[Verse 1]
Picture a stock that moves just two ways
Up or down by the end of trading days
Start at one hundred, simple and clean
Two paths ahead, that's all we've seen
Up factor's one point two, down's zero eight
These movements sealed, we calculate fate
[Chorus]
One period binomial, up and down we go
Risk neutral probability, that's what we need to know
Replicating portfolio, delta shares we hold
One period binomial, watch the story unfold
[Verse 2]
If it goes up, stock hits one twenty
Down it falls to eighty, not plenty
Now we need the risk neutral chance
Not the real world's risky dance
Take risk free rate minus down move
Divide by up minus down, smooth groove
[Chorus]
One period binomial, up and down we go
Risk neutral probability, that's what we need to know
Replicating portfolio, delta shares we hold
One period binomial, watch the story unfold
[Bridge]
Delta equals option change divided by stock change
That's the hedge ratio we arrange
Buy delta shares and borrow cash
Present value makes the splash
Risk free rate discounts it back
Perfect hedge keeps us on track
[Verse 3]
Call option worth fifteen when up
Worth zero when the stock goes down, tough luck
Calculate that risk neutral P
Point four probability
Expected value fifteen times point four
Discount back, that's what options are for
[Chorus]
One period binomial, up and down we go
Risk neutral probability, that's what we need to know
Replicating portfolio, delta shares we hold
One period binomial, watch the story unfold
[Outro]
Two approaches, same result
Risk neutral or replicating, no default
One period model, foundation strong
For complex trees, we sing this song