What are digital (binary) options, and how do their payoffs differ from standard options?

Question

AcadiFi · Accepted Answer

Digital (binary) options are one of the simplest exotic option types to understand but have important applications in structured products and credit derivatives.

## Types of Digital Options

**Cash-or-Nothing Call:** Pays a fixed cash amount Q if S_T > K, otherwise pays $0.
- Payoff = Q if S_T > K; 0 otherwise

**Cash-or-Nothing Put:** Pays Q if S_T < K, otherwise $0.

**Asset-or-Nothing Call:** Pays S_T (the stock price itself) if S_T > K, otherwise $0.
- Payoff = S_T if S_T > K; 0 otherwise

## Payoff Comparison — Whitfield Tech ($75 strike)

```mermaid
flowchart LR
    subgraph Vanilla["Vanilla Call"]
        V1["S_T = $60"] --> V2["Payoff: $0"]
        V3["S_T = $80"] --> V4["Payoff: $5"]
        V5["S_T = $100"] --> V6["Payoff: $25"]
    end
    subgraph Digital["Cash-or-Nothing Call (Q=$10)"]
        D1["S_T = $60"] --> D2["Payoff: $0"]
        D3["S_T = $80"] --> D4["Payoff: $10"]
        D5["S_T = $100"] --> D6["Payoff: $10"]
    end
```

The vanilla call payoff increases with S_T, while the digital payoff is flat at Q once the barrier is crossed.

## Pricing Under BSM

Under risk-neutral pricing, the probability that S_T > K is N(d2). Therefore:

**Cash-or-Nothing Call = Q x e^(-rT) x N(d2)**

This is simply the present value of Q multiplied by the risk-neutral probability of finishing in the money.

## Worked Example

Price a 6-month cash-or-nothing call on Whitfield Tech:
- S = $72, K = $75, Q = $10 (fixed payout)
- r = 4%, σ = 30%, T = 0.5

d1 = [ln(72/75) + (0.04 + 0.045)0.5] / (0.30 x 0.707) = [-0.0408 + 0.0425] / 0.2121 = 0.0080

d2 = 0.0080 - 0.2121 = -0.2041

N(d2) = N(-0.2041) = 0.4191

Digital call = 10 x e^(-0.02) x 0.4191 = 10 x 0.9802 x 0.4191 = **$4.11**

Note this is cheaper than a vanilla call at the same strike because the digital has a capped payoff.

## Decomposition Trick

A standard vanilla call can be decomposed as:

**Vanilla Call = Asset-or-Nothing Call - K x Cash-or-Nothing Call**

This decomposition is sometimes tested on CFA Level II.

## Real-World Applications

- **Credit default swaps** are essentially digital options on credit events
- **Range accrual notes** embed digital options that pay coupons when a rate stays in a range
- **Structured deposit products** often use digital payoffs

**Exam tip:** Focus on the payoff structure and the pricing formula. Know that N(d2) is the risk-neutral probability of exercise.

For more exotic options content, explore our CFA Level II course.

What are digital (binary) options, and how do their payoffs differ from standard options?

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