3.7 Exotic & Path-Dependent Options

Basic Prerequisites: Stochastic Calculus & Itô’s Lemma and Black–Scholes–Merton & Feynman–Kac. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

A vanilla European option’s payoff depends only on the terminal stock price $S_{T}$ . An exotic / path-dependent option’s payoff depends on more of the history: the path’s maximum (lookbacks), its average (Asians), whether it touched a level (barriers), a binary event (digitals), or a second asset / currency (exchange, quanto, basket). This is the first big divide in real structuring desks: the moment the payoff references $S_{T}$ plus any other feature of the path or another asset, three things change at once - (1) the closed form (if one exists) gets far more complicated, (2) the natural pricing tool becomes Monte Carlo over the path, and (3) the model risk shifts from volatility to correlation and the exact specification of path-monitoring.

This folder is the lookup hub for the exotic & path-dependent family. It (a) gives the fast formula lookup below (job #1 of this pillar), and (b) routes you to six sub-pages: raw intuition, barriers & digitals, lookbacks & Asians, compound/chooser/quanto/exchange, failure modes, and advanced extensions (discrete monitoring & MC for path-dependence).

The one-sentence essence. “An exotic’s price is the discounted risk-neutral expectation of a path-functional payoff - and the moment the payoff is not a function of $S_{T}$ alone, you either find a specialized closed form (reflection principle, moment matching) or you simulate the whole path and average.”

2. Mathematical Ground Truth & Lookup

All formulas below are transcribed from Haug (2006) and cross-checked against Shreve Vol II Ch 6–7, Hull Ch 26, and Glasserman Ch 1–3/7–8. Numbers in the check column were re-executed and reproduced exactly in this build (see §3).

Notation: $S$ spot, $X$ strike, $T$ time to expiry, $r$ risk-free, $b$ cost-of-carry ( $b = r - q$ index, $b = 0$ futures, $b = r - r_{f}$ FX), $σ$ vol, $N (\cdot)$ normal CDF, $M (a, b; ρ)$ bivariate normal CDF, $H$ barrier, $K$ rebate/digital payout, $ϕ = + 1$ call/ $- 1$ put, $η = + 1$ down/ $- 1$ up.

Family	Key formula	Verified check
Barrier (Reiner–Rubinstein $A \dots F$ )	$A - C + F$ (down-and-out call), combination table §2.1 (barriers & digitals)	$c_{d o} (H = 95, σ = .25) = 6.7924$ ; $c_{u i} (H = 105) = 8.4482$
In–out parity	$c_{d i} + c_{d o} = c_{vanilla}$ (rebate $K = 0$ )	$3.3368 + 4.5126 = 7.8494$ ✓
Discrete-barrier correction (Broadie–Glasserman–Kou)	$H_{D} = H e^{\pm β σ Δ t}$ , $β = 0.5826$	$H_{D} = 94.027 \Rightarrow$ price $5.9831 \approx$ MC $6.0473$
Cash-or-nothing	$c = K e^{- r T} N (d)$ , $p = K e^{- r T} N (- d)$	put $= 2.6710$
Asset-or-nothing	$c = S e^{(b - r) T} N (d)$ , $p = S e^{(b - r) T} N (- d)$	put $= 20.2069$
Floating-strike lookback call (4.39)	$S e^{(b - r) T} N (a_{1}) - S_{min} e^{- r T} N (a_{2}) + \dots$	$= 25.3534$ (book 25.3533)
Geometric Asian (4.91)	BSM with $σ_{A} = σ / 3$ , $b_{A} = \frac{1}{2} (b - σ^{2} /6)$	put $= 4.6922$
Arithmetic Asian (Turnbull–Wakeman)	moment-match $M_{1}, M_{2} \to$ BSM form	no closed form; MC $= 1.9812$
Put-on-call compound (4.29)	bivariate-normal combo + critical $I$	$= 21.1964$ , $I = 538.3165$
Simple chooser (4.26)	$c (S, X, T_{2}) + e^{(b - r) (T_{2} - t_{1})} p (S, X e^{- b (T_{2} - t_{1})}, t_{1})$	$= 6.1071$
Complex chooser (4.27)	4-term bivariate-normal combo, critical $I$	$= 6.0508$ , $I = 51.1158$ (Haug 4.27)
Margrabe exchange (5.7)	$Q_{1} S_{1} e^{(b_{1} - r) T} N (d_{1}) - Q_{2} S_{2} e^{(b_{2} - r) T} N (d_{2})$	$= 1.5260$
Quanto (5.39)	$E_{p} [S^{} e^{(r_{f} - r - q - ρ σ_{S} σ_{E}) T} N (d_{1}) - X^{} e^{- r T} N (d_{2})]$	call $= 5.3280$
Foreign-equity struck domestic (5.35)	$σ_{ES} = σ_{S}^{2} + σ_{E}^{2} + 2 ρ σ_{E} σ_{S}$	call $= 8.3056$
Kirk spread (5.17)	two-leg BSM with $F = \frac{S _{2} e ^{(b_{2} - r) T}}{S _{2} e ^{(b_{2} - r) T} + X e ^{- r T}}$	call $= 2.1670$

