# American Institute of Mathematical Sciences

January  2018, 3: 2 doi: 10.1186/s41546-018-0027-x

## Arbitrage-free pricing of derivatives in nonlinear market models

 1. Department of Applied Mathematics, Illinois Institute of Technology, Chicago 60616, IL, USA; 2. School of Mathematics and Statistics, University of Sydney, Sydney 2006, NSW, Australia; 3. Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warszawa, Poland

Received  January 29, 2017 Revised  March 15, 2018 Published  April 2018

Fund Project: The research of I. Cialenco and M. Rutkowski was supported by the DVC Research Bridging Support Grant BSDEs Approach to Models with Funding Costs. Part of the research was completed while I. Cialenco and M. Rutkowski were visiting the Institute for Pure and Applied Mathematics (IPAM) at UCLA, which is funded by the National Science Foundation. We would also like to thank the anonymous referees and Stéphane Crépey for their insightful and helpful comments and suggestions, which helped us greatly to improve the final manuscript.

The objective of this paper is to provide a comprehensive study of the no-arbitrage pricing of financial derivatives in the presence of funding costs, the counterparty credit risk and market frictions affecting the trading mechanism, such as collateralization and capital requirements. To achieve our goals, we extend in several respects the nonlinear pricing approach developed in (El Karoui and Quenez 1997) and (El Karoui et al. 1997), which was subsequently continued in (Bielecki and Rutkowski 2015).
Citation: Tomasz R. Bielecki, Igor Cialenco, Marek Rutkowski. Arbitrage-free pricing of derivatives in nonlinear market models. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 2-. doi: 10.1186/s41546-018-0027-x
##### References:
 [1] Albanese, C, Crépey, S:XVA analysis from the balance sheet. Working paper (2017) [2] Albanese, C, Caenazzo, S, Crépey, S:Credit, funding, margin, and capital valuation adjustments for bilateral portfolios. Probability, Uncertainty and Quantitative Risk. 2(7), 1-26 (2017) [3] Bergman, YZ:Option pricing with differential interest rates. Review of Financial Studies. 8, 475-500(1995) [4] Bichuch, M, et al.:Arbitrage-free XVA. Mathematical Finance. 28(2), 582-620 (2018) [5] Bielecki, TR, Rutkowski, M:Valuation and hedging of contracts with funding costs and collateralization. SIAM Journal of Financial Mathematics. 6, 594-655 (2015) [6] Bielecki, TR, Jeanblanc, M, Rutkowski, M:Hedging of defaultable claims. In:R. Carmona, et al. (eds.) Paris-Princeton Lectures on Mathematical Finance 2003, Lecture Notes in Mathematics, Vol. 1847, pp. 1-132. Springer, Berlin Heidelberg New York (2004) [7] Bielecki, TR, Jeanblanc, M, Rutkowski, M:Replication of contingent claims in a reduced-form credit risk model with discontinuous asset prices. Stochastic Models. 22, 661-687 (2006) [8] Bielecki, TR, Jeanblanc, M, Rutkowski, M:Credit Risk Modeling. Osaka University CSFI Lecture Notes Series. Osaka University Press, Osaka (2008) [9] Brigo, D, Pallavicini, A:Non-linear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks. Journal of Financial Engineering. 1, 1-60 (2014) [10] Brigo, D, Buescu, C, Rutkowski, M:Funding, repo and credit inclusive valuation as modified option pricing. Operations Research Letters. 45, 665-670 (2017) [11] Brigo, D, Buescu, C, Francischello, M, Pallavicini, A, Rutkowski, M:Risk-neutral valuation under differential funding costs, defaults and collateralization. Working paper (2018) [12] Burgard, C, Kjaer, M:Partial differential equations representations of derivatives with counterparty risk and funding costs. Journal of Credit Risk. 7, 1-19 (2011) [13] Burgard, C, Kjaer, M:Funding costs, funding strategies. Risk. 12(26), 82-87 (2013) [14] Carassus, L, Pham, H, Touzi, N:No arbitrage in discrete time under portfolio constraints. Mathematical Finance. 11, 315-329 (2001) [15] Carbone, R, Ferrario, B, Santacroce, M:Backward stochastic differential equations driven by càdlàg martingales. Theory of Probability and its Applications. 52, 304-314 (2008) [16] Cheridito, P, Nam, K:BSEs, BSDEs and fixed point problems. Annals of Probability. 45(6A), 3795-3828(2017) [17] Crépey, S:Bilateral counterparty risk under funding constraints-Part I:Pricing. Mathematical Finance. 25, 1-22 (2015a) [18] Crépey, S:Bilateral counterparty risk under funding constraints-Part II:CVA. Mathematical Finance. 25, 23-50 (2015b) [19] Crépey, S, Song, S:Counterparty risk and funding:immersion and beyond. Finance and Stochastics. 20, 901-930 (2016) [20] Crépey, S, Song, S:Invariance times. Annals of Probability. 