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Both the Black-Scholes and lattice models require the use of risk-free interest rates.

9.5.1 Risk-free interest rates in the Black-Scholes model

The risk-free interest rate assumption involves less judgment than the other assumptions required in an option-pricing model. In the US context, the Black-Scholes model typically makes use of the implied rate on the grant date for a traded zero-coupon US Treasury instrument with a term equal to the option's expected term. Zero-coupon bonds are used because they have one payment that will be paid at the end of the expected term to match the period of investment through the time until expected exercise or settlement of the award. For terms greater than one year, Treasury STRIPS should generally be used. However, other estimates of risk-free rates are available (such as swap curves) and may be appropriate. Companies outside the US or companies issuing options with exercise prices denominated in a foreign currency should use an appropriate risk-free instrument in that currency environment in developing a risk-free rate assumption, or may use forward currency exchange rates combined with US risk-free rates. If an option's expected term or an equity award's contractual term falls between two maturities with available risk-free rate data, it is usually appropriate to interpolate a rate from the available maturities.
Implied interest rates on zero-coupon government bonds are based on their traded prices. These are typically reported as bond-equivalent yields based on implied semi-annual compounding (this allows one to compare zero-coupon and coupon-bearing government bonds which make payments semi-annually). To obtain precise results, a company should convert bond-equivalent rates into continuously compounded rates before using them in the Black-Scholes model. Although the difference is usually very small, a company that wishes to omit this step should determine whether the difference is material.

9.5.2 Risk-free interest rates in lattice models

Lattice models require risk-free interest rates for all potential durations until exercise. These rates are obtained by using a yield curve for the relevant instrument as of the grant date. A lattice model will therefore require the yield curve for the entire time period during which employees might exercise their options. Some software packages specify the frequency with which users should input yields over the potential exercise period (e.g., monthly), while others allow users to choose the frequency with which they input a range of yields. These risk-free interest rates are often different in coupon type and compounding frequency from those reported in the financial media. Users should be careful to determine the proper type of rate to input into the modeling software to achieve a zero coupon risk-free rate in the valuation.

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