A digital pays all or nothing across one price, so it cannot be delta hedged near expiry. Dealers replicate it with a call spread and charge for the gap.
A short digital call is hedged by buying a call spread that pays at least as much as the digital everywhere: for a payout Q and a spread width ε, buy Q ÷ ε calls struck at K − ε and sell the same number struck at K. On an illustrative $1,000,000 digital struck at 110 on a stock at 100, worth $348,764 under Black-Scholes, a 2-point spread costs $362,618: an overhedge of $13,854, or 4.0 per cent of the fair price, which is what a dealer charges for not being able to hedge the jump.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A digital (binary) option pays a fixed amount if the underlying finishes above the strike and nothing otherwise. Pricing it is one line, the discounted risk-neutral probability, worked in the probability an option expires in the money. Hedging it is the hard part, because the payoff jumps from zero to the full payout across a single price. This article works the replication a desk actually uses and puts a number on what it costs.
| Input | Value |
|---|---|
| Spot | 100 |
| Digital strike | 110 |
| Risk-free rate, continuous | 4% |
| Volatility | 24% |
| Expiry | 1 year |
| Payout if spot finishes above 110 | $1,000,000 |
d2 = [ln(S ÷ K) + (r − σ² ÷ 2) × T] ÷ (σ × √T) = (−0.0953 + 0.0112) ÷ 0.24 = −0.3505
Digital = Q × e−rT × N(d2) = $1,000,000 × 0.9608 × 0.3630 = $348,764
Excel: =1000000*EXP(-0.04*1)*NORM.S.DIST((LN(100/110)+(0.04-0.24^2/2)*1)/(0.24*SQRT(1)),TRUE)
That is the price at which the digital is worth nothing to either side if it could be hedged continuously. It cannot, which is why no dealer sells at that price.
A vanilla call spread between K − ε and K pays nothing below K − ε, rises linearly between the strikes, and pays ε per unit above K. Scale it by Q ÷ ε and it pays exactly Q whenever the digital pays Q, plus something extra between the strikes, where the digital pays nothing. A dealer who is short the digital therefore holds a position that can never lose at expiry.
Width ε = 2. Quantity = $1,000,000 ÷ 2 = 500,000 calls per leg.
Buy 500,000 calls struck at 108 at 7.9635; sell 500,000 struck at 110 at 7.2382.
Net cost = 500,000 × (7.9635 − 7.2382) = 500,000 × 0.7252 = $362,618
Overhedge = $362,618 − $348,764 = $13,854
At expiry the hedge pays $1,000,000 above 110, exactly matching the digital. Between 108 and 110 it pays up to $1,000,000 while the digital pays nothing; at a finish of 109 it pays $500,000. That band is the $13,854 the dealer has paid for, and it is the margin the client pays on top of the theoretical price.
| Width | Lower strike | Calls per leg | Hedge cost | Overhedge | % of fair price | Centred spread |
|---|---|---|---|---|---|---|
| 1 | 109 | 1,000,000 | $355,641 | $6,878 | 2.0% | $348,777 |
| 2 | 108 | 500,000 | $362,618 | $13,854 | 4.0% | $348,815 |
| 5 | 105 | 200,000 | $384,112 | $35,348 | 10.1% | $349,081 |
The overhedge is almost exactly proportional to the width. A tighter spread is cheaper, but halving the width doubles the number of options per leg, and the gamma it leaves the desk with near expiry is correspondingly larger. The last column is the spread centred on the strike (for example 109 to 111): its cost converges on the fair digital price, $13 above it at a width of 1, which is the numerical proof that a digital is the limit of a call spread, the negative of the derivative of the call price with respect to the strike.
The delta of a digital is Q × e−rT × n(d2) ÷ (S × σ × √T). With spot sitting on the strike, it explodes as expiry approaches:
| Days to expiry | Delta, shares | Stock to hold, $m |
|---|---|---|
| 365 | 14,503 | 1.6 |
| 30 | 52,532 | 5.8 |
| 5 | 129,040 | 14.2 |
| 1 | 288,671 | 31.8 |
On the last day the desk would need $31.8m of stock to hedge a $1m payout, and would have to sell all of it if the price ticks below the strike. Real markets gap, and one gap through 110 costs the whole payout. The call spread converts that cliff into a ramp two points wide, whose risk is ordinary vanilla gamma that the options book already manages. The gamma of the two legs is large near expiry, but bounded.
The digital's sensitivity to volatility changes sign with moneyness, unlike a vanilla option, whose vega is always positive.
| Volatility | Strike 90 | Strike 100 | Strike 110 |
|---|---|---|---|
| 16% | $765,084 | $545,243 | $322,025 |
| 24% | $659,485 | $498,276 | $348,764 |
| 32% | $591,575 | $466,982 | $355,133 |
The out-of-the-money 110 digital gains value as volatility rises, because more paths reach the strike. The in-the-money 90 and the at-the-money 100 lose value, because more paths fall back below it. The call spread hedge inherits the right sign automatically, since it is long the lower strike's vega and short the higher one's. This is also why the volatility smile matters so much for digitals: with skew, the vol at 108 is higher than at 110, and pricing both legs at one flat volatility misprices the hedge.
Pricing rule of thumb from the table: each point of spread width adds about 2 per cent of the fair price on this one-year 110 digital. Quote the width, not just the price.
Price a digital at e−rT × N(d2), then hedge it as a call spread ending at the strike. The fair value here is $348,764; the 2-point hedge costs $362,618, and the $13,854 gap is the real cost of a payoff that cannot be delta hedged. The full Black-Scholes build, with d1 and d2 on their own rows and every Greek checked against a finite difference, is in the free workbook for this case.
A cash-or-nothing call is worth the payout times e^(-rT) times N(d2). For a $1,000,000 payout, strike 110, spot 100, 24 per cent volatility, a 4 per cent rate and one year, d2 is -0.3505, N(d2) is 0.3630 and the discount factor 0.9608, so the price is $348,764. Using N(d1) instead would give $438,142, a common error.
Because they hedge with a call spread that pays at least the digital everywhere, which costs more than the model price. On the illustrative 110 digital a 2-point spread costs $362,618 against a fair $348,764; the $13,854 difference, about 4.0 per cent, is passed to the client as the overhedge.
Only when it is out of the money. On a one-year $1,000,000 digital with spot at 100, the 110 strike rises from $322,025 at 16 per cent volatility to $355,133 at 32 per cent, while the 90 strike falls from $765,084 to $591,575. In or at the money, higher volatility pushes more paths back below the strike.
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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