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binomial

A polynomial with two terms

binomial coefficients

the coefficients of the expansions

Pascal's triangle

any number is sum of the two numbers immediately above it

Fundamental Theorem of Algebra

If P(x) is a polynomial of degree n>or equal to 1. then P(x)=0 has exactly n roots, including multiple and complex roots.

Expand

to expand the power of a bionmial, multiply as needed, then write the polynomial in standard form.

Stock Price Distribution

- value of an option at maturity depends on the value of the underlying instrument at that time

The Binomial Model

assumes that, over a period of time, the price of the underlying asset can move up or down by a specified amount - that is, the asset price follows a binomial distribution

Riskless Hedge

- two possible stock prices (up and down factor)

Finding the value of an option through a risk-free portfolio

- can combine stock and option to replicate bond payoff, so it is a riskless one

The One Period Binomial Model - Assumptions

Euro call option an a non‐div paying stock with exercise price K and maturity T years.

Option Value

Two possible prices depending on state

Derivation

-At start of period,form a portfolio consisting

Derivation continued.

- if portfolio is riskless, can grow by rf rate (otherwise arbitrage opp)

Things to note

- the value of the option doesn't depend on the expected

Risk-Neutral Valuation

- can interpret p in the valuation formula as the

Risk Neutral Valuation steps

- rf = 12%

The Two-Period Binomial Model

Same assumptions as before, but this time there are two

Current Value of the Option and Derivation

f = e^-2rΔt (p^2 fuu +2p(1‐p)fud +(1‐p)^2 fdd )

American Options and the Binomial Model

- no analytical formula for American options like

Valuing American Put Options

Do as if one period model

Arbitrage Opportunities

market price of an option is different from its