Single-Period 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: \[\begin{align*} q_u&=\frac{\exp(rT)-d}{u-d} \\ q_d&=\frac{u-\exp(rT)}{u-d} \end{align*}\]
Replicating Porfolio
Value of portfolio
The value of a portfolio \(h(x,y)\) with \(x\) units of riskless asset and \(y\) units of risky asset is \[V_t^h=xB_t+yS_t,\qquad t=0,T\]
Arbitrage portfolio
The portfolio is arbitrage if
- \(V_o^h=0\)
- \(V_T^h\geq0\) with certainty
- \(V_T^h>0\) with non-zero probability
Attainable payoff
A payoff \(X_T\) is attainable if there exists a replicating portfolio \(h(x,y)\) such that \(V_T^h=X_T\).
Then, a market model is complete if every payoff is attainable.
The one step binomial model is complete and the replicating portfolio h(x,y) is given by \[x=\frac{S_T^uX_T^d-S_T^dX_T^u}{(S_T^u-S_T^d)B_T}\quad\text{and}\quad y=\frac{X_T^u-X_T^d}{S_T^u-S_T^d}\] Note that \(S_T^u=uS_0\) and \(S_T^d=dS_0\). The same goes for \(X_T\).
To prevent arbitrage, according to Arbitrage Pricing Principle II (APP II), \(X_0=V_0^{h(x,y)}\).