Optimal investment in equity and VIX derivatives

Xiangzhen Yan; Yunfan Zhu; Zhenyu Cui; Shuguang Zhang

doi:10.52396/JUSTC-2022-0095

JUSTC > 2023 > 53(2): 6-1-6-14. > DOI: 10.52396/JUSTC-2022-0095 CSTR: 32290.14.JUSTC-2022-0095

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Open Access JUSTC Management Article 20 March 2023

Optimal investment in equity and VIX derivatives

1.
Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei 230026, China
2.
School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, USA

Cite this: JUSTC, 2023, 53(2): 6

https://doi.org/10.52396/JUSTC-2022-0095

CSTR: 32290.14.JUSTC-2022-0095

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Author Bio:
Xiangzhen Yan is currently a Ph.D. student at the School of Management, University of Science and Technology of China. Her research mainly focuses on portfolio optimization and financial risk management

Zhenyu Cui received his Ph.D. from University of Waterloo. He is currently an Associate Professor at the Stevens Institute of Technology. His research interests focus on financial engineering, Monte Carlo simulation, and financial systemic risk
Corresponding author:
Zhenyu Cui, E-mail: zcui6@stevens.edu
Received Date: June 30, 2022
Accepted Date: October 07, 2022
Available Online: March 20, 2023

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Abstract

Abstract

We solve in closed-form the optimal investment strategies in equity and VIX derivatives in a stochastic volatility model with jumps. Our framework includes both complete market and incomplete market cases, when diffusive risk, volatility risk and jump risk are present. VIX derivatives allow for direct exposure to volatility risk compared to equity derivatives. Based on the closed-form formulas, we explicitly determine the portfolio improvements brought by the inclusion of the VIX derivatives and establish that it is theoretically positive. This justifies the economic intuition and observed demand for VIX derivatives in a portfolio management setting. Numerical examples illustrate the results.

Graphical Abstract

Considering the presence of diffusive risk, volatility risk, and jump risk, the inclusion of the VIX derivatives improved the portfolio performance.

Abstract

We solve in closed-form the optimal investment strategies in equity and VIX derivatives in a stochastic volatility model with jumps. Our framework includes both complete market and incomplete market cases, when diffusive risk, volatility risk and jump risk are present. VIX derivatives allow for direct exposure to volatility risk compared to equity derivatives. Based on the closed-form formulas, we explicitly determine the portfolio improvements brought by the inclusion of the VIX derivatives and establish that it is theoretically positive. This justifies the economic intuition and observed demand for VIX derivatives in a portfolio management setting. Numerical examples illustrate the results.
Public Summary
- Consider the optimal investment problem with equity and VIX derivatives in a stochastic volatility model with jumps.
- Solve in analytical closed-form or semi-closed-form the optimal investment strategies for different combinations of equity and VIX derivatives.
- Establish that there is a strict portfolio improvement when including the VIX derivatives into the investment opportunity set.
- Provide a theoretical justification for the demand of VIX derivatives in portfolio management.

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Conflict of Interest

The authors declare that they have no conflict of interest.

① This is common practice in volatility trading, where only the VIX futures exchange traded note (ETN) is invested.

① Note that we corrected for some typos in the formula presented in Ref. [14]. In particular, there is a term ${\rm{e}}^{y\ln(1+\mu)}$ in the definition of $a$ , and there is a typo in the original formula in Ref. [14].

② http://www.cboe.com/micro/vix/vixwhite.pdf

① It is common practice to trade VIX futures, e.g., the VIX short-term futures ETF (ticker VXX), with proper leveraging by borrowing from the bank at a risk-free rate.

Consider the optimal investment problem with equity and VIX derivatives in a stochastic volatility model with jumps. Solve in analytical closed-form or semi-closed-form the optimal investment strategies for different combinations of equity and VIX derivatives. Establish that there is a strict portfolio improvement when including the VIX derivatives into the investment opportunity set. Provide a theoretical justification for the demand of VIX derivatives in portfolio management.

