Option pricing models

One-period risk-neutral pricing

Prices any one-period claim by replication, without needing anyone’s view on where the stock is going.

f0=e−rT[qfu+(1−q)fd],q=erT−du−df_0 = e^{-rT}\big[q f_u + (1-q)f_d\big], \qquad q = \frac{e^{rT} - d}{u - d}

Where

qq
The risk-neutral probability — a weight from the algebra, not a forecast.
u,du, d
Up and down move factors.

Assumptions

  • Requires d<erT<ud < e^{rT} < u, or the stock itself is an arbitrage.

Sanity check. The real probability of an up-move never appears. If yours did, you have not replicated.

Where this is taught