# Hedging and Evaluating Tail Risks via Two Novel Options Based on Type II Extreme Value Distribution

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. The Underlying ”Assets”

#### 3.1. The Construction of the ad hoc Underlying “Assets“

**Type I ad hoc “assets”: daily high-frequency maximum drawdown.**

**Type II ad hoc “assets”: the worst performance of an individual asset within a portfolio.**

#### 3.2. Specification for the Statistical Process of the Ad Hoc “Asset Price“ Dynamic

## 4. The Construction of Tail Risk Options

#### 4.1. The Market Conventions

**Issuance and expiration:**A tail risk option (TRO) is written on the “value“ of tail risk, i.e., the “price“ of an ad hoc “asset“ proposed in Section 3 in a single given day (the “observation day,“ which will be introduced in the next sub-point). Thus, each TRO should be issued at any moment before that day, with a given “strike price“ K, which is a pre-determined value regarding the tail risk in the “observation day.“ The expiration time of a TRO is the closing time of its “observation day.“ The TRO can be issued and traded in the market or over-the-counter (OTC).

**The “observation day“:**An “observation day“ of a TRO is the date that the payoff of this TRO is calculated. Since the payoff of a TRO is computed using a whole day’s trading information on that day (see Section 4.2 for details), this option will be “observed“ all that day. This is the reason why this day is termed “observation day.“ Normally, the “observation day“ of a TRO is the final trading date, i.e., its maturity date.

#### 4.2. The Payoffs

**Type I TRO: high-frequency index option.**If ${Q}_{0,T}$ is defined by Equation (1), i.e., the TRO is written on Type I ad hoc “asset“, thereby, the TRO can be refined as a high-frequency index option with payoff functions:

**Type II TRO: high-dimensional rainbow option.**If ${Q}_{0,T}$ is defined by Equation (2), i.e., the TRO is written on Type II ad hoc “asset“, thereby, the TRO can be refined as a high-dimensional rainbow option written on N assets with payoff functions:

## 5. The Closed-Form Pricing Formulas for TROs

## 6. Simulation Study: Pricing Performance of the TRO Pricing Formulas

## 7. Simulation Study: TRO Price, Volatility, and Tail Index

## 8. Application: Implying the Tail Risks from TRO Prices

#### 8.1. Methodology

#### 8.2. Simulation Example

## 9. Conclusions, Implications, and Future Research Directions

#### 9.1. Conclusions

#### 9.2. Implications for Practice

#### 9.3. Limitations and Future Study Directions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Derivation of the Analytical Pricing Formula for TRO Calls

