# Option Pricing Fundamentals

## Help for option pricing fundamentals in the real options valuation template for Excel

 User Guide Navigation:Business Spreadsheets User Guides: Project Planning and Management Multiple Regression Analysis and Forecasting Investment and Business Valuation Real Options Valuation Portfolio Optimization Portfolio Performance Tracking Customer InvoicingReal Options Valuation Help Topics: Option Pricing Fundamentals The Option to Delay a Project The Option to Expand a Project The Option to Abandon a Project Binomial Option Pricing Game Theory Analysis

An option represents the right to buy (call) or sell (put) a specific quantity of an asset at a fixed price (exercise price) at or before a specified date in the future. This right is not an obligation, therefore the holder can choose not to exercise the purchase or sale and allow the option to expire.

A Call Option gives the holder the right to buy an asset at a specified exercise price at any time before the specified expiry date, for which the holder pays a price for. At the expiry date, if the asset value is less than the exercise price, the asset is not purchased and the option expires worthless. Conversely, if the asset value exceeds the exercise price, the asset is purchased at the exercise price and difference represents the gross profit on the investment. The net profit is the difference between the gross profit and the price paid for the call. The options to Delay or Expand a business case or project are both variations of call options.

A Put Option gives the holder the right to sell an asset at a specified exercise price at any time before the specified expiry date, for which the holder pays a price for. At the expiry date, if the asset value is greater than the exercise price, the asset is not sold and the option expires worthless. Conversely, if the asset value is less than the exercise price, the asset is sold at the exercise price and difference represents the gross profit on the option. The net profit is the difference between the gross profit and the price paid for the put. The option to Abandon a business case or project is a variation of a put option.

The calculation of option value (call or put price) is accomplished in the model by employing modified versions of the Black-Scholes model. It is not the purpose of this paper to detail the mathematical derivation of this calculation; however in broad terms the model employs the use of a 'replicating' portfolio consisting of the underlying asset and risk-free rate to determine the options value. Modifications are made to the model in order to account for:

• Dividends (or cash flows to and from the asset),
• Early exercise of the option before the expiry date, and
• The impact of exercise on the value of the underlying asset.

Determinants of Option Value are essentially the inputs required for such a model. The following table summarizes the key inputs required for the modified Black-Scholes models and their effect on the option values.

 Input Effect of Increase in Input on values for: Call (Delay, Expand) Put (Abandon) Current value of underlying asset UP DOWN Variance in value of the underlying asset UP UP Dividends or cash flows from the asset DOWN UP Exercise Price of the option DOWN UP Time to expiration of the option UP UP Risk free interest rate for life of the option UP DOWN

It should be noted that this model is essentially designed to value options on financially traded assets where inputs are readily available from market data sources. When applying such models to 'real' options, some alternative Adjustments and Assumptions can be made in order to assist application. The following underlying assumptions of the Black-Scholes model illustrate this.

• The value of the asset is continuous (i.e. no price jumps). While this is reasonable to expect for traded assets, non-traded assets often change value erratically. For many real options this will underestimate the value. An alternative is to utilize higher variance estimates to accommodate for such price jumps.
• The variance is known and constant. When applied to long-term real options, the assumption that the price variance is known and does not change over the lifetime of the option is unreasonable. Again, adjustments can be made to variance estimates to accommodate for this.
• Exercise is instantaneous. In many real option scenarios, exercising may take up to several years (e.g. building a new plant under Expansion).  In this case, the life of the option should be reduced to accommodate the time required to exercise.

While making such adjustments to the inputs may seem to weaken the valuation process, it is the understanding of limitations and sensitivity to changes in these inputs that highlights the existence and value of strategic options apparent.

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