Abstract
First introduced by Société Générale Corporate and Investment Banking in 2007, timer options are financial instruments whose payoffs rely on a random date of the exercise related to the realized variance of the underlying asset. This is contrary to vanilla options exercised at a fixed expiration date. However, option holders are vulnerable to credit risks arising from the uncertainty that counterparties may not implement their contractual obligation, particularly in the over-the-counter market. Hence, in this article, motivated by the credit risk model proposed by Johnson and Stulz (JFinac 42:281–300, 1987), we deal with the pricing of the timer option considering the counterparty default risk by utilizing the technique of asymptotic analysis. Moreover, we investigate the pricing accuracy of our analytic formulas, comparing them with the solutions from the Monte Carlo method, and examine the impact of stochastic volatility on the credit risk or the variance budget on the option value based on our pricing formula for vulnerable timer options.
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References
Agarwal, A., Juneja, S., & Sircar, R. (2016). American options under stochastic volatility: Control variates, maturity randomization & multiscale asymptotics. Quantitative Finance, 16(1), 17–30. https://doi.org/10.1080/14697688.2015.1068443
Augustin, P., Sokolovski, V., Subrahmanyam, M. G., & Tomio, D. (2022). In sickness and in debt: The COVID-19 impact on sovereign credit risk. Journal of Financial Economics, 143(3), 1251–1274. https://doi.org/10.1016/j.jfineco.2021.05.009
Bernard, C., & Cui, Z. (2011). Pricing timer options. Journal of Computational Finance, 15(1), 1–37. https://doi.org/10.21314/JCF.2011.228
Carr, P., & Lee, R. (2010). Hedging variance options on continuous semimartingales. Finance and Stochastics, 14(2), 179–207. https://doi.org/10.1007/s00780-009-0110-3
Choi, S.-Y., Veng, S., Kim, J.-H., & Yoon, J.-H. (2022). A Mellin transform approach to the pricing of options with default risk. Computational Economics, 59(3), 1113–1134. https://doi.org/10.1007/s10614-021-10121-w
Fouque, J.-P., Papanicolaou, G., & Sircar, K. R. (1999). Financial modeling in a fast mean-reverting stochastic volatility environment. Asia-Pacific Financial Markets, 6(1), 37–48. https://doi.org/10.1023/A:1010010626460
Fouque, J.-P., Papanicolaou, G., & Sircar, K. R. (2000). Mean-reverting stochastic volatility. International Journal of Theoretical and Applied Finance, 3(01), 101–142. https://doi.org/10.1142/S0219024900000061
Fouque, J.-P., Papanicolaou, G., Sircar, K. R., & Sølna, K. (2011). Multiscale stochastic volatility for equity, interest rate, and credit derivatives. Cambridge: Cambridge University Press.
Guo, C., & Wang, X. (2022). Pricing vulnerable options under correlated skew Brownian motions. Journal of Futures Markets, 42(5), 852–867. https://doi.org/10.1002/fut.22311
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2), 327–343. https://doi.org/10.1093/rfs/6.2.327
Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatilities. The Journal of Finance, 42(2), 281–300. https://doi.org/10.1111/j.1540-6261.1987.tb02568.x
Hung, M.-W., & Liu, Y.-H. (2005). Pricing vulnerable options in incomplete markets. Journal of Futures Markets, 25(2), 135–170. https://doi.org/10.1002/fut.20136
Jeon, J., Yoon, J.-H., & Kang, M. (2016). Valuing vulnerable geometric Asian options. Computers & Mathematics with Applications, 71(2), 676–691. https://doi.org/10.1016/j.camwa.2015.12.038
Johnson, H., & Stulz, R. (1987). The pricing of options with default risk. The Journal of Finance, 42(2), 267–280. https://doi.org/10.1111/j.1540-6261.1987.tb02567.x
Kim, J.-H., & Park, C.-R. (2017). A multiscale extension of the Margrabe formula under stochastic volatility. Chaos, Solitons & Fractals, 97, 59–65. https://doi.org/10.1016/j.chaos.2017.02.006
Kim, D., Yoon, J.-H., & Park, C.-R. (2021). Pricing external barrier options under a stochastic volatility model. Journal of Computational and Applied Mathematics, 394, 113555. https://doi.org/10.1016/j.cam.2021.113555
Kim, D., Choi, S.-Y., & Yoon, J.-H. (2021). Pricing of vulnerable options under hybrid stochastic and local volatility. Chaos, Solitons & Fractals, 146, 110846. https://doi.org/10.1016/j.chaos.2021.110846
Klein, P. (1996). Pricing Black-Scholes options with correlated credit risk. Journal of Banking & Finance, 20(7), 1211–1229. https://doi.org/10.1016/0378-4266(95)00052-6
Li, C. (2016). Bessel processes, stochastic volatility, and timer options. Mathematical Finance, 26(1), 122–148. https://doi.org/10.1111/mafi.12041
Li, M., & Mercurio, F. (2015). Analytic approximation of finite-maturity timer option prices. Journal of Futures Markets, 35(3), 245–273. https://doi.org/10.1002/fut.21659
Ma, J., Deng, D., & Lai, Y. (2015). Explicit approximate analytic formulas for timer option pricing with stochastic interest rates. The North American Journal of Economics and Finance, 34, 1–21. https://doi.org/10.1016/j.najef.2015.07.002
Neuberger, A. (1990). Volatility Trading. Institute of Finance and Accounting: London Business School, working paper.
