Exotic Options Grab Bag: Shouts, Ladders, and Parisians
By this point in the book, Wilmott has been classifying exotic options into tidy categories. Asian options got their own chapter. Lookbacks got their own chapter. Barrier options got their own chapter. But the universe of exotic derivatives is large and growing, and eventually the classification exercise breaks down. Chapter 28 is where Wilmott gives up on neat categories and just throws a bunch of interesting exotics at us. It is a grab bag, and it is fun.
The real goal here is not just listing contracts. It is expanding your toolkit of techniques for pricing new and unfamiliar derivatives in the partial differential equation framework.
Forward-Start Options
A forward-start option is an option that is bought today but does not actually begin until some future date T1. At that future date, the strike price is set, usually to the asset price at that moment. Then the option runs until expiry at time T.
There are two ways to price this. The simple way is elegant. At time T1, you know the stock price S1, and you have an at-the-money option with time T - T1 remaining. The Black-Scholes formula for an at-the-money option gives a value that is proportional to the stock price. So before T1, the forward-start option is just some constant times today’s stock price. Done.
The more complicated way is to introduce a new state variable for the strike level and solve the pricing PDE in higher dimensions. This is unnecessarily complex for the simple forward-start, but the technique of introducing new state variables is powerful for other, more complicated contracts. Wilmott stresses the technique because you will need it later.
Shout Options
Here is a fun one. A shout option is a vanilla call option with a special feature: at any time during its life, the holder can “shout.” When you shout, the strike price resets to the current asset price (if it is higher than the original strike), and you immediately receive the difference between the old and new strike.
Think of it as locking in a partial profit while keeping the upside. You see the stock go up, you shout, you pocket the gain so far, and your option continues but with a new, higher strike. You can only shout once (in the basic version), so the timing matters.
Because there is an optimization problem (when to shout?), you get a free boundary problem just like with American options. You need two value functions: one for the state before shouting and one for after shouting. The “before” function must satisfy a constraint that says you should not shout if it is not yet optimal. When the constraint binds (i.e., the two sides become equal), that is the optimal shout boundary.
Multiple-shout extensions exist and make the problem harder but follow the same logic.
Capped Lookbacks and Asians
We have already seen lookback options (payoff depends on the maximum or minimum of the asset path) and Asian options (payoff depends on the average). You can combine these with caps and floors.
A capped Asian tracks the average, but at each sampling point, the contribution is capped. For example, instead of averaging the stock price, you average the minimum of the stock price and some fixed level. This limits the upside of the average and makes the option cheaper.
The math is straightforward. You modify the stochastic differential equation for the running average to include the cap, and then solve the usual PDE. The cap changes the source term but does not change the overall approach.
Combining Path-Dependent Quantities
This is where things get creative. Why not have an option that depends on both a maximum and an average? Wilmott calls this a lookback-Asian.
But “lookback-Asian” is ambiguous. It could mean:
- The payoff depends on the maximum of the asset AND the average of the asset (two separate path-dependent things computed from the same price series)
- The payoff depends on the maximum of the average (first compute the running average, then track its maximum)
- The payoff depends on the average of the maximum (first compute the running maximum, then average it)
These are all different contracts, and they all require different updating rules at each sampling date. The option value is now a function of four variables (S, time, and two path-dependent quantities), making it computationally intensive but still manageable.
The ordering of operations matters. Taking the max then the average is different from taking the average then the max. This might seem obvious, but when you are staring at a term sheet with complex payoff rules, it is easy to get confused about which quantity is computed first.
Many of these problems have similarity reductions that drop the dimensionality by one. This takes a four-dimensional problem down to three dimensions, which is much more practical for finite-difference methods.
Volatility Options and Variance Swaps
This section is particularly interesting because it connects to a major part of modern derivatives markets: variance swaps.
A volatility option pays off based on the realized historical volatility of the asset. The path-dependent quantity is the statistical variance computed from sampled asset prices along the path.
To value this, you need to track two things: the running variance calculation and the last sampled asset price (because you need it for the next variance update). The option value is a function of four variables: S, the last sampled price, the running variance, and time.
