3.6.4 The Free Boundary & the Linear-Complementarity Problem

Basic Prerequisites: 02 · Optimal-Stopping Theory and BSM · The PDE.

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1. Intuition & Practical Objective

The practical objective: write the American option as a well-posed boundary-value problem whose unknown is a function (the value) and a curve (the exercise boundary), and see why the two clean formulations - variational inequality and linear complementarity - are what a numerical solver actually implements.

The awkwardness is the free boundary $S^{\ast }(t)$. On the “continue” region the value solves the BSM PDE; on the “exercise” region it equals intrinsic. The two regions are separated by a curve we do not know. Three ways to tame it:

1. Guess the boundary, solve, enforce tangency. Solve the PDE on $[S^{\ast },\infty )$ with $v(S^{\ast })=K-S^{\ast }$; then impose smooth pasting $v_S(S^{\ast })=-1$ to pin $S^{\ast }$. This is exactly the perpetual case, where it is closed-form. (This is how we solve the perpetual put in 03 · Analytic Approximations.)
2. Variational inequality (VI). State a single inequality that encodes “value ≥ payoff” and “waiting is not profitable” simultaneously - no explicit boundary.
3. Linear complementarity (LCP). The pointwise algebraic form of the VI: at every $(t,S)$, either the value is above the payoff (and the PDE holds), or the value equals the payoff (and the PDE inequality is slack). This is the form PSOR and penalty methods solve.

One sentence. “American pricing = solve the BSM operator subject to $V\ge$ payoff and making the residual complementary to the constraint - the boundary is whatever curve that complementarity implies.”

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2. Mathematical Ground Truth & Derivations

2.1 The free-boundary formulation (Shreve II §8.4; Björk Prop 21.27)

Let $\mathcal{L}$ be the BSM operator, $\mathcal{L}V=\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}-rV$. On the continuation region $\mathcal{C}=\{(t,S):V(t,S)>$ intrinsic$\}$:

$$\mathcal{L}V=0\text{in }\mathcal{C},V(t,S)=\text{intrinsic on }\partial \mathcal{C},V(T,S)=\text{intrinsic}.$$

The boundary $\partial \mathcal{C}$ is the free boundary $S^{\ast }(t)$ (a curve rising to $K$ at $t=T$ for the put). Two conditions hold across it:

$$\underset{\text{value matching}}{\underbrace{V(t,S^{\ast })=K-S^{\ast }}},\underset{\text{smooth pasting}}{\underbrace{\frac{\partial V}{\partial S}(t,S^{\ast })=-1}}.$$

$V$ and $V_S$ are continuous across $\partial \mathcal{C}$; $V_{SS}$ jumps. Only $C^1$ regularity is required - smooth pasting is a choice (the “smooth fit” heuristic) that the correct solution satisfies, not an assumption.

2.2 The variational inequality (Björk Prop 21.25/21.26; Shreve II 8.3.18–8.3.20)

In the perpetual put the three conditions combine into a single statement with $\mathcal{L}V:=rV-rS{V}_S-\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}$ (the negative of the BSM operator):

$$\text{(i) }V(x)\ge (K-x{)}^{+}\forall x;\text{(ii) }\mathcal{L}V(x)\ge 0\forall x;\text{(iii) at each }x\text{, equality holds in (i) or (ii)}.$$

The unique bounded $C^1$ function satisfying all three is the value $v_{L^{\ast }}$ of 03. In the continuation region $\mathcal{L}V=0$; in the exercise region $V^{\prime }=V^{\prime \prime }=0$, $V=K-x$ so $\mathcal{L}V=rK>0$. The probabilistic twin (Shreve II Thm 8.3.5, Cor 8.3.6): $e^{-rt}{v}_{L^{\ast }}(S_t)$ is a supermartingale, and is a martingale when stopped at the first hitting time $T_{L^{\ast }}$.

