Mathematical Modeling in Finance with Stochastic Processes

2 Binomial Option Pricing Models

2. Replication of the option payouts with the single security and the single
   bond leads to pricing the derivative by arbitrage.

Vocabulary

1. Security: A promise to pay, or an evidence of a debt or property,
   typically a stock or a bond. Also referred to as anasset.


2. Bond: Interest bearing securities, which can either make regular in-
   terest payments, or a lump sum payment at maturity, or both.


3. Stock: A security representing partial ownership of a company, varying
   in value with the value of the company. Also known as shares or
   equities.


4. Derivative: A security whose value depends on or is derived from
   the future price or performance of another security. Also known as
   financial derivatives,derivative securities,derivative products,
   andcontingent claims.


5. A portfolio of the stock and the bond which will have the same value
   as the derivative itself in any circumstance is areplicating portfolio.

Mathematical Ideas

2.1 Single Period Binomial Models

Thesingle period binomial modelis the simplest possible financial model,

yet it contains the elements of all future models. The single period binomial

model is an excellent place to start studying mathematical finance. It is

strong enough to be a somewhat realistic model of financial markets. It is

simple enough to permit pencil-and-paper calculation. It can be compre-

hended as a whole. It is also structured enough to point to natural general-

ization.

The quantifiable elements of the single period binomial financial model

are:

1. A single interval of time, fromt= 0 tot=T.