{\kappa }^{Q}}t}}\right) \nonumber \\&+\frac{({{V}_{0}}-{{\theta }^{Q}})(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{2{{\kappa }^{Q}}}\left( {{e}^{-{{\kappa }^{Q}}t}}-{{e}^{-2{{\kappa }^{Q}}t}}\right) \nonumber \\&\left. +\frac{{{\theta }^{Q}}(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{6{{\kappa }^{Q}}}\left( 1-{{e}^{-3{{\kappa }^{Q}}t}}\right) \right] , \end{aligned}$$

(90)

$$\begin{aligned} E_{0}^{Q}[V{{(t)}^{4}}]=\,&{{e}^{-4{{\kappa }^{Q}}t}}V_{0}^{4}+\frac{4{{\kappa }^{Q}}{{\theta }^{Q}}+6{{\varepsilon }^{2}}}{{{\kappa }^{Q}}}\left[ V_{0}^{3}+\frac{3{{\kappa }^{Q}}{{\theta }^{Q}}+3{{\varepsilon }^{2}}}{{{\kappa }^{Q}}}V_{0}^{2}\right. \nonumber \\&+\frac{3({{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})({{V}_{0}}-\frac{1}{2}{{\theta }^{Q}})(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{{{({{\kappa }^{Q}})}^{2}}} \nonumber \\&-\frac{3({{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})({{V}_{0}}-{{\theta }^{Q}})(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{2{{({{\kappa }^{Q}})}^{2}}}\nonumber \\&\left. -\frac{{{\theta }^{Q}}({{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{2{{({{\kappa }^{Q}})}^{2}}}\right] \left( {{e}^{-3{{\kappa }^{Q}}t}}-{{e}^{-4{{\kappa }^{Q}}t}}\right) \nonumber \\&-\frac{3(4{{\kappa }^{Q}}{{\theta }^{Q}}+6{{\varepsilon }^{2}})({{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})({{V}_{0}}-\frac{1}{2}{{\theta }^{Q}})(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{2{{({{\kappa }^{Q}})}^{3}}}\nonumber \\&\times \left( {{e}^{-2{{\kappa }^{Q}}t}}-{{e}^{-4{{\kappa }^{Q}}t}}\right) \nonumber \\&+\frac{(4{{\kappa }^{Q}}{{\theta }^{Q}}+6{{\varepsilon }^{2}})({{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})({{V}_{0}}-{{\theta }^{Q}})(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{2{{({{\kappa }^{Q}})}^{3}}}\nonumber \\&\times \left( {{e}^{-{{\kappa }^{Q}}t}}-{{e}^{-4{{\kappa }^{Q}}t}}\right) \nonumber \\&+\frac{{{\theta }^{Q}}(4{{\kappa }^{Q}}{{\theta }^{Q}}+6{{\varepsilon }^{2}})({{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})(2{{\kappa }^{Q}}{{\theta }^{Q}}+{{\varepsilon }^{2}})}{8{{({{\kappa }^{Q}})}^{3}}}\left( 1-{{e}^{-4{{\kappa }^{Q}}t}}\right) . \end{aligned}$$

(91)

Based on Eq. (35), calculating the integration ${{G}_{i}}(V(0),m)$ is equivalent to calculating the expectation ${{e}^{r\Delta t}}E_{0}^{Q}[G(V({{t}_{i-1}}),m)]$. Moreover, plugging (37383940414243) into (36), we find $G(V({{t}_{i-1}}),m)$ is only with respect to $V{{({{t}_{i-1}})}^{k}}$, $k=1,\ldots ,m$. Now, we denote ${{A}^{(k)}}={{\left. \frac{{{\partial }^{k}}A(\omega ,\Delta t)}{\partial {{\omega }^{k}}} \right| }_{\omega =0}}$, ${{B}^{(k)}}={{\left. \frac{{{\partial }^{k}}B(\omega ,\Delta t)}{\partial {{\omega }^{k}}} \right| }_{\omega =0}}$, $k=1,\ldots ,m$ to calculate ${{\Gamma }^{(k)}}(0)$, where $A(\omega ,\Delta t)$ and $B(\omega ,\Delta t)$ are given by Proposition 3; $M{{1}_{i}}:=E_{0}^{Q}[V({{t}_{i-1}})]$; $M{{2}_{i}}:=E_{0}^{Q}[V{{({{t}_{i-1}})}^{2}}]$; $M{{3}_{i}}:=E_{0}^{Q}[V{{({{t}_{i-1}})}^{3}}]$; and $M{{4}_{i}}:=E_{0}^{Q}[V{{({{t}_{i-1}})}^{4}}]$. Then we can easily get the following expressions for the numerical computation.