You might ask: “Will the realized volatility just be the volatility we put into the model?” In a constant-volatility Black-Scholes world with infinitely frequent sampling, yes. But with discrete sampling or stochastic volatility, the answer is no, and that is exactly why these contracts are interesting.
Hedging with the One-Over-Strike-Squared Rule
The most famous result in this section is that variance swaps can be hedged with a continuous distribution of vanilla options where you hold an amount proportional to 1/E^2 of options at each strike E.
The derivation is beautiful. You ask: what portfolio of call options has a constant vega (sensitivity to volatility) regardless of where the stock price is? You set up the integral, change variables, and find that the weight function f(E) must satisfy a simple differential equation whose solution is f(E) = k/E^2.
This is why variance swaps became so popular. They can be approximately replicated using liquid vanilla options. No exotic hedging instruments needed. You just buy a strip of puts and calls across many strikes, with weights inversely proportional to the strike squared.
Correlation Swaps and Dispersion Trading
A correlation swap is like a variance swap but for realized correlation between multiple assets. Its payoff depends on how correlated the assets were during the contract’s life.
Dispersion trading exploits the relationship between the volatility of an index and the volatilities and correlations of its components. If you are long options on individual stocks and short options on the index, you are essentially betting that individual stocks will make big moves that cancel out at the index level. This is a play on correlation: you profit when correlation drops and stocks move independently.
Ladder Options
A ladder option is a discretized lookback. Instead of tracking the continuous maximum of the asset price, it only registers when the asset crosses specified levels (the “rungs” of the ladder). For example, if the rungs are at 50, 55, 60, 65, and the asset reaches 58, the registered maximum is 55.
This is cheaper than a continuous lookback because you lose the benefits of the exact maximum. You can decompose ladders into a series of barrier options triggered at each rung, or just treat the payoff as a step function of the tracked maximum.
Parisian Options
Parisian options are a clever modification of barrier options. In a standard barrier option, the barrier triggers instantly when the asset touches it. In a Parisian, the asset must stay beyond the barrier for a prescribed time before the knock-in or knock-out activates.
This fixes two problems with standard barriers. First, the option value is smoother near the barrier, making hedging easier. Second, it is much harder to manipulate. With a standard barrier, a large trader could push the price just past the barrier briefly to trigger a knockout. With a Parisian barrier, they would need to hold it there for, say, ten days. That is a lot harder to manipulate.
The pricing introduces a clock variable tau that tracks how long the asset has been beyond the barrier. When the asset returns inside the barrier, the clock resets to zero. The barrier triggers when tau reaches the threshold omega. You solve the PDE in two regions (inside and outside the barrier) with a continuity condition at the barrier level and a trigger condition at tau = omega.
There is also the Parasian variant where the clock does not reset. In that version, the total cumulative time beyond the barrier is tracked, regardless of whether the asset dips back inside. This makes the triggering even more likely and the option cheaper to buy (for knock-ins) or more expensive to buy protection against (for knock-outs).
The Zoo of Other Exotics
Wilmott rounds out the chapter with a quick list of even more creative structures:
- Balloon options where the quantity increases if barriers are hit
- Break forwards where one side can terminate a forward contract early
- Hawaiian options that combine Asian and American features (naturally)
- Himalayan options where the best-performing stock gets removed from the basket at each sampling date, and the payoff is based on the last stock standing
- HYPER options (High Yielding Performance Enhancing Reversible) that are like American options but flipping between call and put on each exercise
The names are creative. The math for each follows the same general pattern: identify the path-dependent quantities, set up the PDE with appropriate state variables, apply the right boundary and final conditions, and solve.
Key Takeaways
This chapter is really about building confidence. By now you should be able to look at a new exotic contract, identify what path-dependent quantities are needed, set up the governing PDE, and figure out a numerical approach. The specific contracts are interesting, but the techniques are what matter.
The variance swap hedging result (the 1/E^2 rule) is probably the single most practical result in this chapter. Parisian options are the most elegant idea. And the philosophy that you should think of exotics in mathematical terms, rather than trying to decompose them into simpler instruments, is advice that becomes more valuable as contracts get more creative.
Previous post: Derivatives and Stochastic Control: Passport Options
Next post: Reading Real Term Sheets: Equity and FX Derivatives