[2] The Binomial Model (The Multi Period Model)

The multi-period model

(Notatioin)

• time : $t=0,1,2,3,\cdots,T$

• $Z_0, Z_1, \cdots, Z_{T-1} : iid$
  $\begin {cases} P(Z_t=u)=p_u\\P(Z_t=d)=p_d\end {cases}$

• bond dynamics
  $B_t =(1+R)^t$

• stock dynamics
  $\begin {cases} S_{t+1}=S_tZ_t \\ S_0=s\end {cases}$

• $P(S_t=su^jd^{t-j})=\begin{pmatrix} t \\ j \end{pmatrix}p_u^jp_d^{t-j}$
  
  
  

(1) Portfolio (strategy)

A portfolio $h=\{h_t=(x_t, y_t)|\;t=1,2,\cdots,T\}$ with $h_0=h_1$ is a stochastic process and each $h_t$ is a random variable.

$x_t$ is the value of bonds hold on $[t-1,t)$,
$y_t$ is the number of stocks hold on $[t-1,t)$.

(2) Value process $V_t^h$

It means the total asset price of $h$ just before time $t$

$V_t^h=x_t(1+R)+y_tS_t$ is on $[t-1, t)$

(3) Self-financing

A portfolio $h$ is called self-financing if

$x_t(1+R)+y_tS_t = x_{t+1}+y_{t+1}S_t$ for all $t$.

For example, monthly income or spending it for living are not self-financing.

(4) Arbitrage portfolio

A self-financing portfolio is said to be an arbitrage portfolio $h$ if

• first day : $V_0^h=0$
• after : $P(V_T^h \geq 0) = 1$ & $P(V_T^h>0)>0$

(Lemma)

If the multi-period model is arbitrage free, $d \leq 1+R\leq u$.

(5) Contigent claim (Financial derivative)

$\Phi : \mathbb{R} \to \mathbb{R}$ is a contract function from $x$ to $\Phi(x)$.
$X=\Phi(x)$ is called a contigent claim.

(Notation)

$\Pi(t;X)$ is the price of $X$ at time $t$ such that $X = \Phi(x)$.

In the case of European call option, $\Pi(3;X)=\Phi(S_3)=(S_3-K)_+$

(6) Hedging portfolio

A contigent claim $X$ is called reachable if there is a self-financing portfolio $h=\{h_1, h_2, \cdots, h_T\}$ such that $V_T^h=X$. and such $h$ is called hedging portfolio(replicating portfolio). If all claims are reachable, the market is said to be complete.

We want to find hedging portfolio $h_t=(x_t,y_t)$ such that
$\Phi(S_T)=V_T^h.$

We can apply the method of the one-period model repeatedly.

(7) Example

• $T=3,\;\;s=80,\;\;R=1.2$
• $u=1.5,\;\;d=0.5$
• $X=(S_3-80)_+$ : European call option

$\therefore \Pi(0;X) = 17395/432 \fallingdotseq 40.2662$

$\therefore h_0 \fallingdotseq (-23.9005, 0.8021)$