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# American Derivative Securities - PowerPoint PPT Presentation

American Derivative Securities. Presenter:Ting-Hsuan Long. 8.1 Introduction. 8.2 Stopping time. Discrete time Continuous time should be in F(t). Definition8.2.1. A stopping time is a r.v taking values in [0,∞] and satisfying (8.2.1). Remark8.2.2.

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### American Derivative Securities

Presenter:Ting-Hsuan Long

• Discrete time

• Continuous time

should be in F(t)

• A stopping time is a r.v taking values in [0,∞] and satisfying

(8.2.1)

Example8.2.3(First passage time for a continuous process)

• :adapted process with continuous paths

• Show that is a stopping time. Let be given. We need to show thatis in F(t).

Case1:

depending on whether .

In either case,

step(1)

Suppose

In this interval, . is in the set

A=

We have shown that A

If

Let and X has a continuous path, we see that . It follows that .

We have shown that

Under these two step,

Because there are only countably many rational numbers q in [0,t], they can be arranged in a sequence, and the union

is really a union of a sequence of sets in F(t).

We conclude that A