45(6B), 4632-4674 (2017) [21] Crépey, S, Bielecki, TR, Brigo, D:Counterparty Risk and Funding:A Tale of Two Puzzles. Chapman & Hall/CRC Financial Mathematics Series, p. 365. CRC Press, Boca Raton, FL (2014) [22] Crépey, S, Elie, R, Sabbagh, W:When capital is the funding source:The XVA anticipated BSDEs. Working paper (2017) [23] Delbaen, F, Schachermayer, W:The Mathematics of Arbitrage, p. 372. Springer, Berlin Heidelberg New York (2006) [24] Dumitrescu, R, Quenez, M.C, Sulem, A:Game options in an imperfect market with default. SIAM Journal on Financial Mathematics. 8, 532-559 (2017) [25] El Karoui, N, Huang, S.J:A general result of existence and uniqueness of backward stochastic differential equations. In:H. Brezis, et al. (eds.) Backward Stochastic Differential Equations. Pitman Research Notes in Mathematics Series, Vol. 364, pp. 27-36. Addison Wesley Longman, Harlow, Essex (1997) [26] El Karoui, N, Quenez, M.C:Non-linear pricing theory and backward stochastic differential equations. In:B. Biais, et al. (eds.) Financial Mathematics, Lecture Notes in Mathematics, Vol. 1656, pp. 191-246. Springer, Berlin Heidelberg New York (1997) [27] El Karoui, N, Peng, S, Quenez, M.C:Backward stochastic differential equation in finance. Mathematical Finance. 7, 1-71 (1997) [28] Fahim, A, Huang, Y:Model-independent superhedging under portfolio constraints. Finance and Stochastics. 20, 51-81 (2016) [29] Fontana, C:Weak and strong no-arbitrage conditions for continuous financial markets. International Journal of Theoretical and Applied Finance. 18, 1550005 (2015) [30] Fujii, M, Takahashi, A:Derivative pricing under asymmetric and imperfect collateralization and CVA. Quantitative Finance. 13(5), 749-768 (2013) [31] Karatzas, I, Kardaras, K:The numeraire portfolio in semimartingale financial models. Finance and Stochastics. 11, 447-493 (2007) [32] Karatzas, I, Kou, S:On the pricing of contingent claims under constraints. Annals of Applied Probability. 6, 321-369 (1996) [33] Karatzas, I, Kou, S:Hedging American contingent claims with constrained portfolios. Finance and Stochastics. 2, 215-258 (1998) [34] Kardaras, K:Market viability via absence of arbitrage of the first kind. Finance and Stochastics. 16, 651-667 (2012) [35] Kenyon, C, Green, A:Regulatory-compliant derivatives pricing is not risk-neutral. Working paper (2013) [36] Kenyon, C, Green, A:MVA:Initial margin valuation adjustment by replication and regression. Working paper (2014a) [37] Kenyon, C, Green, A:Warehousing credit (CVA) risk, capital (KVA) and tax (TVA) consequences. Working paper (2014b) [38] Korn, R:Contingent claim valuation in a market with different interest rates. Mathematical Methods of Operations Research. 42(3), 255-274 (1995) [39] Mercurio, F:Bergman, Piterbarg and beyond:Pricing derivatives under collateralization and differential rates. In:JA Londono, et al. (eds.) Actuarial Sciences and Quantitative Finance, Springer Proceedings in Mathematics and Statistics, Vol. 135, pp. 65-95. Springer, Berlin Heidelberg New York (2013) [40] Nie, T, Rutkowski, M:Fair bilateral prices in Bergman's model with exogenous collateralization. International Journal of Theoretical and Applied Finance. 18, 1550048 (2015) [41] Nie, T, Rutkowski, M:A BSDE approach to fair bilateral pricing under endogenous collateralization. Finance and Stochastics. 20, 855-900 (2016a) [42] Nie, T, Rutkowski, M:BSDEs driven by a multi-dimensional martingale and their applications to market models with funding costs. Theory of Probability and its Applications. 60, 686-719 (2016b) [43] Nie, T, Rutkowski, M:Fair bilateral pricing under funding costs and exogenous collateralization. Mathematical Finance. 28(2), 621-655 (2018) [44] Pallavicini, A, Perini, D, Brigo, D:Funding, collateral and hedging:uncovering the mechanism and the subtleties of funding valuation adjustments. Working paper (2012a) [45] Pallavicini, A, Perini, D, Brigo, D:Funding valuation adjustment:a consistent framework including CVA, DVA, collateral, netting rules and re-hypothecation. Working paper (2012b) [46] Peng, S, Xu, X:BSDEs with random default time and their applications to default risk. Working paper(2009) [47] Piterbarg, V:Funding beyond discounting:collateral agreements and derivatives pricing. Risk. 23(2), 97-102 (2010) [48] Pulido, S:The fundamental theorem of asset pricing, the hedging problem and maximal claims in financial markets with short sales prohibitions. Annals of Applied Probability. 24, 54-75 (2014) [49] Quenez, MC, Sulem, A:BSDEs with jumps, optimization and applications to dynamic risk measures. Stochastic Processes and their Applications. 123, 3328-3357 (2013) [50] Takaoka, K, Schweizer, M:A note on the condition of no unbounded profit with bounded risk. Finance and Stochastics. 18, 393-405 (2014) [51] Zheng, S, Zong, G:A note on BSDEs and SDEs with time advanced and delayed coefficients. Working paper (2017)

