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
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:
- Calculate the payoff of the option at maturity in both up-move and down-move states.
- Calculate the expected value of the option in one year (probability-weighted average of the payoffs).
- Discount the expected value back to today at the risk-free rate.
Given a risk-free rate of 7%, calculate the value of a 1-year call option with an exercise price of $45.
Follow the three steps:
- Payoff: Up move = $17.50, Down move = $0
- Expected option value in one year: $10.50
- 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.
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.
And that concludes this lesson on the one-period binomial model.
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