Scaling caveats. (1) Most exotics collapse to a BSM-type call/put with adjusted drift/vol (Asians, quantos, exchange) - always reduce to that first. (2) The lookback and barrier closed forms require the reflection principle; a naive “positive-part of the reflected vanilla” misprices the sign of the image term (the verified lookback value needs the + sign in the square bracket - flip it and you get $21.35$ , not $25.35$ ). (3) Formulas flagged [R] in the source (partial-time lookbacks, double barriers, soft barriers, Curran) are reconstructed pointers, not verified - do not quote their digits.

3. Computational Implementation - the formula engine

Stdlib only (math.erf gives the exact normal CDF). The block below is the shared engine; each sub-page embeds a runnable slice that reproduces the numbers above.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - full analysis in 05 · Failure Modes & Practice. In one line each:

Discrete vs continuous monitoring - a barrier can be crossed between monitoring dates; pricing a discrete barrier with the continuous closed form systematically overvalues knock-outs (the Broadie–Glasserman–Kou shift fixes it).
The correlation trap - quantos, baskets and spread options live or die on $ρ$ ; a wrong correlation misprices the convexity terms ( $σ_{2}^{2} T$ , $ρ σ_{1} σ_{2}$ ) that the closed forms carry.
Monte Carlo bias for path-functionals - pathwise Greeks fail for digitals and barriers (the payoff is discontinuous), and MC convergence is $O (n^{- 1/2})$ in the number of paths but slow in the number of monitoring steps - the two are often conflated.

5. References

Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed., 2006)
Shreve, Steven E.: Stochastic Calculus for Finance II
Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
Glasserman, Paul: Monte Carlo Methods in Financial Engineering

6. Connected Graph Bridges

Foundational base: Stochastic Calculus & Itô · Black–Scholes–Merton & Feynman–Kac (the vanilla anchor all exotics reduce to)
Sibling topics: No-Arbitrage & Binomial Trees · The Greeks & Dynamic Hedging · Implied Volatility Surfaces · Heston & SABR
Sub-pages (in-folder): 01 From Zero · 02 Barriers & Digitals · 03 Lookbacks & Asians · 04 Compound/Chooser/Quanto/Exchange · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

3.7 Exotic & Path-Dependent Options

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Lookup

3. Computational Implementation - the formula engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

3.7.1 Exotic & Path-Dependent Options from Zero

3.7.2 Barriers & Digital (Binary) Options

3.7.3 Lookbacks & Asian (Average-Rate) Options

3.7.4 Compound, Chooser, Quanto, Exchange & Spread Options

3.7.5 Failure Modes & Real-World Practice for Exotics

3.7.6 Advanced Extensions