Path-dependent options

Asian, lookback and barrier options pay off on what the spot did along the way, not just where it ended. That is where simulation earns its keep: walk each path, record its average, its extremes or whether it touched the barrier, and average the payoffs. Where a closed form exists it is shown alongside as a check.

Contract

Spot price (S)

Strike (K)

Volatility (σ)

Annualised, 0.20 = 20%

Time to maturity (T)

Years

Risk-free rate (r)

Annualised, 0.05 = 5%

Dividend yield (q)

Annualised, continuous

Call

Put

Try this

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Check the simulator

Choose the geometric Asian and simulate.

What to look for: The simulated price lands within a couple of standard errors of the closed form. When the reference and the estimate agree, the machinery is right and you can trust it where no formula exists.

Free precision

Choose the arithmetic Asian, simulate with the geometric control variate on, then off, at the same paths and seed.

What to look for: The standard error grows by a factor of ten or more when the control is off. The geometric payoff explains almost all the variance of the arithmetic one.

Discrete monitoring bias

Choose a down-and-out barrier call, set the fixings to 10 and simulate, then set them to 500.

What to look for: With few fixings the simulated price sits well above the continuous closed form because the barrier is rarely caught. With many fixings it converges towards the formula.

Payoff

Option type

Pays on the arithmetic average of the spot at each fixing. No closed form; priced with the geometric Asian as a control variate.

Simulation

Paths

Fixings (time steps)

Seed

Antithetic variates

Geometric control variate

Arithmetic-average Asian call

$5.81

± 0.0010 standard error over 50,000 paths

Closed-form reference

none

Vanilla call

$10.45

55.8% of paths paid out

No closed form exists for an arithmetic average. The geometric Asian with the same fixings is used as a control variate.

Control variate: the geometric Asian payoff explains most of the arithmetic payoff's variance (β = 1.032), which is why the standard error is so small.

The payoff is a function of this statistic. For an Asian option it is much less spread out than the terminal spot, which is why Asians are cheaper than vanillas.

Watch the simulation

The first paths of the run draw one at a time, with the running average of the 100 fixings building up beside the spot. At expiry the option pays on the average, not on the final spot, and the chart shows both payoffs. Then the rest of the first 200 paths come in bursts, and beside them the running estimate and its 95% band settle on the price, against the plain European price. It is the run priced above, so it ends on the same estimate.

Playback speed

Path being followedPays at expiry Pays nothing Running averageFixing dates (100)Strike

Estimate 95% band European, Black-Scholes