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.
The binomial model is a discrete-time model that converges to the continuous-time Black-Scholes model when decreasing the timestep delta. It offers fast pricing of European options.
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 $t = 1.$
Figure 2.1: The market at time $t = 0$ and $t = 1.$
We impose the following requirements: $S_0 > 0,$ $r > -1,$ $u > d > 0$ and $0 < p_d, p_u < 1$ and call $\qty{p_d, p_u}$ the historical measure $\probP.$
Definition 2.1 (Portfolio in one-period binomial model): The value $V_t(\pi) = xS_t + yB_t$ of a portfolio $\pi$ is composed of $x \in \R$ units of stock $S$ and $y \in R$ units of bond $B.$
As $t \in \qty{0,1}$ we only have to consider $V_0(\pi) = xS_0 + yB_0$ and $V_1(\pi) = xS_1 + yB_1.$
Definition 2.2 (Arbitrage portfolio for the one-period model): $\pi$ is an arbitrage portfolio iff $V_0 = 0$ and $V_1 \geq 0$ with a positive probability of making profits $\probP(V_1 > 0) > 0.$
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.
Definition 2.3 (NA condition for one-period binomial model): The one-period binomial model has no arbitrage iff $0 < d < 1 + r < u.$
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
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 $X_t.$
Example (Discounted stock price): Take $X_t = \frac{S_t}{(1+r)^t}$ and the risk neutral measure $\Q.$ We know that $\E_{\Q}\qty[\frac{1}{1+r} S_1 \mid S_0] = S_0$ i.e. the discounted stock price is a martingale under the risk-neutral measure for the one-period binomial tree.
Definition 2.5 (Martingale measure): The probability measure $\Q$ is a martingale measure w.r.t. the discounted stock price iff $\E_{\Q}\qty[\frac{1}{1+r} S_1 \mid S_0] = S_0.$
Note (Equivalent measures): We call two measure $\mu, \nu$ equivalent if for all events $\omega \in \Omega$ we have $\mu(\omega) = 0 \iff \nu(\omega) = 0.$
Theorem 2.6 (First fundamental theorem of asset pricing): The NA condition holds iff there exists a martingale measure $\Q$ equivalent to the historical measure $\probP.$
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 $d = 0.92.$Figure 2.3: The possible states at time $t = 0$ and $t = 1.$
At maturity, the payoff is $C_1 = (S_1 - K)^+,$ thus we have $C_1^u = (S_1^u - K)^+ = 9$ and $C_1^d = (S_1^d - K)^+ = 0.$
Figure 2.4: The option values at time $t = 0$ and $t = 1.$
We replicate the option using the underlying assets at disposal, i.e. we consider a portfolio $\pi$ composed of $\Delta$ stocks and $\alpha$ bonds, s.t. $V_t(\pi)$ is characterized by
Solving the linear equations entails $\alpha=-\frac{46}{1.04} \approx-44.23$ and $\Delta=\frac{1}{2}.$ Hence $\pi=\qty{-44.23 \text{ bonds}, 0.5 \text{ stock} }$ replicates the option $C$ and because of the Law of one price this true independently of $t.$ Thus we receive $C_0=V_0(\pi)=\alpha \cdot 1+ \Delta \cdot S_0 \approx 5.77$ as the fair value of the option at time $t=0.$
2.1.1 Completeness
Definition 2.7 (Complete model): A model is considered complete when every contingent claim can be perfectly replicated using only the underlying asset $S$ and the risk-free asset $B.$
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 $S.$
Proposition 2.8 (Completeness of the one-period binomial model): The one-period binomial model is complete as there are at least two assets for two possible future states and the two assets are linearly independent.
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.
Proposition 2.9 (Martingale pricing in the one-period binomial model): In the one-period binomial model, no arbitrage implies that the price $H_0$ of any contingent claim $H_1$ given by $H_0=\frac{1}{1+r} \E_{\Q}\qty[H_1]$ where $\Q \sim \probP$ is the martingale measure given by $q_u=\frac{(1+r)-d}{u-d},$ $ q_d=\frac{u-(1+r)}{u-d}.$
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.
Theorem 2.10: In the absence of arbitrage, the market is complete iff there exists a unique equivalent martingale measure $\Q.$
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.
Figure 2.5: The first two timesteps of the multiperiod binomial model.
We divide our period of interest $[0,T]$ into $N$ equal-length periods $\left[t_k, t_{k+1}\right],$ 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 $\lim_{N \rightarrow \infty} B_N=\lim_{N \rightarrow \infty}\qty(1+r / \frac{N}{T})^N=e^{r T}$ 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. $\probP\qty(S_{k+1} \mid S_0, \ldots, S_k)=\probP\qty(S_{k+1} \mid S_k).$
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 $\probP\qty(S_1^u)=p_u$ and $\probP\qty(S_1^d)=p_d$ 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
Under the context of the multimodal binomial model we refine the definiton of a portfolio/trading strategy.
