3.7.5 Failure Modes & Real-World Practice for Exotics

Basic Prerequisites: Exchange.

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1. Intuition & Practical Objective

The closed forms in this folder are beautiful, and three of their assumptions break in real markets. This page names the failures precisely and shows each in money terms, so a practitioner knows which knob to distrust:

1. Monitoring is discrete, not continuous. The barrier closed forms assume the barrier is watched continuously. Real contracts (and MC) check at discrete dates, so the discrete price is systematically above the continuous one for knock-outs (the grid “misses” touches).
2. Pathwise MC Greeks fail for the step payoffs. Digit/digital and barrier deltas come from a density, not a step - pathwise differentiation gives exactly zero (Glasserman Ch 7). Only the likelihood-ratio (score) method recovers the true Greek.
3. Correlation is a wobbly input. Quanto/exchange/spread prices depend on $\rho$; it is the least-stable market parameter, and a small error propagates through the closed forms (Implied Volatility Surfaces).

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2. Mathematical Ground Truth & Derivations

Discrete vs continuous monitoring. The continuously-monitored down-and-out value $v_c$ uses the first-hitting-time density of the Brownian bridge. A discretely-monitored contract (grid $\Delta t$) only sees the path at grid points, so the knock-out probability is smaller and the value $v_d>v_c$. Broadie–Glasserman–Kou (1995) restore agreement to $O(\Delta t)$ by pricing the continuous formula at the shifted barrier

$$H_D=H{e}^{\pm \beta \sigma \sqrt{\Delta t}},\beta =\frac{\zeta (1/2)}{\sqrt{2\pi }}\approx 0.5826,$$

+ when the barrier is above spot, − when below (Haug §5.6). This is the single most-used “dirty fix” in the exotic-options playbook.

Pathwise vs likelihood-ratio Greeks (Glasserman Ch 7). For a digital $Y=e^{-rT}K1\{S_T>X\}$, the pathwise derivative $dY/d{S}_0$ exists a.s. but equals zero - the indicator is flat almost everywhere, and the genuine delta comes from the strike-crossing that pathwise differentiation misses. The likelihood-ratio method differentiates the density instead: for lognormal $S_T$, the score is $Z/(S_0\sigma \sqrt{T})$, and the LR delta estimator is

$${\hat{\Delta }}_{LR}=e^{-rT}K1\{S_T>X\}\cdot \frac{Z}{S_0\sigma \sqrt{T}},\text{with }\mathbb{E}[{\hat{\Delta }}_{LR}]=K{e}^{-rT}\frac{\phi (d_2)}{S_0\sigma \sqrt{T}}.$$

The rule of thumb (Glasserman §7.2.2): pathwise applies when the payoff is continuous (Lipschitz) in the parameter - which excludes digitals, barriers, and 2nd derivatives.

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3. Computational Implementation - the failures in numbers

Experiment 1 - discrete monitoring overprices a knock-out, BGK fixes it. Stdlib only.

Discrete MC overprices the knock-out by up to ~24% at coarse grids; the BGK-shifted continuous formula lands within MC error of the discrete MC.

Experiment 2 - pathwise delta of a digital fails; likelihood-ratio recovers it.

The pathwise derivative of the digital is identically zero - useless. The likelihood-ratio (score) estimator matches the analytic delta to 4 dp, because it differentiates the density, not the step.

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4. Failure Modes & First-Principles Breakdowns (numbered)

1. Continuous-form assumption (monitoring). Using the continuous closed form for a discretely-monitored barrier misprices by the whole missed-hit probability - Experiment 1 quantifies it, BGK fixes it.
2. Pathwise Greeks on step payoffs. Digit/binary and barrier deltas cannot come from pathwise differentiation (identically zero); use likelihood-ratio or finite differences with common random numbers (Glasserman Ch 7; the central+CRN estimator dominates with RMSE $O(n^{-2/5})$).
3. Correlation instability. Margrabe/quanto/spread depend on $\rho$; it drifts and smiles, and it is less observable than vol. Always re-run the exotic price across a $\rho$-band (sensitivity, not point estimate).
4. Rebate & parity traps. In–out parity holds only at $K=0$; mixing rebated and non-rebated forms (or using American barrier parity, which fails) misprices.
5. MC dimension confusion. MC error is $O(n^{-1/2})$ in paths, but the discrete-monitoring bias is $O(\Delta t)$ in steps. A “converged” MC on too-coarse a grid is confidently wrong - always push $n_{steps}$ and check it changes the answer.

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5. References

• Glasserman, Monte Carlo Methods in Financial Engineering
• Haug, The Complete Guide to Option Pricing Formulas
• Broadie, Glasserman & Kou (1995), “A Continuity Correction for Discrete Barrier Options,” Math. Finance
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: Exchange · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Implied Volatility Surfaces · Heston & SABR