{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:44:17Z","timestamp":1760150657682,"version":"build-2065373602"},"reference-count":33,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,12,7]],"date-time":"2023-12-07T00:00:00Z","timestamp":1701907200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"SeoulTech (Seoul National University of Science and Technology)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In this paper, we study a simplified approach to determine the pricing formula for vulnerable options involving two correlated underlying assets. We utilize an intensity-based model to describe the credit risk associated with these vulnerable options. Without the change of measure technique, we derive pricing formulas for vulnerable options involving two underlying assets based on the probabilistic approach. We provide closed-form pricing formulas for two specific types of options: the vulnerable exchange option and the vulnerable foreign equity option. Finally, we present numerical results to demonstrate the accuracy of our formulas using the Monte-Carlo method and the effect of various parameters on the price of options.<\/jats:p>","DOI":"10.3390\/axioms12121105","type":"journal-article","created":{"date-parts":[[2023,12,7]],"date-time":"2023-12-07T10:45:33Z","timestamp":1701945933000},"page":"1105","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Simplified Approach to the Pricing of Vulnerable Options with Two Underlying Assets in an Intensity-Based Model"],"prefix":"10.3390","volume":"12","author":[{"given":"Geonwoo","family":"Kim","sequence":"first","affiliation":[{"name":"School of Natural Sciences, Seoul National University of Science and Technology, Seoul 01811, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,7]]},"reference":[{"key":"ref_1","first-page":"449","article-title":"On the pricing of corporate debt: The risk structure of interest rates","volume":"29","author":"Merton","year":"1974","journal-title":"J. Financ."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1111\/j.1540-6261.1976.tb01891.x","article-title":"Valuing corporate securities: Some effects of bond indenture provisions","volume":"31","author":"Black","year":"1976","journal-title":"J. Financ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"541","DOI":"10.2307\/2330330","article-title":"The valuation of corporate liabilities as compound options","volume":"12","author":"Geske","year":"1977","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1111\/j.1540-6261.1995.tb05167.x","article-title":"Pricing options on financial securities subject to default risk","volume":"50","author":"Jarrow","year":"1995","journal-title":"J. Financ."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1007\/BF01531332","article-title":"On Cox processes and credit risky securities","volume":"2","author":"Lando","year":"1998","journal-title":"Rev. Deriv. Res."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1765","DOI":"10.1111\/0022-1082.00389","article-title":"Counterparty risk and the pricing of defaultable securities","volume":"56","author":"Jarrow","year":"2001","journal-title":"J. Financ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1111\/j.1540-6261.1987.tb02567.x","article-title":"The pricing of options with default risk","volume":"42","author":"Johnson","year":"1987","journal-title":"J. Financ."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1211","DOI":"10.1016\/0378-4266(95)00052-6","article-title":"Pricing Black-Scholes options with correlated credit risk","volume":"20","author":"Klein","year":"1996","journal-title":"J. Bank. Financ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1080\/14697680500362718","article-title":"Pricing Black\u2013Scholes options with correlated interest rate risk and credit risk: An extension","volume":"5","author":"Liao","year":"2005","journal-title":"Quant. Financ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1016\/j.najef.2018.07.001","article-title":"Pricing of vulnerable options with early counterparty credit risk","volume":"47","author":"Jeon","year":"2019","journal-title":"N. Am. J. Econ. Financ."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"410","DOI":"10.1002\/fut.22064","article-title":"Analytical valuation of Asian options with counterparty risk under stochastic volatility models","volume":"40","author":"Wang","year":"2020","journal-title":"J. Futur. Mark."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2077","DOI":"10.3934\/jimo.2021057","article-title":"Pricing vulnerable options under a jump-diffusion model with fast mean-reverting stochastic volatility","volume":"18","author":"He","year":"2022","journal-title":"J. Ind. Manag. Optim."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"110846","DOI":"10.1016\/j.chaos.2021.110846","article-title":"Pricing of vulnerable options under hybrid stochastic and local volatility","volume":"146","author":"Kim","year":"2021","journal-title":"Chaos Solitons Fractals"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1186\/s13662-023-03783-3","article-title":"An analytical approach to the pricing of an exchange option with default risk under a stochastic volatility model","volume":"2023","author":"Jeon","year":"2023","journal-title":"Adv. Contin. Discret. Model."