What Are Digital Options?
Digital options (also called binary options) are derivatives with a discontinuous payoff: they pay either a fixed amount or nothing at expiry. Unlike vanilla options whose payoff scales with how far in-the-money they finish, digitals have an all-or-nothing character that makes them particularly useful for expressing pure directional views without magnitude risk.
The three types covered by this calculator are all European (exercisable only at expiry):
Cash-or-Nothing
Pays a fixed cash amount k if the spot S finishes above (call) or below (put) strike X at expiry T. The BSM closed-form is:
Call = k · e−rT · N(d₂)
Put = k · e−rT · N(−d₂)
Where d₂ = [ln(S/X) + (b − σ²/2)T] / (σ√T). Note that N(d₂) is the risk-neutral probability of expiring in-the-money — so the price is simply the present value of the expected payout.
The delta of a cash-or-nothing call is always positive but can be very large near-the-money, close to expiry — this "delta spike" makes hedging difficult in practice.
Asset-or-Nothing
Pays the current spot price S (one unit of the underlying) if in-the-money at expiry. The formula:
Call = S · e(b−r)T · N(d₁)
Put = S · e(b−r)T · N(−d₁)
Where d₁ = [ln(S/X) + (b + σ²/2)T] / (σ√T). A plain vanilla call is exactly an asset-or-nothing call minus a cash-or-nothing call with cash amount k = X. This relationship allows market-makers to hedge digitals using vanilla options.
Gap Option
Has two strikes: X₁ (trigger) determines whether the option is in-the-money, and X₂ (payment strike) determines the payout size: S − X₂ for calls, X₂ − S for puts. The formula:
Call = S · e(b−r)T · N(d₁) − X₂ · e−rT · N(d₂)
where d₁ and d₂ use X₁ (not X₂) as the strike. When X₁ = X₂, this reduces to a standard vanilla call. A gap option can generate negative payoffs when X₂ > X₁ for calls — the "gap" refers to this potential discontinuity between trigger and payoff.
Greeks for Digital Options
Delta — the delta of a cash-or-nothing call spikes near-the-money, close to expiry. Mathematically, it becomes a Dirac delta function at expiry, making exact delta hedging impossible in theory. Practitioners typically super-replicate with a tight call spread.
Gamma — can be large and sign-changing. A cash-or-nothing call has positive gamma when OTM and negative gamma when ITM (the opposite of a vanilla call).
Vega — negative for cash-or-nothing calls. Higher vol makes the ITM probability more uncertain (closer to 50-50), which reduces the value of an already-in-the-money bet.
Theta — depends on moneyness and type. ITM digitals gain value as time passes (less chance of reversal); OTM digitals lose value.
Worked Example — Cash-or-Nothing Call
S = $100, X = $100, k = $1, T = 90 days, σ = 20%, r = 4.5%, b = 4.5% (stock option).
- d₂ = [ln(1) + (0.045 − 0.02)×0.247] / (0.20×0.497) = 0.0062 / 0.0994 = 0.062
- N(d₂) = 0.525 (just above 50% — slightly ITM on risk-neutral probability)
- Price = $1 × e−0.045×0.247 × 0.525 = $0.519
If the option is $1 notional, you pay $0.519 today. If AAPL closes above $100 in 90 days, you receive $1. The risk-neutral probability implies a 52.5% chance of finishing above $100.
Applications
- Structured products — barrier notes, capital-protected products, range accruals all embed digital payoffs.
- FX derivatives — one-touch and no-touch options are variants of digitals for FX markets.
- Hedging — a tight vertical call spread (buy call at X, sell call at X + ε) replicates a cash-or-nothing call with payout ε/(option spread).
- Risk indicator — cash-or-nothing prices proxy for risk-neutral probabilities, useful for scenario analysis.
Related Tools
- Black-Scholes Options — vanilla European and American options pricing
- Barrier Options — knock-in and knock-out options with rebate
- FX Options — Garman-Kohlhagen for currency options
- Bond YTM — fixed income analytics