Reading 75

Recall from Quantitative Methods that a binomial model is based on the idea that, over the next period, some value will change to one of two possible values (binomial). To construct a one-period binomial model for pricing an option, we need:

• A value for the underlying at the beginning of the period.
• An exercise price for the option. The exercise price can be different from the value of the underlying. We assume the option expires one period from now.
• Returns that will result from an up-move and a down-move in the value of the underlying over one period.
• The risk-free rate over the period.

For now, we do not need to consider the probabilities of an up-move or a down-move. Later in this reading we will examine one-period binomial models with risk-neutral probabilities.

One-Period Binomial Model for a Call Option

As an example, we can model a call option with an exercise price of $55 on a stock that is currently valued ($S_0$) at $50. Let us assume that in one period the stock's value will either increase ($S_u^1$) to $60 or decrease ($S_d^1$) to $42. We state the return from an up-move ($R_u$) as $60 / $50 = 1.20, and the return from a down-move ($R_d$) as $42 / $50 = 0.84.

The call option will be in the money after an up-move or out of the money after a down-move. Its value at expiration after an up-move, $c_u^1$, is $\max(0,\;\$60-\$55)=\$5$. Its value after a down-move, $c_d^1$, is $\max(0,\;\$42-\$55)=0$.

Now we can use no-arbitrage pricing to determine the initial value of the call option ($c_0$). We do this by creating a portfolio of the option and the underlying stock such that the portfolio will have the same value following either an up-move ($V_u^1$) or a down-move ($V_d^1$) in the stock. To value a call option, we create a portfolio that is short a call (i.e., the call writer) and long a number of shares of the stock that we denote as $h$. We must solve for the $h$ that results in $V_u^1 = V_d^1$.

• The initial value of our portfolio, $V_0$, is $hS_0 - c_0$ (we are short the call option).
• The portfolio value after an up-move, $V_u^1$, is $hS_u^1 - c_u^1$.
• The portfolio value after a down-move, $V_d^1$, is $hS_d^1 - c_d^1$.

In our example, $V_u^1 = h(\$60) - \$5$, and $V_d^1 = h(\$42) - 0$. Setting $V_u^1 = V_d^1$ and solving for $h$:

$h(\$60) - \$5 = h(\$42)$
$h(\$60) - h(\$42) = \$5$
$h = \dfrac{\$5}{\$60 - \$42} = 0.278$

This result, the number of shares of the underlying we would buy for each call option we would write, is known as the hedge ratio for this option.

With $V_u^1 = V_d^1$, the value of the portfolio after one period is known with certainty. This means either $V_u^1$ or $V_d^1$ must equal $V_0$ compounded at the risk-free rate for one period. In this example,

$V_d^1 = 0.278(\$42) = \$11.68$,   or   $V_u^1 = 0.278(\$60) - \$5 = \$11.68$.

Let us assume the risk-free rate over one period is 3%. Then $V_0 = \dfrac{\$11.68}{1.03} = \$11.34$.

Now we can solve for the value of the call option, $c_0$. Recall that $V_0 = hS_0 - c_0$, so $c_0 = hS_0 - V_0$. Here,

$c_0 = 0.278(\$50) - \$11.34 = \$2.56$.

One-Period Binomial Model for a Put Option

We can model a put option using the same technique. The key difference is that the model portfolio consists of long positions in both a put and the underlying stock.

Using the same assumptions — the stock is currently $50, and in one period its value will either increase to $60 (return 1.20) or decrease to $42 (return 0.84) — we value a put option with an exercise price of $48.

The put option will be in the money after a down-move or out of the money after an up-move:

• $p_d^1 = \max(0,\;\$48 - \$42) = \$6$
• $p_u^1 = \max(0,\;\$48 - \$60) = 0$

Again we want to find the hedge ratio $h$, the number of shares at which $V_u^1 = V_d^1$:

• $V_0 = hS_0 + p_0$ (we are long the put).
• $V_u^1 = hS_u^1 + p_u^1$.
• $V_d^1 = hS_d^1 + p_d^1$.

Setting $V_u^1 = V_d^1$:

$h(\$60) + 0 = h(\$42) + \$6$
$h(\$60) - h(\$42) = \$6$
$h = \dfrac{\$6}{\$60 - \$42} = 0.333$

As before, the value of the portfolio one period from now is the same in both states:

$V_u^1 = 0.333(\$60) = \$20$,   or   $V_d^1 = 0.333(\$42) + \$6 = \$20$.

Discounting at the risk-free rate of 3%: $V_0 = \dfrac{\$20}{1.03} = \$19.42$.

Finally, solving for $p_0$ from $V_0 = hS_0 + p_0$:

$p_0 = V_0 - hS_0 = \$19.42 - 0.333(\$50) = \$2.75$.

Another approach to constructing a one-period binomial model involves risk-neutral probabilities of an up-move or a down-move. Consider a share of stock currently priced at $30. The size of the possible price changes, and the probabilities of these changes occurring, are as follows:

• $R_u$ = up-move factor = 1.15
• $R_d$ = down-move factor = $1/R_u = 1/1.15 = 0.87$
• $\pi_U$ = risk-neutral probability of an up-move = 0.715
• $\pi_D$ = risk-neutral probability of a down-move = $1 - \pi_U = 1 - 0.715 = 0.285$

Note that the down-move factor is the reciprocal of the up-move factor, and the probability of a down-move is one minus the probability of an up-move. The one-period binomial tree for the stock is shown in Figure 75.2. The beginning stock value of $30 is to the left, and to the right are the two possible end-of-period stock values: $30 \times 1.15 = \$34.50$ and $30 \times 0.87 = \$26.10$.

The risk-neutral probabilities of an up-move and a down-move are calculated from the sizes of the moves and the risk-free rate:

$\pi_U = \dfrac{1 + R_f - R_d}{R_u - R_d}$    $\pi_D = 1 - \pi_U$

where $R_f$ is the risk-free rate, $R_u$ is the size of an up-move, and $R_d$ is the size of a down-move.

We can calculate the value of an option on the stock by:

• Calculating the payoffs of the option at expiration for the up-move and down-move prices.
• Calculating the expected payoff of the option in one year as the (risk-neutral) probability-weighted average of the up-move and down-move payoffs.
• Calculating the PV of the expected payoff by discounting at the risk-free rate.

In practice, we would construct a binomial model with many short periods and have many possible outcomes at expiration. However, the one-period model is sufficient to illustrate the concept and method.

Note that the actual probabilities of an up-move and a down-move do not enter directly into our calculation of option value. The size of the up-move and down-move, along with the risk-free rate, determines the risk-neutral probabilities we use to calculate the expected payoff at option expiration. Remember, the risk-neutral probabilities come from constructing a hedge that creates a certain payoff. Because their calculation is based on an arbitrage relationship, we can discount the expected payoff based on risk-neutral probabilities, using the risk-free rate.