June  2020, 13(6): 1867-1882. doi: 10.3934/dcdss.2020110

Design of green bonds by double-barrier options

1. 

Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin 300222, China

2. 

College of Science, Tianjin University of Commerce, Tianjin 300134, China

3. 

Department of Mathematics and Statistics, Curtin University, Perth, WA6845, Australia

* Corresponding author: Zhuo Yang

Received  September 2018 Revised  October 2018 Published  September 2019

Fund Project: This project was supported in part by the National Basic Research Program (2012CB955804), the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11771322), the Social Science Fund of Tianjin (TJYJ18023), and the Key Projects of Ministry of Education (DIA160334)

Green finance is an innovative model that can promote sustainable economic development. The green bonds also develop gradually as a part of green finance. The green bonds are designed to fund the projects of positive environmental impact. If the green bonds are superior to other debt securities, they will attract more investors' participation in green energy projects. Thus, the design of green bonds is crucial to the development of green bonds market. This article assumes that the floating rate of green bonds is linked to carbon price, and carbon price is described by a jump diffusion process. The carbon price fluctuation can lead to interest rate fluctuation of green bonds. We set two boundary values of carbon price, and the coupon rate is revalued when the carbon price reaches the boundary. The higher the carbon price is, the higher the coupon rate is to be paid by issuers. Thus, the boundary can impel issuers to boost energy savings and emission-reduction, and the higher interest rate will also attract more investors to invest in green bonds. The lower the carbon price is, the lower the interest rate is to be paid by issuers. Accordingly, the boundary may encourage issuers to boost emission reduction. This design can monitor and incentivize issuers to make more contribution to green finance. Furthermore, the design is characterized by the double-barrier option, such that the interest rate of green bonds can be obtained by double-barrier option pricing. Subsequently, the central difference method and the composite trapezoidal formula are employed to obtain the numerical solution. Finally, we conduct the sensitivity analysis of the model.

Citation: Shuhua Zhang, Zhuo Yang, Song Wang. Design of green bonds by double-barrier options. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1867-1882. doi: 10.3934/dcdss.2020110
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J. JeonJ. H. Yoon and C. R. Park, An analytic expansion method for the valuation of double-barrier options under a stochastic volatility model, Journal of Mathematical Analysis and Applications, 449 (2017), 207-227.  doi: 10.1016/j.jmaa.2016.11.061.

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H. Jiang, The application of barrier option in avoiding foreign exchange risk, Technology and Market, 3 (2009), 24-25. 

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B. Kamrad and P. Ritchken, Multinomial approximating models for options with kstate variables, Manage. Sci., 37 (1991), 1640-1652. 

[23]

J. X. Liu and G. Q. Yang, The comparison of price risk transfer mechanism between agricultural commodity futures market and option market, Economic Forum, 15 (2005), 72-73. 

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R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Finacial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.

[25]

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S. K. Park, Investors as regulators: green bonds and the governance chanllenges of sustainanble finance revolution, 2018, https://heinonline.org/HOL/LandingPage?handle=hein.journals/stanit54&div=5&id=&page=.

[29]

C. Pelizzari, Pricing Inflation Linked Bonds, Quantitave Finance, 2010. https://www.ssoar.info/ssoar/bitstream/handle/document/23318/ssoar-2010-03-pelizzari_et_al-pricing_inflation_linked_bonds.pdf?sequence=1.

[30]

H. Reichelt, Green bonds: A model to mobilise private capital to fund climate change mitigation and adaptation projects, The euromoney environmental finance handbook, 2010.

[31]

A. Sepp, Analytical pricing of the double-barrier options under a doubleexponential jump diffusion processes: Applications of Laplace transform, International Journal of Theoretical and Applied Finance, 7 (2004), 151-175.  doi: 10.1142/S0219024904002402.

[32]

A. R. Stephen, Options and efficiency, Quarterly Journal of Economics, 90 (1976), 121-149. 

[33]

Y. TianE. Roca Akimov and V. Wang, Does the carbon market help or hurt the stock price of electricity companies? Further evidence from the European context, J. Clean. Prod., 112 (2016), 1-8. 

[34]

Toyota Financial Services (TFS) Issues Auto Industry's First-Ever Asset-Backed Green Bond, https://www.prnewswire.com/news-releases/toyota-financial-services-tfs-issues-auto-industrys-first-ever-asset-backed-green-bond-251943831.html

[35]

A. Tripathy, Translating to risk: The legibility of climate change and nature in the green bond market, Economic Anthropology, 4 (2017), 239-250.  doi: 10.1002/sea2.12091.

