What is the Chaffe model formula for DLOM?

Chaffe (1993) prices the DLOM as an at-the-money European put under Black-Scholes, expressed as a percentage of the stock price: DLOM = e^(−rT)·N(−d2) − e^(−qT)·N(−d1), with S = K, d1 = (r − q + σ²/2)T ⁄ (σ√T) and d2 = d1 − σ√T, where σ is volatility, T the restriction period, r the risk-free rate, and q the dividend yield.

What DLOM does the Chaffe model produce at 60% volatility and a 2-year term?

Approximately 27.3% with a 4.5% risk-free rate and no dividends — versus about 18.2% under the Finnerty model at the same inputs. Chaffe consistently produces the higher number because a protective put insures the full starting price while the holder keeps all upside.

Why is the Chaffe model considered an upper bound for DLOM?

Because the protective put overstates the cost of illiquidity: it insures the entire day-one price against any decline while the restricted holder still keeps the upside. Chaffe himself framed the result as an upper indication. At high volatility and long terms the model can exceed 50%, beyond the range of observed restricted stock discounts, so it is used as the ceiling of the model range rather than the conclusion.

Should a 409A valuation use Chaffe or Finnerty for DLOM?

Use both. Standard practice computes Chaffe (upper bound), Finnerty (average-strike), and restricted stock study benchmarks, then documents a concluded DLOM inside that range with an explicit rationale tied to the company's facts — stage, time to liquidity, and shareholder rights.

DLOM · Methodology

The Chaffe Model DLOM: Black-Scholes Put Formula, Worked Example, and Its Role as the Upper Bound

FairValueX Team · 7 min read

Bottom Line: The Chaffe model prices DLOM as the cost of an at-the-money European put — full insurance against any price decline during the restriction period. At 60% volatility and a 2-year term it produces about 27.3%. It is deliberately generous to the discount, which is why professionals treat it as the upper bound of the model range, not the conclusion.

The Intuition

If you hold restricted shares, what would it cost to eliminate the risk of being unable to sell? Chaffe (1993) answered: buy a put struck at today's price, expiring when the restriction lifts. The cost of that put, as a percentage of the share price, is a measure of the marketability discount.

Note what that insurance covers: the entire starting value, against any decline, while you keep all the upside. Real illiquidity costs less than that — a restricted holder isn't guaranteed today's price as a floor. That asymmetry is why Chaffe runs high, and why it pairs naturally with the Finnerty average-strike model as a bracketing pair.

The Formula

                DLOM = e−rT·N(−d2) − e−qT·N(−d1)   (with S = K, per $1 of stock)

                d1 = ( r − q + σ²⁄2 )·T ⁄ ( σ√T )    d2 = d1 − σ√T

✓ S = K — the put is struck at the money; the result reads directly as a percentage of value
✓ σ — equity volatility from the report's peer-group analysis
✓ T — restriction period / expected time to liquidity, in years
✓ r — risk-free rate matched to T (see rate sourcing)
✓ q — dividend yield, typically 0 for venture-stage companies

Worked Example: σ = 60%, T = 2 Years, r = 4.5%

σ√T = 0.60 × 1.4142 = 0.8485
d₁ = (0.045 + 0.18) × 2 ⁄ 0.8485 = 0.45 ⁄ 0.8485 = 0.5303
d₂ = 0.5303 − 0.8485 = −0.3182
N(−d₁) = N(−0.5303) = 0.2980; N(−d₂) = N(0.3182) = 0.6248
DLOM = e^−0.09 × 0.6248 − 0.2980 = 0.9139 × 0.6248 − 0.2980 = 27.3%

Same company through Finnerty: 18.2%. The two models bracket a range of roughly 18–27%, and the concluded discount is documented inside it.

Sensitivity Table

Volatility (σ)	Term (T)	Chaffe DLOM	Finnerty (same inputs)
40%	1.0 yr	13.4%	9.1%
50%	1.5 yr	20.1%	13.6%
60%	2.0 yr	27.3%	18.2%
70%	2.0 yr	32.1%	20.7%
80%	3.0 yr	41.8%	26.3%

Computed from the formulas with r = 4.5%, q = 0. Note the divergence at high σ·T: Chaffe keeps climbing past observed restricted-stock discounts while Finnerty flattens toward its ~32.6% cap — the visual argument for using them as a bracket.

How the Pair Gets Documented

A defensible DLOM exhibit shows: (1) both model outputs from inputs consistent with the rest of the report, (2) restricted stock study benchmarks for context, (3) the concluded discount with a written rationale tied to company facts — stage, expected liquidity path, and any contractual transfer restrictions. The number matters less than the visible reasoning. That structure is exactly what auditor valuation specialists test first.

Related Resources

DLOM exhibits your auditor can reconstruct.

Chaffe and Finnerty side by side, benchmarks, sensitivity, and a documented conclusion — inside every FairValueX 409A binder.

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