3.3.5 Failure Modes & Real-World Practice

Basic Prerequisites: 04 · Greeks & Hedging.

---

1. Intuition & Practical Objective

The BSM model is mathematically beautiful and empirically wrong in three specific ways. This page names them precisely, so a practitioner knows which assumptions to distrust and how the failures show up in money terms. The objective is not cynicism - it is the discipline of knowing exactly where a model is an approximation so the residual risk can be measured and managed.

The three failures, in one line each:

1. Volatility is not constant (it has a smile/surface, and it changes) - so there is no single $\sigma$ to plug in.
2. Hedging is not continuous (costs, discrete rebalancing) - so the “riskless portfolio” isn’t riskless.
3. The underlying does not follow lognormal GBM (jumps, fat tails, stochastic vol) - so $N(d_2)$ is the wrong probability.

---

2. Mathematical Ground Truth & Derivations

Where the assumptions live.

Every BSM result rests on four interlocking assumptions (Hull Ch 15; Shreve II Ch 5):

• (A1) Constant volatility $\sigma$ - the only unknown pricing parameter; enters $d_1,d_2$ and hence every Greek.
• (A2) Continuous, frictionless, unlimited trading - needed for the delta-neutral portfolio to be truly riskless ($dt\to 0$).
• (A3) Lognormal / continuous paths (no jumps) - the stock follows GBM; the market is complete because the Brownian motion fully drives it.
• (A4) Constant risk-free rate $r$, no transaction costs, no dividends during the option’s life (for the basic form).

The gamma–theta identity is the mathematical statement of how (A2) fails:

$$\frac{1}{2}\Gamma S^2{\sigma }^2=-{\Theta }_{\text{driftless}}.$$

Over a rebalance interval the hedged short-call P&L residual is approximately $\frac{1}{2}\Gamma S^2\left[{\left(\frac{\Delta S}{S}\right)}^2-{\sigma }^2\Delta t\right]$ - zero in expectation under $\mathbb{Q}$, but realized with variance proportional to ${\Gamma }^2{S}^4$. This is the irreducible hedging error: the more the market moves, the more gamma bleed.

---

3. Computational Implementation - the failures in numbers

Experiment 1 - discrete hedging P&L. Sell a call, delta-hedge it with $N$ rebalances. Frictionless: mean P&L $\approx 0$ but nonzero dispersion (the gamma residual). With transaction costs: mean P&L goes negative (the cost bleed). Stdlib only.

Frictionless hedging is on average break-even (mean $\approx 0$, the model’s promise) but the per-path dispersion (stdev $\approx 0.94$) is pure gamma/hedging risk. Add transaction costs and the mean bleeds negative - the “riskless portfolio” has become a guaranteed loss. Hedging friction is a first-principles violation of assumption (A2).

Experiment 2 - the constant-vol delusion. The model forces one $\sigma$ for all strikes; real markets show a smile. At a single vol the BSM prices across strikes are:

If the market prices OTM puts richer than this (post-1987 skew), then the implied $\sigma$ differs by strike - direct evidence the constant-vol assumption (A1) is false (see Implied Volatility Surfaces).

---

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

1. The constant-vol delusion (A1 fails). Real implied vols trace a skew/smile, not a flat line. Since $\sigma$ is the only free parameter, forcing one value cannot match all strikes - the model under/overprices systematically by strike and maturity.
2. Continuous-hedging friction (A2 fails). Costs, discrete rebalancing, and gaps turn the “riskless” portfolio into a loss maker, as Experiment 1 shows. There is a fundamental trade: hedge often to control gamma, hedge rarely to control costs.
3. Jumps & fat tails (A3 fails). Real returns jump; $S_T$ is not lognormal. Jumps inject randomness the hedge cannot remove (incomplete markets), so $\mu$-independence breaks and the price is no longer unique. The delta–gamma hedge cannot protect against a discontinuous move.
4. Stochastic/rough volatility. Volatility itself moves randomly (clustering, mean reversion); a deterministic $\sigma$ path is wrong even on average. This is the gateway to Heston & SABR.
5. Rate & dividend simplification (A4). Constant $r$ and no dividends understate term-structure and dividend risk, especially for long-dated and index options.

---

5. References

• Hull, Options, Futures, and Other Derivatives
• Shreve, Stochastic Calculus for Finance II
• Haug, The Complete Guide to Option Pricing Formulas
• Björk, Arbitrage Theory in Continuous Time

---

6. Connected Graph Bridges

• Back: 04 · Greeks & Hedging · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Implied Volatility Surfaces · Heston & SABR