3.7.2 Barriers & Digital (Binary) Options

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and Black–Scholes–Merton & Feynman–Kac.

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1. Intuition & Practical Objective

A barrier option knocks in or out when the spot touches a level $H$; a digital/binary pays a lump sum (cash) or the asset (asset-or-nothing) conditional on a trigger. They are the two most liquid single-asset exotics, and their pricing is pure reflection principle: every closed form is “a vanilla option, plus reflected image terms at $x\mapsto H^2/S$” (Shreve II §7.3). The practical objective: master the barrier building blocks ($A\dots F$), the in–out parity identity that makes barrier pricing trivial once one side is known, and the digital one-liners - then see why discrete monitoring breaks the nice closed forms (the subject of 06 · Advanced Extensions).

Barrier notation (Haug §4.17): down = barrier below spot ($S>H$), up = barrier above spot ($S<H$); in/out = knocked in/out when touched; $\eta =+1$ down / $-1$ up. Eight combinations: down/up $\times$ in/out $\times$ call/put.

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2. Mathematical Ground Truth & Derivations

2.1 The Reiner–Rubinstein building blocks (Haug §4.17.2, eqs 4.51–4.52) - [V-8], the highest-confidence block

With $\varphi =+1$ call / $-1$ put, $\eta =+1$ down / $-1$ up, and the scaling exponents

$$\mu =\frac{b-\frac{1}{2}{\sigma }^2}{{\sigma }^2},\lambda =\sqrt{{\mu }^2+\frac{2r}{{\sigma }^2}},$$ $$x_1=\frac{\ln (S/X)}{\sigma \sqrt{T}}+(1+\mu )\sigma \sqrt{T},x_2=\frac{\ln (S/H)}{\sigma \sqrt{T}}+(1+\mu )\sigma \sqrt{T},$$ $$y_1=\frac{\ln (H^2/(SX))}{\sigma \sqrt{T}}+(1+\mu )\sigma \sqrt{T},y_2=\frac{\ln (H/S)}{\sigma \sqrt{T}}+(1+\mu )\sigma \sqrt{T},z=\frac{\ln (H/S)}{\sigma \sqrt{T}}+\lambda \sigma \sqrt{T},$$

the six blocks are

$$A=\varphi S{e}^{(b-r)T}N(\varphi x_1)-\varphi X{e}^{-rT}N(\varphi x_1-\varphi \sigma \sqrt{T})$$ $$B=\varphi S{e}^{(b-r)T}N(\varphi x_2)-\varphi X{e}^{-rT}N(\varphi x_2-\varphi \sigma \sqrt{T})$$ $$C=\varphi S{e}^{(b-r)T}(\frac{H}{S}{)}^{2(\mu +1)}N(\eta y_1)-\varphi X{e}^{-rT}(\frac{H}{S}{)}^{2\mu }N(\eta y_1-\eta \sigma \sqrt{T})$$ $$D=\varphi S{e}^{(b-r)T}(\frac{H}{S}{)}^{2(\mu +1)}N(\eta y_2)-\varphi X{e}^{-rT}(\frac{H}{S}{)}^{2\mu }N(\eta y_2-\eta \sigma \sqrt{T})$$ $$E=K{e}^{-rT}[N(\eta x_2-\eta \sigma \sqrt{T})-(\frac{H}{S}{)}^{2\mu }N(\eta y_2-\eta \sigma \sqrt{T})]$$ $$F=K\left[\right(\frac{H}{S}{)}^{\mu +\lambda }N(\eta z)+(\frac{H}{S}{)}^{\mu -\lambda }N(\eta z-2\eta \lambda \sigma \sqrt{T})]$$

Combination table (call; puts analogous):

Option	$S>H$	$S<H$
Down-and-in call	$C+E$	$A-B+D+E$
Up-and-in call	$A+E$	$B-C+D+E$
Down-and-out call	$A-C+F$	$B-D+F$
Up-and-out call	$F$	$A-B+C-D+F$

In–out parity (no rebate, $K=0$): $c_{di}+c_{do}=c_{\text{vanilla}}$ - a down-and-in plus down-and-out reproduces the plain call, because exactly one of the two pays. With a rebate $K\ne 0$ there is an extra discounted-rebate term. Verified here: $3.3368+4.5126=7.8494=c_{\text{vanilla}}$.

2.2 Digitals (Haug §4.19)

Cash-or-nothing (pay $K$ if triggered): $c=K{e}^{-rT}N(d)$, $p=K{e}^{-rT}N(-d)$, with $d=[\ln (S/X)+(b-\frac{1}{2}{\sigma }^2)T]/(\sigma \sqrt{T})$.

Asset-or-nothing (deliver $S_T$ if triggered): $c=S{e}^{(b-r)T}N(d)$, $p=S{e}^{(b-r)T}N(-d)$, with $d=[\ln (S/X)+(b+\frac{1}{2}{\sigma }^2)T]/(\sigma \sqrt{T})$.

The delta of a cash-or-nothing is the payoff density at the trigger - this is why pathwise MC Greeks fail for digitals (see 05 · Failure Modes).

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3. Computational Implementation - the barrier engine

Reproduces the Haug-verified values and the parity identity exactly. Stdlib only.

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4. Failure Modes & First-Principles Breakdowns

1. Continuous vs discrete monitoring. The closed forms assume continuous monitoring. Real contracts (and the MC that prices them) check at discrete dates. A continuously-monitored down-and-out is cheaper than a discretely-monitored one, because the continuous form “sees” barrier touches the discrete grid misses. The Broadie–Glasserman–Kou correction (06) restores agreement.
2. Rebate misuse. $E$ and $F$ carry the rebate $K$; in–out parity $c_{di}+c_{do}=c_{vanilla}$ holds only when $K=0$. Mixing rebated and non-rebated forms is a classic desk error.
3. In–out parity is not a free lunch. It holds for the same parameters; American barriers break it, and so does discrete monitoring. Don’t infer the in-side from the out-side without checking the specification matches.
4. Digital delta is a delta spike. The digital price is $K{e}^{-rT}N(d)$; its delta is proportional to the density $n(d)$, concentrated near the trigger. Any hedge or pathwise Greek that ignores the density (a step payoff) is wrong - see 05.

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5. References

• Haug, The Complete Guide to Option Pricing Formulas
• Shreve, Stochastic Calculus for Finance II
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 01 · From Zero
• Forward: 03 · Lookbacks & Asians · Index Hub
• Base: Black–Scholes–Merton & Feynman–Kac · Stochastic Calculus & Itô