Two Period Binomial Model and Put Call Parity

Put Call Parity

Put option value can also be computed using put call parity, which recognizes that the value of a fiduciary call (long call, plus an investment in a zero coupon bond with a face value equal to the strike price) is equal to the value of a protective put (long stock and long put):

Note that both options are on the same underlying stock have the same exercise price, and the same maturity.

In our previous example, PV(X) = 30 / 1.07 = $28.04. We can verify the put call parity as:

Rounding accounts for the slight difference.

Put call parity can be used to create a synthetic instrument that replicates the desired instrument.

For example, a synthetic call can be created by creating a portfolio that combines a long position in the stock, a long position in a put and a short position in a zero coupon bond with a face value equal to the strike price (i.e., borrowing the present value of the exercise price at the risk free rate).

Two Period Binomial Model

Valuing an option using a two period binomial model requires more steps, but uses the same method:

Calculate the stock values at the end of two periods (there are three possible outcomes, because an up then down move gets you to the same place as a down then up move).

Calculate the three possible option payoffs at the end of two periods. Calculate the expected option payoff at the end of two periods (t = 2) using the up and down move probabilities.

Discount the expected option payoff (t = 2) back one period at the risk free rate to find the option values at the end of the first period (t = 1).

Calculate the expected option value at the end of one period (t = 1) using up and down move probabilities. Discount the expected option value at the end of one period (t = 1) back one period at the risk free rate to find the option value today (t = 0).

Let’s look at an example to illustrate the steps involved.

Worked example: Valuing a call option on a stock with a two period model

Suppose you own a stock currently priced at $50 and that a two period European call option on the stock is available with a strike price of $45. The up move factor is 1.25 and the down move factor is 0.80. The risk free rate per period is 7%.

Compute the value of the call option using a two period binomial model.

Solution

First, compute the probability of an up move and a down move, and then compute the theoretical value of the stock at the end of each period:

$$\begin{array}{rl}{\pi }_U& =\frac{1+R_f-D}{U-D}=\frac{1+0.07-0.80}{1.25-0.80}=0.60\\ {\pi }_D& =1-0.60=0.40\end{array}$$

The two period binomial tree for the stock is shown in the following figure.

Two Period Binomial Tree for Stock Price

We know the value of the option at expiration in each state is equal to the stock price minus the exercise price (or zero, if that difference is negative):

$$\begin{array}{rl}{C}^{++}& =\max (0,\$78.13-\$45.00)=\$33.13\\ C^{-+}& =\max (0,\$50.00-\$45.00)=\$5.00\\ C^{+-}& =\max (0,\$50.00-\$45.00)=\$5.00\\ C^{--}& =\max (0,\$32.00-\$45.00)=\$0\end{array}$$

We will approach this problem by using the single period binomial model for each period. Using this method, we can compute the value of the call option in the up state in period one as follows:

$$C^{+}=\frac{E(\text{call option value})}{1+R_f}=\frac{{\pi }_U{C}^{++}+{\pi }_D{C}^{+-}}{1+R_f}=\frac{0.6(\$33.13)+0.4(\$5.00)}{1.07}=\frac{\$21.88}{1.07}=\$20.45$$

The value of the call in the down state at t = 1 is computed as:

$$C^{-}=\frac{E(\text{call option value})}{1+R_f}=\frac{{\pi }_U{C}^{-+}+{\pi }_D{C}^{--}}{1+R_f}=\frac{0.6(\$5.00)+0.4(\$0.00)}{1.07}=\frac{\$3.00}{1.07}=\$2.80$$

Now we know the value of the option in both the up state ($C^{+}$) and the down state ($C^{-}$) one period from now. To get the value of the option today, we simply apply our methodology one more time. Therefore, bringing ($C^{+}$) and ($C^{-}$) back one more period to the present, the value of the call option today is:

$$C=\frac{E(\text{call option value})}{1+R_f}=\frac{{\pi }_U{C}^{+}+{\pi }_D{C}^{-}}{1+R_f}=\frac{0.6(\$20.45)+0.4(\$2.80)}{1.07}=\frac{\$13.39}{1.07}=\$12.51$$

The binomial tree for the call option is shown here:

Two Period Binomial Tree for Option Price