Market Presets:
Digital Option Price
$0.00
Cash-or-Nothing Call
Risk-Neutral Prob $N(d_2)$
0.0%
Probability of Exercise
Binary Delta ($\Delta$)
0.000
Normal Hedging
Call-Spread Replication
$0.00
Spread Width: $\Delta K = $2.00Contract Parameters
Binary Parity & Decomposition
Cash-or-Nothing Call:
$0.00
Cash-or-Nothing Put:
$0.00
Cash Parity Sum (Call + Put):
$0.00
Discounted Cash $Q e^{-rT}$:
$0.00
Asset-or-Nothing Call:
$0.00
Vanilla Call Synthesis ($A_{\text{call}} - K C_{\text{call}}$):
$0.00
Parity Verified: Residual = $0.00
Binary Visualizer
Digital Option theoretical curve vs terminal discontinuous step payoff across spot price $S_0 \in [0.7K, 1.3K]$.
Delta spike and Dirac delta Gamma profile around strike $K$. Notice explosion as maturity $T \to 0$.
Comparison between the pure binary step function and the tight vanilla call spread $[K - \Delta K, K + \Delta K]$.
Binary Greeks & Pin Risk Suite
| Greek | Value | Formula / Scale | Hedging & Pin Risk Interpretation |
|---|---|---|---|
| Delta ($\Delta$) | 0.000 | $\partial V / \partial S$ | Shares per contract required to delta hedge |
| Gamma ($\Gamma$) | 0.0000 | $\partial^2 V / \partial S^2$ | Delta curvature; spikes to $\pm\infty$ at strike $K$ near expiry |
| Vega ($\nu$) | $0.00 | per 1% $\sigma$ | Sensitivity to implied volatility; sign flips around strike |
| Theta ($\Theta$) | $0.00 | per calendar day | Daily time decay; positive OTM, negative ITM |
| Rho ($\rho$) | $0.00 | per 1% $r$ | Interest rate sensitivity discounting future cash payment |
| Probability Density $n(d_2)$ | 0.0000 | $e^{-d_2^2/2} / \sqrt{2\pi}$ | Terminal lognormal density height evaluated at the strike |
5x5 Matrix: Spot $S_0$ vs Volatility $\sigma$
5x5 Matrix: Spot $S_0$ vs Maturity $T$
Digital Option Mastery Quiz
1. What happens to the Delta of an At-The-Money (ATM) cash-or-nothing digital call as time to expiration approaches zero ($T \to 0$)?
Frequently Asked Questions
A digital option (also known as a binary option) is a financial derivative characterized by an all-or-nothing discontinuous payoff structure. Rather than paying a linear intrinsic difference (S - K), a cash-or-nothing contract pays a fixed cash amount Q if the underlying finishes in-the-money, and zero otherwise. An asset-or-nothing contract pays the asset price S_T if in-the-money.
Under the Black-Scholes framework, the price of a cash-or-nothing European digital call is V = Q * exp(-r*T) * N(d2), and for a put is V = Q * exp(-r*T) * N(-d2), where d2 = [ln(S/K) + (r - q - 0.5*sigma^2)*T] / [sigma*sqrt(T)]. N(d2) corresponds to the exact risk-neutral probability of finishing in-the-money.
Because the payoff curve has a discontinuous jump step at strike K, the slope (Delta) near K approaches positive infinity as time-to-expiration T approaches zero, while Gamma behaves as a Dirac delta spike. Market makers attempting dynamic delta hedging face violent rebalancing requirements (pin risk) as the asset price flickers above and below K.
In practice, dealers hedge binary options statically using a tight vanilla bull spread: buying a call at K - epsilon and selling a call at K + epsilon, scaled by Q / (2*epsilon). As epsilon shrinks, the linear payoff ramp between the strikes approximates the discontinuous vertical step function, providing a practical hedging alternative to impossible dynamic delta hedging.