 # One-Period Binomial Model

The law of arbitrage dictates that the value of any two assets (or portfolio of assets) whose payoffs are identical in all possible future scenarios at a given time must also be identical today.

Unlike forward commitments that offer symmetric payoffs at a pre-determined price in the future, contingent claims offer asymmetric payoffs. For this reason, their valuation is a challenge. The binomial model can be used to model the payoffs of contingent claims.

## One-Period Binomial Model

The idea behind the binomial model is that at maturity, an asset’s spot price, $$S_0$$, can either increase to $$S_1^u$$ or decrease to $$S_1^d$$. We do not need to know the asset’s future price in advance since it depends on a random variable’s outcome. The asset price movements, from $$S_0$$ to either $$S_1^u$$ or $$S_1^d$$, can be seen as the outcome of a Bernoulli trial.

Let us denote $$q$$ as the probability of an increase in the asset’s price. Because of the presence of only two possibilities, i.e., an increase or a decrease in the asset’s price, we can denote the probability of a decrease in its price as $$1-q$$, such that the two probabilities will add up to 1.

The gross return when the asset price increases will be:

$$R^u=\frac{S_1^u}{S_0}>1$$

When the asset price decreases, the gross return will be:

$$R^d=\frac{S_1^d}{S_0}<1$$

The difference between $$S_1^u$$ (or $$R^uS_o)$$ and $$S_1^d$$ (or $$R^dS_0$$) is the spread of possible future price outcomes. ### Pricing a European Call Option using a One-Period Binomial Model

Consider a one-year call option with an underlying price of $$S_0$$ and an exercise price of $$X$$. Also, assume that $$S_1^d<X<S_1^u$$ and that the one-period binomial model is equivalent to the time of expiration of one year.

The one-period binomial model gives the underlying asset values in one year, where the option value is defined as a function of the underlying value.

#### At time t=0:

The value of the call option is $$c_0$$. Note that this value is unknown and needs to be calculated. #### At time t=1:

After one year, the option expires. At this time, the value of the option will either be $$c_1^u=\max(0,S_1^u-X)$$ if the underlying price rises to $$S_1^u$$ or $$c_1^d=\max(0,S_1^u-X)$$ if the underlying price falls to $$S_1^d$$.

Intuitively, for the up movement, the call option is in the money, and for the down direction, the option is out of the money. #### Determining the Value of c0

The value of $$c_0$$ is determined by applying replication and no-arbitrage pricing. Replication implies that the value of the option and its underlying asset in any future scenario may be used to construct a risk-free portfolio.

With that in mind, assume at time $$t=0$$, we sell a call option for a price of $$c_0$$ and buy $$h$$ units of the underlying asset. Also, let the value of the portfolio be $$V$$ so that its value at $$t=0$$ is:

$$V_0={\rm hS}_0-c_0$$

The value of the portfolio, if the underlying price moves up, is given by:

$$V_1^u={\rm hS}_1^u-c_1^u=h\times R^u\times S_0-\max{\left(0,S_1^u-X\right)}$$

And for the down movement:

$$V_1^d={\rm hS}_1^d-c_1^d=h\times R^d\times S_0-\max{\left(0,S_1^d-X\right)}$$ Assuming no-arbitrage condition, note that we have established two portfolios with identical payoff profiles at time $$t=1$$. As such, we need to find the value of $$h$$ such that:

$$V_1^u=V_1^d$$

Therefore,

$$\Rightarrow{hS}_1^u-c_1^u={hS}_1^d-c_1^d$$

Making $$h$$ the subject of the formula:

$$h=\frac{c_1^u-c_1^d}{S_1^u-S_1^d}$$

The value $$h$$ is called the hedge ratio. The hedge ratio is a proportion of the underlying that will offset the risk associated with an option.

Since $$V_1^u=V_1^d$$, we can draw two conclusions:

• We can utilize either of the two portfolios to value the option.
• The return $$\frac{V_1^u}{V_0}=\frac{V_1^d}{V_0}=1+r$$.

To avoid arbitrage, the portfolio value at $$t=1$$, ($$V_1={\ V}_1^u=V_1^d$$), must be discounted using a risk-free rate so that:

$$V_0=V_1\left(1+r\right)^{-1}$$

However, recall that $$V_0={hS}_0-c_0$$:

$$\Rightarrow{\rm hS}_0-c_0=V_1\left(1+r\right)^{-1}$$

Making $$c_0$$ the subject of the formula:

$$c_0={\rm hS}_0-V_1\left(1+r\right)^{-1}$$

#### Example: Pricing Call Option Using One Period Binomial Model

A European call option that expires in one year has an exercise price of $70 and an underlying spot price of$60. Use a one-period binomial model to estimate the call option price if the underlying’s spot price is expected to change by 25% in one year. Assume that the risk-free annual rate is 6%.

