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At-the-money one-year call at 30 % volatility: the Black-Scholes-Merton value against the IRS safe-harbor table (Rev. Proc. 2003-68)
| Tool | Option Pricing |
| Source | U.S. Internal Revenue Service (2003). Revenue Procedure 2003-68: Valuation of Stock Options for Purposes of Sections 280G and 4999. Internal Revenue Service, Washington, D.C. Table, p. 12; Section 5 example, pp. 9–10. |
| Level | Professional |
Problem
The revenue procedure gives a safe-harbor value of a stock option as the stock price times a factor read from a table of Black-Scholes values for three volatility classes, fourteen ratios of stock price to exercise price and eleven terms. For low volatility, an exercise price equal to the stock price (spread 0 %) and a term of 12 months the factor is 13.6 % of the stock price. Compute the value of the call from the formula, with its put, the sensitivities and put-call parity, and compare the table's factors.
Inputs to Enter
- Analysis: Black-Scholes-Merton (European); option: call
- Stock price 100, strike 100, time to expiry 1 yr, risk-free rate 5 %, dividend yield 1 %, volatility 30 % (continuous compounding)
- The table does not print its assumptions; a least-squares fit to its 462 cells gives volatilities of 30 / 50 / 70 % for the Low / Medium / High classes, an interest rate of 5 % and a dividend yield of 1 %
- Quantity 100 shares; no market price, no cash dividends
Answers
- d1 = [ln(100/100) + (0.05 − 0.01 + 0.30²/2) · 1] / (0.30 · √1) = 0.28333; d2 = −0.016667; N(d1) = 0.611539, N(d2) = 0.493351
- Call = 100 e^−0.01 · 0.611539 − 100 e^−0.05 · 0.493351 = 13.6164, i.e. 13.6 % of the stock price (book: 13.6 %, Low volatility, spread 0 %, 12 months)
- Put = 9.7344; parity c − p = 3.8820 = 100 e^−0.01 − 100 e^−0.05 = 99.0050 − 95.1229
- Call sensitivities per share: delta 0.6055, gamma 0.01265, vega 0.3794 per 1 % of volatility, theta −0.0204 per day, rho 0.4693 per 1 % of rate; position of 100 shares 1,361.64
- With the fitted assumptions the formula reproduces all 462 printed factors after rounding to 0.1 % (for example Medium volatility, spread 0 %, 60 months: 45.8 %; High, −60 %, 120 months: 59.5 %)
- Section 5 example of the procedure: stock 50, exercise price 25 (spread 100 %), low volatility, 36 months: factor 54.8 %, 27.40 per share, 1,096,000 for 40,000 shares; recomputed at 12 months 51.5 %, 25.75 per share, 1,030,000 (book: the same figures)
Notes
The tool's default inputs are this case; press Submit. The volatilities, interest rate and dividend yield of the table are inferred from its numbers, not quoted from the procedure. The table is a tax safe harbor for sections 280G and 4999 and nothing here is tax or investment advice. The binomial, employee-option, strategy and Monte Carlo analyses of the tool are checked against an independent numpy / scipy reference (test_option_pricing.php), not against a printed example.