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1

He meant to compute the swaption price given by (2.4a), that is, \begin{align*} V_{opt} = L_0 E\left((R_s(\tau)-R_{fix})^+ \mid \mathcal{F}_0 \right). \end{align*} Under the swap measure (i.e., with $L_t$ as the numeraire), the swap rate process $\{R_s(t), t \ge 0\}$ is a martingale, and is assumed to be of the form \begin{align*} R_s(\tau) = R_s^0 e^{\sigma ...

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Let's recall the definition of a Martingale first: it is a stochastic process $X(t)$ that has the following property: let $0 \leq t < T$ two real numbers. Let $\mathcal{F}_t$ be a filtration for the process $X$ at time $t$. We have then: $$\mathbb{E}[X(T)|\mathcal{F}_t] = X(t)$$ Now, if you use Black's model, you describe your asset price using a ...

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It seems that uniformly integrable martingales, as described by quasi, account for a specific class of strict local martingales. A martingale on $[0,\infty)$ that is not uniformly integrable (like geometric Brownian motion) is a uniformly integrable martingale on $[0,t]$ for every $t\in [0,\infty)$. Consequently, mapping $[0,\infty]$ onto $[0,1]$ as done ...

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