show all references

##### References:
 [1] Albanese, C, Crépey, S:XVA analysis from the balance sheet. Working paper (2017) [2] Albanese, C, Caenazzo, S, Crépey, S:Credit, funding, margin, and capital valuation adjustments for bilateral portfolios. Probability, Uncertainty and Quantitative Risk. 2(7), 1-26 (2017) [3] Bergman, YZ:Option pricing with differential interest rates. Review of Financial Studies. 8, 475-500(1995) [4] Bichuch, M, et al.:Arbitrage-free XVA. Mathematical Finance. 28(2), 582-620 (2018) [5] Bielecki, TR, Rutkowski, M:Valuation and hedging of contracts with funding costs and collateralization. SIAM Journal of Financial Mathematics. 6, 594-655 (2015) [6] Bielecki, TR, Jeanblanc, M, Rutkowski, M:Hedging of defaultable claims. In:R. Carmona, et al. (eds.) Paris-Princeton Lectures on Mathematical Finance 2003, Lecture Notes in Mathematics, Vol. 1847, pp. 1-132. Springer, Berlin Heidelberg New York (2004) [7] Bielecki, TR, Jeanblanc, M, Rutkowski, M:Replication of contingent claims in a reduced-form credit risk model with discontinuous asset prices. Stochastic Models. 22, 661-687 (2006) [8] Bielecki, TR, Jeanblanc, M, Rutkowski, M:Credit Risk Modeling. Osaka University CSFI Lecture Notes Series. Osaka University Press, Osaka (2008) [9] Brigo, D, Pallavicini, A:Non-linear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks. Journal of Financial Engineering. 1, 1-60 (2014) [10] Brigo, D, Buescu, C, Rutkowski, M:Funding, repo and credit inclusive valuation as modified option pricing. Operations Research Letters. 45, 665-670 (2017) [11] Brigo, D, Buescu, C, Francischello, M, Pallavicini, A, Rutkowski, M:Risk-neutral valuation under differential funding costs, defaults and collateralization. Working paper (2018) [12] Burgard, C, Kjaer, M:Partial differential equations representations of derivatives with counterparty risk and funding costs. Journal of Credit Risk. 7, 1-19 (2011) [13] Burgard, C, Kjaer, M:Funding costs, funding strategies. Risk. 12(26), 82-87 (2013) [14] Carassus, L, Pham, H, Touzi, N:No arbitrage in discrete time under portfolio constraints. Mathematical Finance. 11, 315-329 (2001) [15] Carbone, R, Ferrario, B, Santacroce, M:Backward stochastic differential equations driven by càdlàg martingales. Theory of Probability and its Applications. 52, 304-314 (2008) [16] Cheridito, P, Nam, K:BSEs, BSDEs and fixed point problems. Annals of Probability. 45(6A), 3795-3828(2017) [17] Crépey, S:Bilateral counterparty risk under funding constraints-Part I:Pricing. Mathematical Finance. 25, 1-22 (2015a) [18] Crépey, S:Bilateral counterparty risk under funding constraints-Part II:CVA. Mathematical Finance. 25, 23-50 (2015b) [19] Crépey, S, Song, S:Counterparty risk and funding:immersion and beyond. Finance and Stochastics. 20, 901-930 (2016) [20] Crépey, S, Song, S:Invariance times. Annals of Probability. 45(6B), 4632-4674 (2017) [21] Crépey, S, Bielecki, TR, Brigo, D:Counterparty Risk and Funding:A Tale of Two Puzzles. Chapman & Hall/CRC Financial Mathematics Series, p. 365. CRC Press, Boca Raton, FL (2014) [22] Crépey, S, Elie, R, Sabbagh, W:When capital is the funding source:The XVA anticipated BSDEs. Working paper (2017) [23] Delbaen, F, Schachermayer, W:The Mathematics of Arbitrage, p. 372. Springer, Berlin Heidelberg New York (2006) [24] Dumitrescu, R, Quenez, M.C, Sulem, A:Game options in an imperfect market with default. SIAM Journal on Financial Mathematics. 8, 532-559 (2017) [25] El Karoui, N, Huang, S.J:A general result of existence and uniqueness of backward stochastic differential equations. In:H. Brezis, et al. (eds.) Backward Stochastic Differential Equations. Pitman Research Notes in Mathematics Series, Vol. 364, pp. 27-36. Addison Wesley Longman, Harlow, Essex (1997) [26] El Karoui, N, Quenez, M.C:Non-linear pricing theory and backward stochastic differential equations. In:B. Biais, et al. (eds.) Financial Mathematics, Lecture Notes in Mathematics, Vol. 1656, pp. 191-246. Springer, Berlin Heidelberg New York (1997) [27] El Karoui, N, Peng, S, Quenez, M.C:Backward stochastic differential equation in finance. Mathematical Finance. 