Minimax theorems for American options without time-consistency

Abstract

In this paper, we give sufficient conditions guaranteeing the validity of the well-known minimax theorem for the lower Snell envelope. Such minimax results play an important role in the characterisation of arbitrage-free prices of American contingent claims in incomplete markets. Our conditions do not rely on the notions of stability under pasting or time-consistency and reveal some unexpected connection between the minimax result and path properties of the corresponding process of densities. We exemplify our general results in the case of families of measures corresponding to diffusion exponential martingales.

Appendix A: Paths of nearly sub-Gaussian random fields

Let $(\Theta ,d)$ be some totally bounded semimetric space with diameter $\Delta$. For $\delta , \varepsilon > 0$, the symbols $\mathcal{D}( \delta ,d)$ and $N(\Theta ,d;\varepsilon )$ are used in an analogous manner as the notations $\mathcal{D}(\delta ,d_{\Theta })$ and $N(\Theta ,d_{\Theta };\varepsilon )$ from Sect. 3. We call a centered stochastic process $(X^{\theta })_{\theta \in \Theta }$ a nearly sub-Gaussian random field with respect to$d$ if there is some $C \geq 1$ with

$$ \mathbb{E}\big[\exp \big(\lambda (X^{\theta }- X^{\vartheta })\big) \big]\leq C\exp \big(\lambda ^{2}~d(\theta ,\vartheta )^{2}/2\big) \qquad \mbox{for $\theta , \vartheta \in \Theta $ and $\lambda >0$}. $$

(A.1)

Note that by symmetry, condition (A.1) also holds for arbitrary $\lambda \in \mathbb{R}$. For $C = 1$, this definition reduces to the ordinary notion of sub-Gaussian random fields. For further information on sub-Gaussian random fields, see e.g. [12, Sect. 2.3]. By a suitable change of the semimetric, we may describe any nearly sub-Gaussian random field as a sub-Gaussian random field.

Lemma A.1

If$(X^{\theta })_{\theta \in \Theta }$is a nearly sub-Gaussian random field with respect to$d$, then it is a sub-Gaussian random field with respect to$\overline{d} := \varepsilon d$for some$\varepsilon > 1$.

Proof

Let $C > 1$ be such that $(X^{\theta })_{\theta \in \Theta }$ satisfies (A.1). Then $\varepsilon := \sqrt{12(2 C + 1)}$ is as required (cf. [12, Lemma 2.3.2]). □

The following properties of sub-Gaussian random fields are fundamental.

Proposition A.2

Let$X = (X^{\theta })_{\theta \in \Theta }$be a nearly sub-Gaussian random field on some probability space$(\overline{\Omega },\overline{ \mathcal{F}},\overline{\mathrm{P}})$with respect to$d$. If$\mathcal{D}(\Delta ,d) < \infty$, then$X$admits a separable version, and each separable version of$X$has$\overline{\mathrm{P}}$-almost surely bounded and$d$-uniformly continuous paths. In particular, for any separable version$\widehat{X}$and for every$\overline{\theta } \in \Theta$, there is some random variable$U^{\overline{\theta }}$on$(\overline{\Omega },\overline{\mathcal{F}},\overline{\mathrm{P}})$such that

$$ \sup _{\theta \in \Theta }\widehat{X}^{\theta }\leq U^{\overline{ \theta }} + \widehat{X}^{\overline{\theta }}\quad \overline{ \mathrm{P}}\textit{-a.s.} \quad \,\,\,\, \textit{and} \quad \,\,\,\, \mathbb{E}_{\overline{\mathrm{P}}}[\exp (p U^{\overline{\theta }})] < \infty \quad \textit{for every}~p\in (0,\infty ). $$

Proof

In view of Lemma A.1, we may assume without loss of generality that $X$ is a sub-Gaussian random field with respect to $d$. It is already known (see [12, Theorem 2.3.7]) that $X$ admits a separable version, and that each such version has $\overline{\mathrm{P}}$-almost surely bounded and $d$-uniformly continuous paths. Now fix any separable version $\widehat{X}$ of $X$ and an arbitrary $\overline{\theta }\in \Theta$. We have

$$ \sup _{\theta \in \Theta }\widehat{X}^{\theta }\leq \widehat{X}^{\overline{ \theta }} + \sup _{\theta \in \Theta }|\widehat{X}^{\theta } - \widehat{X}^{\overline{\theta }}|, $$

and the process $(|\widehat{X}^{\theta } - \widehat{X}^{\overline{ \theta }}|)_{\theta \in \Theta }$ is separable due to the separability of $\widehat{X}$. Then we may find some at most countable subset $\Theta _{0}\subseteq \Theta$ such that

$$\begin{aligned} \sup _{\theta \in \Theta }|\widehat{X}^{\theta } - \widehat{X}^{\overline{ \theta }}| = \sup _{\theta \in \Theta _{0}}|\widehat{X}^{\theta } - \widehat{X}^{\overline{\theta }}| \qquad \overline{\mathrm{P}}\text{-a.s.} \end{aligned}$$