Øksendal, B. (2003). Stochastic Differential Equations. Berlin: Springer.
Ramm, A. G. (2001). A simple proof of the Fredholm alternative and a characterization of the Fredholm operators. The American Mathematical Monthly, 108(9), 855–860. https://doi.org/10.1080/00029890.2001.11919820
Renault, E., & Touzi, N. (1996). Option hedging and implied volatilities in a stochastic volatility model. Mathematical Finance, 6(3), 279–302. https://doi.org/10.1111/j.1467-9965.1996.tb00117.x
Sawyer, N. (2007). SG CIB launches timer options. Risk. Jul 01.
Saunders, D. (2009). Pricing timer options under fast mean-reverting stochastic volatility. Canadian Applied Mathematics Quarterly, 17(4), 737–53.
Umar, Z., Polat, O., Choi, S.-Y., & Teplova, T. (2022). The impact of the Russia-Ukraine conflict on the connectedness of financial markets. Finance Research Letters, 48, 102976. https://doi.org/10.1016/j.frl.2022.102976
Wang, X. (2022). Pricing vulnerable options with stochastic liquidity risk. The North American Journal of Economics and Finance, 60, 101637. https://doi.org/10.1016/j.najef.2021.101637
Xie, Y., & Deng, G. (2022). Vulnerable European option pricing in a Markov regime-switching Heston model with stochastic interest rate. Chaos, Solitons & Fractals, 156, 111896. https://doi.org/10.1016/j.chaos.2022.111896
Yang, S.-J., Lee, M.-K., & Kim, J.-H. (2014). Pricing vulnerable options under a stochastic volatility model. Applied Mathematics Letters, 34, 7–12. https://doi.org/10.1016/j.aml.2014.03.007
Yin, J., Han, B., & Wong, H. Y. (2022). COVID-19 and credit risk: A long memory perspective. Insurance: Mathematics and Economics, 104, 15–34. https://doi.org/10.1016/j.insmatheco.2022.01.008
Yoon, J.-H., & Kim, J.-H. (2015). The pricing of vulnerable options with double Mellin transforms. Journal of Mathematical Analysis and Applications, 422(2), 838–857. https://doi.org/10.1016/j.jmaa.2014.09.015
Zheng, W., & Zeng, P. (2016). Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model. Applied Mathematical Finance, 23(5), 344–373. https://doi.org/10.1080/1350486X.2017.1285242
Zhu, S.-P., & Chen, W.-T. (2011). Pricing perpetual American options under a stochastic-volatility model with fast mean reversion. Applied Mathematics Letters, 24(10), 1663–1669. https://doi.org/10.1016/j.aml.2011.04.011
Funding
The research by J.-H. Yoon was supported by the National Research Foundation of Korea (NRF) grants funded by the Korean government (MSIT) (No. 2022R1A5A1033624 and No. 2023R1A2C1006600), the work of S.-Y. Choi was supported by the NRF grant funded by the Korean government (MSIT) (No. 2021R1F1A1046138), and the research of D. Kim was supported by BK21 FOUR Program by Pusan National University Research Grant, 2021-2022.
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Kim, D., Ha, M., Choi, SY. et al. Pricing of Vulnerable Timer Options. Comput Econ (2023). https://doi.org/10.1007/s10614-023-10469-1
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DOI: https://doi.org/10.1007/s10614-023-10469-1