Datar–Mathews method for real option valuation

The Datar–Mathews method ^[1] (DM method^[2]) is a method for real options valuation. The method provides an easy way to determine the real option value of a project simply by using the average of positive outcomes for the project. The method can be understood as an extension of the net present value (NPV) multi-scenario Monte Carlo model with an adjustment for risk aversion and economic decision-making. The method uses information that arises naturally in a standard discounted cash flow (DCF), or NPV, project financial valuation. It was created in 2000 by Professor Vinay Datar, Seattle University, and Scott H. Mathews, Technical Fellow, The Boeing Company.

Method

Fig. 1 Typical project cash flow with uncertainty

The mathematical equation for the DM method is shown below. The method captures the real option value by discounting the distribution of operating profits at µ, the market risk rate, and discounting the distribution of the discretionary investment at r, risk-free rate, before the expected payoff is calculated. The option value is then the expected value of the maximum of the difference between the two discounted distributions or zero. Fig. 1.

• S_T is a random variable representing the future benefits, or operating profits at time T. The present valuation of S_T uses μ, a discount rate consistent with the risk level of S_T. μ is the required rate of return for participation in the target market, sometimes termed the hurdle rate.
• X_T is a random variable representing the strike price. The present valuation of X_T uses r, the rate consistent with the risk of investment, X_T . In many generalized option applications, the risk-free discount rate is used. However other discount rates can be considered, such as the corporate bond rate, particularly when the application is a risky corporate product development project.
• C₀ is the real option value for a single stage project. The option value can be understood as the expected value of the difference of two present value distributions with an economically rational threshold limiting losses on a risk-adjusted basis.

The differential discount rate for μ and r implicitly allows the DM Method to account for the underlying risk. If μ > r, then the option will be risk-averse, typical for both financial and real options. If μ < r, then the option will be risk-seeking. If μ = r, then this is termed a risk-neutral option, and has parallels with NPV-type analyses with decision-making, such as decision trees. The DM Method gives the same results as the Black–Scholes and the binomial lattice option models, provided the same inputs and the discount methods are used. This non-traded real option value therefore is dependent on the risk perception of the evaluator toward a market asset relative to a privately held investment asset.

The DM method is advantageous for use in real option applications because unlike some other option models it does not require a value for sigma (a measure of uncertainty) or for S₀ (the value of the project today), both of which are difficult to derive for new product development projects; see further under real options valuation. Finally, the DM method uses real-world values of any distribution type, avoiding the requirement for conversion to risk-neutral values and the restriction of a lognormal distribution;^[3] see further under Monte Carlo methods for option pricing.

Extensions of the method for other real option valuations have been developed such as contract guarantee (put option), Multi-Stage (compound option), Early Launch (American option), and others.

Implementation

Fig. 2A Net profit present-value distribution

Fig. 2B Rational decision distribution

Fig. 2C Payoff distribution and option value

Fig. 3 Range option calculation procedure

The method may be implemented using Monte-Carlo simulation, or in a simplified, approximate form (the DM range option).

Using simulation, for each sample, the engine draws a random variable from both S_T and X_T, calculates their present values, and takes the difference.^[4][5] Fig. 2A. The difference value is compared to zero, the maximum of the two is determined, and the resulting value recorded by the simulation engine. Here, reflecting the optionality inherent in the project, a forecast of a net negative value outcome corresponds to an abandoned project, and has a zero value. Fig. 2B. The resulting values create a payoff distribution representing the economically rational set of plausible, discounted value forecasts of the project at time t₀.

When sufficient payoff values have been recorded, typically a few hundred, then the mean, or expected value, of the payoff distribution is calculated. Fig. 2C. The option value is the expected value, the first moment of all positive NPVs, of the payoff distribution.

A simple interpretation is:

where operating profit and launch costs are the appropriately discounted range of cash flows to time t₀. ^[6]

The sampled distributions may take any form, although the triangular distribution is often used, as is typical for low data situations. Here, the mean value corresponds to the “most likely” scenario, typically the same as for the NPV case. Two other scenarios, “pessimistic” and “optimistic”, represent plausible deviations from the most likely scenario (often modeled as approximating a 1-out-of-20, or 1-out-of-10 likelihood). This range of probabilistic cases tends to be within the organizational memory bounds of the corporation.

An approximate but conservative option value, termed the DM Range Option, can be estimated simply using range estimates of the present values of operating profit and launch costs.^[7] Fig. 3. As described, a range is an estimate of a maximum, most-likely (or mode) and minimum (or Optimistic, Most-Likely, Pessimistic) values that circumscribe a triangular distribution. The two distributions are then combined, and, similar to the approach for simulation described, the expected value is the first moment of all positive NPVs. Here, using equations from triangular distributions, the mean of the launch cost distribution is calculated. The present value imputed net profit distribution is the difference between the operating profit distribution and the mean value of the launch cost distribution. In one implementation, the approximate option value is the product of the mean and the probability of the payoff distribution right triangle, the positive value right tail. The DM Range Option requires no simulation. This approach is useful for early-stage estimates of project option value when there has not been sufficient time or resources to gather the necessary quantitative information required for a complete cash flow simulation, or in a portfolio of projects when simulation of all the projects is too computationally demanding.^[8] If the launch cost is a scalar value, then the range option value calculation is exact. The range option method is similar to the fuzzy method for real options.

Interpretation

Under certain constraints, the framework of a project investment problem structured for the Datar–Mathews Method can be converted to an equivalent framework structured for the Black–Scholes formula. Figure 4, Left. The Black–Scholes (as well as the binomial lattice) option pricing model is constrained to a lognormal distribution for the asset value, S, typical of traded financial options, and requires a value for S₀, the asset value at time t₀, and sigma (σ₀), a measure of volatility of the asset. Assume a project investment problem at time T, and a forecasted lognormal asset value distribution with mean S_T and standard deviation σ_T. The equivalent Black–Scholes values are:

The terms N(d₁) and N(d₂) are applied in the calculation of the Black–Scholes formula, and are expressions related to operations on lognormal distributions;^[9] see section "Interpretation" under Black–Scholes. The Datar–Mathews method does not use N(d₁) or N(d₂), but instead typically solves the option problem by means of Monte Carlo simulation applicable to many different types of distributions inherent in real option contexts. When the Datar–Mathews method is applied to assets with lognormal distributions, it becomes possible to visualize graphically the operation of N(d₁) and N(d₂).

Fig. 4 Left: Comparison of Black–Scholes and Datar–Mathews frameworks. Right: Detail of tail distribution at t₀

N(d₂) is a measure of the area of the tail of the distribution relative to that of the entire distribution, e.g. the probability of tail of the distribution, at time t₀. The tail of the distribution is delineated by X_t0 = X_Te ^− rT, the present value of the strike price. Figure 4, Right. The true probability of expiring in-the-money in the real (“physical”) world is calculated at time T, the launch date, measured by area of the tail of the distribution delineated by X_T. N(d₁) is the value of the option payoff relative to that of the asset; N(d₁) = [MT × N(d₂)]/S₀, where MT is the mean of the tail at time t₀. Using the DM Method, the value of a call option can be understood as C₀ = (MT − X_t0) × N(d₂).

References

External links

• A Practical Method for Valuing Real Options: The Boeing Approach