I asked claude to help with risk neutral probabilities and it started with the 1 period binomial model

Imagine a stock today trades at $100. In one period, it will either go up to $110 or down to $90. There’s also a risk-free bond paying 5% interest.

Suppose you want to price a derivative that pays $10 if the stock goes up, and $0 if it goes down.

Question for you: if you wanted to figure out the fair price of this derivative today, what’s one approach you might try — even a naive one?

Let’s set it up properly with algebra instead of guessing. You want a portfolio of:

such that the portfolio’s payoff matches the derivative in both states.

Question: can you write down the two equations — one for the “up” state, one for the “down” state — that $\Delta$ and $B$ must satisfy?

(Recall: up state stock = $110, down state stock = $90, bond grows to $1.05B.)

my attempt

$\Delta S_0 + B$ at time 0

$\Delta 110 + 1.05B = 10$ at time 1

$\Delta 90 + 1.05B = 0$ at time 1

\[\Delta = \frac{1}{2}\]