On pricing contingent claims in a two interest rates jump-diffusion model via market completions

Abstract

Abstract. This paper deals with the problem of hedging contingent claims in the framework of a two factors jump-diffusion model with different credit and deposit rates. The upper and lower hedging prices are derived for European options by means of auxiliary completions of the initial market.

On investment and minimization of shortfall risk for a diffusion model with jumps and two interest rates via market completion

“…Investment problem. In parallel to the hedging agent who wants to find the optimal strategy to hedge his claim f T (see Kane and Melnikov [13]), an investing agent has to find the optimal strategy that allows him to maximize the expected utility of his terminal wealth. Assume a given utility function U : R + → R is concave, non-decreasing, continuously differentiable, and such that…”

“…In the setting of a two-interest-rates financial market with the above assumptions, the optimal proportions invested are defined by (3.15) [13] the optimal strategy and initial capital required to hedge perfectly a claim f (S 1 T ). We also derived in the previous section the optimal investment strategy and terminal wealth of an expected utility maximization problem.…”

“…In reality, the credit rate is always higher than the deposit rate. Such a market constraint brings new difficulties in the problems of hedging, investing and shortfall risk minimization (see Bergman [4], Kane and Melnikov [13], Korn [15], Bart [3], and also Cvitanić and Karatzas [8], Cvitanić [6,7], Föllmer and Kramkov [11], Karatzas and Shreve [14], Cvitanić, Pham and Touzi [9], Soner and Touzi [21] regarding other market constraints).…”

“…As in Korn [15], Kane and Melnikov [13], we use a methodology of completions to solve the investment problem with logarithmic utility function and a shortfall risk minimization in a financial market consisting of a diffusion model with jumps and two interest rates.…”

Pricing Contingent Claims in a Two-Interest-Rate Multi-Dimensional Jump-Diffusion Model via Market Completion

“…(Refer to [12] for detailed insights). However, our focus shifts to a more realistic financial market scenario, introducing a two-interest-rate model where the credit rate surpasses the deposit rate, aligning more closely with real-world financial markets (as discussed in [13]). In this paper, we consider a multi-dimensional model featuring m + 2 securities, encompassing two risk-free assets, d stocks driven by a d-dimensional Brownian motion, and m − d stocks influenced by an (m − d)-dimensional Poisson process.…”