References (19)

References

[1]	Merton R C. Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 1971, 3: 373–413. DOI: 10.1016/0022-0531(71)90038-X
[2]	Ye W, Wu B, Chen P. Pricing VIX derivatives using a stochastic volatility model with a flexible jump structure. Probability in the Engineering and Informational Sciences, 2023, 37 (1): 245–274. DOI: 10.1017/S0269964821000577
[3]	Cui Z, Lee C, Liu M. Valuation of VIX derivatives through combined Ito-Taylor expansion and Markov chain approximation. Hoboken, NJ: Stevens Institute of Technology, 2021.
[4]	Chen T, Zhuo H. Pricing VIX options with realized volatility. Journal of Futures Markets, 2021, 41 (8): 1180–1200. DOI: 10.1002/fut.22201
[5]	Yin F, Bian Y, Wang T. A short cut: Directly pricing VIX futures with discrete-time long memory model and asymmetric jumps. Journal of Futures Markets, 2021, 41 (4): 458–477. DOI: 10.1002/fut.22183
[6]	Yuan P. Time-varying skew in VIX derivatives pricing. Management Science, 2021, 68 (10): 7065–7791. DOI: 10.1287/mnsc.2021.4168
[7]	Mencía J, Sentana E. Valuation of VIX derivatives. Journal of Financial Economics, 2013, 108 (2): 367–391. DOI: 10.1016/j.jfineco.2012.12.003
[8]	Simon D P. Trading the VIX futures roll and volatility premiums with VIX options. Journal of Futures Markets, 2017, 37 (2): 184–208. DOI: 10.1002/fut.21788
[9]	Simon D P, Campasano J. The VIX futures basis: Evidence and trading strategies. The Journal of Derivatives, 2014, 21 (3): 54–69. DOI: 10.3905/jod.2014.21.3.054
[10]	Chen H C, Chung S L, Ho K Y. The diversification effects of volatility-related assets. Journal of Banking & Finance, 2011, 35 (5): 1179–1189. DOI: 10.1016/j.jbankfin.2010.09.024
[11]	Black K H. Improving hedge fund risk exposures by hedging equity market volatility, or how the VIX ate my kurtosis. The Journal of Trading, 2006, 1 (2): 6–15. DOI: 10.3905/jot.2006.628190
[12]	Daigler R T, Rossi L. A portfolio of stocks and volatility. The Journal of Investing, 2006, 15 (2): 99–106. DOI: 10.3905/joi.2006.635636
[13]	Carr P, Jin X, Madan D P. Optimal investment in derivative securities. Finance and Stochastics, 2001, 5 (1): 33–59. DOI: 10.1007/s007800000023
[14]	Liu J, Pan J. Dynamic derivative strategies. Journal of Financial Economics, 2003, 69 (3): 401–430. DOI: 10.1016/S0304-405X(03)00118-1
[15]	Bates D S. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. The Review of Financial Studies, 1996, 9 (1): 69–107. DOI: 10.1093/rfs/9.1.69
[16]	Lian G H, Zhu S P. Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance, 2013, 36: 71–88. DOI: 10.1007/s10203-011-0124-0
[17]	Escobar M, Ferrando S, Rubtsov A. Robust portfolio choice with derivative trading under stochastic volatility. Journal of Banking & Finance, 2015, 61: 142–157. DOI: 10.1016/j.jbankfin.2015.08.033
[18]	Moreira A, Muir T. Volatility-managed portfolios. The Journal of Finance, 2017, 72 (4): 1611–1644. DOI: 10.1111/jofi.12513
[19]	Gatheral J, Jaisson T, Rosenbaum M. Volatility is rough. Quantitative Finance, 2018, 18 (6): 933–949. DOI: 10.2139/ssrn.2509457

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Figure 1. Sensitivity of portfolio weights for the complete market case. The lines are in the same style of Fig. 1 in Ref. [14] to make comparison, i.e., the $y$ -axes are the optimal portfolio weights $\phi$ (dash line), $\psi^{(1)}$ (solid line), $\psi^{(2)}$ (dot line), and the remaining portion invested in risk-free asset (dashed-dot line).

Figure 2. Sensitivity of certainty equivalent wealth $W^{\rm S}$ for different values of $\mu$ . The $y$ -axes represent the certainty equivalent wealth levels for respectively the three cases $\mu=-0.1,\,\mu=-0.15$ , and $\mu=-0.25$ .

Figure 3. Sensitivity of certainty equivalent wealth $W^{\rm V}$ for different values of $\mu$ . The $y$ -axes represent the certainty equivalent wealth levels for respectively the three cases $\mu=-0.1,\;\mu=-0.15$ , and $\mu=-0.25$ .

Figure 4. Sensitivity of certainty equivalent wealth $W^{\rm S}$ for different values of $\dfrac{\lambda^{\rm Q}}{\lambda}$ . The $y$ -axes represent the certainty equivalent wealth levels for respectively the three cases $\dfrac{\lambda^{\rm Q}}{\lambda}=1$ , $\dfrac{\lambda^{\rm Q}}{\lambda}=3$ and $\dfrac{\lambda^{\rm Q}}{\lambda}=5$ .

