The Binomial Asset Pricing Model: One-Period Binomial Model

In this note, we model stock prices in discrete time, assuming that at each step the price per share of stock will be one of two possible values determined by the outcome of a coin toss.

Let $S_0$ be the initial stock price. We introduce two numbers  $u$ and $d$  with $0<d<u$ such that the stock price will be either $dS_0$ or $uS_0$ at the next period. Suppose that we are tossing a coin and when we get a Head ($H$) the stock price goes up and when we get a Tail ($T$), the price goes down. Denote the price at time $1$ by $S_1(H)=uS_0$ if the toss results in $H$ and by $S_1(T)=dS_0$ if the toss results in $T$. After the second toss the price will be one of
\begin{align*}
 S_2(HH)&=uS_1(H)=u^2S_0,\ S_2(HT)=dS_1(H)=duS_0,\\
 S_2(TH)&=uS_1(T)=udS_0,\ S_2(TT)=dS_1(T)=d^2S_0
\end{align*}
Suppose that the 3rd toss is the last one. Then
$$\Omega=\{HHH, HHT, HTH, THH, THT, TTH, TTT\}$$
is the set of all possible outcomes of the three tosses. $\Omega$ is called the sample space for the experiment and each $\omega\in\Omega$ is called a sample point.

We now introduce a money market with interest $r$. We take $r$ to be the interest rate for both borrowing and lending. One dollar invested in the money market at time $0$ will yield $(1+r)$ dollars at time $1$. One dollar borrowed from the money market at time $0$ will result in a debt of $(1+r)$ dollars at time $1$.

In this model, we assume no arbitrage. So, we require that $0<d<1+r<u$ for one-period binomial model. If $d\geq 1+r$, one can start with no money and borrow from the money market to buy the stock. Even in the case of a tail, the stock at time $1$ will be worth enough to pay off the money market debt. This provides an arbitrage. On the other hand, if $u\leq r+1$, one can sell the stock short and invest the proceeds in the money market. Even in the case of a head, the cost of replacing it at time $1$ will be less than or equal to the value of the money market investment. This again provides an arbitrage.

We now consider a European call option, which confers on its owner the right but not the obligation to buy one share of the stock at time $1$ for the strike price of $K$. We assume that $S_1(T)<K<S_1(H)$. If we get a tail, the option expires worthless. If we get a head, the option can be exercised and yields a profit of $S_1(H)-K$. This situation can be described as saying the option at time $1$ is worth $(S_1-K)^+$ where $(S_1-K)^+=\max\{S_1(\omega_1)-K,0\}$.

Example. Let $S_0=4$, $u=2$, $d=\frac{1}{2}$, and $r=\frac{1}{4}$. Then $S_1(H)=8$ and $S_1(T)=2$. Suppose the strike price of the European call option is $K=5$. Suppose we begin with the initial wealth $X_0=1.20$ and buy $\Delta_0=\frac{1}{2}$ shares of stock at time $0$. Since the stock costs $4$ per share at time $0$, we must use our initial wealth $X_0=1.20$ and borrow an additional $0.80$ to buy $\frac{1}{2}$ shares of stock. This leaves us with a cash position $X_0-\Delta_0S_0=-0.80$ (i.e. a debt of $0.80$ to the money market). At time $1$, our cash position will be $(1+r)(X_0-\Delta_0S_0)=-1$. On the other hand, at time $1$ we will have stock valued at either $\frac{1}{2}S_1(H)=4$ or $\frac{1}{2}S_1(T)=1$. If the coin toss results in a head, the value of our portfolio of stock and money market account at time $1$ will be
 $$X_1(H)=\frac{1}{2}S_1(H)+(1+r)(X_0-\Delta_0S_0)=3$$
 If the coin toss results in a tail, the value of our portfolio at time $1$ will be
 $$X_1(T)=\frac{1}{2}S_1(T)+(1+r)(X_0-\Delta_0S_0)=0$$
 In either case, the value of the portfolio agrees with the value of the option at time $1$, which is either $(S_1(H)-5)^+=3$ or $(S_1(T)-5)^+=0$. We have replicated the option by trading in the stock and money markets.

In the one-period model, a derivative security is a security that pays $V_1(H)$ at time $1$ if the coin toss results in $H$ and pays $V_1(T)$ at time $1$ if the coin toss results in $T$. We want to determine the price $V_0$, i.e. how much the option is worth at time $0$. Suppose that we begin we begin with wealth $X_0$ and buy $\Delta_0$ shares of stock at time $0$. This leaves us with a cash position $X_0-\Delta_0S_0$. The value of our portfolio of stock and money market account at time $1$ is
\begin{align*}
 X_1&=\Delta_0S_1+(1+r)(X_0-\Delta_0S_0)\\
 &=(1+r)X_0+\Delta_0(S_1-(1+r)S_0)
\end{align*}
We want to choose $X_0$ and $\Delta_0$ so that $X_1(H)=V_1(H)$ and $X_1(T)=V_1(T)$. Replication of the derivative security requires that
\begin{align*}
 (1+r)X_0+\Delta_0(S_1(H)-(1+r)S_0)&=V_1(H) \tag{1}\\
 (1+r)X_0+\Delta_0(S_1(T)-(1+r)S_0)&=V_1(T) \tag{2}
\end{align*}
Subtracting (2) from (1), we have
$$V_1(H)-V_1(T)=\Delta_0(S_1(H)-S_1(T))$$
Solving this for $\Delta_0$ we obtain
$$
 \Delta_0=\frac{V_1(H)-V_1(T)}{S_1(H)-S_1(T)} \tag{3}
$$
(3) is called the delta-hedging formula. (1) along with (3) can be solved for $X_0$ as
$$
X_0=\frac{1}{1+r}\left[\frac{1+r-d}{u-d}V_1(H)+\frac{u-(1+r)}{u-d}V_1(T)\right] \tag{4}
$$
Introducing new variables $a$ and $b$ such that $u=1+a$ and $d=1+b$, (4) can be written as
$$
 X_0=\frac{1}{1+r}\left[\frac{r-b}{a-b}V_1(H)+\frac{a-r}{a-b}V_1(T)\right] \tag{5}
$$
To summarize, if an agent begins with wealth $X_0$ given by (5) and at time $0$ buys $\Delta_0$ shares of stock given by (3), then at time $1$, if the coin toss results in $H$, the agent will have a portfolio worth $V_1(H)$, and if the coin toss results in $T$, the portfolio will be worth $V_1(T)$. The agent hedged a short position in the derivative security. The derivative security that pays $V_1$ at time $1$ should be priced at
$$
 V_0=\frac{1}{1+r}\left[\frac{r-b}{a-b}V_1(H)+\frac{a-r}{a-b}V_1(T)\right] \tag{6}
$$
at time $0$. This price allows the seller to hedge the short position in the claim.

