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When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each... more
When pricing OTC contracts in the presence of additional risk factors and costs, such as credit risk and funding and collateral costs, the starting “clean price” is modified additively by valuation adjustments (XVAs) that account for each factor or cost in isolation, while seemingly ignoring the combined effects. Instead, risk factors and costs can be jointly accounted for ab initio in the pricing mechanism at the level of cash flows, and this “adjusted cash flow" approach leads to a nonlinear valuation formula. While for practitioners this made more sense because it showed which discount factor is used for which cash flow (recall the multi-curve environment post-crisis), for academics, the focus was on checking that the resulting nonlinear valuation formula is consistent with the theoretical arbitrage-free “replication approach” that we also analyse in the paper. We formulate specific reasonable assumptions, which ensure that the valuation formulae obtained by the two approach...
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The goal is to re-examine and extend the findings from the recent paper by Dumitrescu et al. [11] who studied game options within the nonlinear arbitrage-free pricing approach developed in El Karoui and Quenez [16]. We consider the setup... more
The goal is to re-examine and extend the findings from the recent paper by Dumitrescu et al. [11] who studied game options within the nonlinear arbitrage-free pricing approach developed in El Karoui and Quenez [16]. We consider the setup introduced in Kim et al. [26] where contracts of an American style were examined. We give a detailed study of unilateral pricing, hedging and exercising problems for the counterparties within a general nonlinear setup. We also present a BSDE approach, which is used to obtain more explicit results under suitable assumptions about solutions to doubly reflected BSDEs.
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Research Interests: Applied Mathematics, Economics, Mathematical Finance, Credit Risk, Variational Inequality Problems, and 11 moreHedging, Mathematical, Convertible bond, Variational Inequalities, Convertible Bonds, Boolean Satisfiability, Markov Process, Process Model, Finance and Investment Banking Area, Finance and Investment Banking, and variational inequality
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This paper departs from the usual methods for pricing contracts with counterparty risk found in the existing literature. In effect, typically, these models, first, do not account for either systemic effects or ‘at first default’... more
This paper departs from the usual methods for pricing contracts with counterparty risk found in the existing literature. In effect, typically, these models, first, do not account for either systemic effects or ‘at first default’ contagion, second, postulate that the contract value at default equals either the default-free value or the pre-default value, and third, do not take margin agreements into account. Instead, we propose a general framework which allows for the CVA computation under bilateral counterparty risk of a contract in the presence of systemic and rightor wrong-way risks, and under alternative settlement conventions and margin agreements. ∗The research of M. Rutkowski was supported under Australian Research Council’s Discovery Projects funding scheme (DP0881460). The authors thank J.P. Laurent for enlightening discussions and comments.
∗This work was completed during our visit to the Isaac Newton Institute for Mathematical Sciences in Cambridge. We thank the organizers of the programme Developments in Quantitative Finance for the kind invitation. †The research of T.R.... more
∗This work was completed during our visit to the Isaac Newton Institute for Mathematical Sciences in Cambridge. We thank the organizers of the programme Developments in Quantitative Finance for the kind invitation. †The research of T.R. Bielecki was supported by NSF Grant 0202851 and Moody’s Corporation grant 5-55411. ‡The research of M. Jeanblanc was supported by Zeliade, Ito33, and Moody’s Corporation grant 5-55411. §The research of M. Rutkowski was supported by the 2005 Faculty Research Grant PS06987.
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The aim of this work is to demonstrate, with the help of multiplicative systems introduced by Meyer in [15], that for any given positive F-supermartingale G such that G∞ = 0, there exists a random time τ on some extension of the filtered... more
The aim of this work is to demonstrate, with the help of multiplicative systems introduced by Meyer in [15], that for any given positive F-supermartingale G such that G∞ = 0, there exists a random time τ on some extension of the filtered probability space such that the Azéma supermartingale associated with τ is given by G. This construction is subsequently extended to the case of several correlated random times with predetermined marginal conditional distributions.
I n t h e i r r e c e n t work J .M.Har r i son and L.A.Shepp \_2~\ cons i d e r a s t o c h a s t i c equa t ion whioh i n c l u d e s i n p a r t i c u l a r the loo a l tinte a t the point 0 of a se aimer t i n g a l e . Por ' h i... more
I n t h e i r r e c e n t work J .M.Har r i son and L.A.Shepp \_2~\ cons i d e r a s t o c h a s t i c equa t ion whioh i n c l u d e s i n p a r t i c u l a r the loo a l tinte a t the point 0 of a se aimer t i n g a l e . Por ' h i s equat i o n they g ive a s u f f i c i e n t and necessary c o n d i t i o n s f o r the e x i s t e n c e of a unique s t r i o t s o l u t i o n · Moreover they i d e n t i f y i t s s o l u t i o n as a d i f f u s i o n process known under the name of a skew Brownian motion· In t h i s note we in t roduce a s l i g h t l y more g e n e r a l c l a s s of equa t ions by d i s t i n g u i s h i n g the lower and upper l o o a l t imes a t the point 0 of a semimar t inga le · I n o o n t r a s t t o the r e s u l t s of , [ 2 ] we show t h a t f o r a o e r t a i n choice of parameters the considered equa t ion possess i n f i n i t e l y many s t r i c t s o l u t i o n s · There fore our equa t ion may serve a s a simple example of the d i f f u s i o n equa t ion f o r whioh the non-uniqueness of a s t r i o t s o l u t i o n i s v a l i d .
