Abstract
We consider the problem of computing the credit value adjustment (CVA) of a European option in presence of the wrong way risk in a default intensity setting. Namely we model the asset price evolution as solution to a linear equation that might depend on different stochastic factors and we provide an approximate evaluation of the option’s price, by exploiting a correlation expansion approach, introduced in Antonelli and Scarlatti (Finance Stoch 13:269–303, 2009). We also extend our theoretical analysis to include some further value adjustments, for instance due to collateralization and funding costs. Finally, in the CVA case, we compare the numerical performance of our method with the one recently proposed by Brigo and Vrins (Eur J Oper Res 269:1154–1164, 2018) and Brigo et al. (Innovations in insurance, risk and asset management, WSPC proceedings, 2018), in the case of a call option driven by a GBM correlated with a CIR default intensity. We additionally compare with the numerical evaluations obtained by other methods.
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References
Abramowitz, M., & Stegun, I. (1972). Handbook of mathematical functions. New York: Dover Publ. Inc.
Albanese, C., & Lawi, S. (2005). Laplace transforms for integrals of Markov processes. Markov Process and Related Fields, 11, 677–724.
Antonelli, F., Ramponi, A., & Scarlatti, S. (2010). Exchange option pricing under stochastic volatility: A correlation expansion. Review of Derivatives Research, 13, 45–73.
Antonelli, F., Ramponi, A., & Scarlatti, S. (2016). Random time forward-starting options. International Journal of Theoretical and Applied Finance, 19, 8.
Antonelli, F., Ramponi, A., & Scarlatti, S. (2019). Real analyticity in multi-factor Black & Scholes models for option prices and the Faà di Bruno formula(work in progress, preprint upon request to the authors).
Antonelli, F., & Scarlatti, S. (2009). Pricing options under stochastic volatility: A power series approach. Finance and Stochastics, 13, 269–303.
Bielecki, T. R., Crepey, S., & Brigo, D. (2014). Counterparty risk and funding: A tale of two puzzles. Boca Raton: Chapman and Hall/CRC.
Bielecki, T. R., Jeanblanc, M., & Rutkowski, M. (2009). Valuation and hedging of credit derivatives. Lecture notes CIMPA- UNESCO Morocco School.
Bielecki, T. R., & Rutkowski, M. (2002). Credit risk: Modeling. Valuation and Hedging, Berlin: Springer.
Bo, L., Capponi, A., & Chen, P. (2019). Credit portfolio selection with decaying contagion intensities. Matematical Finance, 29, 137–173.
Bo, L., & Ceci, C. (2019). Locally risk-minimizing hedging of counterparty risk for portfolio of credit derivatives. Applied Mathematics & Optimization. https://doi.org/10.1007/s00245-018-9549-y.
Brigo, D., Beuscu, C., & Rutkowski, M. (2017). Funding, repo and credit inclusive valuation as modified option pricing. Operations Research Letters, 45(6), 665–670.
Brigo, D., Francischello, M., & Pallavicini, A. (2016). Analysis of nonlinear valuation equations under credit and funding effects. In Innovations in derivatives martkets. Springer proceedings in mathematics & statistics (Vol. 165).
Brigo, D., Hvolby, T., & Vrins, F. (2018). Wrong-way risk adjusted exposure: Analytical approximations for options in default intensity models. In Innovations in insurance, risk and asset management, WSPC proceedings.
Brigo, D., Liu, Q. D., Pallavicini, A., & Sloth, D. (2016). Nonlinear valuation under collateralization, credit risk, and funding costs. In Innovations in derivatives martkets. Springer proceedings in mathematics & statistics, Vol. 165.
Brigo, D., Morini, M., & Pallavicini, A. (2013). Counterparty credit risk, collateral and funding: With pricing cases for all asset classes. Hoboken: Wiley.
Brigo, D., & Vrins, F. (2018). Disentangling wrong-way risk: Pricing credit valuation adjustment via change of measures. European Journal of Operational Research, 269, 1154–1164.
Canabarro, E., & Duffie, D. (2003). Measuring and marking counterparty risk? Asset/Liability Management of Financial Institutions. London: Euromoney Books.
Capponi, A., Pagliarani, S., & Vargiolu, T. (2014). Pricing vulnerable claims in a Levy driven model. Finance and Stochastics, 18, 755–789.
Cherubini, U. (2013). Credit valuation adjustment and wrong way risk. Quantitative Finance Letters, 1, 9–15.
Cherubini, U., & Luciano, E. (2003). Pricing vulnerable options with copulas. Journal of Risk Finance, 5, 27–39.
Duffie, D., & Huang, M. (1996). Swap rates and credit quality. The Journal of Finance, 51(3), 687–720.
Duffie, D., & Singleton, K. J. (1999). Modeling term structures of defaultable bonds. Review Financial Studies, 12, 687–720.
Fard, F. A. (2015). Analytical pricing of vulnerable options under a generalized jump-diffusion model. Insurance Mathematics and Economics, 60, 19–28.
