ASC 815 provides limited guidance on the valuation of derivative instruments and hedges. Adams Capital’s experience valuing simple options to complex derivatives with changing terms helps owners and management understand the financial reporting challenges associated with ASC 815. For simple options, a binomial option model may be appropriate since it is flexible and capable of handling both European and American options. For more complex derivatives, a Monte Carlo analysis will be customized to model each specific term accurately. Leveraging extraordinary math skills, Adams Capital designs custom models for each client addressing specific instrument terms. We do not use a black box approach. While developed for fair value reporting, clients and investors frequently adopt the Adams Capital derivative models for their own business management, trading, and to understand better market risk.
Binomial Option Model
The binomial option pricing model assumes that each period can lead to two outcomes in the price of the underlying asset. Each outcome in the current period is contingent upon the outcomes of the prior periods, as would be the case in a decision tree. Each outcome indicates that the price of the underlying stock has either increased (an “Up” state) or decreased (a “Down” state) at each period. The degree of expected increase and the degree of expected decrease in each instance lead to an implied volatility. Implied volatility is also a function of the percentage of periods in which an “Up” state is expected and the percentage in which a “Down” state is expected. The probability of an “Up” state or a “Down” state occurring is calculated within the model as a function of the two potential price states. This probability is called a “risk neutral” probability function.
A binomial option pricing model belongs to a broader category of option pricing models called “lattice” models. Lattice models are so named because when the potential price outcomes over time are represented graphically, they look like an interconnecting network, or lattice.
A straightforward way to execute the calculation of the call option price using a binomial option model is to calculate the various outcomes in the lattice. In a binomial model, two lattices are used; one that calculates the stock price in the various scenarios, and one that calculates the corresponding call option price for each stock price scenario. The lattice tree method is useful for relatively small numbers of periods. After a larger number of periods, approximately 15-20, the use of a computer program to calculate more complex formulas is considered to be more efficient.
One advantage of the binomial option model compared to a Black-Scholes model is that it can be modified easily to accommodate an early exercise decision that is consistent with an American option. The Black-Scholes model relies on the assumption that the subject option is European.
The binomial option model relies on five key inputs: the underlying stock price, the strike price, the volatility, the risk-free rate, and the time to maturity. The number of steps represents the number of periods in the lattice, and the concluded value will converge as the number of steps increase.
Monte Carlo Simulation
The Monte Carlo simulation was named after the city in Monaco where games of chance involve repetitive events with known probabilities. Although there were a number of isolated and undeveloped applications of Monte Carlo simulation principles at earlier dates, the modern application of Monte Carlo methods date from the 1940’s during work on the atomic bomb.
Monte Carlo simulations rely on the process of explicitly representing uncertainties by specifying inputs as probability distributions. If the inputs describing a system are uncertain, the prediction of future performance is necessarily uncertain. That is, the result of any analysis based on inputs represented by probability distributions is itself a probability distribution.
Whereas the result of a single simulation of an uncertain process or system outcome is a qualified statement the result of a Monte Carlo simulation is a quantified probability. Such a result is typically much more useful to decision-makers who utilize the simulation results.
In order to compute the probability distribution of predicted performance, it is necessary to translate the input uncertainties into uncertainties in the results. A variety of methods exist for quantifying uncertainty. Monte Carlo simulation is an accepted and common technique for incorporating the uncertainty in the various aspects of a system outcome into the analysis of predicted performance.
In a Monte Carlo simulation, the entire process is simulated a large number of times. Each simulation is referred to as a realization of the system, or a trial. For each realization, all of the uncertain parameters are sampled. For example, a single random value is selected from the specified distribution describing each parameter. The system is then simulated through time such that the performance of the system can be computed. This results in a large number of separate and independent results, each representing a possible “future” for the system or one possible path the system may follow through time. The results of the independent system realizations are assembled into probability distributions of possible outcomes. Thus, the outputs are not single values, but probability distributions.
As a simple example of a Monte Carlo simulation, consider calculating the probability of a particular sum of the throw of two dice (with each die having values one through six). In this particular case, there are 36 combinations of dice rolls. Based on this, you can manually compute the probability of a particular outcome. For instance, there are six different ways that the dice could sum to seven. Hence, the probability of rolling seven is equal to 6 divided by 36, or 0.167.
Instead of computing the probability in this way, however, we could instead throw the dice 100 times and record how many times each outcome occurs. If the dice totaled seven 18 times (out of 100 rolls), we would conclude that the probability of rolling seven is approximately 0.18 (18%). The more times we rolled the dice, the more likely our result is illustrative of realistic behavior. To improve efficiency beyond rolling dice a hundred times, we use a computer to simulate rolling the dice 10,000 times.
There are a wide variety of instances in which a Monte Carlo simulation is particularly useful in finance. One example is the valuation of complex securities for which no analytical pricing formula exists. By calculating a quantity of interest thousands of times, an estimate of the expected value may be determined with a high level of certainty even though the underlying variables are highly variable. The following steps provide a general overview of the implementation of a Monte Carlo simulation.
- Specify the quantities of interest in terms of underlying variables and the starting values of those underlying variables.
- Specify distributional assumptions for the risk factors that drive the underlying variables.
- Using a computer program or spreadsheet, draw random values of each risk factor.
- Calculate the underlying variables using the random observations generated in Step 3.
- Compute the quantities of interest.
- Iteratively go back to Step 3 until a specified number of trials is completed. Finally, produce statistics for the simulation.
Steps 3 and 4 utilize random values between 0 and 1 determined by a random number generator. Random observations from any distribution can be produced using the uniform random variable with endpoints 0 and 1. Consider the inverse transformation method of producing random observations. Suppose we are interested in obtaining random observations for a random variable, X, with cumulative distribution function F(x). F(x) evaluated at x is a number between 0 and 1, or r, corresponding to a cumulative probability. An inverse function of F(x) may be defined as F-1(r). The inverse function returns a random outcome, x, based on a given probability, r.
In Step 6 we produce statistics for the simulation to estimate the expected outcome for the quantities of interest.