Binomial Model
- Risk assets: prices can go up or down
- Riskless assets: stable interest
Notations
- \(B_t\): value of risk-free asset at time \(t\)
- \(V_t\): value of risky asset at time \(t\)
- \(u\): factor to make upward movement
- \(d\): factor to make downward movement
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 \[S_0=E_{\mathbb{Q}}[\exp(-rT)S_T]\]
A one-period binomial model is arbitrage free if and only if there exists a unique equivalent martingale measure Q: \[\begin{align*} q_u&=\frac{\exp(rT)-d}{u-d} \\ q_d&=\frac{u-\exp(rT)}{u-d} \end{align*}\]