Let
$$
p=\frac{r-b}{a-b}\ \mathrm{and}\ q=\frac{a-r}{a-b} \tag{7}
$$
Then $p+q=1$, and (5) and (6) can be written as
$$
V_0=X_0=\frac{1}{1+r}[pV_1(H)+qV_1(T)] \tag{8}
$$
Alternatively, (8) can be written as
$$
 V_0=\frac{1}{1+r}[p(S_0(a+1)-K)^++(1-p)(S_0(b+1)-K)^+] \tag{9}
$$
Can $p$ be interpreted as a probability? Suppose that the stock price goes up from $S_0$ to $uS_0$ with probability (Note this $q$ is different from $q$ in (7)) $q$ and it goes down from $S_0$ to $dS_0$ with probability $1-q$. If investors were risk-neutral, the expected rate of return on the stock would be the riskless interest rate, so we have
$$q(uS_0)+(1-q)(dS_0)=(1+r)S_0$$
Solving this equation for $q$, we obtain
$$q=\frac{1+r-d}{u-d}=p$$
Hence, the value of the call can be interpreted as the expectation of its discounted future value in a risk-neutral world.

A Glossary of Finance:

1. Arbitrage: Arbitrage is the simultaneous purchase and sale of the same or similar asset in different markets in order to profit from tiny differences in the asset's listed price. It exploits short-lived variations in the price of identical or similar financial instruments in different markets or in different forms. 
2. derivative:  The term derivative refers to a type of financial contract whose value is dependent on an underlying asset, group of assets, or benchmark. A derivative is set between two or more parties that can trade on an exchange or over-the-counter (OTC). These contracts can be used to trade any number of assets and carry their own risks. Prices for derivatives derive from fluctuations in the underlying asset. These financial securities are commonly used to access certain markets and may be traded to hedge against risk. Derivatives can be used to either mitigate risk (hedging) or assume risk with the expectation of commensurate reward (speculation). Derivatives can move risk (and the accompanying rewards) from the risk-averse to the risk seekers. 
3. European call option: A European option is a version of an options contract that limits execution to its expiration date. In other words, if the underlying security such as a stock has moved in price, an investor would not be able to exercise the option early and take delivery of or sell the shares. Instead, the call or put action will only take place on the date of option maturity.
4. hedge: To hedge, in finance, is to take an offsetting position in an asset or investment that reduces the price risk of an existing position. A hedge is therefore a trade that is made with the purpose of reducing the risk of adverse price movements in another asset. Normally, a hedge consists of taking the opposite position in a related security or in a derivative security based on the asset to be hedged. Derivatives can be effective hedges against their underlying assets because the relationship between the two is more or less clearly defined. Derivatives are securities that move in correspondence to one or more underlying assets. They include options, swaps, futures, and forward contracts. The underlying assets can be stocks, bonds, commodities, currencies, indexes, or interest rates. It's possible to use derivatives to set up a trading strategy in which a loss for one investment is mitigated or offset by a gain in a comparable derivative. 
5. Option:  The term option refers to a financial instrument that is based on the value of underlying securities such as stocks, indexes, and exchange traded funds (ETFs). An options contract offers the buyer the opportunity to buy or sell-depending on the type of contract they hold-the underlying asset. Unlike futures, the holder is not required to buy or sell the asset if they decide against it. Each options contract will have a specific expiration date by which the holder must exercise their option. The stated price on an option is known as the strike price. Options are typically bought and sold through online or retail brokers. 
6. Short position: A short, or a short position, is created when a trader sells a security first with the intention of repurchasing it or covering it later at a lower price. A trader may decide to short a security when she believes that the price of that security is likely to decrease in the near future. There are two types of short positions: naked and covered. A naked short is when a trader sells a security without having possession of it. However, that practice is illegal in the U.S. for equities. It is banned fully in India and other countries. A covered short is when a trader borrows the shares from a stock loan department; in return, the trader pays a borrowing rate during the time the short position is in place. In the futures or foreign exchange markets, short positions can be created at any time. 
7. Short selling: Short selling is an investment or trading strategy that speculates on the decline in a stock or other security's price. It is an advanced strategy that should only be undertaken by experienced traders and investors.