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The paper is devoted to a specific optimization problem associated with the hedging of contingent claims in continuous-time incomplete models of financial markets. Generally speaking, we place ourselves within the standard framework of... more
The paper is devoted to a specific optimization problem associated with the hedging of contingent claims in continuous-time incomplete models of financial markets. Generally speaking, we place ourselves within the standard framework of the theory of continuous trading, as exposed in Harrison and Pliska [13]. Our aim is twofold. Firstly, we present a relatively concise exposition of the risk-minimizing methodology (due essentially to Follmer and Sondermann [12], Follmer and Schweizer [11] and Schweizer [33]) in a multi-dimensional continuous-time framework. Let us mention here that this approach is based on the specific kind of minimization of the additional cost associated with a hedging strategy at all times before the terminal date T. Secondly, we provide some new results which formalize some concepts introduced in Hofman et a/.[l5], in particular, the general results of the first, part are specialized to the case of multi-dimensional Ito processes. Finally, in Section 6 the general theory is illustrated by means of an example dealing with the risk-minimizing hedging of a stock index option in an incomplete framework. This example is motivated bv the work of Lamberton and Lapeyre [22] who have! solved a related, but simpler, problem of a risk-minimizing hedging under the martingale measure.
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Purpose This study aims to render a fundamental assessment of the Basel II internal ratings-based (IRB) approach by taking readings of the Australian banking sector since the implementation of Basel II and comparing them with signals from... more
Purpose This study aims to render a fundamental assessment of the Basel II internal ratings-based (IRB) approach by taking readings of the Australian banking sector since the implementation of Basel II and comparing them with signals from macroeconomic indicators, financial statistics and external credit ratings. The IRB approach to capital adequacy for credit risk, which implements an asymptotic single risk factor (ASRF) model, plays an important role in protecting the Australian banking sector against insolvency. Design/methodology/approach Realisations of the single systematic risk factor, interpreted as describing the prevailing state of the Australian economy, are recovered from the ASRF model and compared with macroeconomic indicators. Similarly, estimates of distance-to-default, reflecting the capacity of the Australian banking sector to absorb credit losses, are recovered from the ASRF model and compared with financial statistics and external credit ratings. With the impleme...
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A single period, zero-sum, multi-player game is constructed. Each player can either exit the game for a fixed payoff or stay and split the remaining payoff with the other non-exiting players. The emphasis is put on the rivalrous nature of... more
A single period, zero-sum, multi-player game is constructed. Each player can either exit the game for a fixed payoff or stay and split the remaining payoff with the other non-exiting players. The emphasis is put on the rivalrous nature of the payoffs, meaning that the sum of all payoffs is fixed, but the exact allocation is based on the players’ decisions. The value at which Nash and optimal equilibria are attained is shown to be unique and it is constructed explicitly.
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The purpose of th is work i s to consider the concept of a general stochastic equation and to Investigate the problem of existence and unicity of i t s solution. Thus our aim i s twofold. F i r s t l y , we intend to study various types... more
The purpose of th is work i s to consider the concept of a general stochastic equation and to Investigate the problem of existence and unicity of i t s solution. Thus our aim i s twofold. F i r s t l y , we intend to study various types of operators which map the c lass of adapted, r ight continuous and possessing l e f t l imits processes into i t s e l f . These examinations lead to the equation of the form
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We extend, to the case of more general coefficients, the result concerning the rate of convergence of the local times of solutions of stochastic differential equations with the Lipschitz continuous coefficients Nous etendons au cas des... more
We extend, to the case of more general coefficients, the result concerning the rate of convergence of the local times of solutions of stochastic differential equations with the Lipschitz continuous coefficients Nous etendons au cas des coefficients plus generaux le resultat concernant le taux de convergence des temps locaux des solutions d'equations differentielles stochastiques a coefficients lipschitziens
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under appropriate conditions on a family B(t), t e R , of processes of finite variation, a semimartinagle X and operators G,H. Namely to ensure the existence and unicity of a global solution (by a solution we mean a strong solution) we... more
under appropriate conditions on a family B(t), t e R , of processes of finite variation, a semimartinagle X and operators G,H. Namely to ensure the existence and unicity of a global solution (by a solution we mean a strong solution) we assume certain continuity and smoothness property of a family B(t) and a suitable form of a Lipschitz condition on G,H. Definitions and lemmas which lead to a main result (i.e. Theorem 12) are given in the first part of a paper. The second part is devoted to a proof of this theorem only. In the next one we consider the problem of existence of a solution under weakened assumption on operators G and H. Namely the possibility of occurence of explosions and/or oscillations of a solution is examined. For related results we refer to the recent paper of H.Doss and E.Lenglart [3] (see also P.Protter [11] ). At last in the final section we give examples and counterexamples. ( 1 )
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We consider the behaviour of the Black-Scholes implied volatility in the small time to expiry limit under the condition of no arbitrage. We present formulae describing the exact asymptotics of the implied volatility in this situation.... more
We consider the behaviour of the Black-Scholes implied volatility in the small time to expiry limit under the condition of no arbitrage. We present formulae describing the exact asymptotics of the implied volatility in this situation. Some basic properties of the time-scaled implied volatility are also derived. It is established that there exist arbitrage free markets in which implied volatility
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ABSTRACT
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Various probabilistic techniques, which are used in the modeling of derivative securities (in particular, zero-coupon bonds) that are subject to default risk are presented in a systematic way. A large class of existing models of the... more
Various probabilistic techniques, which are used in the modeling of derivative securities (in particular, zero-coupon bonds) that are subject to default risk are presented in a systematic way. A large class of existing models of the defaultable term structure is covered by our analysis, in addition, some new ideas are presented.