2. The Binomial Model

We are motivated by the fact that no static replicating portfolio of European options and other products exists. We thus have to develop a dynamic replicating portfolio based on an appropriate model.

2.1 One-Period Binomial Model

Our market consists of two basic instruments, a stock $S$ and bond $B$, with a starting state at $t = 0$ and two possible states at $t = 1$.

We call the measure $\P$ with the probabilities $p_d$ and $p_u$ the historical measure and impose the following requirements:

• non-negative stock price: $S_0 > 0$
• lower bound on interest rate: $r > -1$
• positive, ordered stock price movements: $u > d > 0$
• non-zero probabilities: $0 < p_d, p_u < 1$

As $t \in \cb{0,1}$ we only have to consider $\pi_0 = xS_0 + yB_0$ and $\pi_1 = xS_1 + yB_1$.

Hence an arbitrage portfolio is a strategy where you can make money out of nothing. We aim at building models that are arbitrage-free and search for so called NA, i.e. no-arbitrage, conditions for the model parameters to ensure this.

As NA implies $1 + r \in (d, u)$ we can write $1+r$ as a convex combination of $d$ and $u$, i.e. $1+r = q_d d + q_u u$ where $$q_u = \frac{(1+r) - d}{u-d} \qand q_d = \frac{u - (1+r)}{u-d}$$ and $q_u + q_d = 1$.

We can interpret $\cb{q_u, q_d}$ as a new probability measure

We note that $$\E_{\Q}\bk{\frac{1}{1+r} S_1} = \frac{1}{1+r} S_0(q_d d + q_u u) = S_0$$ In other words, the discounted expectation under the risk-neutral measure of the stock price tomorrow is equal to the stock price today.

Recall that a process $(X_t)_{t\geq0}$ is a martingale under a probability measure $\mu$ if $\E_{\mu}[X_{t+1} \mid X_t] = X_t$, i.e. if the estimated value for the future value at $t+1$ is its current value $X_t$.

Martingale measures are extremely useful as their existence guarantees NA and they will give us the no-arbitrage price of any asset.

Recall that our aim is to price options. We provide an example of a dynamic replication in the context of the one-period binomial model.

Example (Pricing European options): Consider a European call option on a stock $S$ with maturity $T = 1$ and strike $K = 101$. Let $S_0 = 100$, $r = 4\%$, $u = 1.1$ and $d = 0.92$.

At maturity, the payoff is $C_1 = (S_1 - K)^+$, thus we have $C_1^u = (S_1^u - K)^+ = 9$ and $C_1^d = (S_1^d - K)^+ = 0$.

We replicate the option using the underlying assets at disposal, i.e. we consider a portfolio $\pi$ composed of $\alpha$ stocks and $\beta$ bonds, s.t. $\pi_t$ is characterized by $$\begin{align*} & \pi_1^u=110 \alpha + 1.04 \beta=9=C_1^u \\ & \pi_1^d=92 \alpha + 1.04 \beta=0=C_1^d \end{align*}$$ Solving the linear equations entails $\beta=-\frac{46}{1.04} \approx-44.23$ and $\alpha=\frac{1}{2}$. Hence $\pi_t=\frac{1}{2}S_t-\frac{46}{1.04}B_t$ replicates the option $C_t$ and because of the Law of One Price this is true independently of $t$. Thus we receive $C_0=\pi_0= \frac{1}{2} S_0 + \frac{46}{1.04} B_0 \approx 5.77$ as the fair value of the option at time $t=0$.

2.1.1 Completeness

In other words, completeness ensures that for any derivative security with a given payoff $K$ at maturity, we can construct a self-financing portfolio that exactly matches $K$ using only the two basic instruments $B$ and $S$.

Completeness is an attractive feature because we get a hedging portfolio for each contingent claim. However, it also means that contingent claims are superfluous since we can trade in the stock/bond and replicate it.

In practice the market is not complete, i.e., it is not possible to hedge any contingent claim only trading a risk-free bond and the stock. The model is overly simplistic.

2.1.2 Martingale Pricing

We summarize the aforementioned results.

In other words, under a martingale measure, the price of any contingent claim is given by the expectation of its discounted payoff. Hence, this theorem allows to price a contingent claim without dealing with replication.

This allows to link replication and completeness to the martingale measure. It is often easier to work with martingale measures than with portfolios.

2.2 Multiperiod Binomial Model

The multiperiod binomial model improves upon the one-period model. From $t_0 = 0$ to $t = T$ it allows more than one move of the underlying stock $S$ and more than two states in the economy. We achieve this by increasing the number of periods in the binomial model.

We divide our period of interest $[0,T]$ into $N$ equal-length periods $\bk{t_k, t_{k+1}}$, with $t_k= \frac{k T}{N}$ for $0 \leq k \leq N$. At each time $t_k$ we hace the two assets with prices $B_k$ and $S_k$.