7, 1-71 (1997) [28] Fahim, A, Huang, Y:Model-independent superhedging under portfolio constraints. Finance and Stochastics. 20, 51-81 (2016) [29] Fontana, C:Weak and strong no-arbitrage conditions for continuous financial markets. International Journal of Theoretical and Applied Finance. 18, 1550005 (2015) [30] Fujii, M, Takahashi, A:Derivative pricing under asymmetric and imperfect collateralization and CVA. Quantitative Finance. 13(5), 749-768 (2013) [31] Karatzas, I, Kardaras, K:The numeraire portfolio in semimartingale financial models. Finance and Stochastics. 11, 447-493 (2007) [32] Karatzas, I, Kou, S:On the pricing of contingent claims under constraints. Annals of Applied Probability. 6, 321-369 (1996) [33] Karatzas, I, Kou, S:Hedging American contingent claims with constrained portfolios. Finance and Stochastics. 2, 215-258 (1998) [34] Kardaras, K:Market viability via absence of arbitrage of the first kind. Finance and Stochastics. 16, 651-667 (2012) [35] Kenyon, C, Green, A:Regulatory-compliant derivatives pricing is not risk-neutral. Working paper (2013) [36] Kenyon, C, Green, A:MVA:Initial margin valuation adjustment by replication and regression. Working paper (2014a) [37] Kenyon, C, Green, A:Warehousing credit (CVA) risk, capital (KVA) and tax (TVA) consequences. Working paper (2014b) [38] Korn, R:Contingent claim valuation in a market with different interest rates. Mathematical Methods of Operations Research. 42(3), 255-274 (1995) [39] Mercurio, F:Bergman, Piterbarg and beyond:Pricing derivatives under collateralization and differential rates. In:JA Londono, et al. (eds.) Actuarial Sciences and Quantitative Finance, Springer Proceedings in Mathematics and Statistics, Vol. 135, pp. 65-95. Springer, Berlin Heidelberg New York (2013) [40] Nie, T, Rutkowski, M:Fair bilateral prices in Bergman's model with exogenous collateralization. International Journal of Theoretical and Applied Finance. 18, 1550048 (2015) [41] Nie, T, Rutkowski, M:A BSDE approach to fair bilateral pricing under endogenous collateralization. Finance and Stochastics. 20, 855-900 (2016a) [42] Nie, T, Rutkowski, M:BSDEs driven by a multi-dimensional martingale and their applications to market models with funding costs. Theory of Probability and its Applications. 60, 686-719 (2016b) [43] Nie, T, Rutkowski, M:Fair bilateral pricing under funding costs and exogenous collateralization. Mathematical Finance. 28(2), 621-655 (2018) [44] Pallavicini, A, Perini, D, Brigo, D:Funding, collateral and hedging:uncovering the mechanism and the subtleties of funding valuation adjustments. Working paper (2012a) [45] Pallavicini, A, Perini, D, Brigo, D:Funding valuation adjustment:a consistent framework including CVA, DVA, collateral, netting rules and re-hypothecation. Working paper (2012b) [46] Peng, S, Xu, X:BSDEs with random default time and their applications to default risk. Working paper(2009) [47] Piterbarg, V:Funding beyond discounting:collateral agreements and derivatives pricing. Risk. 23(2), 97-102 (2010) [48] Pulido, S:The fundamental theorem of asset pricing, the hedging problem and maximal claims in financial markets with short sales prohibitions. Annals of Applied Probability. 24, 54-75 (2014) [49] Quenez, MC, Sulem, A:BSDEs with jumps, optimization and applications to dynamic risk measures. Stochastic Processes and their Applications. 123, 3328-3357 (2013) [50] Takaoka, K, Schweizer, M:A note on the condition of no unbounded profit with bounded risk. Finance and Stochastics. 18, 393-405 (2014) [51] Zheng, S, Zong, G:A note on BSDEs and SDEs with time advanced and delayed coefficients. Working paper (2017)
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