. There is a second part to Theorem 8.2.4 (optional...

. There is a second part to Theorem 8.2.4 (optional sampling), which says the following. Theorem 8.8.1 (Optional sampling - Part II). Let $X(t), t \geq 0$, be $a$ submartingale, and let $\tau$ be a stopping time. Then $\mathbb{E} X(t \wedge \tau) \leq \mathbb{E} X(t)$. If $X(t)$ is a supermartingale, then $\mathbb{E} X(t \wedge \tau) \geq \mathbb{E} X(t)$. If $X(t)$ is a martingale, then $\mathbb{E} X(t \wedge \tau)=\mathbf{E} X(t)$. The proof is technical and is omitted. The idea behind the statement about submartingales is the following. Submartingales tend to go up. Since $t \wedge \tau \leq t$, we would expect this upward trend to result in the inequality $\mathbb{E} X(t \wedge \tau) \leq$ $\mathbb{E} X(t)$. When $\tau$ is a stopping time, this intuition is correct. Once we have Theorem 8.8.1 for submartingales, we easily obtain it for supermartingales by using the fact that the negative of a supermartingale is a submartingale. Since a martingale is both a submartingale and a supermartingale, we obtain the equality $\mathbb{E} X(t \wedge \tau)=\mathbb{E} X(t)$ for martingales. Use Theorem 8.8.1 and Lemma 8.5.1 to show in the context of Subsection 8.5.1 that $$ \widetilde{\mathbb{E}}\left[e^{-r T}(S(T)-K)^{+}\right]=\max _{\tau \in \mathcal{T}_{0, T}} \widetilde{\mathbb{E}}\left[e^{-r \tau}(S(\tau)-K)^{+}\right], $$ where as usual we interpret $e^{-r \tau}(S(\tau)-K)^{+}$to be zero if $\tau=\infty$. The righthand side is the American call price analogous to Definition 8.4.1 for the American put price. The left-hand side is the European call price.

. There is a second part to Theorem 8.2.4 (optional sampling), which says the following.

Theorem 8.8.1 (Optional sampling - Part II). Let $X(t), t \geq 0$, be $a$ submartingale, and let $\tau$ be a stopping time. Then $\mathbb{E} X(t \wedge \tau) \leq \mathbb{E} X(t)$. If $X(t)$ is a supermartingale, then $\mathbb{E} X(t \wedge \tau) \geq \mathbb{E} X(t)$. If $X(t)$ is a martingale, then $\mathbb{E} X(t \wedge \tau)=\mathbf{E} X(t)$.

The proof is technical and is omitted. The idea behind the statement about submartingales is the following. Submartingales tend to go up. Since $t \wedge \tau \leq t$, we would expect this upward trend to result in the inequality $\mathbb{E} X(t \wedge \tau) \leq$ $\mathbb{E} X(t)$. When $\tau$ is a stopping time, this intuition is correct. Once we have Theorem 8.8.1 for submartingales, we easily obtain it for supermartingales by using the fact that the negative of a supermartingale is a submartingale. Since a martingale is both a submartingale and a supermartingale, we obtain the equality $\mathbb{E} X(t \wedge \tau)=\mathbb{E} X(t)$ for martingales.

Use Theorem 8.8.1 and Lemma 8.5.1 to show in the context of Subsection 8.5.1 that
$$
\widetilde{\mathbb{E}}\left[e^{-r T}(S(T)-K)^{+}\right]=\max _{\tau \in \mathcal{T}_{0, T}} \widetilde{\mathbb{E}}\left[e^{-r \tau}(S(\tau)-K)^{+}\right],
$$
where as usual we interpret $e^{-r \tau}(S(\tau)-K)^{+}$to be zero if $\tau=\infty$. The righthand side is the American call price analogous to Definition 8.4.1 for the American put price. The left-hand side is the European call price.

Key Concepts

American vs. European Option Pricing

In financial mathematics, European options can only be exercised at maturity, while American options can be exercised at any time up to maturity. A fundamental insight, particularly for American call options on non-dividend-paying stocks, is that early exercise is suboptimal, leading to the equivalence in price between the American option and its European counterpart. This result can be established using tools like the Optional Sampling Theorem by showing that the maximum expected discounted payoff obtained by optimally choosing a stopping time is equal to the European option's fixed-time payoff.

Risk-Neutral Valuation

Risk-neutral valuation is a method used to price financial derivatives under a measure where the discounted asset prices are martingales. In this framework, one can compute the price of a derivative as the expected discounted payoff under the risk-neutral measure. This concept is key in demonstrating the equivalence between different option pricing formulations, such as the equality of the European call price and the supremum over discounted payoffs corresponding to American options when early exercise does not add value.

Martingales and Their Variants (Submartingales and Supermartingales)

Martingales are stochastic processes that, informally, maintain a constant expected value over time given current information, while submartingales and supermartingales exhibit trends in one direction (upward and downward, respectively). These properties are fundamental for pricing in financial mathematics because they capture the notion of 'fair game' or directional tendency. They provide the framework upon which optional sampling arguments and risk-neutral pricing are built.

Stopping Times

A stopping time is a random variable that represents the time at which a given stochastic process exhibits a certain behavior, determined solely by information up to that time. Stopping times are an essential concept in the study of stochastic processes and are central to applying the Optional Sampling Theorem, as the theorem's conclusions hold when stopping rules are based on the natural filtration of the process.

Optional Sampling Theorem

The Optional Sampling Theorem allows one to replace a fixed time with a random time (a stopping time) in the evaluation of expectations, under suitable conditions on the process (typically being a martingale or a sub/supermartingale). This theorem is crucial in stochastic process theory, as it ensures that the expected value of a process evaluated at a stopping time behaves in a predictable way relative to its evolution in time, preserving inequalities that occur in submartingales or supermartingales.

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