The binomial option pricing model is a popular method used in quantitative finance for pricing options. It is based on the assumption that the underlying asset follows a binomial distribution, where the stock price can either go up or down with a certain probability over a given time period. The model allows us to value options by determining the probabilities of different outcomes and calculating the expected value of the option at expiration.
The basic steps in the binomial option pricing model are as follows:
1. Divide the time period into equal intervals, such as months, weeks, days, or even hours.
2. Calculate the probability of the underlying asset going up or down for each time interval. This probability is often estimated using historical data or implied volatility.
3. Calculate the up and down movements of the stock price for each time interval, which will depend on the size of the up and down movements as well as the time interval.
4. Construct a binomial tree that shows all possible price paths of the underlying asset over the relevant time period.
5. Work out the payoff of the option at expiration for each possible price path on the tree.
6. Move backwards along the tree to calculate the option price at each earlier time point, taking the expected value of the option at each node. This expected value is calculated by weighting the possible future values by their associated probabilities of occurring.
For example, suppose we have a stock that is currently trading at $100, and we want to price a call option with a strike price of $110 that expires in 3 months. We assume that the stock price can either go up by 10% or down by 10% with equal probability in each month. We construct a binomial tree as shown below:
110
121 99
133.1 108.9 108.9 89.1
146.41 119.79 119.79 97.29 119.79 97.29
S0 = $100 S3 = ?Each node on the tree represents a possible price of the stock in 3 months, depending on whether it goes up or down in each of the three months. We can then work out the option payoff at expiration for each node, which is the maximum of zero and the difference between the stock price and the strike price. For example, at the top node where the stock goes up in each of the three months, the option payoff is $0, since the stock price is below the strike price of $110.
We can then move backwards along the tree to calculate the option price at each of the earlier nodes, taking the expected value of the option at each node. For example, at the nodes where the stock price is $119.79 and $97.29, the expected value of the option is:
Expected value of option = [0.5 x (Value of option if stock goes up) + 0.5 x (Value of option if stock goes down)]
= [0.5 x (119.79 - 110) + 0.5 x (0)]
= $4.895We continue this process all the way back to the initial node, where the expected value of the option is the final option price. In this case, we find that the expected value of the option is $7.778. Therefore, the fair price of the call option using the binomial option pricing model is $7.778.