[36]

Y. P. XiaoX. F. WangX. L. Wang and Z. C. Wu, Trading wind power with barrier option, Applied Energy, 182 (2016), 232-242.  doi: 10.1016/j.apenergy.2016.08.123.

[37]

United Nations Climate Change Conference, 2015, https://www.politiekemonitor.nl/9353000/1/j9tvgajcor7dxyk_j9vvioaf0kku7zz/vjmhg41ub7pp.

show all references

References:
[1]

As You Sow, 2014. http://www.gogreenbonds.org/new-report-green-bonds-in-brief-risk-reward-and-opportunity/

[2]

N. B. Behmiri and M. Maner, The role of outliers and oil price shocks on volatility of metal prices, Resour. Policy., 46 (2015), 139-150. 

[3]

F. Black and M. Scholes, The pricing of options and corporate liabilities, The Journal of Political Economy, 81 (1973), 637-654.  doi: 10.1086/260062.

[4]

F. Black and J. C. Cox, Valuing corprate securities: Some effects of bond indenture provisions, The Journal of Finance, 2 (1976), 351-366. 

[5]

D. A. Bloch, The pricing of carbon bonds and other index-linked carbon derivatives, Register to Save Articles to Your Library, 2014, 16pp, http://ssrn.com/abstract=1763168. doi: 10.2139/ssrn.1763168.

[6]

M. Burke, F. González, P. Baylis, S. H.Neal, C. Baysan, S. Basu and S. Hsiang, Higher temperatures increase suicide rates in the United States and Mexico, Nature Climate Change, 8 (2018), 723–729, https://www.nature.com/articles/s41558-018-0222-x. doi: 10.1038/s41558-018-0222-x.

[7]

N. CaiN. Chen and X. W. Wan, Pricing double-barrier options under a flexible jump diffusion model, Operation Research Letters, 37 (2009), 163-167.  doi: 10.1016/j.orl.2009.02.006.

[8]

P. K. Chugan, Y. Mungra and K. Mehta, Challenges and policy implications for marketing green bonds, Consumer Behaviour and Contemporary Marketing Strategy, 2017,371–384.

[9]

G. DaskalakisD. Psychoyios and R. N. Markellos, Modeling CO2 emissions allowance prices and derivatives: Evidence from the European trading scheme., J. Bank. Financ., 33 (2009), 1230-1241.  doi: 10.1016/j.jbankfin.2009.01.001.

[10]

S. Dietz, A. Bowen, C. Dixon and P. Gradwell, "Climate value at risk" of global financial assets, Nature Climate Change., 6 (2016), 676–679, https://www.nature.com/articles/nclimate2972.pdf. doi: 10.1038/nclimate2972.

[11]

A. Dutta, Modeling and forecasting the volatility of carbon emission market: The role of outliers, time-varying jumps and oil price risk, Journal of Cleaner Production, 172 (2018), 2773-2781.  doi: 10.1016/j.jclepro.2017.11.135.

[12]

M. Flaherty, A. Gevorkyan, S. Radpour and W. Semmler, Financing climate policies through climate bonds - A three stage model and empirics, Research in International Business and Finance, (2016), 1–12.

[13]

G. Fusai and M. C. Recchioni, Analysis of quadrature methods for pricing discrete barrier options, J. Econom. Dynam. Control, 31 (2007), 826-860.  doi: 10.1016/j.jedc.2006.03.002.

[14]

G. FusaiI. D. Abrahams and C. Sgarra, An exact analytical solution for discrete barrier options, Finance and Stochastics, 10 (2006), 1-26.  doi: 10.1007/s00780-005-0170-y.

[15]

F. GaoJ. Guo and Q. B. Zhao, The barrier options-based research of the value of government guarantees in frastructure projects, Forecasting, 2 (2007), 76-80. 

[16]

M. GoldmanH. Sosin and M. Gatto, Path dependent options: Buy at the low, sell at the high, Journal of Finance, 34 (1979), 1111-1127. 

[17]

P. HartmannV. A. Ibáñez and F. J. F. Sainz, Green branding effects on attitude: Functional versus emotional positioning strategies, Marketing Intelligence & Planning, 23 (2005), 9-29.  doi: 10.1108/02634500510577447.

[18]

G. Inderst, C. Kaminker and F. Stewart, Defining and measuring green investments: Implications for institutional investors, asset allocations, oecd working papers on finance, Insurance and Private Pensions, 2012, 1–55.

[19]

J. JeonJ. H. Yoon and C. R. Park, An analytic expansion method for the valuation of double-barrier options under a stochastic volatility model, Journal of Mathematical Analysis and Applications, 449 (2017), 207-227.  doi: 10.1016/j.jmaa.2016.11.061.