To calculate the bond price in the multiperiod binomial model, we assume that the annual interest on the bond is $r$. Hence the one-period interest rate is $\tilde{r}=r \frac{T}{N}$ assuming that $T$ is measured in years. Thus $B_{k+1}=(1+\tilde{r}) B_k$. Note that when $N \rightarrow \infty$, we have $\lim_{N \rightarrow \infty} B_N=\lim_{N \rightarrow \infty}\pa{1+r / \frac{N}{T}}^N=e^{r T}$ through continuous compounding.

Note that the stock price is set to $S_k$ at time $t \in [t_k, t_{k+1})$, i.e. the function of stock prices over time is step-wise.

2.2.1 Stock Distributional Properties

The probabilistic evolution of the stock does not depend on its past, only on its present, i.e. $\P\pa{S_{k+1} \mid S_0, \ldots, S_k}=\P\pa{S_{k+1} \mid S_k}$.

We derive the distribution of the stock price by induction. At time $t_1$, $S_1$ can take $1+1$ values: $u S_0$ and $d S_0$ with probabilities $\P\pa{S_1^u}=p_u$ and $\P\pa{S_1^d}=p_d$ respectively. Hence at time $t_N = T$, $S_N$ can take $N+1$ values: $u^i d^{N-i} S_0$, $\forall 0 \leq i \leq N$ with probabilities $$\mathbb{P}\left(S_N=S_0 u^i d^{N-i}\right)=\binom{N}{i} p_u^i p_d^{N-i}$$ respectively.

2.2.2 Trading Strategy

Under the context of the multimodal binomial model we refine the definiton of a portfolio/trading strategy.

We are interested in strategies where we do not need to inject/remove capital.

In other words, when rebalancing our portfolio $\cb{x_k, y_k} \rightarrow \cb{x_{k+1}, y_{k+1}}$, the value of our portfolio does not change. The value of the portfolio can only change due to movements in the assets $B$ and $S$.

2.2.3 No Arbitrage

We extend the definition of the arbitrage portfolio to the multiperiod model.

Thus the $N$-period model has the same non arbitrage condition and properties as the one-period model. We can therefore define the martingale measure in the multiperiod model.

In other words, a measure $\Q$ is a martingale measure in the $N$-period model iff it is a martingale measure for each sub-period model.

From the one period model we therefore know that NA implies the existence of an equivalent martingale measure $\Q$, that $q_u=\frac{(1+\tilde{r})-d}{u-d}$, $q_d=\frac{u-(1+\tilde{r})}{u-d}$ define such measure and that the measure is unique.

We extend the market completeness property to the general binomial model.

In other words, market completeness implies that e.g. a simple European security can always be hedged.

2.2.4 Binomial Option Pricing Formula

In particular, we have

H_0=\E_{\Q}\qty[\frac{1}{(1+\tilde{r})^N} H(S_N) \mid S_0]

At time $t_N$ there are $N+1$ nodes defined e.g. by the number of times $i$ the stock goes up. There are

\# \qty{S_N = u^i d^{N-i} S_0} = \binom{N}{i}

2.2.5 Rewriting the Call Option Formula

We consider now a European call option with maturity after $N$ timesteps. For the claim to be exercisable, we require the option to be in the money, i.e. $S_0 u^j d^{N-j}>K$. Hence, we can determine the minimal number of up-moves for the underlying as

A=\left\lfloor\frac{\ln \left(K /\left(S_0 d^N\right)\right)}{\ln (u / d)}\right\rfloor+1

\begin{align*} C_0& = S_0 \sum_{j=A}^N\binom{N}{j}\qty(\frac{q u}{R})^j \qty(\frac{(1-q) d}{R})^{N-j}-\frac{K}{R^N} \sum_{j=A}^N\binom{N}{j} q^j(1-q)^{N-j} \\ &= S_0 \sum_{j=A}^N\binom{N}{j} {q^{\prime}}^j \qty(1-q^{\prime})^{N-j}-\frac{K}{R^N} \sum_{j=A}^N\binom{N}{j} q^j(1-q)^{N-j} \\ &= S_0 \Psi(A, N, q^{\prime}) - K R^{-N} \Psi(A, N, q) \end{align*}

Recall that we have defined our timestep delta to be $h = \frac{T}{N}$. One can show that in the limit $h \to 0$, i.e. $N \to \infty$, the aforementioned rewritten call option formula converges to $$C_0=S_0 \mathcal{N}(d_1)-K e^{-r T} \mathcal{N}(d_2)$$ TODO where $\mathcal{N}$ is the standard normal density function and

\begin{align*} d_1&=\frac{\ln \qty(S_0 / K)+\qty(r+\frac{1}{2} \sigma^2) T}{\sigma \sqrt{T}} \\ d_2&=d_1-\sigma \sqrt{T} \end{align*}