Hence $U^{\overline{\theta }} := \sup _{\theta \in \Theta _{0}}|X^{ \theta } - X^{\overline{\theta }}|$ defines a random variable on $(\overline{\Omega },\overline{\mathcal{F}},\overline{\mathrm{P}})$ satisfying

$$ U^{\overline{\theta }} = \sup _{\theta \in \Theta }|\widehat{X}^{ \theta } - \widehat{X}^{\overline{\theta }}| \qquad \overline{\mathrm{P}}\mbox{-a.s.} $$

It remains to show that $\mathbb{E}_{\overline{\mathrm{P}}}[\exp (p U ^{\overline{\theta }})] < \infty$ for $p\in (0,\infty )$. So fix $p\in (0,\infty )$. First, observe that $\widehat{X}$ is again a sub-Gaussian random field. Thus by [12, Lemma 2.3.1], we have

$$ \mathbb{E}_{\overline{\mathrm{P}}}\bigg[\exp \bigg(\Big(\frac{X^{ \theta }-X^{\vartheta }}{\sqrt{6}~ d(\theta ,\vartheta )}\Big)^{2} \bigg)\bigg]\leq 2 \qquad \mbox{for}~\theta , \vartheta \in \Theta ,~d(\theta ,\vartheta ) \not = 0. $$

Hence we may apply the results from [25] with respect to the totally bounded semimetric $\overline{d} := \sqrt{6}d$. Note that $(\widehat{X}^{\theta })_{\theta \in \Theta }$ is also separable with respect to $\overline{d}$, and that $\overline{ \Delta } = \sqrt{6}\Delta$ for the diameter $\overline{\Delta }$ with respect to $\overline{d}$. Since $N(\Theta ,\overline{d};\varepsilon )\leq N(\Theta ,d;\varepsilon /\sqrt{6})$ for every $\varepsilon > 0$, we obtain for every $\delta > 0$ that

$$ \int _{0}^{\delta }\sqrt{\ln N(\Theta ,\overline{d};\varepsilon )}~d \varepsilon \leq \int _{0}^{\delta }\sqrt{\ln N(\Theta ,d;\varepsilon /\sqrt{6})}~d\varepsilon = \sqrt{6} {\mathcal{D}}(\delta / \sqrt{6},d). $$

Then in view of [25, Corollary 3.2], we may find some constant $C > 0$ such that

$$ \overline{\mathrm{P}}\big[ U^{\overline{\theta }} > xC\sqrt{6}{\mathcal{D}}( \Delta ,d)\big] \leq 2\exp (-x^{2}/2) \qquad \mbox{for}~x\geq 1. $$

Furthermore, setting $\widehat{C} := C\sqrt{6}{\mathcal{D}}(\Delta ,d)$, we may observe that

$$\begin{aligned} \int _{1}^{\infty }\overline{\mathrm{P}}\big[ U^{\overline{\theta }} > x\widehat{C}\big]\exp (xp\widehat{C})~dx & \leq \int _{1}^{\infty } 2 \exp (-x^{2}/2)\exp (xp\widehat{C})~dx \\ & \leq 2\sqrt{2\pi }\exp (p^{2}\widehat{C}^{2}/2). \end{aligned}$$

Then applying the change of variables formula several times, we obtain

$$\begin{aligned} \int _{\exp (p\widehat{C})}^{\infty }\overline{\mathrm{P}}\big[ \exp (pU ^{\overline{\theta }}) > y\big]~dy &= p\widehat{C}\int _{1}^{\infty }\overline{ \mathrm{P}}\big[ U^{\overline{\theta }} > \widehat{C}u\big]\exp (p \widehat{C}u)~du < \infty . \end{aligned}$$

Hence

$$ \mathbb{E}_{\overline{\mathrm{P}}}[\exp (pU^{\overline{\theta }})] = \int _{0}^{\infty }\overline{\mathrm{P}}\big[ \exp (pU^{\overline{ \theta }}) > y\big]~dy < \infty $$

which completes the proof. □