International regulation deals with the two model families accepted for option valuations:

1. Closed form models, the most common and accepted example of which is the Black-Scholes-Merton (“B&S”) equation.
2. Lattice models, such as the binomial model, the trinomial model, Monte-Carlo model etc., that simulate employee conduct.

U.S GAAP sets that the entity shall consider factors that willing market participants, operating in an informed manner, would consider when selecting the model to be implemented in the pricing of options. For example, most employee stock options have a long lifetime, they are usually exercisable during the period between the vesting date and the end of the options-lifetime are usually exercised early. These factors will be examined when the fair value of the options is estimated at their grant date.

In practice, the most common models for option valuation are Black-Scholes, binomial, and Monte Carlo simulation, as follows:

Black-Scholes theory suggests that financial instruments such as stocks or futures follow a predictable pattern in their price movements, characterized by a continuous and smooth progression with consistent levels of change and risk.

B&S Modell – This model bases its calculations on several key factors: the fluctuating nature of the asset’s value, the asset’s current market price, the predetermined price at which the option can be exercised, the time remaining until the option’s expiration, and the prevailing interest rate unaffected by market risks. By integrating these elements, the Black-Scholes formula provides a logical framework for option sellers to determine appropriate pricing for their offerings.

The advantage of this model lies in its simplicity and ease of use. On the other hand, this model takes simplistic assumptions which sometimes do not fully model the complexity of reality.

Binomial Modell – The binomial pricing model evaluates the progression of an option’s primary underlying factors over specific time intervals. This evaluation is conducted using a binomial tree, which charts the potential price changes of the underlying asset until the option’s expiration. Each point in this tree signifies a conceivable price for the asset at different moments.

The assessment process begins at the end of the tree, examining each possible outcome at the expiration date, and then retraces steps back to the starting point, which is the valuation date. At each step of this backward journey, the model calculates the option’s value at that specific time, effectively building a comprehensive picture of the option’s valuation journey over time.

Binominal tree:

The advantages of this model include the ability to examine complex options and the possibility of converting the options during their lifespan. However, in certain cases, this model does not fully represent reality.

Monte Carlo simulation:

Consider the Monte Carlo simulation as similar to repeatedly rolling dice. Imagine a beginner at the craps table, uncertain of the likelihood of rolling a six through various combinations like four and two, or three and three. Specifically, what’s the probability of getting a ‘hard six’ (two threes)? By rolling the dice numerous times, ideally millions, we can obtain a broad spectrum of outcomes, which helps us understand the frequency of a ‘hard six’ in a roll of six. This method should be executed both efficiently and swiftly, which is precisely the strength of a Monte Carlo simulation.

In financial and other sectors, the Monte Carlo simulation has wide-ranging uses. It’s particularly valuable in option pricing, where it generates a multitude of random paths for an underlying asset’s price, each path leading to a different potential outcome. The results from these paths are then adjusted to their present value and averaged, resulting in the calculated price of the option.

The main advantage of Monte Carlo simulation is its ability to model complex situations and unique conditions of options. The disadvantage is the significant complexity of the model which requires substantial input in the framework of value assessment.