Digital and Binary Options Explained

Digital options (also called binary options) provide a fixed payoff if the underlying asset finishes above or below a strike price at expiration. Unlike vanilla options with linear payoffs, digital options have an all-or-nothing payoff structure that creates unique hedging challenges and is commonly embedded in structured products.

Definition and Key Concepts

Digital Option Types

Cash-or-Nothing Structure

Payoff formula:

• Call: Pays $X if S > K at expiry, else $0
• Put: Pays $X if S < K at expiry, else $0

Example:

• Strike: $100
• Cash payout: $10
• At expiry:
  • Stock at $101 → Call pays $10
  • Stock at $99 → Call pays $0

Asset-or-Nothing Structure

Payoff formula:

• Call: Pays S if S > K at expiry, else $0
• Put: Pays S if S < K at expiry, else $0

Example:

• Strike: $100
• At expiry:
  • Stock at $110 → Asset-or-nothing call pays $110
  • Stock at $99 → Asset-or-nothing call pays $0

Relationship to Vanilla Options

Decomposition: Vanilla call = Asset-or-nothing call - K × Cash-or-nothing call

This relationship enables replication and arbitrage-free pricing.

How It Works in Practice

Pricing Digital Options

Black-Scholes formula for cash-or-nothing call: Price = e^(-rT) × N(d2) × Cash Payout

Where:

• d2 = [ln(S/K) + (r - σ²/2)T] / (σ√T)
• N(d2) = probability of finishing in-the-money

Example pricing:

• Spot: $100
• Strike: $105
• Time: 3 months
• Volatility: 20%
• Risk-free rate: 5%
• Cash payout: $10

Calculation: d2 = [ln(100/105) + (0.05 - 0.04/2) × 0.25] / (0.20 × 0.5) d2 = [-0.0488 + 0.0075] / 0.10 = -0.413 N(d2) = 0.340

Price = e^(-0.05 × 0.25) × 0.340 × $10 = $3.36

Delta of Digital Options

Digital delta is discontinuous at strike:

Hedging challenge: Delta jumps from 0 to very large near expiry.

Vega of Digital Options

Digital vega can be positive or negative:

Reason: Higher volatility increases chance of crossing strike in both directions.

Worked Example

Hedging a Digital Call

Position:

• Short cash-or-nothing call
• Strike: $100
• Cash payout: $1,000,000
• Expiry: 1 week
• Current spot: $99.50
• Implied vol: 15%

Initial hedge: N(d2) ≈ 0.42 → 42% probability of paying out

Hedge approach 1: Call spread approximation Buy vanilla call spread to approximate digital:

• Buy 1,000 × $99 calls
• Sell 1,000 × $101 calls

Spread payoff:

• At $99 or below: $0
• At $100: $1,000 × 100 × ($100-$99) = $100,000
• At $101 or above: $1,000 × 100 × $2 = $200,000

Issue: Spread doesn't perfectly match $1M binary payout.

Better hedge: Tighter spread with more contracts

• Buy 5,000 × $99.50 calls
• Sell 5,000 × $100.50 calls
• Payoff at $100: 5,000 × 100 × $0.50 = $250,000

Still imperfect—digital hedging is inherently challenging.

Scenario Analysis

Digital call P/L scenarios:

The critical zone is within 1% of strike near expiration.

Embedded in Structured Products

Example: Range accrual note

• Pays 6% coupon if S&P stays between 4,500-5,500 each day
• Each day is effectively a digital option

Decomposition: Daily coupon = 6%/252 × [Digital(S>4,500) × Digital(S<5,500)]

Valuation: Sum of 252 double digital options, each paying 2.4 bps.

Risks, Limitations, and Tradeoffs

Pin Risk

Issue: As expiration approaches, stocks near the strike create extreme risk.

Example:

• Strike: $100
• Payout: $1M
• Spot: $99.95 with 1 hour to expiry
• Delta: ~500,000 shares (massive hedge requirement)

Market makers:

• May manipulate spot to favorable side
• Experience large P/L swings in final hours
• Often cap digital notionals for risk management

Discontinuous Payoff

Hedging challenge:

Model Risk

Digital option pricing is model-sensitive:

Common Pitfalls

Practical Applications

Use Cases

Regulatory Considerations

Retail binary options:

• Banned in EU (ESMA) and UK (FCA) for retail
• Restricted in US (only on regulated exchanges)
• High fraud risk in unregulated offerings

Institutional digitals:

• Traded OTC under ISDA
• Embedded in structured products
• Require sophisticated risk management

Checklist and Next Steps

Pre-trade checklist:

• Determine strike and payout amount
• Calculate theoretical price
• Assess pin risk near expiration
• Plan hedging approach
• Size position appropriately
• Document in ISDA format

Risk management checklist:

• Monitor distance to strike
• Track delta as expiration approaches
• Set maximum notional limits
• Plan for expiration day procedures
• Consider early termination rights

Hedging checklist:

• Set up call/put spread approximation
• Calculate spread width vs. accuracy tradeoff
• Monitor hedge error
• Adjust hedge as spot moves
• Plan expiration day exit

Related articles:

• For barrier options, see Barrier Options: Knock-In and Knock-Out Structures
• For Asian options, see Asian and Lookback Option Structures