Solution

Step 1: Determine the Call Option’s Value at Maturity t=1

At maturity, the value can either be $$c_1^u$$ if the price of the underlying goes up or $$c_1^d$$ if the price of the underlying goes down.

The spot price at maturity, if the price goes up by 25%, will be:

$$S_1^u=\frac{125}{100}\times60=75$$

and the gross return will be:

$$R^u=\frac{S_1^u}{S_o}=\frac{75}{60}=1.25$$

If the price goes down by 25%, the spot price at maturity will be:

$$S_1^d=\frac{75}{100}\times60=45$$

and the gross return will be:

$$R_d=\frac{S_1^d}{S_o}=\frac{45}{60}=0.75$$

If the underlying’s price goes up (the call option expires in the money):

$$c_1^u=\max\ \left(0,\ S_1^u-X\right)=\max\ \left(0,\ 75-70\right)=5$$

If the underlying’s price goes down, and the call option expires out of the money:

$$c_1^d=\max\ \left(0,\ S_1^d-X\right)=\max\ \left(0,\ 45-70\right)=0$$

Step 2: Determining h, the Hedge Ratio

The hedge ratio is given by

\begin{align} h&=\frac{c_1^u-c_1^d}{S_1^u-S_1^d}\\&=\frac{5-0}{75-45}=0.167\end{align}

The hedge ratio of 0.167 implies that we either need to buy 0.167 units of the underlying for every call option we sell or sell 6 call options for each underlying asset to equate the portfolio values at maturity ($$t=1$$). Therefore, the portfolio values when the price of the underlying increases and decreases respectively is:

\begin{align}V_1^u&=\left(0.167\times75\right)-5=7.5\\V_1^d&=\left(0.167\times45\right)-0=7.5\end{align}

The portfolio values are the same, implying that either of the portfolios can be used to value the derivative.

Step 4: Determining $$\mathbf{V_0}$$

We can obtain $$V_0$$ by discounting $$V_1$$ at the risk-free discount rate. Remember that $$V_1=V_1^u=V_1^d$$:

$$V_o=7.5(1+{0.05)}^{-1}=7.14$$

Step 5: Determining $$\mathbf{c_0}$$

Recall that:

$$c_0={\rm hS}_0-V_1\left(1+r\right)^{-1}=hS_o-V_0$$

Therefore,

$$c_o=hS_o-V_0=\left(0.167\times60\right)-7.14=2.88$$

Note: The hedging approach can be used to value many derivatives, not just call options, provided the derivative’s value depends on the underlying asset’s value at contract maturity, i.e., $$t=1$$

## Pricing a European Put Option One-period Binomial Model

For put options, the same explanations we gave under the call option apply, albeit with a different replication strategy. Consider the following diagram: Under the put option, the hedge ratio is given by

$$h=\frac{p_1^u-p_1^d}{S_1^d-S_1^u}$$

Note that the formula remains the same as in the call option, except for the change of notations. Replication in pricing put option using a one-period binomial model involves buying the put option and $$h$$ units of the underlying so that:

$$V_0=p_0+hS_0$$

Therefore,

$$p_0=V_0-hS_0$$

## Question

Consider a one-year put option on a non-dividend paying stock with an exercise price of $50. The current stock price is$45. The stock price is expected to go up or down by 20%. Calculate the non-arbitrage price of the put option if the risk-free rate of return is 4%.

A. 0

B. $5.38 C.$14.00

Solution

Consider the following diagram: From the above results, the hedge ratio is given by:

$$h=\frac{p_1^u-p_1^d}{S_1^u-S_1^d}=\frac{0-14}{54-36}=-0.7778$$

We need to calculate $$V_1={\ V}_1^u=V_1^d$$, which are:

\begin{align}V_1^u&={\rm hS}_1^u+p_1^u=0.7778\times54+0=\42.00\\V_1^d&={\rm hS}_1^d+p_1^d=0.7778\times36+14=\42.00\end{align}

Next, we can either use $$V_1^u$$ or $$V_1^d$$ to calculate the value of $$V_0$$:

$$V_0=V_1\left(1+r\right)^{-1}=42\left(1.04\right)^{-1}=\40.38$$

Therefore, the put option value is:

$$p_0=V_0-{\rm hS}_0=40.38-0.7778\times45=\5.38$$

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