Consistent pricing of VIX and equity derivatives with the 4/2 stochastic volatility plus jumps model

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• 作者：Wei Lin^a ; ^{weilin1991@zju.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; ^{mathslin@126.com" class="auth_mail" title="E-mail the corresponding author} ; Shenghong Li^a ; ^{shli@zju.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Xingguo Luo^b ; ^{xgluo@zju.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Shane Chern^a ; ¹ ; ^{shanechern@zju.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; ^{chenxiaohang92@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Stochastic volatility ; 4/2 Model ; VIX derivatives ; Transform
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：778-797
• 全文大小：615 K

文摘

In this paper, we develop a 4/2 stochastic volatility plus jumps model, namely, a new stochastic volatility model including the Heston model and 3/2 model as special cases. Our model is highly tractable by applying the Lie symmetries theory for PDEs, which means that the pricing procedure can be performed efficiently. In fact, we obtain a closed-form solution for the joint Fourier–Laplace transform so that equity and realized-variance derivatives can be priced. We also employ our model to consistently price equity and VIX derivatives. In this process, the quasi-closed-form solutions for future and option prices are derived. Furthermore, through adopting data on daily VIX future and option prices, we investigate our model along with the Heston model and 3/2 model and compare their different performance in practice. Our result illustrates that the 4/2 model with an instantaneous volatility of the form class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306357&_mathId=si1.gif&_user=111111111&_pii=S0022247X16306357&_rdoc=1&_issn=0022247X&md5=9ed9d2ad44136d93ddc75258e5e852ac">class="imgLazyJSB inlineImage" height="17" width="113" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16306357-si1.gif">class="mathContainer hidden">class="mathCode">$(a\sqrt{V_t}+b/\sqrt{V_t})$ for some constants class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16306357&_mathId=si2.gif&_user=111111111&_pii=S0022247X16306357&_rdoc=1&_issn=0022247X&md5=bbb7542aef93f84b2e06343abcc8451f" title="Click to view the MathML source">a,bclass="mathContainer hidden">class="mathCode">$a,b$ presents considerable advantages in pricing VIX derivatives.