Differentiating Option Pricing Models
Option pricing models attempt to set a current theoretical value. Models use certain fixed contractual terms – factors such as underlying price, strike and days until expiration – along with forecasts (or assumptions) for factors like implied volatility, to compute the theoretical value for a specific option or a warrant at a certain point in time. Variables will fluctuate over the life of the option or warrant, and the option/warrant position’s theoretical value will adapt to reflect these changes.
Options are derivative contracts that give the holder the right, but not the obligation, to buy or sell the underlying instrument at a specified price on or before a specified future date. Warrants are derivative securities that give the holder the right to purchase securities (usually equity) from the issuer at a specific price within a certain time frame. Warrants are often included in a new debt issue as a “sweetener” to entice investors.
There are three commonly used option pricing models for valuing options or warrants:
| Black-Scholes Merton Model (“BSM”) |
| Binomial Option Pricing Model (“BOPM”) |
| Monte-Carlo Simulation Model (“Monte-Carlo”) |
Black-Scholes Merton Model (“BSM”)
BSM was developed by three economists – Fischer Black, Myron Scholes and Robert Merton. BSM is a closed-form solution, or single equation, used to calculate the theoretical price of European put and call options (options are expected to exercise at the term equal to the expected life).
The model makes certain assumptions, including:
- The options can only be exercised at expiration
- No dividends are paid out during the life of the option
- Efficient markets (i.e., market movements cannot be predicted)
- No commissions
- The risk-free rate and volatility of the underlying are known and constant
- Follows a lognormal distribution; that is, returns on the underlying are normally distributed
The BSM is the most well-known and generally accepted as a valuation tool for options and warrants. The BSM is appropriate when the terms of the options or warrants are simple. The BSM may not be appropriate to value options or warrants on a stand-alone basis if the terms of the options/warrants or the issuer’s capital structure are complex.
Binomial Option Pricing Model (“BOPM”)
The BOPM is a variation of the original BSM. John Carrington Cox, Stephen Ross and Mark Edward Rubenstein developed the BOPM. The BOPM uses a time-discrete framework that considers the underlying instrument over a period of time in the equity share price is varied over time. The BOPM is useful for analyzing American style options, which can be exercised at any time up to expiration as opposed to European options that can only be exercised upon expiration.
The model consists of a lattice for a number of time steps between the valuation and expiration date. Each node in the lattice represents a possible stock price at a given point in time. Valuation is performed iteratively, starting at each of the ending nodes, and then works backwards through the lattice tree towards the first node. The value computed at each stage is the value of the option at that point in time. This value is discounted to calculate a present value of the option.
In very basic terms, the model involves three steps:
| The creation of the binomial price tree |
| Option value calculated at each final node |
| Option value calculated at each preceding node |
The BOPM is computationally more involved than the BSM and it may increase precision, particularly for longer-term options or warrants.
Monte Carlo Simulation Model (“MC”)
MC calculates the value of an option or warrant with multiple sources of uncertainty or with complicated features. MC is named after the city in Monaco, where the primary attractions are casinos that have games of chance. Casino games, like roulette, dice, and slot machines, exhibit random behavior. MC is based on the idea that by looking at the results of many discrete scenario outcomes given certain constraints, one may draw conclusions on the average outcome.
Monte Carlo methods vary, but tend to follow a particular pattern:
- Define a domain of possible inputs
- Generate inputs randomly from a probability distribution over the domain
- Perform a deterministic computation on the inputs
- Aggregate the results
The price of the option is its discounted expected value. The technique applied is (1) to generate a large number of possible (but random) price paths for the underlying (or underlyings) via simulation, (2) calculate the associated exercise value (i.e. “payoff”) of the option for each path, (3) average these payoffs and (4) discount to today.
MC simulations are more complex to implement than using a BSM. In addition, formula constituting the simulation algorithm may need to be significantly adjusted to apply to options or warrants.
In conclusion, option pricing models are used to value options or warrants. The application of the right methodology is based on all relevant facts. It is critical to consult with a valuation specialist when the need to value options or warrants arises.
Dave Yoon, CPA
201-265-2800
[email protected]
Travis Gerstacker
609-520-1188
[email protected]
To ensure compliance with U.S. Treasury rules, unless expressly stated otherwise, any U.S. tax advice contained in this communication is not intended or written to be used, and cannot be used, by the recipient for the purpose of avoiding penalties that may be imposed under the Internal Revenue Code.