Definition 2.11 (Trading strategy): A trading strategy/portfolio strategy $\pi_t$ is a discrete time stochastic process that is composed of $x_k \in \R$ units of stock $S$ and $y_k \in \R$ units of bond $B$ at time $t_k.$ We are only allowed to change our portfolio just after the stock price has moved, i.e. at times $t_k^+,$ thus $\pi_t=\left(x_t, y_t\right)=\left(x_k, y_k\right)$ for $t \in(k-1, k).$ $x_k$ and $y_k$ are only allowed to depend on $S_0, \ldots, S_{k-1}.$ The value of the portfolio is therefore
$$
V_t = \begin{cases}
V_k=x_k S_k+y_k B_k & \text{for } t = t_k \\
V_{k^{+}}=x_{k+1} S_k+y_{k+1} B_k & \text{for } t \in (t_k, t_{k+1})
\end{cases}
$$
We are interested in strategies where we do not need to inject/remove capital.
Definition 2.12 (Self-financing trading strategy): A trading strategy is self-financing if we have $x_k S_k+y_k B_k=V_k=V_{k+}=x_{k+1} S_k+y_{k+1} B_k$ for all $k=0, \ldots, N-1.$
In other words, when rebalancing our portfolio $\qty(x_k, y_k) \rightarrow \qty(x_{k+1}, y_{k+1}),$ 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.
Definition 2.13 (Arbitrage portfolio): A self-financing trading strategy $\pi$ is an arbitrage iff $V_0 = 0,$ $\probP(V_N \geq 0)$ and $\probP(V_N>0)>0.$
Definition 2.14 (NA condition in the multiperiod binomial model): The following statements are equivalent:
There is no arbitrage in the multiperiod binomial model
There is no arbitrage in each one-period sub model
We have $0 < d < 1 + \tilde{r} < u$
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.
Definition 2.15 (Martingale measure in the multiperiod binomial model): A martingale measure or risk-netral measure is a probability measure $\Q$ such that $\E_{\Q}\qty[\frac{1}{1+\tilde{r}} S_{k+1} \mid S_k ]=S_k$ for all $k \in \qty{0, \ldots, N}.$
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.
Theorem 2.16 (Market completeness for the binomial model): The binomial model is complete. In particular, let $V_N$ be the payoff of a simple European security and define
$$
V_k =(1+\tilde{r})^k \ \E_{\Q}\qty[(1+\tilde{r})^{-N} V_N \mid \mathcal{F}_k ]
$$
and
$$
\Delta_{k+1} =\Delta_{k+1}(S_k) =\frac{V_{k+1}(S_{k+1}=u S_k)-V_{k+1}(S_{k+1}=d S_k)}{u S_k-d S_k}
$$
Starting with initial wealth $V_0=\E_{\Q}\qty[(1+\tilde{r})^{-N} V_N \mid \mathcal{F}_0],$ the value of the self-financing portfolio process $\Delta_0, \ldots, \Delta_{N-1}$ is the process $V_1, \ldots, V_N.$ TODO
In other words, market completeness implies that e.g. a simple European security can always be hedged.
2.2.4 Binomial Option Pricing Formula
Definition 2.17 (Binomial algorithm): Under no-arbitrage, the price $H_k^i$ at time $t_k$ of any contingent claim $H$ with payoff $H_N=H(S_N)$ at time $T$ is given by the following algorithm
for all $i \in \qty{0, \ldots, N}$ and $k \in \qty{N-1, \ldots, 0}.$
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 $\# \qty{S_N = u^i d^{N-i} S_0} = \binom{N}{i}$ paths leading to this node. All paths are independent of each other and have risk-neutral probability $q_u^i q_d^{N-i},$ hence
We can thus formulate the binomial option pricing formula.
Definition 2.18: The price at time $t = 0$ of a European contigent claim $H$ with payoff $H(S_N)$ at time $T$ is given in the multiperiod binomial model by the binomial option pricing formula
$$
H_0=\frac{1}{(1+\tilde{r})^N} \sum_{i=0}^N\binom{N}{i} q_u^i q_d^{N-i} H(S_0 u^i d^{N-i})
$$
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
$$
A=\left\lfloor\frac{\ln \left(K /\left(S_0 d^N\right)\right)}{\ln (u / d)}\right\rfloor+1
$$
For simplification we write $1 + \tilde{r} = R.$ Then, we can rewrite the formula for the call option as
where $q^{\prime} = \frac{qu}{R}$ and $\Psi(a, n, p)$ is the complimentary binomial distribution function. More specifically, $\Psi(a, n, p)$ describes the probability of getting at least $a$ heads out of $n$ tosses if the probability for heads is $p.$
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
