Bielecki and Rutkowski introduced and studied a generic nonlinear market model, which includes several risky assets, multiple funding accounts, and margin accounts. In this paper, we examine the pricing and hedging of contract from the perspective of both the hedger and the counterparty with arbitrary initial endowments. We derive inequalities for unilateral prices and we study the range of fair bilateral prices. We also examine the positive homogeneity and monotonicity of unilateral prices with respect to the initial endowments. Our study hinges on results from Nie and Rutkowski for backward stochastic differential equations (BSDEs) driven by continuous martingales, but we also derive the pricing partial differential equations (PDEs) for path‐independent contingent claims of a European style in a Markovian framework.
We establish some well-posedness and comparison results for BSDEs driven by one-and multi-dimensional martingales. On the one hand, our approach is largely motivated by results and methods developed in Carbone et al. [3] and El Karoui and Huang [7]. On the other hand, our results are also motivated by the recent developments in arbitrage pricing theory under funding costs and collateralization. A new version of the comparison theorem for BSDEs driven by a multi-dimensional martingale is established and applied to the pricing and hedging BSDEs studied in Bielecki and Rutkowski [1] and Nie and Rutkowski [25]. This allows us to obtain the existence and uniqueness results for unilateral prices and to demonstrate the existence of no-arbitrage bounds for a collateralized contract when both agents have non-negative initial endowments.
We consider a class of linear-quadratic-Gaussian mean-field games with a major agent and considerable heterogeneous minor agents in the presence of mean-field interactions. The individual admissible controls are constrained in closed convex subsets Γ k of R m . The decentralized strategies for individual agents and consistency condition system are represented in an unified manner through a class of mean-field forward-backward stochastic differential equations involving projection operators on Γ k . The well-posedness of consistency system is established in both the local and global cases by the contraction mapping and discounting method respectively. Related ε−Nash equilibrium property is also verified.
We examine the pricing and hedging of general contracts in an extension of the market model proposed by [B-1995]. We study both problems from the perspectives of the hedger and the counterparty with arbitrary initial endowments. We derive inequalities satisfied by unilateral prices of a contract and we give the range for its fair bilateral prices. Our study hinges on results for backward stochastic differential equations (BSDEs) driven by multi-dimensional continuous martingales obtained in Nie & Rutkowski (2014b). We also derive the pricing partial differential equations (PDEs) for path-independent contingent claims of European style in a Markovian framework.
Results from Nie and Rutkowski [12,14] are extended to the case of the margin account, which may depend on the contract's value for the hedger and/or the counterparty (recall that the collateral was given exogenously in [12,14]). The present work generalizes also the papers by Bergman [1], Mercurio [11] and Piterbarg [16]. Using the comparison theorems for BSDEs, we derive inequalities for the unilateral prices and we give the range for its fair bilateral prices. We also establish results yielding the link to the market model with a single interest rate. In the case where the collateral amount is negotiated between the counterparties, so that it depends on their respective unilateral values, the backward stochastic viability property studied by Buckdahn et al. [4] is used to derive the bounds on fair bilateral prices.
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