[20]

L. Jiang, C. Xu, X. Ren and S. Li, The Mathematical Model and Case Study for the Price of Financial Derivatives, (in Chinese), High Education Press, Beijing, 2008.

[21]

H. Jiang, The application of barrier option in avoiding foreign exchange risk, Technology and Market, 3 (2009), 24-25. 

[22]

B. Kamrad and P. Ritchken, Multinomial approximating models for options with kstate variables, Manage. Sci., 37 (1991), 1640-1652. 

[23]

J. X. Liu and G. Q. Yang, The comparison of price risk transfer mechanism between agricultural commodity futures market and option market, Economic Forum, 15 (2005), 72-73. 

[24]

R. Merton, Option pricing when underlying stock returns are discontinuous, Journal of Finacial Economics, 3 (1976), 125-144.  doi: 10.1016/0304-405X(76)90022-2.

[25]

R. C. Merton, Theory of rational option pricing, Bell Journal of Economics and Management Science, 4 (1973), 141-183.  doi: 10.2307/3003143.

[26]

Nielsen, Global consumers are willing to put their money where their heart is when it comes to goods and services from companies comitted to social responsibility, 2014, http://www.nielsen.com/apac/en/press-room/2014/global-consumers-are-willing-to-put-their-money-where-their-hearts-are.html

[27]

M. Northrop, Is the post-fossil fuel era now inevitable?, 2016, https://www.rbf.org/news/michael-northrop-is-the-post-fossil-fuel-era-now-inevitable

[28]

S. K. Park, Investors as regulators: green bonds and the governance chanllenges of sustainanble finance revolution, 2018, https://heinonline.org/HOL/LandingPage?handle=hein.journals/stanit54&div=5&id=&page=.

[29]

C. Pelizzari, Pricing Inflation Linked Bonds, Quantitave Finance, 2010. https://www.ssoar.info/ssoar/bitstream/handle/document/23318/ssoar-2010-03-pelizzari_et_al-pricing_inflation_linked_bonds.pdf?sequence=1.

[30]

H. Reichelt, Green bonds: A model to mobilise private capital to fund climate change mitigation and adaptation projects, The euromoney environmental finance handbook, 2010.

[31]

A. Sepp, Analytical pricing of the double-barrier options under a doubleexponential jump diffusion processes: Applications of Laplace transform, International Journal of Theoretical and Applied Finance, 7 (2004), 151-175.  doi: 10.1142/S0219024904002402.

[32]

A. R. Stephen, Options and efficiency, Quarterly Journal of Economics, 90 (1976), 121-149. 

[33]

Y. TianE. Roca Akimov and V. Wang, Does the carbon market help or hurt the stock price of electricity companies? Further evidence from the European context, J. Clean. Prod., 112 (2016), 1-8. 

[34]

Toyota Financial Services (TFS) Issues Auto Industry's First-Ever Asset-Backed Green Bond, https://www.prnewswire.com/news-releases/toyota-financial-services-tfs-issues-auto-industrys-first-ever-asset-backed-green-bond-251943831.html

[35]

A. Tripathy, Translating to risk: The legibility of climate change and nature in the green bond market, Economic Anthropology, 4 (2017), 239-250.  doi: 10.1002/sea2.12091.

[36]

Y. P. XiaoX. F. WangX. L. Wang and Z. C. Wu, Trading wind power with barrier option, Applied Energy, 182 (2016), 232-242.  doi: 10.1016/j.apenergy.2016.08.123.

[37]

United Nations Climate Change Conference, 2015, https://www.politiekemonitor.nl/9353000/1/j9tvgajcor7dxyk_j9vvioaf0kku7zz/vjmhg41ub7pp.

Figure 1.  Index-linked green bonds issued by World Bank
Figure 2.  The carbon prices and its rates of return from 2012 to 2018
Figure 3.  The green bond interest rates when it has no barriers and it has barriers
Figure 4.  The green bond interest rates when the barrier changes in non barriers and barriers
Figure 5.  The effects of parameters $ r $ and $ {\sigma} $ on the green bond interest rate
Figure 6.  The effects of parameters $ {\lambda} $ on the green bond interest rate
Table 1.  Parameters' values
ParametersValues Parameters Values
$r$ 0.019 $\delta$ 0.5
$\sigma$ 0.106 $\mu$ -0.06
$\lambda$ 0.04
ParametersValues Parameters Values
$r$ 0.019 $\delta$ 0.5
$\sigma$ 0.106 $\mu$ -0.06
$\lambda$ 0.04
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