What Is Portfolio Churning?
“Churning” is the practice of excessive trading in a client’s account to generate commission income for the broker, without legitimate investment purpose. Proving churning in court requires establishing that (a) the broker controlled the account, (b) the trading was excessive, and (c) the broker acted with intent to generate commissions rather than serve the client’s interest.
Quantitative finance provides the tools to establish point (b) with mathematical rigor. By computing the profit probability of each trade with and without commissions, we can measure the exact impact of commission costs on the client’s expected outcomes.
Quantitative Framework: Black-Scholes Profit Probability
Under the geometric Brownian motion model with risk-free rate $r = 1.74\%$ (13-week T-Bill at origination), the probability that a call option expires in the money—net of commissions and sales charges—is given by:
| Stock | Price | Implied Vol | Pr (no comm.) | Pr (with comm.) |
|---|---|---|---|---|
| AA | $48.97 | 34.68% | 1.81% | 1.60% |
| BB | $29.72 | 41.86% | 8.00% | 6.57% |
| HH | $25.21 | 15.08% | 22.65% | 15.20% |
| II | $43.82 | 12.79% | 27.15% | 19.13% |
| JJ | $12.94 | 38.98% | 25.39% | 19.11% |
The average profit probability without commissions was 12.48%. With commissions: 9.52%. A reduction of 2.96 percentage points that, for an already low-probability trade, represents a 23.7% relative decline in expected profitability.
The Relative Commission Impact
The raw 2.96-percentage-point reduction understates the true impact. Consider stock EE: a call option with strike 30, 264 days to expiry, priced at $55 per contract. The commission was $100 per contract—nearly double the option premium.
The break-even stock price without commissions was $30.55. With commissions: $31.55. This means the stock needed to rise an additional 3.3% just to cover transaction costs—before the client could begin to profit.
Using Monte Carlo simulation with 0.3 assumed correlation between SPX constituent stocks, the joint probability that the overall portfolio generates a profit (across all positions simultaneously) falls to 8.35%. In other words, there was less than a 1-in-12 chance that the portfolio as constructed would be profitable—a fact that was knowable at the time of trade execution.
Expert Witness Implications
This analysis demonstrates several principles relevant to securities litigation and regulatory enforcement:
- Commission impact is quantifiable. Black-Scholes profit probability with and without costs provides an objective measure of commission burden.
- Joint probability matters. Individual trades may have non-trivial profit probabilities, but the portfolio-level probability (accounting for correlation) may be negligible.
- The relative impact exceeds the absolute impact. A 3-point reduction in profit probability from 12% to 9% is a 24% relative decline—far more dramatic than the raw numbers suggest.
- Implied volatilities at trade time establish the information available to the broker. These are market-observable quantities—the analysis does not require hindsight.
Derived from From Equations to Capital research program, by Mourad E. Mazouni, PhD, PMP. View Volume I →