A taxonomy of contracts whose payoff depends on the whole path — barriers, Asians, lookbacks — and the dimensionality penalty that comes with them.
A vanilla European call has a payoff that depends only on $S_T$. Everything else is, in Wilmott's language, exotic. Chapter 11 organises the zoo along six axes — a useful way to read any new payoff specification.
How does the contract depend on time? One exercise date (European), one of many (Bermudan), any time (American), or multiple payoff dates (cliquet)?
Discrete dividends on the underlying, coupons on the contract, rebates on knock-out events. Any cash that moves between counterparties before $T$.
Does the payoff use only $S_T$ (none), or some functional of the whole path? Average (Asian), extreme (lookback), barrier-cross (knock-out)?
One underlying or many? How many state variables does the pricing PDE need — the underlyings plus any path-dependent "augmenting" variables?
Is the underlying itself an option? Compound options (calls on calls) and chooser options live here. Pricing nests one BS calculation inside another.
Does someone — the holder, the issuer — get to choose? American exercise, callable bonds, voluntary conversion. Turns the PDE into a free-boundary problem.
A given exotic typically sits at a single point in this space. "Bermudan callable convertible bond" reads off as: discrete time, multiple cashflows (coupons), embedded decisions (call and conversion), two underlyings (rates and equity). Each axis costs you something computationally.
The temporal structure of an option has more dimensions than "European vs American."
American and Bermudan turn the PDE into a free-boundary problem — the optimal exercise boundary $S^*(t)$ is part of the answer.
The total number of "events" — observation dates, exercise dates, cashflow dates — sets the cost of any discretised pricing scheme. A daily-observed Asian over a year is $252$ observations; an annual cliquet over $5$ years is $5$ resets but each is a separate forward-start option.
Anywhere between today and $T$, money can move. The PDE has to absorb each cashflow as a boundary condition at that instant.
At each ex-dividend date $t_d$, $S$ drops by $D_d$. The option value is continuous across the jump: $V(S, t_d^-) = V(S - D_d, t_d^+)$.
If the option pays a coupon stream $c(S,t)$, the PDE picks up an extra term: $V_t + \tfrac{1}{2}\sigma^2 S^2 V_{SS} + rSV_S - rV + c = 0$.
A knock-out can pay a cash rebate $R$ when it knocks out — consolation money to the holder, set at $V = R$ on the barrier.
Every cashflow is, structurally, just a boundary condition. The whole machinery of derivative pricing is the BS PDE solved on a domain with the right conditions at the right places. Exotic contracts are not exotic equations — they are familiar equations with elaborate boundaries.
Wilmott's most important distinction in this chapter. Both apply to path-dependent payoffs — the question is whether you can fold the path dependence into the existing state space.
Need an extra state variable to track the path-dependent quantity. The PDE moves from $V(S,t)$ to $V(S, I, t)$ where $I$ is the running average / max / sum.
Three-dimensional PDE. Computationally an order of magnitude harder.
The history changes the boundary or stops the contract entirely, but doesn't add a continuous state variable.
Two-dimensional PDE with modified boundary. Same cost as vanilla pricing.
The distinction tells you, before you write a line of code, whether the problem is one curse-of-dimensionality worse than vanilla pricing or merely a more elaborate boundary condition. It also tells you whether closed-form solutions are likely to exist — weak path dependence often admits them; strong rarely does.
A small gallery of structures that illustrate the taxonomy in action.
An option on an option. Pay $K_1$ at $T_1$ to receive a vanilla option struck at $K_2$ maturing at $T_2 > T_1$. Closed-form (Geske 1979) using bivariate normal CDFs.
Used to defer commitment — pay for optionality on optionality.
At date $T_c$, the holder picks: this option is now a call, or it is now a put. From $T_c$ onward it's a vanilla.
Worth more than a vanilla, less than a straddle — the holder pays for delayed commitment.
Pay a coupon each day that $S$ lies in $[L, U]$. Total payoff at $T$ is the sum of in-range days $\times$ coupon.
Weakly path dependent — an extra accumulator state, but no extra continuous variable.
Each of these can be priced by nesting Black–Scholes once or twice, or by a 2D PDE in $(S,t)$ — nothing the toolkit doesn't already handle. They're "exotic" only in payoff, not in maths.
A barrier option is a vanilla option that turns on (in) or off (out) if the underlying touches a level $B$ before maturity. There are eight standard variants, generated by three binary choices.
| call | put | |
|---|---|---|
| up-and-in | activates when $S$ hits $B>S_0$ | activates when $S$ hits $B>S_0$ |
| up-and-out | cancels when $S$ hits $B>S_0$ | cancels when $S$ hits $B>S_0$ |
| down-and-in | activates when $S$ hits $B| activates when $S$ hits $B | |
| down-and-out | cancels when $S$ hits $B| cancels when $S$ hits $B | |
For any pair sharing barrier, strike and maturity: $\text{vanilla} = \text{up-and-in} + \text{up-and-out}$. Either both knock-out paths cancel and the in pays nothing, or the knock-out cancels and the in activates — in both cases exactly one of the two pays. So pricing one of the pair gives you the other for free.
