Control Variates In Option Pricing

Price an option by Monte Carlo and every simulated path adds noise around the true answer — run 10,000 paths and the standard error only shrinks as $1/\sqrt{N}$, so halving the error means quadrupling the paths. A control variate shrinks that error for free, using information you already have: if a second, closely related payoff has a known closed-form price, the gap between its simulated average and its true value shows exactly how much noise this batch of random paths generated — and that noise can be subtracted straight out of the option you actually care about.

Racing with a friend whose time you already know

Imagine timing your own 10K with an unreliable stopwatch, while a friend with an identical pace ran the same route yesterday and you know their official, exactly-measured time. If your stopwatch clocks your friend's replayed pace at 2% slower than their known time, that 2% is almost certainly your stopwatch's error — so you correct your own measured time by the same 2%. A control variate works the same way: simulate a payoff whose true price you already know, see how far the simulation average missed it by, and use that miss to correct the price of the option you don't.

$$\hat{C}_{\text{cv}} = \hat{C} - \beta\left(\hat{Y} - \mathbb{E}[Y]\right)$$

In plain English: $\hat{C}$ is the raw Monte Carlo average for the option you're pricing, $Y$ is the payoff of a related instrument whose true expected value $\mathbb{E}[Y]$ is known in closed form, $\hat{Y}$ is that same instrument's simulated average from the identical random paths, and $\beta$ is a scaling factor (chosen to minimise variance; $\beta=1$ is a common simple choice when $C$ and $Y$ move closely together). Whatever error $\hat{Y}$ shows relative to its known truth gets subtracted, scaled by $\beta$, from the option price you actually wanted.

Worked example 1: an Asian call controlled by a European call. Price an Asian call (payoff on the average stock price, no closed form) from 5 simulated paths for a stock starting at $100, using an ordinary European call on the same paths (payoff on the final price, which has a closed form) as the control. Say the 5 simulated Asian payoffs are $8, 11, 6, 14, 9$ (average $\hat C = 9.60$), the European payoffs on those same paths are $10, 15, 5, 18, 12$ (average $\hat Y = 12.00$), and the true Black-Scholes price of that European call is $\mathbb{E}[Y]=11.20$. With $\beta=1$: $\hat{C}_{\text{cv}} = 9.60 - (12.00 - 11.20) = 9.60 - 0.80 = 8.80$, i.e. $8.80. The European control overshot its known truth by $0.80 on this batch — almost certainly these 5 paths trending a bit high overall — so that same $0.80 is subtracted from the Asian estimate too, since both payoffs share the same paths and the same random error.

The Asian and European payoffs above come from the exact same simulated paths — that shared randomness is why an error in one is informative about the error in the other.

Worked example 2: how much variance this removes. The corrected estimator's variance is roughly $\text{Var}(\hat C) - \beta^2\text{Var}(\hat Y)\,\rho^2$, where $\rho$ is the correlation between the two payoffs across paths. If $\rho=0.9$ (plausible, since both depend on the same path), the variance reduction factor is $1-\rho^2=1-0.81=0.19$ — the corrected estimator's variance is only 19% of the uncorrected one's, equivalent to roughly $1/0.19 \approx 5.3\times$ as many raw simulations, for free.

What this means in practice

Control variates are the workhorse variance-reduction technique whenever a complex, closed-form-free payoff (Asian, basket, barrier options under stochastic volatility) is close in behaviour to something simpler that does have a closed form — a plain vanilla call is the most common choice, and combining a control variate with antithetic sampling (see Antithetic Variates and Control Variates) is standard practice for production pricers.