3.6.2 Optimal-Stopping Theory

Basic Prerequisites: 01 · From Zero and Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

The practical objective: replace the vague instruction “exercise at the best time” with a mathematically precise object - the smallest supermartingale that dominates the payoff - and read the optimal exercise region off it. This gives both the value and the exercise rule, and explains in one stroke why calls and puts behave so differently.

Why supermartingales? A discounted price process under $\mathbb{Q}$ is a martingale. Add the right to stop and the value must be a process you can never make money by stopping - i.e. a supermartingale - that stays above the payoff (you can always exercise). The smallest such process is the cheapest thing that dominates the payoff: that is the price. Björk calls this the Snell envelope; Shreve calls it Definition 6.1(b). They are the same theorem.

One picture. Below the exercise boundary the value process equals the payoff (you are at the supermartingale’s “floor”); above it, the value process is a pure martingale (waiting is fair, so keep waiting). The boundary is where the two regions meet.

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2. Mathematical Ground Truth & Derivations

2.1 Stopping times and optimal-stopping value

A random time $\tau \in \{0,\dots ,n\}$ is a stopping time if $\{\tau =k\}\in {\mathcal{F}}_k$ for all $k$ - the decision uses no look-ahead (Shreve I §5.2). The American value is (Shreve II §8.1; Björk §21.2)

$$V_k=(1+r{)}^k\underset{\tau \in {\mathcal{T}}_k}{\max }\tilde{\mathbb{E}}\left[(1+r{)}^{-\tau }{G}_{\tau }|{\mathcal{F}}_k\right],$$

where ${\mathcal{T}}_k$ is the set of stopping times with $\tau \ge k$. For a continuous-time diffusion with reward $\Phi$, the value is $V(t,x)={\sup }_{\tau }{\mathbb{E}}_{t,x}[\Phi (\tau ,X_{\tau })]$.

2.2 Three equivalent characterisations (Shreve I Def 6.1; Björk Thm 21.12/21.23)

1. Optimal-stopping value - the max over $\tau$ above.
2. Smallest supermartingale. $\{(1+r{)}^{-k}{V}_k\}$ is the smallest supermartingale dominating $\{G_k\}$ (the Snell envelope of the discounted payoff).
3. Optimal exercise time. ${\tau }^{\ast }=\min \{k:V_k=G_k\}$ is optimal; the stopped process $V^{{\tau }^{\ast }}$ is a martingale on $[k,n]$ (Björk Prop 21.15).

The backward recursion that computes all of this is exactly the tree step from 01 · From Zero:

$$V_n=G_n,V_k=\max \{G_k,\mathbb{E}[V_{k+1}\mid {\mathcal{F}}_k]\}\text{(undiscounted form, Bjo¨rk Prop 21.7)}.$$

2.3 The drift test and the convexity test (Björk Props 21.2–21.5)

• Supermartingale payoff ⇒ stop now ($\hat{\tau }=0$); submartingale ⇒ stop late ($\hat{\tau }=T$); martingale ⇒ every $\tau$ is optimal.
• Convex increasing $g$ of a submartingale is a submartingale ($\text{mart}\to \text{mart}$ for linear, $\text{sub}$ for convex, $\text{super}$ for concave).

2.4 The call/put asymmetry - a two-line proof each

Call, no dividend (Björk §21.6.1; Shreve I Cor 2.25). Under $\mathbb{Q}$, $e^{-rt}{S}_t$ is a martingale; $g(x)=(x-K{)}^{+}$ is convex increasing with $g(0)=0$. Hence $e^{-rt}g(S_t)=\max (e^{-rt}{S}_t-e^{-rt}K,0)$ is a submartingale (a convex increasing function of a martingale, with the driftless discounted strike adding an upward push). Submartingale ⇒ stop late ⇒ $\hat{\tau }=T$ ⇒ the American call equals the European call.

Put (Shreve II §8.3). $g(x)=(K-x{)}^{+}$ is decreasing in $x$; $e^{-rt}(K-S_t)$ is a supermartingale, so the value lies strictly above the European value and exercise can be optimal. This is the structural reason puts - and dividend-paying calls - need the free-boundary machinery.

Call with dividends. Between dividend dates the American call satisfies the BSM equation; it is only ever optimal to exercise immediately before a dividend date $t_j$ (Shreve II §8.5), giving the backward recursion: the American call equals a European call expiring at $t_j$ until $S(t_j-)-K>c_j(t_j,(1-a_j)S(t_j-))$, else hold to $T$.

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3. Computational Implementation - the theory, checked on a lattice

Two experiments. (A) Reproduce Shreve’s two-step example and expose the early-exercise node. (B) Test the submartingale argument: a dividend-free call should have zero nodes where intrinsic beats continuation; adding a yield should create an exercise region. Stdlib only.

The no-dividend call has exactly zero nodes where intrinsic beats continuation - the submartingale property made visible. Add a 5% yield and $55769$ nodes flip to exercise. The asymmetry between calls and puts is not a convention; it is a theorem, and the lattice obeys it.

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4. Failure Modes & First-Principles Breakdowns

1. “No early exercise” applied to puts. The submartingale argument is specific to convex increasing payoffs with $g(0)=0$ and no dividends. Puts are decreasing payoffs and violate every hypothesis - early exercise is the rule, not the exception.
2. Using the physical measure. The Snell envelope is a $\mathbb{Q}$-object (discounted prices are $\mathbb{Q}$-martingales). Computing ${\mathbb{E}}^{\mathbb{P}}[e^{-r\tau }g(S_{\tau })]$ prices the wrong thing; the drift $\mu$ contaminates the exercise rule.
3. Assuming the smallest-supermartingale characterisation is automatically the price. It is - because the market is complete (single Brownian driver). In an incomplete market the supermartingale envelope gives only a range, not a unique price (Björk Ch 15) - the bridge to jump/vol models.
4. Ignoring the freedom in ${\tau }^{\ast }$. Multiple stopping times can be optimal; ${\tau }^{\ast }=\min \{k:V_k=G_k\}$ is a smallest optimal one, and different but equivalent rules give the same value. Comparing two different exercise rules can be a false test of a pricing bug.

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5. References

• Björk, Arbitrage Theory in Continuous Time
• Shreve, Stochastic Calculus for Finance I
• Shreve, Stochastic Calculus for Finance II
• Glasserman, Monte Carlo Methods in Financial Engineering

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Analytic Approximations · 04 · Free Boundary
• Theory base: Stochastic Calculus & Itô · Fundamental Theorems of Asset Pricing