The article about synthetic forwards described the restrictions on option prices and a fundamental principle called put-call parity. The next logical step in the study of options and the Black-Scholes pricing model is the construction of a binary tree.

Let's consider an example to build a binary tree, which is based on the assumption that the underlying asset is traded at $100 and can either rise by 15% or fall by 15% within one period of time, as shown in Figure 1 below.

**Figure 1
**

**Binary tree**

Let's assume that the trader sells at-the-money call option, which expires at the end of period 1. The trader's goal is to determine the fair value of the call option at the present time (period 0). Let us name this fair value C. In period 1, the call option price is either $15 or zero, which is shown in Figure 2.

**Fig. 2. Value of the call**

**Binary tree - example**

The price of a call option in the period 0 can be set by calculating the price of hedging a short call option position. For example, the interest rate is currently 10%. The trader creates the next portfolio in the period 0:

- short call option and receive premium C

- purchase of the underlying asset (number of shares equals to option delta Δ)

- we acquire shares partly by receiving a C premium and getting a loan B at 10%

We get two equations:

(Δ x $115) - (B x 1.1) = $15

(Δ x $85) - (B x 1.1) = $0

The first equation implies a 15% increase in the share price to $115, and the trader must pay the intrinsic value of the option $15 on a short call option. In order for the trader not to generate a loss of $15, the value of shares in the portfolio minus credit and interest must be $15.

The second equation implies that the stock falls by 15%, therefore the option price in period 1 is equal to zero. Therefore, the share price in the portfolio minus credit and interest should be $0. If we combine the two equations, we get:

(Δ x $115) - (Δ x $85) = $15

Consequently, Δ = 0.5. The resulting value of 0.5 is the option delta, also among the traders it is called the hedge factor. If a trader sells a call option for a certain amount of shares, trader will need to buy exactly half of the shares to neutralize the risk of price movements in the underlying asset. This type of hedge is called a delta hedge, and the result of the delta hedge is a delta-neutral position.

Note that an option delta can be directly calculated as follows:

Δ = (Cu – Cd)/(Su – Sd)

Cu - option value if share price increases

Cd - option value in case of share price decrease

Su - share price in case of rally

Sd - share price in case of downfall

In this example:

Δ = ($15 – $0) / ($115 – $85)= 0,5

Since the option delta has already been calculated, we will substitute it in either of the two equations to find the size of loan required to purchase the share. Using the second equation:

(Δ x $85) – (B x 1.1) = 0

(0.5 x $85) – (B x 1.1) = 0

B = $38,64

That leaves one last step. In order for the trader not to fix losses in the period 0, the cost of buying shares (in our case, half of the share) in the portfolio must be covered by credit B and the premium received from the short call. In our example, the delta is 0.5 and the credit B is 38.64. Hence:

C + B = Δ x Spot price of the share

C + 38.64 = 0.5 x $100

C = $11,36

**Formula for calculating the value of a call**

The option value calculations shown above can be reduced to one formula:

p = (1 + r – d)/(u – d)

r - interest rate for one period (10%)

d - a factor that will lower the share price (0.85)

u - a factor that will raise the share price (1.15)

From the example above:

p = (1,1 – 0,85)/(1,15 – 0,85)= 0,83

The cost of a call C option per share is expressed in the following equation:

Cu - option value if share price increases

Cd - option value in case of share price decrease

Ergo:

C = (0.83 x $15 + 0.17 x $0)/1.1= $11.32.

Note that the call option price is calculated as a weighted average value and is based on the idea that the risk from the option can be fully hedged. It follows from these assumptions that there is an 83% chance that the stock will rise to $115 by the time of expiry (i.e. the option price will reach $15). While the probability of the share falling to $85 (i.e. the option will depreciate) is equal to 17%. The average of two potential payouts, weighted on the basis of the probabilities of each scenario, is discounted one period back at an interest rate of 10% to calculate the value of the option in period 0.

These probabilities are widely used in academic research to describe a risk-neutral environment in which the risk from an option can be accurately reduced to zero by creating a delta-neutral portfolio. It is very important not to confuse probability with an analyst's forecast of the share price in the future. It should be noted that the expected share price in period 1 in the example is higher than $100, as the risk-free interest rate is at 10%. The expected price is the forward price of the underlying asset, which is calculated as $100 x 1.1 = $110. That is why the probability of the share price rising to $115 (83%) is much higher than the probability of falling to $85 (17%).