The binary tree that was built in the article “Binary tree (Part I)” is a simplified form of the real option pricing model used by investment banks. The basic model assumes that the share price will either rise to $115 or fall to $85 during one period.
In reality, firstly, the asset price rises and falls in small steps, building up a whole movement dynamics over time. Secondly, the factors u and d that change the share price from the spot value are used only for illustration. In reality, the developers of option models try to derive the most appropriate values for these factors by calculating the volatility of the stock price (Read How to calculate historical volatility?). It follows from logical considerations that, all else equal, a higher volatility of the underlying asset will lead to a larger deviation of the price from the current level, and therefore to a higher value of the option.
To show how the volatility can be dealt with, in this article we will build a 3-step binary tree to find the fair value of a European stock option that does not pay dividends.
Spot stock price S = $300
Strike price X = $250
Risk-free interest rate r = 10% per annum
Time to Expiration t = 0.25 years
Volatility σ = 40%
To construct a tree that compares the volatility of the underlying asset with factors u and d, we will use the following formulas proposed by American academics Cox, Ross and Rubinstein in 1979:
e - mathematical constant (approximately equal to 2.71828)
σ - volatility, standard deviation of daily returns of the price of the underlying asset, expressed in annual terms (in our case equals to 0.4).
t - time to expiration of the option in years (0.25 years or 3 months)
n - number of steps in a binary tree (three steps)
Calculate u and d:
The total duration of the option is divided into three equal periods. Figure shows how u and d values are used to build the binary tree. For example, the first upward price movement of an asset having a value of $337 obtained by multiplying $300 by the value of u. The value of $267 is calculated by multiplying $300 by the value of d. If the share price reaches the value of $267 in period 1, it can either rise to $300 in period 2 ($267 multiplied by u), or it can further fall to $238.
Building a call option tree
The next step is to build a tree that represents the value of a call option with a strike of $250 based on the share price movements. The easiest way is to start the tree is from the end - moment of expiration, the value of the call option is determined exclusively by the intrinsic value of the option, as shown in figure below.
For example, in the upper right corner of the underlying asset price tree the stock price is $424. In this case, a call with an exercise price of $250 will cost $174.
C – represents the value of the option in the period 0
Cu – option price when the share price rises by one level
Cuu - option price when the share price rises by two levels
Cd - option price when the share price falls by one level
Cdd - option price when the share price falls by two levels
Cud - option price when the share price grows by one level and then falls by one level.
The cost of a call option in each period can be obtained by dividing a three-step tree into three trees with one step for easy calculations. To start with, we need to calculate the probabilities p and (1 - p) using factors u and d.
1 – p = 0,4926
Cuu equals the discounted value of the sum:
1) call option if the share rises to $424 (i.e. $174) times the probability of the share growth to $424 over the period and
2) call option if the share rises to $337 (i.e. $87) times the probability of the share growth to $337 over the period.
The equation is as follows:
Cuu = [($174 x 0.5074) + ($87 x 0.4926)] x e-0.1 x 0.25/3 = $130
The completed three-step binary tree is shown below. The current fair value of a call option with a expiration in 3 months is approximately $61. If we found the value of an American put option, it would be critical to check at every step so that the price does not fall below the intrinsic value of the option. This situation is possible due to the fact that the American put is sometimes advantageous to execute in advance.