2.3 The linear-complementarity problem (the solver’s form)

For a finite-maturity American put on a grid, conditions (i)–(iii) become, at every node $(t_i,S_j)$:

$$\boxed{\min (V(t_i,S_j)-\text{intrinsic}(S_j),\mathcal{L}V(t_i,S_j))=0}$$

or equivalently the complementarity system

$$V\ge g,\mathcal{L}V\ge 0,(V-g)\cdot (\mathcal{L}V)=0.$$

Discretise $\mathcal{L}$ (implicit Euler / Crank–Nicolson) and you get, at each time step, an algebraically messy linear system $AV\ge b$, $V\ge g$, with complementarity $(V-g{)}^{\top }(AV-b)=0$ - solved by PSOR (projected SOR) or by a penalty method (Duffy Ch 27–29). This is precisely the “1-factor American” row of the numerical-methods scheme table - the implementation detail lives in Numerical Methods · FDM; this page owns the formulation.

2.4 Why the boundary is where it is - the drift test (Björk Prop 21.28)

It is never optimal to stop where the payoff alone is a submartingale: if ${\partial }_t\Phi +\mu {\Phi }_x+\frac{1}{2}{\sigma }^2{\Phi }_{xx}>0$ then $(t,x)\in \mathcal{C}$. Intuitively you never exercise while the immediate reward is expected to grow faster than the discounting can erode it - the boundary is pushed out to the point where the two balance.

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3. Computational Implementation - smooth pasting, complementarity and the perpetual limit

Three checks. (A) The boundary $L^{\ast }$ is the maximiser over constant boundaries of the value - smooth pasting as a first-order condition. (B) Long-dated CRR converges to the perpetual closed form. (C) The complementarity residual $\min (V-\text{payoff},\mathcal{L}V)=0$ holds at every $x$, computed by finite differences with the negative-BSM convention. Stdlib only.

Read the residual table: on the exercise side ($x=60,75$) the value equals payoff (first bracket $=0$) and $\mathcal{L}V=rK=10>0$; on the continuation side ($x=80,100,130$) the value is strictly above payoff and $\mathcal{L}V=0$ (to finite-difference accuracy). The minimum is zero everywhere - the complementarity system, verified. And the finite-maturity American put climbs monotonically toward the perpetual $13.9719$ as $T\to \infty$, exactly the limit Björk Prop 21.30 predicts.

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4. Failure Modes & First-Principles Breakdowns

1. Sign confusion in the VI. Shreve writes $\mathcal{L}V=rV-rS{V}_S-\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}$, the negative of the usual BSM operator; it is $\ge 0$ (not $\le$) and equals $rK$ in the exercise region. Writing the wrong sign turns a correct solver into a subtly wrong one with no obvious symptom.
2. Smooth pasting is a consequence, not an assumption. Imposing $V_S(S^{\ast })=-1$ with an otherwise wrong ansatz produces a boundary that satisfies the tangency but not the PDE - a plausible-looking wrong answer. The VI formulation (no explicit boundary) is the safe statement.
3. $V_{SS}$ discontinuity breaks naive schemes. The gamma jump at $S^{\ast }$ is precisely what makes Crank–Nicolson ring near the strike; Rannacher startup / extrapolated implicit Euler suppress it (Numerical Methods · Failure Modes).
4. Freezing the boundary or the “exercise-if-deep-ITM” heuristic. Both are sub-optimal policies; they return a value below the true one and, worse, an inconsistent delta. The boundary must be solved, not guessed.
5. The free boundary is a full curve, not a point. For a finite-maturity put $S^{\ast }(t)$ rises to $K$ at $t=T$; discretising only the current boundary (a perpetuity) misprices intermediate maturities.

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5. References

• Shreve, Stochastic Calculus for Finance II
• Björk, Arbitrage Theory in Continuous Time
• Duffy, Finite Difference Methods in Financial Engineering
• Wilmott, Howison & Dewynne, The Mathematics of Financial Derivatives

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6. Connected Graph Bridges

• Back: 03 · Analytic Approximations · Index Hub
• Forward: 05 · Failure Modes · 06 · Advanced Extensions
• Implementation (linked, not duplicated): FDM: penalty & PSOR · BSM · PDE