Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Filtration Reduction and Completeness in Jump-Diffusion Models

Karen Grigorian, Robert Jarrow

TL;DR

This paper tackles derivative pricing in arbitrage-free but incomplete jump-diffusion markets by using an uplifted EMM framework based on filtration reduction. By projecting the original market to a fictitious complete market with a unique EMM and invoking a consistency condition, it uplifts the fictitious EMM back to the original market, ensuring non-hedged risks carry no risk premium. It develops two main approaches: complete neglect (ignore certain jump risks) and partial neglect (aggregate certain jump sizes and hedge on average), and provides explicit constructions, uniqueness theorems, and a general batching procedure for jumps. The results yield a practical pricing rule where the cost of constructing a hedge in the fictitious market equals the price in the original market, with hedging errors characterized when non-priced risks are not fully hedged; the framework extends to time-varying coefficients and both discrete and continuous jump sizes, with potential applications in markets where traders hedge only a subset of risks.

Abstract

This paper studies the pricing and hedging of derivatives in frictionless and competitive, but incomplete jump-diffusion markets. A unique equivalent martingale measure (EMM) is obtained using filtration reduction to a fictitious complete market. This unique EMM in the fictitious market is uplifted to the original economy using the notion of consistency. For pedagogical purposes, we begin with simple setups and progressively extend to models of increasing generality.

Filtration Reduction and Completeness in Jump-Diffusion Models

TL;DR

This paper tackles derivative pricing in arbitrage-free but incomplete jump-diffusion markets by using an uplifted EMM framework based on filtration reduction. By projecting the original market to a fictitious complete market with a unique EMM and invoking a consistency condition, it uplifts the fictitious EMM back to the original market, ensuring non-hedged risks carry no risk premium. It develops two main approaches: complete neglect (ignore certain jump risks) and partial neglect (aggregate certain jump sizes and hedge on average), and provides explicit constructions, uniqueness theorems, and a general batching procedure for jumps. The results yield a practical pricing rule where the cost of constructing a hedge in the fictitious market equals the price in the original market, with hedging errors characterized when non-priced risks are not fully hedged; the framework extends to time-varying coefficients and both discrete and continuous jump sizes, with potential applications in markets where traders hedge only a subset of risks.

Abstract

This paper studies the pricing and hedging of derivatives in frictionless and competitive, but incomplete jump-diffusion markets. A unique equivalent martingale measure (EMM) is obtained using filtration reduction to a fictitious complete market. This unique EMM in the fictitious market is uplifted to the original economy using the notion of consistency. For pedagogical purposes, we begin with simple setups and progressively extend to models of increasing generality.
Paper Structure (33 sections, 30 theorems, 92 equations)

This paper contains 33 sections, 30 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Filtration Reduction with Complete Neglect
  4. Three Stocks, One Brownian Motion, and Three Poisson Processes
  5. The Original Market
  6. No-Arbitrage and An Incomplete Market
  7. The Fictitious Market
  8. The Uplifted EMM
  9. $n$ Stocks, One Brownian Motion, and $M$ Poisson Processes
  10. The Original Market
  11. The Fictitious Market
  12. The Uplifted EMM: Issues of Uniqueness
  13. $n$ Stocks, $D$ Brownian Motions, and $M$ Poisson Processes
  14. The Original Market
  15. The Fictitious Market
  16. ...and 18 more sections

Key Result

Proposition 1

Let $y_{1},...,y_{M}$ be a finite set of nonzero numbers, and let $p(y_{1}),...,p(y_{M})$ be positive numbers summing to 1. Giventhe constant $\lambda>0$, let $\overline{N}_{1},...,\overline{N}_{M}$ be independent Poisson processes, each $\overline{N}_{m}$ having intensity $\lambda p(y_{m})$. Define

Theorems & Definitions (41)

  • Proposition 1
  • Corollary 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • Definition 8
  • Proposition 9
  • proof
  • ...and 31 more