Filipovic, D., & Mayerhofer, E. (2009). Affine diffusion processes: Theory and applications. Radon Series on Computational and Applied Mathematics, 8, 1–40.
Frey, R., Embrechts, P., & McNeil, A. J. (2015). Quantitative risk management: Concepts, techniques and tools (revised ed.). Princeton: Princeton University Press.
Gapeev, P. V. (2014). Some extensions of Norros’ lemma in models with several defaults. Inspired by finance. In Yu. M. Kabanov, M. Rutkowski, & Th. Zariphopoulou (Eds.), The Musiela Festschrift (pp. 273–281). Springer, Berlin.
Gapeev, P. V., Jeanblanc, M., Li, L., & Rutkowski, M. (2010). Constructing random measures with given survival processes and applications to valuation of credit derivatives. In C. Chiarella & A. Novikov (Eds.), Contemporary quantitative finance, essays in Honour of Eckhard Platen (pp. 255–280). Berlin: Springer.
Glasserman, P., & Yang, L. (2016). Bounding wrong way risk in cva calculations. Mathematical Finance, 28, 268–305.
Glau, K., Grbac, Z., Scherer, M., & Zagst, R. (Eds.). (2016). Innovations in derivatives markets. In Springer proceedings in mathematics & statistics (Vol. 165).
Gregory, J. (2012). Counterparty credit risk and credit value adjustment. Hoboken: Wiley.
Hull, J., & White, A. (1995). The impact of default risk on the prices of options and other derivative securities. Journal of Banking & Finance, 19, 299–322.
Hull, J., & White, A. (2012). CVA and wrong way risk. Financial Analyst Journal, 68, 58–69.
Kao, L. J. (2016). Credit valuation adjustment of cap and floor with counterparty risk: A structural pricing model for vulnerable European options. Review of Derivatives Research, 19, 41–64.
Jarrow, R., & Turnbull, S. (1995). Pricing derivatives on financial securities subject to credit risk. Journal of Finance, 50, 53–85.
Johnson, H., & Stulz, R. (1987). The pricing of options with default risk. Journal of Finance, 42, 267–280.
Knaber, P., & Angermann, L. (2003). Numerical methods for elliptic and parabolic partial differential equations. Berlin: Springer.
Klein, P. (1996). Pricing Black–Scholes options with correlated credit risk. Journal of Banking & Finance, 20, 1211–1229.
Klein, P., & Inglis, M. (1999). Valuation of European options subject to financial distress and interest rate risk. Journal of Derivatives, 6, 44–56.
Klein, P., & Inglis, M. (2001). Pricing vulnerable European option’s when the option payoff can increase the risk of financial distress. Journal of Banking & Finance, 25, 993–1012.
Kim, Y. J., & Kunimoto, N. (1999). Pricing options under stochastic interest rates: A new approach. Asia-Pacific Financial Markets, 6, 49–70.
Niu, H., & Wang, D. (2016). Pricing vulnerable options with correlated jump-diffusion processes depending on various states of the economy. Quantitative Finance, 16(7), 1129–1145.
Lando, D. (1998). On Cox processes and credit risky securities. Review of Derivatives Research, 2, 99–120.
Lando, D. (2004). Credit risk modeling. Princeton: Princeton University Press.
Lepage, T., Lawi, S., Tupper, P., & Bryant, D. (2006). Continuous and tractable models for the variation of evolutionary rates. Mathematical Biosciences, 199, 216–233.
Nikeghbali, A. (2006). An essay on the general theory of stochastic processes. Probability Surveys, 3, 345–412.
Pykhtin, M., & Rosen, D. (2010). Pricing counterparty risk at the trade level and CVA allocations. Journal of Credit Risk, 6, 3–38.
Prayoga, A., & Privault, N. (2017). Pricing CIR yield options by conditional moment matching. Asia-Pacific Financial Markets, 24(1), 19–38.
Rabinovitch, R. (1989). Pricing stock and bond options when the default-free rate is stochastic. The Journal of Financial and Quantitative Analysis, 24(4), 447–457.
Rosen, D., & Saunders, D. (2012). CVA the wrong way. Journal Risk Management Financial Institutions, 5, 252–272.
Tian, L., Wang, G., Wang, X., & Wang, Y. (2014). Pricing vulnerable options with correlated credit risk under jump-diffusion processes. Journal of Futures Markets, 34, 957–979.
Zhu, S., & Pykhtin, M. (2007). A guide to modeling counterparty credit risk. GARP Risk Review. July/August 2007.
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The authors gratefully acknowledge the anonymous referees for the careful reading and the constructive comments, which led to an improved version of the paper.
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Antonelli, F., Ramponi, A. & Scarlatti, S. CVA and vulnerable options pricing by correlation expansions. Ann Oper Res 299, 401–427 (2021). https://doi.org/10.1007/s10479-019-03367-z
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DOI: https://doi.org/10.1007/s10479-019-03367-z
Keywords
- Credit value adjustment
- Vulnerable options
- Counterparty credit risk
- Wrong way risk
- XVA
- Affine processes
- Duhamel principle