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 \(t\).

Arbitrage-free condition

\[d<\exp(r\delta t)<u\] for each sub-period of length \(\delta t\).

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: \[\begin{align*} q_u&=\frac{\exp(r\delta t)-d}{u-d} \\ q_d&=\frac{u-\exp(r\delta t)}{u-d} \end{align*}\]

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 \(X_{t-\delta t}=E_{\mathbb{Q}}[\exp(-r\delta t)X_t\ |\ F_{t-\delta t}]\).