In-out parity, the reflection formula for the knock-in, and the barrier shift that daily monitoring requires, worked on one at-the-money option.
A down-and-out call is a vanilla call minus a down-and-in call with the same strike and barrier. On a one-year at-the-money option with spot 100, volatility 24 per cent and a rate of 4 per cent, the vanilla is worth 11.45, the down-and-in 2.84, so a knock-out at 90 prices at 8.62, about 24.8 per cent cheaper. If the barrier is only checked at the daily close, the fair price rises to 8.99.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The inputs are the Kestrel 100 option that runs through Quantitative Finance: a one-year at-the-money call whose Black-Scholes price of 11.454068 the book builds from d1 and d2. The only addition here is a barrier at 90, ten per cent below spot, illustrative of the knock-out level on a structured note or a cheapened hedge.
| Input | Value |
|---|---|
| Spot, S | 100 |
| Strike, K | 100 |
| Barrier, H (down, no rebate) | 90 |
| Maturity, T | 1 year |
| Risk-free rate, r (no dividend) | 4.0% |
| Volatility | 24% |
| Vanilla call (Black-Scholes) | 11.4541 |
Every path either touches 90 before expiry or it does not. A down-and-in call pays the vanilla payoff on the paths that touch; the down-and-out pays it on the paths that never do. Hold both and you hold the vanilla, whatever happens. That is in-out parity:
Down-and-out = vanilla call − down-and-in call
The down-and-in has a compact closed form when the barrier sits at or below the strike, which is the usual case for a down-and-out call. It comes from the reflection principle: the value of a path that has touched H equals the value of its mirror image across H, which is why the barrier appears squared.
λ = (r + σ²/2) / σ²
y = ln(H² / (S·K)) / (σ√T) + λσ√T
Down-and-in = S·(H/S)2λ·N(y) − K·e−rT·(H/S)2λ−2·N(y − σ√T)
Excel, with S in B1, K in B2, H in B3, T in B4, r in B5, vol in B6, lambda in B8 and y in B9:
=B1*(B3/B1)^(2*B8)*NORM.S.DIST(B9,TRUE)-B2*EXP(-B5*B4)*(B3/B1)^(2*B8-2)*NORM.S.DIST(B9-B6*SQRT(B4),TRUE)
With σ = 0.24, σ²/2 = 0.0288, so λ = (0.04 + 0.0288) / 0.0576 = 1.1944. The log term is ln(8,100 / 10,000) = ln(0.81), and dividing by σ√T = 0.24 then adding λσ√T gives y = −0.5913. Subtract 0.24 and the second argument is −0.8313.
| Step | Value |
|---|---|
| λ | 1.1944 |
| y | −0.5913 |
| N(y) | 0.2771 |
| N(y − σ√T) | 0.2029 |
| (H/S) to the power 2λ | 0.7775 |
| (H/S) to the power 2λ − 2 | 0.9599 |
| Discount factor e−rT | 0.9608 |
| First term: 100 × 0.7775 × 0.2771 | 21.5477 |
| Second term: 100 × 0.9608 × 0.9599 × 0.2029 | 18.7110 |
| Down-and-in call | 2.8367 |
Then the parity step: 11.4541 − 2.8367 = 8.6174 for the down-and-out. On a lot of 10,000 options that is 86,174 against 114,541 for the vanilla, a saving of 28,367, or 24.8 per cent of the premium.
The saving is not free money. It is the market value of the paths on which the buyer loses the option: every scenario in which the stock dips to 90 at some point and then recovers above 100 by expiry. A buyer who believes in a sharp dip followed by a rally is selling exactly the scenario they expect.
The closed form assumes the barrier is watched continuously. Most term sheets check it once a day, at the official close. Fewer observations mean fewer knock-outs, so the option is worth more. Broadie, Glasserman and Kou showed that a discretely monitored barrier is priced well by the continuous formula with the barrier shifted away from spot by a factor of e−0.5826·σ·√Δt.
Shifted barrier = 90 × e−0.5826 × 0.24 × √(1/252) = 89.21
Rerun the same formula at 89.21 and the daily-monitored down-and-out is worth 8.99, 0.37 more than the continuous price, a gap of 4.3 per cent. On a trading desk that difference is real money: price the note with the continuous formula and the bank is selling the option too cheaply.
| Barrier | Continuous | Daily close | Share of vanilla |
|---|---|---|---|
| 70 | 11.44 | 11.45 | 99.9% |
| 80 | 11.13 | 11.19 | 97.2% |
| 85 | 10.38 | 10.54 | 90.6% |
| 90 | 8.62 | 8.99 | 75.2% |
| 95 | 5.29 | 5.98 | 46.2% |
The discount is strongly non-linear. Moving the barrier from 70 to 80 saves very little; moving it from 90 to 95 takes the price from 8.62 to 5.29, under half the vanilla. The gap between continuous and daily pricing also widens as the barrier approaches spot, because a barrier close to spot is more likely to be crossed between two closes.
| Volatility | Vanilla | Down-and-out | Share of vanilla |
|---|---|---|---|
| 16% | 8.41 | 7.58 | 90.2% |
| 20% | 9.93 | 8.20 | 82.6% |
| 24% | 11.45 | 8.62 | 75.2% |
| 28% | 12.99 | 8.92 | 68.7% |
| 32% | 14.52 | 9.13 | 62.9% |
From 20 to 28 per cent volatility the vanilla gains 3.06; the down-and-out gains only 0.72. Higher volatility adds upside but also pushes more paths through the barrier, and the two effects nearly cancel. The knock-out carries far less vega than the vanilla it resembles, which matters to anyone hedging it with listed options.
Price the vanilla, price the knock-in with the reflection formula, subtract. Then adjust the barrier for the monitoring the contract actually uses. On the Kestrel 100 inputs, a barrier 10 per cent below spot takes the premium from 11.45 to 8.62 continuous, or 8.99 on daily closes. The vanilla leg of this calculation is built cell by cell in the free Kestrel 100 workbook, and the d1 and d2 arithmetic it relies on is worked in the Black-Scholes price by hand.
A knock-in and a knock-out with the same strike, barrier and maturity add up to the vanilla option, because every path either touches the barrier or does not. With no rebate, down-and-out plus down-and-in equals the vanilla call. In the worked case 8.62 plus 2.84 gives the vanilla price of 11.45, so pricing either barrier option prices the other.
Because the holder gives up every path that touches the barrier before expiry, including paths that later recover and finish in the money. With spot 100, strike 100, volatility 24 per cent and a barrier at 90, those paths are worth 2.84, so the knock-out costs 8.62 instead of 11.45, a discount of 24.8 per cent.
Use the Broadie-Glasserman-Kou correction: move a down barrier further from spot by the factor e to the power of minus 0.5826 times volatility times the square root of the observation interval, then use the continuous formula. With 24 per cent volatility and 252 daily observations, a barrier of 90 becomes 89.21 and the knock-out rises from 8.62 to 8.99.
Much less. Higher volatility raises the upside but also sends more paths through the barrier. Between 20 and 28 per cent volatility the vanilla in the worked case gains 3.06, while the down-and-out with a barrier at 90 gains only 0.72, so a vanilla hedge overstates its volatility exposure.
This article is one calculation from Quantitative Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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