Option Parameters

$
Current price of underlying asset
$
Binary payoff threshold
years
e.g., 0.25 = 3 months
%
Enter as percentage (e.g., 5 for 5%)
%
Enter as percentage (e.g., 2 for 2%)
%
Enter as percentage (e.g., 20 for 20%)
$
Amount paid if option expires ITM

Model Assumptions

  • European-style exercise (at expiration only)
  • Continuous dividend yield supported
  • Log-normal asset price distribution
  • Constant volatility and interest rates
  • No transaction costs or taxes
  • Frictionless markets

Binary Option Prices

Cash-or-Nothing Call --
Cash-or-Nothing Put --
Asset-or-Nothing Call --
Asset-or-Nothing Put --
Vanilla Option Reference
Vanilla Call --
Vanilla Put --
Intermediate Values
d1 --
d2 --
N(d1) --
N(d2) --
N(-d1) --
N(-d2) --
Invariant Checks
Cash Call + Cash Put: --
Asset Call + Asset Put: --

Payoff at Expiration

Cash-or-Nothing Call
Cash-or-Nothing Put
Asset-or-Nothing Call
Vanilla Call

Formulas (Hull Ch. 26.10)

d1 = [ln(S/K) + (r - q + σ²/2)T] / (σ√T)
d2 = d1 - σ√T
Cash-or-Nothing Call Q × e-rT × N(d2)
Cash-or-Nothing Put Q × e-rT × N(-d2)
Asset-or-Nothing Call S × e-qT × N(d1)
Asset-or-Nothing Put S × e-qT × N(-d1)

Understanding Binary Options

What Are Binary Options?

Binary options (also called digital options) have all-or-nothing payoffs. Unlike vanilla options that pay the difference between asset price and strike, binary options pay a fixed amount if they expire in the money, or nothing if they expire out of the money.

Cash-or-Nothing vs Asset-or-Nothing

Cash-or-Nothing

Fixed cash payout
Pays a fixed amount Q if the option expires in the money. Cash call pays Q if S > K; cash put pays Q if S < K.

Asset-or-Nothing

Asset value payout
Pays the underlying asset value S if in the money. Asset call pays S if S > K; asset put pays S if S < K.

Relationship to Vanilla Options

Binary options are building blocks for vanilla options. A vanilla call can be decomposed as:

Vanilla Option Decomposition
Vanilla Call = Asset-or-Nothing Call - K × (Cash-or-Nothing Call / Q)

This decomposition shows that buying a vanilla call is equivalent to receiving the asset when ITM, minus paying the strike when ITM.

Key Invariants

Pricing Identities:
  • Cash Call + Cash Put = Q × e-rT: The sum equals the discounted payoff, since exactly one will pay.
  • Asset Call + Asset Put = S × e-qT: The sum equals the forward-adjusted asset price.
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Frequently Asked Questions

Vanilla options have variable payoffs based on how far in the money they finish (S - K for calls). Binary options have fixed, all-or-nothing payoffs: they pay a predetermined amount if ITM, or zero if OTM. This makes binary options simpler but with discontinuous payoff profiles.

N(d2) represents the risk-neutral probability that the option expires in the money. Cash-or-nothing options pay a fixed amount contingent on this probability, so they use N(d2). Asset-or-nothing options, which pay the asset value, use N(d1) because it incorporates the asset's expected value conditional on finishing ITM.

Dividend yield (q) reduces the forward price of the asset, making calls less valuable and puts more valuable. For asset-or-nothing options, dividends also affect the e-qT discount factor applied to the asset delivery. Higher dividends lower asset-or-nothing call prices and raise asset-or-nothing put prices.

Binary options are used for hedging threshold risks (e.g., a portfolio that loses value if an index falls below a level), building structured products, and as theoretical building blocks for understanding vanilla option decomposition. They provide directional exposure with known maximum loss and gain.

Yes, approximately. A tight bull call spread (long call at K, short call at K+epsilon) replicates a cash-or-nothing call payoff as epsilon approaches zero. However, exact replication requires infinitely tight spreads, which isn't practical due to transaction costs and limited strike availability.

In the continuous Black-Scholes model, the probability of S exactly equaling K is zero, so this edge case doesn't affect pricing. In practice, exchange-traded binary options have explicit rules for at-the-money expiration. This calculator follows the theoretical convention where calls pay at S > K and puts pay at S < K.
Disclaimer

This calculator is for educational purposes only and assumes European-style options with continuous dividend yield and constant volatility. Actual binary options trading involves additional factors like market liquidity, discrete dividends, and exchange-specific rules. This tool should not be used for trading decisions.

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