A “Complex Security” can refer to a number of different financial securities including: 1) preferred stock; 2) convertible debt; 3) options; and 4) warrants. What makes one security more complex over the other? Simply put, it is the number of unknown variables or conditions that exist that drive the behavior of that security and ultimately its value.

Let’s look at an example of a “complex security” that most people are very familiar with – equity options. Equity options are commonly granted to employees by companies as a form of incentive. An option gives the holder the right to purchase or sell a stated number of shares of stock at a fixed price (also known as: exercise or strike price) within a predetermined period. The holder of the option, however, is not obligated to exercise the option^{[1]}.

The value of a stock option can be broken down into two components, intrinsic value and time value. Intrinsic value is the difference between the value of the stock and the exercise price. For example, if the share price is $90 and the exercise price of the option is $100, the intrinsic value is zero. Conversely, if the share price is $100 and the exercise price of the option is $90, the intrinsic value is $10. In general, as the value of the underlying stock increases, the value of the call option also increases. For privately-held companies that issue equity options the primary unknown variable is the price of the underlying stock.

The Black-Scholes model is the most widely used and best known theoretical option model for valuing options. The primary inputs of the Black-Scholes model are 1) stock price; 2) exercise price; 3) risk-free rate; 4) volatility; and 5) time to expiration. However, the Black-Scholes model has a number of limitations including:

- It is more appropriate for options with expiration dates that are relatively short;
- It assumes the short term rate is known and constant through time;
- The distribution of possible stock prices at the end of any infinite interval is lognormal;
- It assumes that the stock pays no dividends;
- It assumes the option can be exercised only at maturity; and
- It assumes no commissions or other transaction costs are incurred to buy or sell the stock or the option.

The Black-Scholes may not be appropriate to use for complex securities that have a fairly long time until expiration or for complex securities that have various triggering events (i.e. unpredictable conditions that will drive their behavior). A lattice model may be a more appropriate model to use when greater flexibility is required to predict exercise behavior and various market and performance conditions. Whereas the Black-Scholes model is a ‘closed form’ method that calculates the price of an option based on a formula of certain parameters, the lattice method is an ‘opened form’ method that requires the prediction of certain behaviors and parameters with distinct probabilities at certain points in the future.

Let’s take a look at an example of when a security becomes more complex to value and how a lattice model can better accommodate the flexibility required to value the security. We recently worked with a client to determine the value of a number of convertible debt securities issued by the company. Convertible debt securities are hybrid instruments since they exhibit characteristics of both debt and equity securities. What made these securities vastly more complex to value were the various events around when and how the convertible debt would convert (“Triggering Events”) and their corresponding payout structures. The convertible debt securities could be 1) held to maturity; 2) convert into an existing round of preferred financing or into a new round of preferred financing; or 3) receive two times the principal amount of the note if there was a change in control of the company.

In order to value the convertible debt securities, we had to determine:

- Under what conditions would the debt securities convert into existing preferred stock and when;
- Under what conditions would the company raise new capital and when;
- Under what conditions would there be a change of control and when; and
- What was the probability of default.

The value of the debt securities was different under each scenario and each scenario was dependent on several uncertain variables including timing and conversion ratios. The probability of each scenario was also dependent upon several internal and external factors outside the control of the company.

We utilized a binomial model to value the straight bond, a call option price lattice for the conversion option, and calculated the present value of any future payments of principal and interest in the case of a change in management or a company acquisition. Once each scenario was modeled, we probability weighted each scenario based on discussions with management, in order to determine the overall value of the security.

The lattice model is a decision tree model that allows for an infinite number of possible outcomes over time. In the illustration below, *S* is the value of the security and *q* is the probability of an upward movement to *Su* and *1-q* is the probability of downward movement to *Sd* more complex and integrated model is required.

Each node represents the value of the security and considers the probabilities of various triggering events and incorporates the value of the security of such an event occurs. Lattice modeling is a powerful analytic tool and when set up correctly can provide more accurate results. However, despite their many uses, lattice models remain a neglected tool in the appraisal process, primarily because they are complex to administer. When determining if a lattice model is appropriate to use, be sure that you are working with a valuation firm that has experience in valuing complex securities and knows how to set up a lattice model properly.

[1] There are two types of options: call options and put options. *Call options *provide the holder the right to *buy *a security. *Put options* provide the holder the right to *sell *a security