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### Binomial Theorem

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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