Arbitrage-free pricing of derivatives in nonlinear market models

Abstract

which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.Abstract 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 , which was subsequently continued in (Bielecki and Rutkowski 2015).

Risk-Neutral Valuation Under Differential Funding Costs, Defaults and Collateralization

“…First, we extend the "adjusted cash flows" approach by incorporating external funding costs. Then, this generalized "adjusted cash flows" approach is reconciled with the classic, albeit typically nonlinear in the present framework, replication paradigm ( [3,8]), thereby validating the "adjusted cash flows" approach as a sound way to price contracts in practice. Furthermore, we show that for claims that can be replicated, the explicit expression of the cash flows adjustments that need to be operated in the valuation under some martingale measure chosen for the pricing purposes is no longer the result of astute insights into the contract time-line, but an immediate outcome of the replication approach.…”

American options in nonlinear markets

“…The goal of this work is to re-examine and extend the findings from the recent paper by Dumitrescu et al [26] who applied the nonlinear pricing approach developed in El Karoui and Quenez [32]. In contrast to [26] where a particular model with a single jump of the underlying asset was studied, we place ourselves within the setup of a general nonlinear arbitrage-free market with possibly discontinuous asset prices, as introduced in Bielecki et al [9,12] and we examine unilateral acceptable prices for American contracts. We obtain results regarding the pricing, hedging, break-even times and rational exercise times using results on backward stochastic differential equations (BSDEs) from Nie and Rutkowski [62,63], but without explicitly specifying the dynamics of underlying risky assets and funding accounts.…”

“…In the context of counterparty credit risk, Crépey [18,19] and Crépey et al [20] analyzed the pricing and hedging of the CVA (Credit Valuation Adjustment) term of the price for European options under funding constraints through nonlinear BSDEs and quasi-linear PDEs. A more systematic study was undertaken by Bielecki and Rutkowski [12] (see also the follow-up work by Bielecki et al [9]) who introduced a generic nonlinear trading model for collateralized contracts and attempted to develop a unified framework for the nonlinear approach to hedging and pricing of over-the-counter (OTC) financial contracts in the spirit of the seminal work by El Karoui and Quenez [32].…”

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We study unilateral valuation problems for American options within the framework of a general nonlinear market by extending results from Bielecki et al [9,12] who examined contracts of European style. A BSDE approach is used to establish more explicit pricing, hedging and exercising results when solutions to reflected BSDEs have additional desirable properties.

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Binary funding impacts in derivative valuation

“…These general practices forced us to reexamine existing theory. Definitions of fair value under entity-specific parameters such as default risk and funding spreads are introduced by Bielecki and Rutkowski (2015), Bielecki, Cialenco, and Rutkowski (2018), Bichuch, Capponi, and Sturm (2018), and Kim, Nie, and Rutkowski (2018). In addition, many other pricing methodologies have been developed for collective adjustment, see, for example, Piterbarg (2010), Wu (2015), Li and Wu (2016), Brigo, Capponi, Pallavicini, and Papatheodorou (2011), Crépey (2015aCrépey ( , 2015b.…”