Finance & Securities: Valuation of Complex Instruments

Gnarus Experts have developed a suite of Equity and Stock Option Valuation (ESOVAL®) models. These models, based on state-of-the-art lattice- and Monte Carlo simulation approaches, are cutting-edge tools for the valuation of employee stock options and other compensation instruments typically awarded by present-day employers. These instruments include traditional employee stock options (ESOs) as well as complex non-traditional instruments, such as capped, indexed, barrier, hurdle rate and market-based ESOs and stock appreciation rights.

Clients Use Gnarus’ ESOVAL® models for:

  • Reporting – Estimating the fair value of compensation instruments for SEC reporting purposes;
  • Audits– Defending our estimation results before auditors;
  • Instrument Evaluation – Helping firms and their advisors determine the instrument(s) that best meet their compensation design goals; and
  • Litigation – Serving as consultants or experts in disputes involving complex instruments.

The ESOVAL® Advantage

Gnarus’ ESOVAL® models offer a vast improvement over alternative models. As required in FAS 123R, our models are designed to reflect the complex features of the compensation instruments they value using methods recommended in the literature. Our models should be contrasted with traditional models that are based on simple modifications to models originally designed to value traded, rather than the non-traded instruments awarded in compensation programs. As a result, our estimates of fair value are more accurate and tend to be significantly lower than those provided by the older models. Advantages include:

  • Accurately reflects instrument characteristics
  • Provides a wide range of stochastic processes
  • Possesses significant computational power
  • Reflects input uncertainty
  • Meets regulatory requirements
  • Passes big four reviews/audits

Gnarus ESOVAL Models

To perform these valuations, we have developed a suite of state-of-the-art lattice- and Quasi Monte Carlo-based Equity and Stock Option Valuation (ESOVAL®).

These models are designed to value virtually any complex instrument or real option asset accurately. These assets typically have path dependent (Asian and look-back) and complex payoff mechanisms and decisions that can be made at virtually any point in time (“American” exercise). They also typically require the use of sophisticated stochastic processes to describe uncertainty in asset prices, model parameters, including volatility, dividend yields and correlations and other sources of uncertainty, such as weather.

Fair values produced by ESOVAL are accurate and significantly lower than Modified Black-Scholes.

The models traditionally used to value these assets are the Black-Scholes (BS) model, lattice-based Hull and White (H&W) model and traditional Monte Carlo simulation-based models (collectively “traditional models”). These models either can’t reflect or have difficulty reflecting three or more of the characteristics discussed above.

There are also significant differences between how the ESOVAL® and traditional models are calibrated. The ESOVAL® models are calibrated by adjusting parameters affecting exercise and termination behavior (e.g., risk aversion, lack of diversification and termination rates) until the calibration metric(s) output by the model equal those estimated from company data. As recommended in the literature, the calibration metrics are predicted under the assumption that a company’s stock price will increase at the risk-adjusted return. This procedure ensures that the ESOVAL® models are accurately calibrated.

The method used to calibrate the traditional models, such as the BS and H&W models, involves inputting a single calibration metric into each of the models. These models assume that stock prices increase at the lowest return possible – the risk-free rate. A drawback with this procedure is that there is no checking to ensure the model is correctly calibrated. For example, the BS model is calibrated by substituting an instrument’s Expected Term for its Contractual Term.

The H&W model is calibrated by assuming that exercise will occur whenever the stock price equals or exceeds the product of the exercise price and a calibration metric usually termed the “Exercise Multiple,” or EM. The EM is computed by averaging the ratios of the stock price at exercise to the strike price. As noted in the original H&W paper, when the options are subject to vesting restrictions, this procedure will generally cause the EM to be overstated. This coupled with the assumption that stock prices will increase at the risk-free rate will typically result in the H&W model overstating an instrument’s fair value.

To estimate the inputs required by the ESOVAL models our experts use advanced statistical techniques (hazard rate and GARCH-based methods ) to estimate model inputs. As required in FAS 123R, these methods are designed to reflect expected future changes in model inputs. The methods typically used to estimate the inputs for the traditional models are not designed to reflect expected future changes in model inputs.

The models and estimation methods our experts use to value the instruments awarded in compensation programs comply with both domestic (FAS 123R) and international standards (IFRS2) standards and have passed reviews and audits by the Big Four accounting firms. In addition, all of the ESOVAL® models have been shown to produce results that are close to those produced by well-known models in the literature. Gnarus’ ESOVAL® models have diverse and in-depth capabilities compared to more traditional models.

Who Uses these Models?

Public and Private Companies: Our experts use the ESOVAL® models to value instruments and real options for public and private companies, law firms and compensation design firms. We help public and private companies meet their reporting obligations by estimating the fair value (cost to the firm) of the employee stock options and other instruments awarded in their compensation programs. Our experts also help these companies with the valuation of traded and other types of non-traded instruments and real options.

Our experts also perform various ad hoc analyses associated with instruments awarded in compensation programs, including conducting modification and re-pricing analyses and the estimating cash flows from ESO exercises.

Gnarus professionals also provide similar assistance to private companies; however, when performing the valuations, we use models and estimation techniques that reflect the unique challenges faced by these companies. These challenges include limited or non-existent data for calculating volatility and correlation estimates and the need to reflect the constraint that the fair values for all equity-based instruments, including ESOs and common stock, must equal the total equity value of the firm. It should be noted that failure to reflect this constraint will overstated instrument fair values.

Law Firms: Gnarus experts assist law firms and their clients by preparing and defending expert reports that provide damage estimates for the three types of assets discussed above.

The damage estimates are used in such legal actions as backdating investigations, wrongful termination cases and disputes concerning the value of complex capital assets. Before developing our damage estimates, Gnarus experts work closely with outside counsel to determine the appropriate framework and model(s) to use to estimate values for both the actual and “but-for” worlds. For the reasons discussed above, our ESOVAL® models are generally viewed as producing more accurate and more defensible damage estimates than those produced by the models used by opposing experts.

Our fair value-based damages estimates for instruments awarded in compensation programs are generally lower than those produced by opposing experts. Moreover, in addition to providing fair value-based damage estimates, we also provide damage estimates based on the value of a particular instrument to employees (VTE). VTE is arguably a more appropriate measure of damages suffered by an employee than is fair value, which measures the cost of an instrument to a firm. It also results in damage estimates that are usually substantially lower than those based on fair value. It should be noted that the models typically used by opposing experts are not designed to estimate VTE.

Compensation Design Firms: Lastly, Gnarus experts help compensation design firms evaluate alternative instruments designs. Our models can provide virtually any measure of cost to the firm, value to the employees, efficiency and incentive strength

The measures we use to evaluate alternative instrument designs include fair value, value to employees (VTE), efficiency (measured by the difference between fair value and VTE), incentive strength (measured by the change in the value of an instrument to employees from a one dollar change in the price of the underlying asset), and exchange ratios (ratios of the minimum number of units of one type of instrument [complex market-based option] required for an employee to be willing to give up one unit of another type of instrument [traditional ESO]). To assure consistency, our experts use the same model to estimate all five measures.

For more information regarding the valuation of complex instruments, please send an email to Ron Rudkin at