## References

- Zhao, Z.; Zhang, Z.; Chen, R. Modeling maxima with autoregressive conditional Fréchet model. J. Econom.
**2018**, 207, 325–351. [Google Scholar] [CrossRef] - Bhansali, V. Tail Risk Hedging: Creating Robust Portfolios for Volatile Markets; McGraw-Hill Education: New York, NY, USA, 2014. [Google Scholar]
- Fisher, R.A.; Tippett, L.H.C. Limiting forms of the frequency distribution of the largest or smallest member of a sample. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1928; Volume 24, pp. 180–190. [Google Scholar]
- Jang, J.W.; Krvavych, Y. Arbitrage-free premium calculation for extreme losses using the shot noise process and the Esscher transform. Insur. Math. Econ.
**2004**, 35, 97–111. [Google Scholar] [CrossRef] - Shrivastava, U.; Dawar, G.; Dhingra, S.; Rani, M. Extreme value analysis for record loss prediction during volatile market. Manag. Sci. Eng.
**2011**, 5, 19–25. [Google Scholar] - Harrison, J.M.; Pliska, S.R. Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Their Appl.
**1981**, 11, 215–260. [Google Scholar] [CrossRef][Green Version] - Kroner, K.F.; Ng, V.K. Modeling asymmetric comovements of asset returns. Rev. Financ. Stud.
**1998**, 11, 817–844. [Google Scholar] [CrossRef] - Harris, R.D.; Küçüközmen, C.C. The empirical distribution of UK and US stock returns. J. Bus. Financ. Account.
**2001**, 28, 715–740. [Google Scholar] [CrossRef] - Harris, R.D.; Coskun Küçüközmen, C. The empirical distribution of stock returns: Evidence from an emerging European market. Appl. Econ. Lett.
**2001**, 8, 367–371. [Google Scholar] [CrossRef] - Bollerslev, T.; Todorov, V.; Xu, L. Tail risk premia and return predictability. J. Financ. Econ.
**2015**, 118, 113–134. [Google Scholar] [CrossRef][Green Version] - Van Oordt, M.R.; Zhou, C. Systematic tail risk. J. Financ. Quant. Anal.
**2016**, 51, 685–705. [Google Scholar] [CrossRef] - Zhang, Z.; Huang, J. Extremal financial risk models and portfolio evaluation. Comput. Stat. Data Anal.
**2006**, 51, 2313–2338. [Google Scholar] [CrossRef] - Bhansali, V. Tail risk management. J. Portf. Manag.
**2008**, 34, 68–75. [Google Scholar] [CrossRef] - Kelly, B.; Jiang, H. Tail risk and asset prices. Rev. Financ. Stud.
**2014**, 27, 2841–2871. [Google Scholar] [CrossRef][Green Version] - Poon, S.H.; Rockinger, M.; Tawn, J. Extreme value dependence in financial markets: Diagnostics, models, and financial implications. Rev. Financ. Stud.
**2004**, 17, 581–610. [Google Scholar] [CrossRef] - Agarwal, V.; Ruenzi, S.; Weigert, F. Tail risk in hedge funds: A unique view from portfolio holdings. J. Financ. Econ.
**2017**, 125, 610–636. [Google Scholar] [CrossRef][Green Version] - Chen, Y.; Wang, Z.; Zhang, Z. Mark to market value at risk. J. Econom.
**2019**, 208, 299–321. [Google Scholar] [CrossRef] - Spulbar, C.; Trivedi, J.; Birau, R. Investigating abnormal volatility transmission patterns between emerging and developed stock markets: A case study. J. Bus. Econ. Manag.
**2020**, 21, 1561–1592. [Google Scholar] [CrossRef] - Lin, L.; Zhou, Z.; Jiang, Y.; Ou, Y. Risk spillovers and hedge strategies between global crude oil markets and stock markets: Do regime switching processes combining long memory and asymmetry matter? N. Am. J. Econ. Financ.
**2021**, 57, 101398. [Google Scholar] [CrossRef] - Trivedi, J.; Spulbar, C.; Birau, R.; Mehdiabadi, A. Modelling volatility spillovers, cross-market correlation and co-movements between stock markets in European Union: An empirical case study. Bus. Manag. Econ. Eng.
**2021**, 19, 70–90. [Google Scholar] [CrossRef] - Fasanya, I.O.; Oyewole, O.; Adekoya, O.B.; Odei-Mensah, J. Dynamic spillovers and connectedness between COVID-19 pandemic and global foreign exchange markets. Econ.-Res.-Ekon. IstražIvanja
**2021**, 34, 2059–2084. [Google Scholar] [CrossRef] - Wang, L.; Xu, T. Bidirectional Risk Spillovers between Exchange Rate of Emerging Market Countries and International Crude Oil Price–Based on Time-varing Copula-CoVaR. Comput. Econ.
**2021**, 58, 1–32. [Google Scholar] - Go, Y.H.; Lau, W.Y. Extreme risk spillovers between crude palm oil prices and exchange rates. N. Am. J. Econ. Financ.
**2021**, 58, 101513. [Google Scholar] [CrossRef] - Yang, Y.; Ma, Y.R.; Hu, M.; Zhang, D.; Ji, Q. Extreme risk spillover between chinese and global crude oil futures. Financ. Res. Lett.
**2021**, 40, 101743. [Google Scholar] [CrossRef] - Zhao, W.L.; Fan, Y.; Ji, Q. Extreme risk spillover between crude oil price and financial factors. Available online: https://www.sciencedirect.com/science/article/abs/pii/S1544612321003457 (accessed on 10 August 2021).
- Nguyen, L.H.; Chevapatrakul, T.; Yao, K. Investigating tail-risk dependence in the cryptocurrency markets: A LASSO quantile regression approach. J. Empir. Financ.
**2020**, 58, 333–355. [Google Scholar] [CrossRef] - Xu, Q.; Zhang, Y.; Zhang, Z. Tail-risk spillovers in cryptocurrency markets. Financ. Res. Lett.
**2021**, 38, 101453. [Google Scholar] [CrossRef] - Moratis, G. Quantifying the spillover effect in the cryptocurrency market. Financ. Res. Lett.
**2021**, 38, 101534. [Google Scholar] [CrossRef] - Guo, Y.; Li, P.; Li, A. Tail risk contagion between international financial markets during COVID-19 pandemic. Int. Rev. Financ. Anal.
**2021**, 73, 101649. [Google Scholar] [CrossRef] - Abuzayed, B.; Al-Fayoumi, N. Risk spillover from crude oil prices to GCC stock market returns: New evidence during the COVID-19 outbreak. North Am. J. Econ. Financ.
**2021**, 58, 101476. [Google Scholar] [CrossRef] - Bhansali, V.; Davis, J.M. Offensive risk management II: The case for active tail hedging. J. Portf. Manag.
**2010**, 37, 78–91. [Google Scholar] [CrossRef] - Coles, S.; Bawa, J.; Trenner, L.; Dorazio, P. An Introduction to Statistical Modeling of Extreme Values; Springer: London, UK, 2001; Volume 208, p. 208. [Google Scholar]
- Hansen, B.E. Autoregressive conditional density estimation. Int. Econ. Rev.
**1994**, 35, 705–730. [Google Scholar] [CrossRef] - Brunnermeier, M.K.; Sannikov, Y. A macroeconomic model with a financial sector. Am. Econ. Rev.
**2014**, 104, 379–421. [Google Scholar] [CrossRef][Green Version] - Hattori, M.; Schrimpf, A.; Sushko, V. The response of tail risk perceptions to unconventional monetary policy. Am. Econ. J. Macroecon.
**2016**, 8, 111–136. [Google Scholar] [CrossRef] - Gilli, M. An application of extreme value theory for measuring financial risk. Comput. Econ.
**2006**, 27, 207–228. [Google Scholar] [CrossRef][Green Version] - Feng, W.; Wang, Y.; Zhang, Z. Can cryptocurrencies be a safe haven: A tail risk perspective analysis. Appl. Econ.
**2018**, 50, 4745–4762. [Google Scholar] [CrossRef]