The fair delivery price of variance swaps is given by

$$\begin{aligned} \textit{NA}\left[ {{G}_{1}}(V(0),2)+\sum \limits _{i=2}^{N}{{{G}_{i}}(V(0),2)} \right] , \end{aligned}$$

(92)

where

$$\begin{aligned}&{{G}_{i}}(V(0),2)=-\left[ {{({{B}^{(1)}})}^{2}}M{{2}_{i}}+(2{{A}^{(1)}}{{B}^{(1)}}+{{B}^{(2)}})M{{1}_{i}}+{{({{A}^{(1)}})}^{2}}+{{A}^{(2)}}\right] ,\\&{{G}_{1}}(V(0),2)=-\left[ {{({{B}^{(1)}})}^{2}}V_{0}^{2}+(2{{A}^{(1)}}{{B}^{(1)}}+{{B}^{(2)}}){{V}_{0}}+{{({{A}^{(1)}})}^{2}}+{{A}^{(2)}}\right] . \end{aligned}$$

The fair delivery price of skewness swaps is given by

$$\begin{aligned} \textit{NA}\left[ {{G}_{1}}(V(0),3)+\sum \limits _{i=2}^{N}{{{G}_{i}}(V(0),3)} \right] , \end{aligned}$$

(93)

where

$$\begin{aligned} {{G}_{i}}(V(0),3)=&j\Big [{{({{B}^{(1)}})}^{3}}M{{3}_{\text {i}}}\\&+3{{B}^{(1)}}({{A}^{(1)}}{{B}^{(1)}}+{{B}^{(2)}})M{{2}_{\text {i}}} +(3{{A}^{(2)}}{{B}^{(1)}}+3{{({{A}^{(1)}})}^{2}}{{B}^{(1)}}\\&+3{{A}^{(1)}}{{B}^{(2)}}+{{B}^{(3)}})M{{1}_{\text {i}}}+{{({{A}^{(1)}})}^{3}} +3{{A}^{(1)}}{{A}^{(2)}}+{{A}^{(3)}}\Big ], \\ {{G}_{1}}(V(0),3)=&j\Big [{{({{B}^{(1)}})}^{3}}V_{0}^{3}+3{{B}^{(1)}}({{A}^{(1)}}{{B}^{(1)}}+{{B}^{(2)}})V_{0}^{2} \\&+(3{{A}^{(2)}}{{B}^{(1)}}+3{{({{A}^{(1)}})}^{2}}{{B}^{(1)}}+3{{A}^{(1)}}{{B}^{(2)}}+{{B}^{(3)}}){{V}_{0}}+{{({{A}^{(1)}})}^{3}}\\&+3{{A}^{(1)}}{{A}^{(2)}}+{{A}^{(3)}}\Big ]. \end{aligned}$$

The fair delivery price of kurtosis swaps is given by

$$\begin{aligned} \textit{NA}\left[ {{G}_{1}}(V(0),4)+\sum \limits _{i=2}^{N}{{{G}_{i}}(V(0),4)} \right] , \end{aligned}$$

