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Review of Quantitative Finance and Accounting

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29-05-2021 | Original Research

Analytical pricing formulae for vulnerable vanilla and barrier options

Authors: Liang-Chih Liu, Chun-Yuan Chiu, Chuan-Ju Wang, Tian-Shyr Dai, Hao-Han Chang

Published in: Review of Quantitative Finance and Accounting | Issue 1/2022

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Abstract

This paper proposes analytically vulnerable vanilla option pricing formulae that simultaneously consider the premature default, the correlation between the underlying asset and the issuer’s asset, and other outstanding debts of the issuer. Our pricing formulae can be easily extended to solve the problem of pricing vulnerable barrier options, which has been rarely studied before. We show that previous studies on pricing (non)-vulnerable vanilla options and barrier options are degenerate cases of our formulae. We conduct numerical experiments to analyze the relations among the financial/contract parameters and counterparty risk, and also empirically evaluate vulnerable vanilla warrants on the TAIEX issued by Capital Securities with detailed studies of parameter calibrations to examine the robustness of our approach.

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TAIEX is the abbreviation for Taiwan Capitalization Weighted Stock Index, a stock market index for companies traded on the Taipei Stock Exchange.

We thank a referee for reminding us of this contract feature.

This assumption is mild since derivatives usually play tiny roles in many firms’ debt obligations as in Klein (1996).

Note that \(\lim _{x\rightarrow \infty }N_2(x, y, \rho ) = N_1(y)\) for any arbitrary constants \(\rho \) and y.

Note that the pricing formula for vulnerable vanilla calls in Klein (1996) is derived under Merton’s model, whereas the formula in Sec. 3 of this paper is derived under the first-passage model.

Note that \(\lim _{x\rightarrow \infty }N_2(x, y, \rho ) = N_1(y)\) for any arbitrary constants \(\rho \) and y.

These two parameters are not required for pricing vulnerable vanilla options.

See http://www.taifex.com.tw/cht/3/optDailyMarketReport.

See https://www.ba.org.tw/Taibor/TaiborList?type=DayHistory.

Note that TAIEX options can still be treated as the otherwise almost identical non-vulnerable counterparts of our target warrants, due to a small average interval \(\tau = 30\) min.

Black F, Cox J (1976) Valuing corporate securities: some effects of bond indenture provisions. J Finance 42(2):351–367CrossRef

Crosbie P, Bohn J (2002) Modeling default risk. Working paper KMV Corp

Dai TS, Wang CJ, Lyuu YD (2013) A multiphase, flexible, and accurate lattice for pricing complex derivatives with multiple market variables. J Futures Mark 33(9):795–826CrossRef

Fard FA (2015) Analytical pricing of vulnerable options under a generalized jump-diffusion model. Insur Math Econ 60:19–28CrossRef

Haworth H, Reisinger C, Shaw W (2008) Modelling bonds and credit default swaps using a structural model with contagion. Quant Finance 8(7):669–680CrossRef

Hull JC (2014) Options, futures, and other derivatives, 9th edn. Prentice Hall, Hoboken

Hull JC, White AD (1995) The impact of default risk on the prices of options and other derivative securities. J Bank Finance 19(2):299–322CrossRef

Hull JC, White AD (2001) Valuing credit default swaps II: modeling default correlations. J Deriv 8(3):12–21CrossRef

Hung MW, Liu YH (2005) Pricing vulnerable options in incomplete markets. J Futures Mark Futures Options Other Deriv Products 25(2):135–170

Jeon J, Yoon JH, Kang M (2016) Valuing vulnerable geometric Asian options. Comput Math Appl 71(2):676–691CrossRef

Johnson H, Stulz R (1987) The pricing of options with default risk. J Finance 42(2):267–280CrossRef

Kemna AG, Vorst AC (1990) A pricing method for options based on average asset values. J Bank Finance 14(1):113–129CrossRef

Kiesel R, Scherer M (2007) Dynamic credit portfolio modelling in structural models with jumps. Universität Ulm, preprint

Klein P (1996) Pricing Black–Scholes options with correlated credit risk. J Bank Finance 20(7):1211–1229CrossRef

Klein P, Inglis M (1999) Valuation of European options subject to financial distress and interest rate risk. J Deriv 6(3):44–56CrossRef

Klein P, Inglis M (2001) Pricing vulnerable European options when the option’s payoff can increase the risk of financial distress. J Bank Finance 25(5):993–1012CrossRef

Levendorskiĭ S (2014) Method of paired contours and pricing barrier options and CDSs of long maturities. Int J Theor Appl Finance 17(5):1450033CrossRef

Liao SL, Huang HH (2005) Pricing Black–Scholes options with correlated interest rate risk and credit risk: an extension. Quant Finance 5(5):443–457CrossRef

Lipton A, Savescu I (2013) CDSs, CVA and DVA—a structural approach. Risk 26(4):56–61

Lipton A, Savescu I (2014) Pricing credit default swaps with bilateral value adjustments. Quant Finance 14(1):171–188CrossRef

Lipton A, Sepp A (2009) Credit value adjustment for credit default swaps via the structural default model. J Credit Risk 5(2):123–146CrossRef

Ma Y, Shrestha K, Xu W (2017) Pricing vulnerable options with jump clustering. J Futures Mark 37(12):1155–1178CrossRef

Ma C, Yue S, Wu H, Ma Y (2020) Pricing vulnerable options with stochastic volatility and stochastic interest rate. Comput Econ 56:391–429CrossRef

Merton RC (1973) Theory of rational option pricing. Bell J Econ Manag Sci 4(1):141–183CrossRef

Merton RC (1974) On the pricing of corporate debt: the risk structure of interest rates. J Finance 29(2):449–470

Niu H, Wang D (2016) Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy. Quant Finance 16(7):1129–1145CrossRef

Rubinstein M, Reiner E (1991) Breaking down the barriers. Risk 4(8):28–35

Sarkar S (2003) Early and late calls of convertible bonds: theory and evidence. J Bank Finance 27(7):1349–1374CrossRef

Shreve S (2004) Stochastic calculus for finance (II): continuous-time models. Springer, New YorkCrossRef

Tian L, Wang G, Wang X, Wang Y (2014) Pricing vulnerable options with correlated credit risk under jump-diffusion processes. J Futures Mark 34(10):957–979CrossRef

Wang X (2017) Differences in the prices of vulnerable options with different counterparties. J Futures Mark 37(2):148–163CrossRef

Wang W, Wang W (2010) Pricing vulnerable options under a Markov-modulated regime switching model. Commun Stat Theory Methods 39(19):3421–3433CrossRef

Yang SJ, Lee MK, Kim JH (2014) Pricing vulnerable options under a stochastic volatility model. Appl Math Lett 34:7–12CrossRef

Zhou C (2001) An analysis of default correlations and multiple defaults. Rev Financ Stud 14(2):555–576CrossRef

Title: Analytical pricing formulae for vulnerable vanilla and barrier options
Authors: Liang-Chih Liu
Chun-Yuan Chiu
Chuan-Ju Wang
Tian-Shyr Dai
Hao-Han Chang
Publication date: 29-05-2021
Publisher: Springer US
Published in: Review of Quantitative Finance and Accounting / Issue 1/2022
Print ISSN: 0924-865X
Electronic ISSN: 1573-7179
DOI: https://doi.org/10.1007/s11156-021-00990-5

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