# Valuing Derivatives with a One-Period Binomial Model | CFA Level I Derivatives

A binomial model is a simple yet effective method to estimate the value of derivatives, like options, over time. It assumes that an underlying asset’s value will change to one of two possible values over a period: an up move or a down move.

## Constructing a Binomial Model

To create a binomial model, we need:

• Initial value of the underlying asset
• Size of up move and down move
• Probabilities of up move and down move

### EXAMPLE

A stock is currently priced at \$50. An analyst wants to estimate the expected price after one year. She estimates an up move of 25% and a down move of 20%. The up-factor (U) is 1.25, and the down-factor (D) is 0.8. The probability of an up move is 0.6, and a down move is 0.4. What is the expected price after one year?

Step 1: Calculate the expected price in case of up and down moves.

• Up move: \$50 × 1.25 = \$62.50
• Down move: \$50 × 0.8 = \$40

Step 2: Multiply the expected prices with their probabilities and sum them up to get the expected price after one year.

• Expected price: (\$62.50 × 0.6) + (\$40 × 0.4) = \$53.50

Analysts usually estimate up move sizes based on stock volatility, and the risk-neutral pseudo probability of an up move is calculated using the size of the moves and the risk-free rate.

## Valuing Options Using a Binomial Model

To determine the value of an option using the binomial model, follow these three steps:

1. Calculate the payoff of the option at maturity in both up-move and down-move states.
2. Calculate the expected value of the option in one year (probability-weighted average of the payoffs).
3. Discount the expected value back to today at the risk-free rate.

### EXAMPLE

Given a risk-free rate of 7%, calculate the value of a 1-year call option with an exercise price of \$45.

1. Payoff: Up move = \$17.50, Down move = \$0
2. Expected option value in one year: \$10.50
3. Discounted value of the call option: \$9.81

For a put option, the steps are the same. The put option value can also be calculated using the put-call parity relationship if the equivalent call option price is available.

EXAMPLE

## Hedge Ratio: A Key Component

From the one-period binomial tree, we can calculate an important ratio called the hedge ratio. Denoted with small letter H, it is calculated as:

H = (C+ – C-) / (S+ – S-)

With this hedge ratio, you can long H units of the shares to hedge against a short position in a call option. The hedge ratio works similarly for put options, just with a negative hedge ratio value.

## Using the Hedge Ratio to Calculate No-Arbitrage Value

The hedge ratio can also help us determine the no-arbitrage value of a relevant option. By adhering to the hedge ratio, the portfolio value will be the same regardless of whether the stock goes up or down, ensuring a risk-free arbitrage opportunity. In order to have no arbitrage opportunity, the loan repayment must equal the ending portfolio value, allowing us to calculate the no-arbitrage value of a call or put option.

EXAMPLE

And that concludes this lesson on the one-period binomial model.

## ✨ Visual Learning Unleashed! ✨ [Premium]

Elevate your learning with our captivating animation video—exclusive to Premium members! Watch this lesson in much more detail with vivid visuals that enhance understanding and make lessons truly come alive. 🎬