Define θ_r to ensure bounded local correlation in joint calibration

Construct a concrete, implementable definition of the remainder function θ_r(t,s,x) within the joint calibration framework for local-correlation models such that the induced local-correlation function ρ(t,s,x) = [θ_z(t,sx) + θ_r(t,s,x)] / [η(t,s) ψ(t,x)] remains bounded in the interval [-1,1] for all times and states, thereby enabling a practical joint calibration to quanto corrections and composite option quotes.

Background

The paper proposes a joint calibration strategy for quanto corrections and composite options by introducing the auxiliary function θ(t,s,x) = ρ(t,s,x) η(t,s) ψ(t,x), and decomposing it into θ_z(t,z) determined from composite option local volatility and a remainder θ_r(t,s,x) that is constrained to satisfy the quanto-correction calibration equation and a structural conditional expectation constraint.

While θ_z(t,z) is fully determined by market-observable quantities, θ_r(t,s,x) remains unspecified and must be chosen to meet the calibration constraints. The author notes that, in practice, finding a suitable definition for θ_r is nontrivial because it must also ensure that the resulting local correlation function ρ(t,s,x) stays within the admissible bounds [-1,1].