One Period Binomial Model

As the underlying value determines the option payoff, if we know the outcome of the underlying, we know the value of the option. If the underlying is above the exercise price at expiration, then the payoff is S_T – X for calls and zero for puts. The converse is true if the underlying is below the exercise price at expiration. The derivation of an option pricing model requires the specification of a model of random processes that describe the movements in the underlying.

The Binomial Model for Stocks

A model with two possible outcomes is a binomial model. We start with the underlying at S₀ and let the price move up to S₁⁺ and down to S₁^–. We don’t know which outcome will occur, but we can assign probabilities. Assuming the probability of the move to S₁⁺ is q, then the probability of moving to S₁^– is 1 – q.

$$
\begin{array}
\hline
{} & {\small q } & { S }_{ 1 }+ \\
{ S }_{ 0 } & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{} & {\small 1-q} & { S }_{ 1 }- \\
\end{array} $$

We then specify the returns implied by these moves up and down as factors u and d where u = S₁⁺/S₀ and d = S₁^–/S₀.

$$
\begin{array}
\hline
{} & {\small q } & { S }_{ 0 }u \\
{ S }_{ 0 } & {\Huge \begin{matrix} \diagup \\ \diagdown \end{matrix} } & {} \\
{} & {\small 1-q} & { S }_{ 0 }d \\
\end{array} $$

Deriving the Value of a Call Option Using a Binomial Model

We now consider a European call option with price c_o today and price c₁⁺ and c₁^– at expiration. Assume we sell a call and buy h units of the underlying asset with portfolio value at inception V₀ = hS₀ – c_o. At time 1, the portfolio will either be worth:

$$ V_1^+ = hS_1^+ – c_1^+; or $$

$$ V_1^- = hS_1^-  – c_1^-$$

If the portfolio was hedged then:

$$ V_1^+ = V_1^- $$

Which could be re-written as:

$$ hS_1^+ – c_1^+= hS_1^-  – c_1^- $$

Where \(h = \frac{c_1^+ – c_1^- }{S_1^+ – S_1^-}\)

We also know that a perfectly hedged portfolio will earn the risk-free rate so:

$$ V_1^+ or \quad V_1^- = V_0(1+r)$$

We can finally obtain the formula for the option price as:

$$ c_0 = \frac{πc_1^+ + (1-π)c_1^-}{1+r}$$

Where \(π = \frac{1 + r – d}{u – d}\)

How do we Interpret this Equation?

Having worked through all of the above, we have arrived at an equation for the value of a call option today, which takes the form of an expected future value (the numerator) discounted at the risk-free rate (the denominator). The volatility of the underlying is an important factor in determining the value of the option. If the volatility increases, the difference between S₁⁺ and S₁^– increases which widen the range between c₁⁺ and c₁^– leading to a higher option value.

How did q become π?

We note that our actual probabilities of q and (1 – q) are not used, but instead, we have π and (1 – π), which are called risk-neutral probabilities. If the option is trading at a price too high relative to the formula, investors can sell the call, buy h shares of the underlying and earn a return in excess of the risk-free rate while funding the transaction by borrowing at the risk-free rate. This action will put downward pressure on the call price until it conforms with the model price once more.

The Value of a Put Option Using a Binomial Model

Following the same methodology as above, we can derive a model for a put option as follows:

$$p_0 = \frac{πp_1^+ + (1-π)p_1^-}{1+r}$$

Where \(π = \frac{1 + r – d}{u – d}\)

Question

Which factors are the most relevant to determine an option’s value using a binomial pricing model?

A. The probability that the underlying will go up or down, the risk-free rate, and the initial value of the option

B. The risk-free rate, the volatility of the underlying, and the exercise price

C. The probability that the underlying will go up or down, the risk-free rate, and the risk-neutral probability

Solution

The correct answer is B.

The probability that the underlying will go up or down is not a factor in determining the price of an option using a binomial model because we derive it from the formula \(π = \frac{1 + r – d}{u – d}\).

The volatility of the underlying asset is an important factor, as is the risk-free rate, the risk-neutral probability, and the exercise price.