Figure 5. Sensitivity of certainty equivalent wealth $W^{\rm V}$ for different values of $\dfrac{\lambda^{\rm Q}}{\lambda}$ . The $y$ -axes represent the certainty equivalent wealth levels for respectively the three cases $\dfrac{\lambda^{\rm Q}}{\lambda}=1$ , $\dfrac{\lambda^{\rm Q}}{\lambda}=3$ and $\dfrac{\lambda^{\rm Q}}{\lambda}=5$ .

Figure 6. Portfolio improvement in $R$ in model BZN. The $y$ -axes represent the return of certainty equivalent wealth levels.

Figure 7. Portfolio improvement in $W$ in model BZN. The $y$ -axes represent the certainty equivalent wealth levels.

References

[1]	Merton R C. Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 1971, 3: 373–413. DOI: 10.1016/0022-0531(71)90038-X
[2]	Ye W, Wu B, Chen P. Pricing VIX derivatives using a stochastic volatility model with a flexible jump structure. Probability in the Engineering and Informational Sciences, 2023, 37 (1): 245–274. DOI: 10.1017/S0269964821000577
[3]	Cui Z, Lee C, Liu M. Valuation of VIX derivatives through combined Ito-Taylor expansion and Markov chain approximation. Hoboken, NJ: Stevens Institute of Technology, 2021.
[4]	Chen T, Zhuo H. Pricing VIX options with realized volatility. Journal of Futures Markets, 2021, 41 (8): 1180–1200. DOI: 10.1002/fut.22201
[5]	Yin F, Bian Y, Wang T. A short cut: Directly pricing VIX futures with discrete-time long memory model and asymmetric jumps. Journal of Futures Markets, 2021, 41 (4): 458–477. DOI: 10.1002/fut.22183
[6]	Yuan P. Time-varying skew in VIX derivatives pricing. Management Science, 2021, 68 (10): 7065–7791. DOI: 10.1287/mnsc.2021.4168
[7]	Mencía J, Sentana E. Valuation of VIX derivatives. Journal of Financial Economics, 2013, 108 (2): 367–391. DOI: 10.1016/j.jfineco.2012.12.003
[8]	Simon D P. Trading the VIX futures roll and volatility premiums with VIX options. Journal of Futures Markets, 2017, 37 (2): 184–208. DOI: 10.1002/fut.21788
[9]	Simon D P, Campasano J. The VIX futures basis: Evidence and trading strategies. The Journal of Derivatives, 2014, 21 (3): 54–69. DOI: 10.3905/jod.2014.21.3.054
[10]	Chen H C, Chung S L, Ho K Y. The diversification effects of volatility-related assets. Journal of Banking & Finance, 2011, 35 (5): 1179–1189. DOI: 10.1016/j.jbankfin.2010.09.024
[11]	Black K H. Improving hedge fund risk exposures by hedging equity market volatility, or how the VIX ate my kurtosis. The Journal of Trading, 2006, 1 (2): 6–15. DOI: 10.3905/jot.2006.628190
[12]	Daigler R T, Rossi L. A portfolio of stocks and volatility. The Journal of Investing, 2006, 15 (2): 99–106. DOI: 10.3905/joi.2006.635636
[13]	Carr P, Jin X, Madan D P. Optimal investment in derivative securities. Finance and Stochastics, 2001, 5 (1): 33–59. DOI: 10.1007/s007800000023
[14]	Liu J, Pan J. Dynamic derivative strategies. Journal of Financial Economics, 2003, 69 (3): 401–430. DOI: 10.1016/S0304-405X(03)00118-1
[15]	Bates D S. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. The Review of Financial Studies, 1996, 9 (1): 69–107. DOI: 10.1093/rfs/9.1.69
[16]	Lian G H, Zhu S P. Pricing VIX options with stochastic volatility and random jumps. Decisions in Economics and Finance, 2013, 36: 71–88. DOI: 10.1007/s10203-011-0124-0
[17]	Escobar M, Ferrando S, Rubtsov A. Robust portfolio choice with derivative trading under stochastic volatility. Journal of Banking & Finance, 2015, 61: 142–157. DOI: 10.1016/j.jbankfin.2015.08.033
[18]	Moreira A, Muir T. Volatility-managed portfolios. The Journal of Finance, 2017, 72 (4): 1611–1644. DOI: 10.1111/jofi.12513
[19]	Gatheral J, Jaisson T, Rosenbaum M. Volatility is rough. Quantitative Finance, 2018, 18 (6): 933–949. DOI: 10.2139/ssrn.2509457

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