Multi-Period Binomial Model
Denote \(h(x_t,y_t)\) as a portfolio of \(x_t\) units of riskless asset and \(y_t\) units of risky asset from time \(t-\delta t\) to \(t\).
Risk-Neutral Probability (Martingale)
A probability measure \(\mathbb{Q}\) consisting of an upward-movement probability \(q_u\) and a downward-movement probability \(q_d\), is called a martingale probability or risk-neutral probability if \[S_{t-\delta t}=E_{\mathbb{Q}}[\exp(-r\delta t)S_t\ |\ F_{t-\delta t}]\]
A one-period binomial model is arbitrage free if and only if there exists a unique equivalent martingale measure Q: \[\begin{align*} q_u&=\frac{\exp(r\delta t)-d}{u-d} \\ q_d&=\frac{u-\exp(r\delta t)}{u-d} \end{align*}\]
Replicating Porfolio
Value of portfolio
The value of a portfolio \(h\) with \(x_t\) units of riskless asset and \(y_t\) units of risky asset at time \(t-\delta t\) is \[V_t^h=x_tB_t+y_tS_t\]
Self-financing portfolio
\[x_tB_{\delta t}+y_tS_t=V_t=x_{t+\delta t}B_0+y_{t+\delta t}S_t\]
The multi step binomial model is complete and the replicating portfolio \(h(x_t,y_t)\) is given by \[x_t=\frac{S_t^uX_t^d-S_t^dX_t^u}{(S_t^u-S_t^d)B_{\delta t}}\quad\text{and}\quad y_t=\frac{X_t^u-X_t^d}{S_t^u-S_t^d}\]
Risk Neutral Pricing Formula
If the binomial model is arbitrage free, then the arbitrage free price of a derivative at time \(t-\delta t\) with payoff \(X_t\) is given by \(X_{t-\delta t}=E_{\mathbb{Q}}[\exp(-r\delta t)X_t\ |\ F_{t-\delta t}]\).
For American options, check the payoff when exercise at each time point and pick the higher one.