What is a binary option?

A binary option (also called a digital option) pays a fixed amount if it expires in the money, or nothing if it expires out of the money. Unlike vanilla options with variable payoffs, binary options have all-or-nothing payoffs at expiration.

What is the difference between cash-or-nothing and asset-or-nothing options?

A cash-or-nothing option pays a fixed cash amount Q if it expires in the money. An asset-or-nothing option pays the value of the underlying asset S if it expires in the money. Both pay zero if out of the money.

How are binary options priced?

Binary options are priced using Black-Scholes-style formulas. Cash-or-nothing options are priced as Q times the discounted probability of finishing in the money. Asset-or-nothing options use the risk-neutral expected value of receiving the asset when in the money.

Does this calculator support dividend-paying stocks?

Yes. This calculator supports continuous dividend yield (q), which adjusts the forward price and affects both d1 calculations and the asset-or-nothing pricing through the e^(-qT) discount factor.

What are binary options used for?

Binary options are used for hedging specific price thresholds, structured product construction, and as building blocks for replicating more complex payoffs. They provide directional exposure with capped risk and reward.

Binary Option Calculator: Cash-or-Nothing & Asset-or-Nothing Pricing

Option Parameters

Asset Price (S)

Current price of underlying asset

Strike Price (K)

Binary payoff threshold

Time to Expiration (T)

years

e.g., 0.25 = 3 months

Risk-Free Rate (r)

Enter as percentage (e.g., 5 for 5%)

Dividend Yield (q)

Enter as percentage (e.g., 2 for 2%)

Volatility (σ)

Enter as percentage (e.g., 20 for 20%)

Cash Payoff (Q)

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.

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.

Binary Option Calculator

Option Parameters

Model Assumptions

Binary Option Prices

Vanilla Option Reference

Intermediate Values

Invariant Checks

Payoff at Expiration

Formulas (Hull Ch. 26.10)

Understanding Binary Options

What Are Binary Options?

Cash-or-Nothing vs Asset-or-Nothing

Cash-or-Nothing

Asset-or-Nothing

Relationship to Vanilla Options

Key Invariants

Frequently Asked Questions

Disclaimer

Related Calculators

Binary Option Calculator

Option Parameters

Model Assumptions

Binary Option Prices

Vanilla Option Reference

Intermediate Values

Invariant Checks

Payoff at Expiration

Formulas (Hull Ch. 26.10)

Understanding Binary Options

What Are Binary Options?

Cash-or-Nothing vs Asset-or-Nothing

Cash-or-Nothing

Asset-or-Nothing

Relationship to Vanilla Options

Key Invariants

Frequently Asked Questions

What is the difference between binary options and vanilla options?

Why do cash-or-nothing options use N(d2) instead of N(d1)?

How does dividend yield affect binary option prices?

What are binary options used for in practice?

Can I replicate binary options with vanilla options?

What happens at exactly S = K at expiration?

Disclaimer

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