Beyond the vanilla payoff

A vanilla option pays a function of the spot at expiry. Exotics change the payoff or make it depend on the path. The first kind, digitals, still have a closed form. The second kind sometimes does, and when it does not, simulation is the tool of choice because a simulated path carries all the information a payoff could ask for.

Digitals

A cash-or-nothing call pays a fixed amount if the spot ends above the strike. Its price is the discounted risk-neutral probability of that event, \(e^{-rT} N(d_2)\), the same \(N(d_2)\) that appears in the vanilla formula. An asset-or-nothing call pays the spot itself and is worth \(S e^{-qT} N(d_1)\). A vanilla call is an asset-or-nothing call minus \(K\) cash-or-nothing calls, which is one way to read the Black-Scholes formula.

Barriers

A knock-out option dies if the spot touches a barrier; a knock-in only comes alive if it does. Since every path either touches the barrier or does not, knock-in plus knock-out equals the vanilla for the same barrier: the in-out parity. With continuous monitoring there are closed forms based on the reflection principle. Real contracts watch the barrier at discrete times, daily fixings say, which makes a touch less likely: knock-outs are then worth a little more and knock-ins a little less than the continuous formula says. Broadie, Glasserman and Kou showed that shifting the barrier by \(0.5826\,\sigma\sqrt{\Delta t}\) corrects most of the gap.

Asian options

An Asian option pays on the average spot over a set of fixings. Averaging damps the noise, so the average is less volatile than the terminal spot and Asians are cheaper than vanillas. If the average is geometric, it is lognormal and there is an exact formula; for \(n\) equally spaced fixings the variance of its logarithm is \(\sigma^2 \Delta t\,(n+1)(2n+1)/(6n)\), roughly a third of the vanilla's. The arithmetic average, which is what contracts actually use, has no closed form.

Lookbacks

A floating-strike lookback call pays the terminal spot minus the lowest spot seen, so it is never out of the money: you bought at the low. A fixed-strike lookback call pays the highest spot seen minus the strike. Both are expensive, since they hand the holder perfect hindsight. Continuous-monitoring formulas exist; with discrete fixings the extremes are less extreme and the option is worth less.

Simulation and control variates

The simulator walks each path fixing by fixing, records the average, the extremes and whether the barrier was touched, and averages the discounted payoffs. Its error falls like one over the square root of the number of paths, which is slow. For the arithmetic Asian there is a famous shortcut: the geometric Asian payoff on the same path is highly correlated with the arithmetic one and has a known expectation, so subtracting \(\beta\,(X - E[X])\) from every payoff removes most of the noise. This is the Kemna-Vorst control variate, and it typically cuts the standard error by a factor of ten or more for free.

To see it happen, watch the simulation on the path-dependent page: the paths draw one at a time with what the payoff watches (a barrier checked only on its fixing dates, the running average of an Asian, the running maximum or minimum of a lookback), then the running price settles against the plain European price.