Gap Option Pricing & Payoff Discontinuity Lab

Q: Can a gap option have a negative price, and why?

Yes. A gap call option can have a negative theoretical price if the payoff strike K2 is set significantly higher than the trigger hurdle K1 (K2 > K1). When triggered (S_T > K1), if S_T happens to land between K1 and K2, the payoff (S_T - K2) is negative, requiring the holder to make a cash payment to the writer. Because the expected payoff can be negative, the upfront premium can be negative.

Market Presets:

Gap Option Price

$0.00

Call Option

Trigger Probability $N(d_2)$

0.0%

Hurdle $K_1 = $100

Jump Discontinuity at $K_1$

$0.00

Step jump payoff

Vanilla Baseline ($K_1$)

$0.00

Gap Spread: $0.00

Contract Parameters

Option Type

Current Spot Price ($S_0$)

Trigger Hurdle ($K_1$)

Payoff Strike ($K_2$)

Maturity ($T$ yrs)

Volatility ($\sigma$ %)

Risk-Free Rate ($r$ %)

Dividend Yield ($q$ %)

Bonus call: When triggered at $100, payoff jumps by +$10 immediately.

Gap Option Greeks

Delta ($\Delta$)	0.0000
Gamma ($\Gamma$)	0.0000
Vega ($\mathcal{V}$)	0.0000
Theta ($\Theta$ / day)	0.0000
Rho ($\rho$)	0.0000

Pin Risk: Gamma contains a digital binary Dirac delta spike $\frac{e^{-rT}(K_1 - K_2) N'(d_2)}{\sigma^2 S_0^2 T}$.

Gap Option Value (Pre-expiry) Discontinuous Terminal Payoff Vanilla Call/Put ($K_1$)

Observe the rapid oscillation and extreme spike in Delta near trigger hurdle $K_1$.

Linear relationship between Gap Option Value and $K_2$. Notice crossing into negative territory when $K_2 \gg K_1$.

Analytical Component Decomposition

Component	Formula Expression	Economic Interpretation	Value ($)
Asset Leg	$S_0 e^{-q T} N(\pm d_1)$	Present value of underlying asset received if triggered	$0.00
Cash Leg	$K_2 e^{-r T} N(\pm d_2)$	Present value of cash paid if triggered	$0.00
Digital Cash Adjustment	$(K_1 - K_2) e^{-r T} N(\pm d_2)$	Value of embedded binary cash-or-nothing tranche	$0.00
Net Gap Option Price	$\text{Asset Leg} - \text{Cash Leg}$	Theoretical fair value upfront premium	$0.00

Spot ($S_0$) vs. Volatility ($\sigma$) Grid

Payoff Strike ($K_2$) vs. Expiry ($T$) Grid

Gap Option Theory & Mathematical Formulation

1. Analytical Closed-Form Formulas

For a European gap option with trigger hurdle $K_1$ and payoff strike $K_2$, the risk-neutral expected payoffs under geometric Brownian motion are:

$$C_{\text{gap}} = S_0 e^{-q T} N(d_1) - K_2 e^{-r T} N(d_2)$$ $$P_{\text{gap}} = K_2 e^{-r T} N(-d_2) - S_0 e^{-q T} N(-d_1)$$

Where the standardized normal parameters $d_1$ and $d_2$ are evaluated exclusively using the trigger hurdle $K_1$:

$$d_1 = \frac{\ln(S_0 / K_1) + (r - q + 0.5\sigma^2)T}{\sigma \sqrt{T}}, \quad d_2 = d_1 - \sigma \sqrt{T}$$

2. The Binary Decomposition & Negative Prices

A gap call option can be decomposed into a vanilla call with strike $K_1$ plus a cash-or-nothing binary call paying $(K_1 - K_2)$:

$$C_{\text{gap}}(K_1, K_2) = C_{\text{vanilla}}(K_1) + (K_1 - K_2) e^{-r T} N(d_2)$$

When $K_1 = K_2$, the binary term vanishes and the gap option collapses to a standard vanilla option.

If $K_2 > K_1$ for a call, the binary term is negative. If $K_2$ is sufficiently large, the negative cash commitment dominates the asset leg, driving the overall option premium below zero!

Executive Compensation

Corporate incentives frequently award options that only become exercisable if share price surpasses a demanding hurdle ($K_1 = \$120$), but once cleared, payoff starts from an in-the-money base ($K_2 = \$100$).

Debt Covenants & Restructuring

In structured finance, debt restructurings use gap puts where bondholders receive compensatory equity if asset value falls below a covenant floor ($K_1$), settled against a fixed par recovery claim ($K_2$).

Pin Risk & Delta Hedging

Near expiration, if $S_T \approx K_1$, the contract experiences a cliff discontinuity of $(K_1 - K_2)$. The option's Delta swings wildly between large positive and negative numbers, exposing writers to severe pin risk.

Gap Option Mastery Quiz

1. Under what condition can a European gap call option have a negative theoretical price?

Frequently Asked Questions

A gap option is an exotic European derivative with two distinct strike prices: a trigger hurdle (K1) that determines whether the option pays off, and a payoff strike (K2) that determines the amount paid. For a call, the payoff is (S_T - K2) if S_T > K1, and 0 otherwise. For a put, the payoff is (K2 - S_T) if S_T < K1, and 0 otherwise.

Yes. A gap call option can have a negative theoretical price if the payoff strike K2 is set significantly higher than the trigger hurdle K1 (K2 > K1). When triggered (S_T > K1), if S_T happens to land between K1 and K2, the payoff (S_T - K2) is negative, requiring the holder to make a cash payment to the writer. Because the expected payoff can be negative, the upfront premium can be negative.

The closed-form formula for a European gap call option is V_call = S_0 * exp(-q*T) * N(d1) - K2 * exp(-r*T) * N(d2), where d1 = [ln(S_0 / K1) + (r - q + 0.5*sigma^2)*T] / [sigma*sqrt(T)] and d2 = d1 - sigma*sqrt(T). Notice that the d1 and d2 terms are calculated using trigger hurdle K1, but the cash discount term multiplies the payoff strike K2.

At expiration, if the asset price S_T crosses K1, the payoff experiences an instant vertical jump discontinuity of magnitude (K1 - K2). As time-to-maturity T approaches 0, the Delta near K1 exhibits extreme spikes, and Gamma approaches positive or negative infinity (pin risk), making dynamic delta hedging practically impossible close to the barrier.