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Quantitative Finance · Glossary

What is Variational inequality, free-boundary problem, smooth pasting?

Also known as: variational inequality · free-boundary problem · smooth pasting

Definition 10.4 Quantitative Methods · Chapter 10 — Optimal Stopping and Impulse Control

The display above is a variational inequality. Written as “solve LV=rV\mathcal LV = rV on an unknown region DD with V=gV = g on its boundary”, it is a free-boundary problem: the boundary ∂D\partial D is part of the solution. The extra condition that fixes it, V′=g′V' = g' on ∂D\partial D where gg is smooth, is smooth pasting (or smooth fit).

Taking profit on a drifting position (= 0.5, = 2 bp a day, closing cost 5 bp, discount 2% a day). The value of waiting (solid) lies above the value of closing now (dashed) until b* = c + 1/, where it meets it with the same slope: smooth pasting. Data: closed form, checked by the chapter’s grid solver.
Figure 10.1. Taking profit on a drifting position (μ=0.5\mu = 0.5, σ=2\sigma = 2 bp a day, closing cost 5 bp, discount 2% a day). The value of waiting (solid) lies above the value of closing now (dashed) until b∗=c+1/θb^* = c + 1/\theta, where it meets it with the same slope: smooth pasting. Data: closed form, checked by the chapter’s grid solver.
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