Alexander S. Cherny

Markets As A Counterparty: An Introduction To Conic Finance

Markets are modeled as a counterparty accepting at zero cost a set of cash flows that are closed under addition, scaling and contain the nonnegative cash flows. Formulas are then provided for bid and ask prices in terms of this marketed cone. Additionally closed forms are obtained when parametric concave distortions introduced in Cherny and Madan (2009) define the marketed claims. Finally explicit expressions price call and put options at bid and ask. Three applications illustrate. The first estimates the movement of the cone through the financial crisis using data on bid and ask prices for S&P 500 index options. It is observed that the cone contracted significantly in 2008 and slowly opened up thereafter. The second application documents the improvements possible in terms of reduced ask prices by hedging at a flat Black-Scholes volatility even when the underlying assumptions for replication are violated. The third application considers a number of structured products written on daily returns to an underlying asset price and illustrates the use of our closed form expressions for the ask price as an objective function in designing hedges.

Singular stochastic differential equations

Introduction.- 1. Stochastic Differential Equations.- 2. One-Sided Classification of Isolated Singular Points.- 3. Two-Sided Classification of Isolated Singular Points.- 4. Classification at Infinity and Global Solutions.- 5. Several Special Cases.- Appendix A: Some Known Facts.- Appendix B: Some Auxiliary Lemmas.- Rferences.- Index of Notation.- Index of Terms.

Coherent Measurement of Factor Risks

We propose a new procedure for the risk measurement of large portfolios. It employs the following objects as the building blocks: - coherent risk measures introduced by Artzner, Delbaen, Eber, and Heath; - factor risk measures introduced in this paper, which assess the risks driven by particular factors like the price of oil, SP - risk contributions and factor risk contributions, which provide a coherent alternative to the sensitivity coefficients. We also propose two particular classes of coherent risk measures called Alpha V@R and Beta V@R, for which all the objects described above admit an extremely simple empirical estimation procedure. This procedure uses no model assumptions on the structure of the price evolution. Moreover, we consider the problem of the risk management on a firms level. It is shown that if the risk limits are imposed on the risk contributions of the desks to the overall risk of the firm (rather than on their outstanding risks) and the desks are allowed to trade these limits within a firm, then the desks automatically find the globally optimal portfolio.

On Stochastic Integrals up to Infinity and Predictable Criteria for Integrability

The first goal of this paper is to give an adequate definition of the stochastic integral

Dilatation monotone risk measures are law invariant

\int_0^\infty {{H_s}} {\text{d}}{X_s},(*)

General Arbitrage Pricing Model: II – Transaction Costs

where \(H = (H_t)_{t\ge0}\) is a predictable process and \(X = (X_t)_{t\ge0}\) is a semimartingale. We consider two different definitions of (*): as a stochastic integral up to infinity and as an improper stochastic integral.

Pricing and hedging European options with discrete-time coherent risk

We define the capital allocation and the risk contribution for discrete-time coherent risk measures and provide several equivalent representations of these objects. The formulations and the proofs are based on two instruments introduced in the paper: a probabilistic notion of the extreme system and a geometric notion of the generator. These notions are also of interest on their own and are important for other applications of coherent risk measures. All the concepts and results are illustrated by JP Morgans Risk Metrics model.

General Arbitrage Pricing Model: I – Probability Approach

We prove that on an atomless probability space, every dilatation monotone convex risk measure is law invariant. This result, combined with the known ones, shows the equivalence between dilatation monotonicity and important properties of convex risk measures such as law invariance and second-order stochastic monotonicity.