**Figure 2.**The tail risk option prices with different strike prices K and different tail indexes $\alpha $ when: (

**a**) $\sigma =0.06$ for calls (low volatility); (

**b**) $\sigma =0.06$ for puts (low volatility); (

**c**) $\sigma =0.135$ for calls (medium volatility); (

**d**) $\sigma =0.135$ for puts (medium volatility); (

**e**) $\sigma =0.21$ for calls (high volatility); (

**f**) $\sigma =0.21$ for puts (high volatility).

**Figure 3.**The tail risk option prices with different strike prices K and different Fréchet volatilities $\sigma $ when: (

**a**) $\alpha =2$ for calls (high tail risk); (

**b**) $\alpha =2$ for puts (high tail risk); (

**c**) $\alpha =5$ for calls (medium tail risk); (

**d**) $\alpha =5$ for puts (medium tail risk); (

**e**) $\alpha =8$ for calls (low tail risk); (

**f**) $\alpha =8$ for puts (low tail risk).

**Figure 4.**The simulated daily maximum negative losses data $\left\{{Q}_{t}\right\}$ generated from the AcF(1,1) model and corresponding tail risk index dynamic.

**Figure 5.**The p-p plots: (

**a**) simulated and calibrated $\left\{\widehat{{\alpha}_{t}}\right\}$ for calls; (

**b**) simulated and calibrated $\left\{\widehat{{\alpha}_{t}}\right\}$ for puts; (

**c**) simulated and calibrated $\left\{\widehat{{\sigma}_{t}}\right\}$ for calls; (

**d**) simulated and calibrated $\left\{\widehat{{\sigma}_{t}}\right\}$ for puts.

**Figure 6.**The dynamic of: (

**a**) the simulated tail index; (

**b**) the simulated Fréchet volatility; (

**c**) the tail index implied by calls; (

**d**) the Fréchet volatility implied by calls; (

**e**) the tail index implied by puts; (

**f**) the Fréchet volatility implied by puts.

RMSE_{call} | RMSE_{put} | MAE_{call} | MAE_{put} | |
---|---|---|---|---|

$\alpha =3$ | $1.7539\times {10}^{-4}$ | $4.5043\times {10}^{-5}$ | $4.3008\times {10}^{-5}$ | $1.4291\times {10}^{-5}$ |

$\alpha =4$ | $5.6705\times {10}^{-5}$ | $2.4401\times {10}^{-5}$ | $1.6823\times {10}^{-5}$ | $7.8508\times {10}^{-6}$ |

$\alpha =5$ | $1.3918\times {10}^{-5}$ | $7.2705\times {10}^{-6}$ | $7.2705\times {10}^{-6}$ | $3.5977\times {10}^{-6}$ |

$\alpha =6$ | $1.6123\times {10}^{-5}$ | $1.0564\times {10}^{-5}$ | $6.5398\times {10}^{-6}$ | $3.0115\times {10}^{-6}$ |

$\alpha =7$ | $9.3093\times {10}^{-6}$ | $7.6468\times {10}^{-6}$ | $4.3364\times {10}^{-6}$ | $3.4457\times {10}^{-6}$ |

average | $5.6358\times {10}^{-5}$ | $2.0314\times {10}^{-5}$ | $1.5595\times {10}^{-5}$ | $6.4393\times {10}^{-6}$ |

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Lin, H.; Liu, L.; Zhang, Z. Hedging and Evaluating Tail Risks via Two Novel Options Based on Type II Extreme Value Distribution. *Symmetry* **2021**, *13*, 1630.
https://doi.org/10.3390/sym13091630

**AMA Style**

Lin H, Liu L, Zhang Z. Hedging and Evaluating Tail Risks via Two Novel Options Based on Type II Extreme Value Distribution. *Symmetry*. 2021; 13(9):1630.
https://doi.org/10.3390/sym13091630

**Chicago/Turabian Style**

Lin, Hang, Lixin Liu, and Zhengjun Zhang. 2021. "Hedging and Evaluating Tail Risks via Two Novel Options Based on Type II Extreme Value Distribution" *Symmetry* 13, no. 9: 1630.
https://doi.org/10.3390/sym13091630