Notes On Arbitrage Asset Pricing

The Binomial Model

Lets begin with the one Period Model:

Model Description Time is denoted by 𝑡 and by definition we have two points in time $t=0(today)$, $t=1(tomorrow)$.

In the model we have two assets a bond and a stock, At time t the price of a bond is denoted by $B_t$, and the price of one share of the security is denoted by $S_t$.Thus we have two price processes $B$ and $S$.

The bond price process is deterministic and given by:
\(𝐵_0=1, \\ 𝐵_1=1+𝑅\)

The constant R is the spot rate for the period and we can also interpret the existence of the bond as the existence of a bank with R as its rate of interest.

The stock price process is a stochastic process and its dynamical behavior is given as:

\[S_0 = s, \quad S_1 = \begin{cases} s \cdot u, & \text{with probability } p_u,\\ s \cdot d, & \text{with probability } p_d. \end{cases}\]

we will often write it as: \(\begin{cases} S_0 = s,\\ S_1 = s \cdot Z \end{cases}\)

Where Z is a stochastic variable defined as:

\[Z = \begin{cases} u, & \text{with probability } p,\\ d, & \text{with probability } p_d \end{cases}\]

If we consider a portfolio $ℎ=(𝑥,𝑦)$.This portfolio has a deterministic market value at $𝑡=0$ and a stochastic value at $t=1$.

The value process can be defined as \(𝑉_𝑡^ℎ=𝑥𝐵_𝑡+𝑦𝑆_𝑡,𝑡=0,1 \\ 𝑉_0^ℎ=𝑥+𝑦𝑠, \\ 𝑉_1^ℎ=𝑥(1+𝑅)+𝑦𝑠𝑧\)

An arbitrage portfolio is a portfolio h with the properties: \(𝑉_0^ℎ=0 𝑉_1^ℎ>0, \text{with probability} 1\)

The existence of an arbitrage portfolio is equivalent to a case of mispricing on the market. Hence it can now be investigated when a given model is arbitrage free.

To determine if the above model is arbitrage free then 𝑑≤(1+𝑅)≤𝑢,We can show that this implied absence of arbitrage.

Example: $𝑉_0^ℎ=0=𝑥𝐵_0+𝑦𝑆_𝑜$

• A single period binary model with a single stock is viable if: $𝑆_1^𝑑<(1+𝑟) 𝑆_0<𝑆_1^𝑢$

• To see that this portfolio cannot be an arbitrage portfolio for y > 0 : \(𝑥𝐵_1+𝑦𝑆_1^𝑑<𝑥𝐵_1+𝑦(1+𝑟) \\ 𝑆_0<𝑥𝐵_1+𝑦𝑆_1^𝑢\)
• The middle term from the above inequality is equal to:

\[𝑥(1+𝑟) 𝐵_0+𝑦(1+𝑟) 𝑆_0=(1+𝑟) 𝑉_0^(𝑥,𝑦)=0\]

The other values in the two scenarios at time 1.Thus $𝑉_1^(𝑥,𝑦) (𝑢)<0<𝑉_1^(𝑥,𝑦) (𝑑)$ -So here (x,y) is not an arbitrage opportunity. Hence the model is viable.

We will continue with the arbitrage pricing as theory in another write up.