(94)

where

$$\begin{aligned} {{G}_{i}}(V(0),4)=\,&{{({{B}^{(1)}})}^{4}}M{{4}_{i}}+(4{{A}^{(1)}}{{B}^{(1)}}+6{{B}^{(2)}}){{({{B}^{(1)}})}^{2}}M{{3}_{i}} \\&+(4{{B}^{(1)}}{{B}^{(3)}}+3{{({{B}^{(2)}})}^{2}}+12{{A}^{(1)}}{{B}^{(1)}}{{B}^{(2)}}+6{{A}^{(2)}}{{({{B}^{(1)}})}^{2}}\\&+6{{({{A}^{(1)}})}^{2}}{{({{B}^{(1)}})}^{2}})M{{2}_{i}} \\&+({{B}^{(4)}}+4{{A}^{(1)}}{{B}^{(3)}}+4{{({{A}^{(1)}})}^{3}}{{B}^{(1)}}+6{{A}^{(2)}}{{B}^{(2)}}+6{{({{A}^{(1)}})}^{2}}{{B}^{(2)}}\\&+12{{A}^{(1)}}{{A}^{(2)}}{{B}^{(1)}}+4{{A}^{(3)}}{{B}^{(1)}})M{{1}_{i}} \\&+{{A}^{(4)}}+4{{A}^{(1)}}{{A}^{(3)}}+3{{({{A}^{(2)}})}^{2}}+6{{({{A}^{(1)}})}^{2}}{{A}^{(2)}}+{{({{A}^{(1)}})}^{4}}, \\ {{G}_{1}}(V(0),4)=\,&{{({{B}^{(1)}})}^{4}}V_{0}^{4}+(4{{A}^{(1)}}{{B}^{(1)}}+6{{B}^{(2)}}){{({{B}^{(1)}})}^{2}}V_{0}^{3} \\&+(4{{B}^{(1)}}{{B}^{(3)}}+3{{({{B}^{(2)}})}^{2}}+12{{A}^{(1)}}{{B}^{(1)}}{{B}^{(2)}}+6{{A}^{(2)}}{{({{B}^{(1)}})}^{2}}\\&+6{{({{A}^{(1)}})}^{2}}{{({{B}^{(1)}})}^{2}})V_{0}^{2} \\&+({{B}^{(4)}}+4{{A}^{(1)}}{{B}^{(3)}}+4{{({{A}^{(1)}})}^{3}}{{B}^{(1)}}+6{{A}^{(2)}}{{B}^{(2)}}+6{{({{A}^{(1)}})}^{2}}{{B}^{(2)}}\\&+12{{A}^{(1)}}{{A}^{(2)}}{{B}^{(1)}}+4{{A}^{(3)}}{{B}^{(1)}}){{V}_{0}} \\&+{{A}^{(4)}}+4{{A}^{(1)}}{{A}^{(3)}}+3{{({{A}^{(2)}})}^{2}}+6{{({{A}^{(1)}})}^{2}}{{A}^{(2)}}+{{({{A}^{(1)}})}^{4}}. \end{aligned}$$

B.2 Numerical Codes

At first, we use Maple to calculate $A^{(k)}$, $B^{(k)}$, $M1_i$, $M2_i$, $M3_i$ and $M4_i$, and then we plug them into Eqs. (92)–(94). We use the NIG model as our example and we only provide the results for daily sampling ($N=252$) in Figs. 4, 5, 6, 7 and 8.

Fig. 4
Step 1: input formulas in Maple based on the NIG model

Fig. 5
Step 2: input parameters in Maple based on the NIG model (daily sampling)

Fig. 6
Step 3: calculate $A^{(k)}$ and $B^{(k)}$ in Maple based on the NIG model (daily sampling)

Fig. 7
Step 4: calculate $M1_i$, $M2_i$, $M3_i$ and $M4_i$ in Maple based on the NIG model (daily sampling)

Fig. 8
Step 5: calculate ${{G}_{i}}(V(0),2)$, ${{G}_{i}}(V(0),3)$ and ${{G}_{i}}(V(0),4)$ in Maple based on the NIG model (daily sampling)

Finally, we transfer these numerical expressions from Maple into Matlab, in order to calculate the final results. See Fig. 9.

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Wenli Zhu
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Xinfeng Ruan
- 1
- 2
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1.School of Economic MathematicsSouthwestern University of Finance and EconomicsChengduPeople’s Republic of China
2.Department of Accountancy and Finance, Otago Business SchoolUniversity of OtagoDunedinNew Zealand

Pricing Swaps on Discrete Realized Higher Moments Under the Lévy Process

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