Exotic options: barriers, Asians and digitals

So far every option in this track paid a function of the final price $S_T$ alone. This lesson looks at options whose payoff is either non-standard or depends on the whole price path: digital, barrier and Asian options. It uses the pricing rule from "Risk-neutral probabilities", the formula from "The Black–Scholes formula" and the simulation method from "Monte Carlo option pricing", and it shows where the static replication of "Static replication: Breeden–Litzenberger and Carr–Madan" stops working.

Exotic options

An option is called exotic when it is not a plain European or American call or put. There are two ways to depart from the standard contract. The payoff can be a different function of $S_T$, such as a fixed cash amount. Or the payoff can be path-dependent: it depends on prices at dates before $T$, for example on the maximum or the average of the path.

The difference matters for hedging. A European payoff $f(S_T)$ can be built statically from bonds, a forward and vanilla options, as the Carr–Madan result showed. A path-dependent payoff cannot in general: two paths that end at the same $S_T$ can pay different amounts, so no portfolio of instruments that pay a function of $S_T$ can match it. The pricing rule is unchanged, though. The price is still the discounted $\mathbb{Q}$-expectation of the payoff $X$:

Digital options

A digital (cash-or-nothing) call pays 1 at $T$ if $S_T > K$ and nothing otherwise. Its payoff is the indicator $\mathbf{1}\{S_T > K\}$, a step instead of a ramp. The expected payoff is the risk-neutral probability of finishing above the strike, which "The Black–Scholes formula" identified as $N(d_2)$. Discounting gives

With $S_0 = K = 100$, $r = 5\%$, $\sigma = 20\%$ and $T = 1$, $d_2 = 0.03/0.2 = 0.15$, $N(d_2) \approx 0.5596$ and $D_0 \approx 0.5323$. The digital is still a European payoff. It is the limit of a tight call spread from "Combining positions": long $1/h$ calls struck at $K$ and short $1/h$ calls struck at $K + h$, with $h \to 0$.

Its delta shows why digitals are hard to hedge. Differentiating $D_0$ with respect to $S_0$ gives $\Delta = e^{-rT}\varphi(d_2)/(S_0\sigma\sqrt{T})$, where $\varphi$ is the standard normal density. At the money the factor $1/\sqrt{T}$ makes it blow up as expiry approaches: about $0.019$ with one year left, $0.069$ with one month and $0.381$ with one day. Near expiry the hedge must swing between holding almost nothing and holding a large position as the price crosses $K$.

Barrier options

A barrier option is a vanilla option that is switched off or on when the price touches a level $B$. A knock-out option dies, paying nothing, if the barrier is touched before $T$; a knock-in option comes to life only if it is touched. The most common example is the up-and-out call with $B > K$. Writing $M_T$ for the maximum of the path up to $T$, its payoff is

The payoff is never larger than the vanilla payoff, so the up-and-out call is cheaper than the vanilla. It loses exactly the paths that matter most to a call holder: those that rise far. As $B$ falls toward $K$ its value goes to zero, because a path that finishes above $K \ge B$ must have touched $B$. As $B$ rises the barrier is touched less often and the price approaches the vanilla price.

Volatility acts in two directions. Higher $\sigma$ makes large final payoffs more likely, which raises the vanilla price, but it also makes touching $B$ more likely. With a barrier close to the spot the second effect wins, and the up-and-out call can get cheaper as $\sigma$ rises.

In–out parity

Hold an up-and-in call and an up-and-out call with the same $K$, $B$ and $T$. On every path exactly one of them is alive, and the live one pays $(S_T - K)^+$. Together they pay the vanilla payoff on every path, so by the law of one price

Pricing one of the two barrier options therefore prices the other.

Discrete and continuous monitoring

A contract must say how the barrier is checked. Textbook formulas assume continuous monitoring: the price is watched at every instant. Real contracts often use discrete monitoring, for example at each daily close. A path can cross $B$ between two observation dates and come back unnoticed, so a discretely monitored knock-out is knocked out less often and is worth more than its continuously monitored version. The figure below monitors 100 dates in the year.

