CFA Level I Derivatives - Binomial Model for Pricing Options

Summary

This video explains how the binomial model is used to estimate the future price of an asset, like a stock, and how this model can be applied to value options. The binomial model assumes two possible outcomes for a stock's price: an up move and a down move. By calculating the probabilities of these moves, analysts can estimate future stock prices and option payoffs. The video walks through a practical example using a stock priced at $50 and explains how to calculate the value of a call option using risk-neutral probabilities and discounting methods.

Takeaways

• 📈 The binomial model is used to estimate the future value of an asset by considering two possible outcomes: an up move or a down move.
• 💲 To construct a binomial model, you need the initial asset value, the up and down move sizes, and the probabilities of each move.
• 📊 In the example, a stock priced at $50 has an up move factor of 1.25 and a down move factor of 0.8.
• 🎲 The analyst estimates the probability of an up move at 0.6 and a down move at 0.4.
• 📉 The expected stock price after one year is calculated as $53.50 by multiplying the potential prices with their respective probabilities.
• 📐 The size of the up move is often based on the stock's volatility, while the probabilities are based on risk-neutral pseudo probabilities.
• 🔢 The pseudo probability of an up move is calculated using the risk-free rate, in this case, 7%, yielding a probability of 0.6 for the up move.
• 💼 Binomial models can be used to value options by estimating future asset prices and option payoffs in different scenarios.
• 📅 The process for valuing a call option involves calculating the payoff at maturity, the expected option value, and discounting it to the present.
• 💰 In the example, the value of a one-year call option with an exercise price of $45 is calculated to be $9.81.

Q & A

What is the basic premise of the binomial model in financial analysis?

The binomial model is based on the idea that the value of an asset will change to one of two possible values in the next period: one for an upward move and one for a downward move.

What information is required to construct a binomial model?

To construct a binomial model, you need to know the beginning asset value, the size of the up and down moves, and the probabilities of each move occurring.

In the provided example, what are the up and down factors for the stock priced at $50?

The up factor (U) is 1.25, indicating a 25% increase, and the down factor (D) is 0.8, which is the reciprocal of the up factor.

How are the probabilities of up and down moves estimated in this example?

The probability of an up move is estimated at 0.6, and since the events are not mutually exclusive, the probability of a down move is 0.4.

What is the expected stock price after one year, based on the given probabilities?

The expected stock price after one year is $53.50, calculated by multiplying the up and down prices by their respective probabilities and summing the results.

How does an analyst estimate the size of an up move?

The size of an up move is often estimated based on the volatility of the stock.

What is a risk-neutral pseudo-probability in the binomial model?

The risk-neutral pseudo-probability is a probability calculated based on the size of the moves and the risk-free rate. It is not the actual probability but is used for pricing options in a risk-neutral framework.

How do you calculate the risk-neutral pseudo-probabilities for an up move and down move?

Using the given formula, the risk-neutral pseudo-probability of an up move (π_U) and a down move (π_D) can be calculated. In this example, π_U is 0.6, and π_D is 0.4.

What are the three steps for valuing an option using the binomial model?

The steps are: 1) Calculate the payoff of the option at maturity for both up and down moves, 2) Calculate the expected value of the option one year from now using the pseudo-probabilities, and 3) Discount the expected value back to today using the risk-free rate.

What is the value of a one-year call option with an exercise price of $45, given the example data?

The value of the one-year call option is $9.81, calculated using the binomial pricing approach.