Option pricing models formulas

Replication and risk-neutral pricing

One-step risk-neutral price

f_0=e^{-rT}\big(qf_u+(1-q)f_d\big),\qquad q=\frac{e^{rT}-d}{u-d}

Replicate the payoff in both states; $q$ is a pricing weight, not the forecast probability of an up move.

Remember

Price by replication: match the payoff in every state and the prices must match today.

Black–Scholes: what it says and what breaks it

The formula

C = S_0 \Phi(d_1) - Ke^{-rT}\Phi(d_2)

With the two arguments given by:

The put formula

P = Ke^{-rT}\Phi(-d_2) - S_0\,\Phi(-d_1)

It follows from the call formula and put–call parity, and it reads the same way: the strike received, weighted by the risk-neutral probability of finishing below it, minus the stock given up, weighted by that probability under the stock measure.

With a dividend yield

C = S_0e^{-qT}\Phi(d_1) - Ke^{-rT}\Phi(d_2), \quad d_{1,2} = \frac{\ln(S_0/K) + (r - q \pm \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt T}

Merton’s extension replaces spot by its dividend-discounted value everywhere. Equivalently, it is Black’s formula on the forward $F = S_0e^{(r - q)T}$ , discounted at $r$ .

Remember

$C = S_0\Phi(d_1) - Ke^{-rT}\Phi(d_2)$ , with $\Phi(d_2)$ the risk-neutral exercise probability and $\Phi(d_1)$ the delta.

Binomial trees, backward induction and early exercise

The Cox–Ross–Rubinstein parameterisation

u = e^{\sigma\sqrt{\Delta t}}, \qquad d = \frac{1}{u}, \qquad q = \frac{e^{r\Delta t} - d}{u - d}

Choosing $d = 1/u$ makes the tree recombine, so $n$ steps give $n+1$ terminal nodes rather than $2^n$ paths.

Remember

$u = e^{\sigma\sqrt{\Delta t}}$ , $d = 1/u$ , and the tree recombines.

Numerical pricing: Monte Carlo, finite differences and when to use which

Pathwise delta for a call

\Delta = e^{-rT}\,\mathbb{E}\!\left[\mathbf{1}\{S_T > K\}\,\frac{S_T}{S_0}\right]

Differentiate the payoff along each path instead of bumping: under geometric Brownian motion $\partial S_T/\partial S_0 = S_T/S_0$ . The estimator uses the same paths as the price and has no bump-size error, but needs a payoff that is differentiable almost everywhere.

Remember

Path-dependent means Monte Carlo; early exercise means a lattice or PDE.

Beyond Black–Scholes: local, stochastic and jump models

Dupire’s local volatility

\sigma_{\text{loc}}^2(K, T) = \frac{\partial_T C + rK\,\partial_K C}{\tfrac{1}{2} K^2\,\partial_{KK} C}

Given call prices for every strike and expiry (and no dividends), there is exactly one volatility function of spot and time that reproduces all of them. It is read directly off the surface’s slopes and curvature, which is why local volatility fits perfectly — and why noisy quotes make it unstable.

Heston’s stochastic volatility

dS = \mu S\,dt + \sqrt{v}\,S\,dW_1, \quad dv = \kappa(\theta - v)\,dt + \xi\sqrt{v}\,dW_2, \quad dW_1dW_2 = \rho\,dt

Variance follows its own mean-reverting process. The correlation $\rho$ sets the skew, the volatility of variance $\xi$ the curvature, and $\kappa$ and $\theta$ the term structure. It has a semi-closed-form price via characteristic functions, which is why it became the standard stochastic-volatility benchmark.

Remember

Local volatility fits the surface exactly and has unrealistic dynamics.

Replication and risk-neutral pricing

Black–Scholes: what it says and what breaks it

Binomial trees, backward induction and early exercise

Numerical pricing: Monte Carlo, finite differences and when to use which

Beyond Black–Scholes: local, stochastic and jump models

Detailed formula cards