Asian options

An Asian option pays on the average price over a set of monitoring dates $t_1, \dots, t_n$ instead of the final price. The arithmetic-average Asian call pays

An average moves less than the price itself: early prices in the average have had little time to drift away, and their errors partly cancel. For averaging over the whole period the volatility of the average is roughly $\sigma/\sqrt{3}$, about 11.5% when $\sigma = 20\%$. Lower volatility means a smaller time value, so the Asian call is cheaper than the vanilla. Averaging also makes the contract hard to manipulate by moving the price on a single day.

A sum of log-normal prices is not log-normal, so the arithmetic Asian call has no closed form. The geometric average is log-normal and has a Black–Scholes-type formula; since it moves closely with the arithmetic one, it is a natural control variate in the sense of "Monte Carlo option pricing".

Pricing by Monte Carlo

For path-dependent payoffs a single draw of $S_T$ is not enough: we need the whole path at the monitoring dates. Under $\mathbb{Q}$ the stock follows geometric Brownian motion with drift $r$, and the exact solution steps from one date to the next with $\Delta t = T/n$:

1. Simulate $N$ paths at the $n$ monitoring dates with independent standard normal $Z$'s.
2. On each path record what the payoff needs: $S_T$, the running maximum $M_T$, the average $A_T$.
3. Compute each path's payoff, average over the $N$ paths and multiply by $e^{-rT}$.

The same paths price all the contracts at once; only the payoff function changes. The standard error behaves as described in "Monte Carlo option pricing".

A worked example

Take $S_0 = 100$, $K = 100$, $r = 5\%$, $T = 1$, $\sigma = 20\%$, 100 monitoring dates, barrier $B = 130$, and simulate 20 000 paths (seed 1). The estimates are:

• Vanilla call: $10.37$; the Black–Scholes price is $10.45$, within about one standard error ($\approx 0.10$).
• Asian call: $5.77$, a little over half the vanilla price.
• Up-and-out call with $B = 130$: $3.69$, so the up-and-in call is worth $10.37 - 3.69 = 6.68$ by in–out parity.
• Digital call paying 1: $0.530$, against the closed form $e^{-rT}N(d_2) \approx 0.532$.

The barrier removes almost two thirds of the vanilla value, because a 30% rise within a year is not unusual at 20% volatility.

The figure uses these settings: $S_0 = 100$, $K = 100$, $r = 5\%$, $T = 1$, with 20 000 paths simulated under $\mathbb{Q}$ and monitored at 100 dates. The sliders set the volatility $\sigma$ (10% to 50%) and the barrier level $B$ (105 to 250), and "New sample" draws new paths. The chart plots the up-and-out call price against $B$ from 105 to 250, with the vanilla Black–Scholes price as a horizontal line and a vertical line at the chosen $B$. The readouts give the Monte Carlo vanilla call, the Asian call, the up-and-out call at $B$ and the digital call paying 1. Push $B$ down toward $K$ and watch the barrier price fall to zero. Then raise $\sigma$: the vanilla gets dearer, while at $B = 130$ the up-and-out call gets cheaper, from about $3.69$ at 20% to about $1.88$ at 30%.

Path-dependent options rarely have closed forms, but they are priced by the same rule as any other claim: simulate paths under $\mathbb{Q}$, evaluate the payoff on each path, average and discount.

Summary

• An exotic option has a non-standard payoff or one that depends on the path; path-dependent payoffs cannot in general be replicated statically with vanillas.
• A digital call pays 1 if $S_T > K$; its price is $e^{-rT}N(d_2)$ and its delta spikes near expiry at the money.
• An up-and-out call is cheaper than the vanilla, falls to zero as $B$ approaches $K$, and satisfies $C_0^{\text{in}} + C_0^{\text{out}} = C_0$; discrete monitoring knocks out less often.
• Averaging lowers volatility, so an arithmetic Asian call is cheaper than the vanilla and has no closed form.
• Monte Carlo prices all of them from the same simulated paths at the monitoring dates.