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Kim, G. (2020). Valuation of Exchange Option with Credit Risk in a Hybrid Model. Mathematics, 8.","DOI":"10.3390\/math8112091"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1699","DOI":"10.1080\/13504851.2020.1713980","article-title":"Valuing vulnerable options with two underlying assets","volume":"27","author":"Wang","year":"2020","journal-title":"Appl. Econ. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/s13662-021-03486-7","article-title":"Closed-form pricing formula for foreign equity option with credit risk","volume":"2021","author":"Kim","year":"2021","journal-title":"Adv. Differ. Equations"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Jeon, J., and Kim, G. (2021). Power Exchange Option with a Hybrid Credit Risk under Jump-Diffusion Model. Mathematics, 10.","DOI":"10.3390\/math10010053"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1007\/s11147-022-09192-0","article-title":"Pricing vulnerable basket spread options with liquidity risk","volume":"26","author":"Dong","year":"2023","journal-title":"Rev. Deriv. Res."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1007\/s00780-004-0122-y","article-title":"Valuation of credit default swaps and swaptions","volume":"8","author":"Jamshidian","year":"2004","journal-title":"Financ. Stochastics"},{"key":"ref_21","first-page":"25","article-title":"Credit default swap valuation with counterparty risk","volume":"74","author":"Leung","year":"2005","journal-title":"Kyoto Econ. Rev."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1007\/s11424-017-5126-8","article-title":"Pricing credit derivatives under fractional stochastic interest rate models with jumps","volume":"30","author":"Zhang","year":"2017","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_23","first-page":"412890","article-title":"Total return swap valuation with counterparty risk and interest rate risk","volume":"12","author":"Wang","year":"2014","journal-title":"Abstr. Appl. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/s13226-020-0405-9","article-title":"The pricing of total return swap under default contagion models with jump-diffusion interest rate risk","volume":"51","author":"Wang","year":"2020","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.insmatheco.2014.10.007","article-title":"Analytical pricing of vulnerable options under a generalized jump\u2013diffusion model","volume":"60","author":"Fard","year":"2015","journal-title":"Insur. Math. Econ."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1017\/S0269964816000292","article-title":"Analytical valuation of vulnerable options in a discrete-time framework","volume":"31","author":"Wang","year":"2017","journal-title":"Probab. Eng. Informational Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.chaos.2017.05.012","article-title":"Explicit formula for the valuation of catastrophe put option with exponential jump and default risk","volume":"101","author":"Koo","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.cam.2019.01.019","article-title":"Pricing vulnerable power exchange options in an intensity based framework","volume":"355","author":"Pasricha","year":"2019","journal-title":"J. Comput. Appl. Math."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"113412","DOI":"10.1016\/j.cam.2021.113412","article-title":"Analytical valuation of vulnerable European and Asian options in intensity-based models","volume":"393","author":"Wang","year":"2021","journal-title":"J. Comput. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1111\/j.1540-6261.1978.tb03397.x","article-title":"The value of an option to exchange one asset for another","volume":"33","author":"Margrabe","year":"1978","journal-title":"J. Financ."},{"key":"ref_31","unstructured":"Kwok, Y.K. (2008). Mathematical Models of Financial Derivatives, Springer."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1142\/S0219024900000127","article-title":"Currency-translated foreign equity options with path dependent features and their multi-asset extensions","volume":"3","author":"Kwok","year":"2000","journal-title":"Int. J. Theor. Appl. Financ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1023\/A:1022822004204","article-title":"Contingent claims on foreign assets following jump-diffusion processes","volume":"6","author":"Martzoukos","year":"2003","journal-title":"Rev. Deriv. Res."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/12\/1105\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:34:57Z","timestamp":1760132097000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/12\/1105"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,12,7]]},"references-count":33,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["axioms12121105"],"URL":"https:\/\/doi.org\/10.3390\/axioms12121105","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2023,12,7]]}}}