Closed-form prices exist for all eight under GBM (Rubinstein–Reiner 1991), but they're long — products of normal CDFs at modified arguments involving $\ln(B/S_0)$ and $\ln(B^2/(S_0 K))$. In practice traders use them as a sanity check; for stochastic-vol models you go to PDE or Monte Carlo.
A knock-out barrier is the cleanest example of weak path dependence and the cleanest PDE problem after vanilla.
Take an up-and-out call with strike $K$ and barrier $B > K$. The option value $V(S, t)$ satisfies the Black–Scholes PDE
$$\frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0$$
on the strip $\{0 \le S < B,\; 0 \le t \le T\}$, with boundary conditions
$$V(S, T) = \max(S - K, 0)$$
$$V(B, t) = 0 \;\;\text{for all }t$$
(or $V = R$, the rebate.)
$$V(0, t) = 0$$ (call worthless at zero spot)
Solving on the truncated domain is no harder than vanilla — same finite-difference scheme, just stop one grid point earlier in $S$. The same PDE on $\{S > B\}$ with terminal $\max(S-K,0)$ would price a down-and-out call.
For the corresponding knock-in, use in–out parity: $V_{\text{in}} = V_{\text{vanilla}} - V_{\text{out}}$. No second PDE solve needed.
Real contracts often monitor the barrier only at discrete observation times (daily close, monthly fixing). Continuous-monitoring closed forms overprice the knock-out / underprice the knock-in. The Broadie–Glasserman–Kou correction shifts the barrier by $\sigma\sqrt{\Delta t}\cdot 0.5826$ to recover the discrete price — a beautiful little adjustment.
The payoff depends on a time average of $S$, not the terminal value. Two axes:
$$\text{payoff} = \max(A - K, 0)$$
The average $A$ replaces the spot in a vanilla payoff. Cheaper than a vanilla because the average is less volatile than $S_T$.
$$\text{payoff} = \max(S_T - A, 0)$$
The average replaces the strike. Pays off when the terminal price exceeds the rolling average — a relative-performance call.
The average can be arithmetic (the natural one)
$$A_{\text{arith}} = \tfrac{1}{T}\!\int_0^T S_u\,du$$
or geometric
$$A_{\text{geom}} = \exp\!\left(\tfrac{1}{T}\!\int_0^T \ln S_u\,du\right).$$
The geometric average of a lognormal is itself lognormal, with
$$\mu_g = \tfrac{1}{2}(r - \tfrac{1}{2}\sigma^2)T, \quad \sigma_g^2 = \tfrac{1}{3}\sigma^2 T.$$
So geometric Asians admit a Black–Scholes-style closed form (Kemna–Vorst 1990). The arithmetic average is not lognormal — no closed form, and either Monte Carlo or the 3D PDE approach is required. A standard trick uses the geometric Asian as a control variate in MC, slashing variance by $90$%+.
Cheaper than vanillas with the same strike, and far less manipulable around expiry — you can't push a stock for one print and book a profit. Common on commodity contracts where the relevant price is the monthly average, not a single fixing.
A few of the wilder corners of the path-dependent zoo.
Payoff $\max(S_T - m, 0)$ with $m = \min_{0\le t \le T} S_t$, the running minimum. Strongly path dependent: extra state $m$.
"Buy at the low, sell at the maturity." Closed form under GBM (Goldman–Sosin–Gatto 1979) — rare for strong-path-dep options.
Knock-out only if $S$ spends more than $\tau$ continuous units of time beyond the barrier — a single brief touch is no longer fatal.
Designed to defeat market manipulation. Extra state: time spent in the excursion. 3D PDE.
Knock-out occurs gradually over a range $[B_1, B_2]$ rather than at a point. Notional fades in (or out) linearly across the band.
Smooths the delta hedge — vanilla barriers have a delta that jumps wildly near $B$. Used in retail structured products.
Each is a deliberate response to a problem with simpler contracts — barriers can be manipulated (Parisians fix that), discrete monitoring is awkward (Asians soften it), barrier hedging is unstable (soft barriers tame it). The exotic universe grows by patching the failings of the previous generation.
Simulate GBM paths under risk-neutral dynamics:
$$S_{t+\Delta t} = S_t \exp\!\left[(r - \tfrac{1}{2}\sigma^2)\Delta t + \sigma\sqrt{\Delta t}\,Z\right], \quad Z \sim N(0,1).$$
A path is killed when it crosses the barrier $B$ (under continuous monitoring approximated by daily steps). Knock-out price is the discounted expectation of $\max(S_T - K, 0)$ over the surviving paths; knock-in is over the killed paths that ended in the money.
Watch the knock-out price fall (and the knock-in price rise) as the barrier approaches the spot. As $B \to \infty$ for an up barrier, the knock-out converges to the vanilla. For a tight up-and-out call ($B$ just above $K$) the price collapses — almost every winning path is killed before payoff.