Reading Real Term Sheets: Equity and FX Derivatives

Theory is nice, but at some point you have to look at real contracts. Chapter 29 of Wilmott’s book takes a collection of actual term sheets for equity and FX derivatives and walks through them one by one. The goal is practical: can you look at a piece of paper describing some exotic contract and figure out how to price and hedge it?

These are real term sheets that Wilmott received while they were still being finalized. Some have blanks and question marks where the final parameters had not been set yet. That messiness is part of the point. Real contracts are not textbook-clean.

Contingent Premium Put

The first term sheet is a put option on the S&P 500 with a twist. It is bought today, but additional premiums must be paid if the index rises above certain trigger levels. The holder starts by paying less upfront than for a vanilla put. In exchange, they accept the risk of paying more later if the market goes up.

From a pricing perspective, this is not really exotic at all. You can decompose it into a long vanilla put and three short one-touch (American binary) calls, each with a different trigger level. In the Black-Scholes world, there are explicit formulas for all of these. Add them up and you have the price.

A customer buys this because the upfront cost is lower. If the market drops (which is when they want the put to pay off), the trigger levels are never hit and the extra premiums are never owed. They only pay more if the market goes up, at which point they probably do not need the put anyway. It is a reasonable trade-off.

The lesson here: before reaching for numerical methods, check if the contract can be taken apart into simpler pieces that you already know how to price. If it can, do that. It is faster and more accurate.

Basket Options

Simple Basket

The next term sheet is an option on three exchange rates that pays off the second-highest performing out of the three. With only three underlyings, you can use either Monte Carlo or finite-difference methods. Since the underlyings are exchange rates, you use lognormal random walks with adjusted drift rates to account for foreign interest rates.

For this particular contract, there is actually a closed-form formula involving a three-dimensional integral. All you need to do is numerical integration, which is fast.

The practical warning: there are three correlations to input, and correlations between exchange rates are not very stable. Proceed with caution.

Twenty-Stock Basket with Averaging

The second basket is more challenging. It has 20 underlying stocks from different countries, making it also a quanto (cross-currency) contract. The payoff involves averaging across time (making it an Asian option too) and averaging across stocks.

How many dimensions is this? One per stock, plus exchange rates, plus 20 dimensions for the running averages, plus time. There is absolutely no way you are solving this with finite differences. Monte Carlo simulation is the only feasible approach.

With 20 factors, you have 20 volatilities and 190 correlations to estimate. Wilmott’s advice: many of those correlations will be unstable and unreliable. Consider simplifying. For example, group all US pharmaceutical stocks into a single “index” to reduce the number of parameters.

One saving grace: the averaging in this contract is a smoothing operation. It makes the option value less sensitive to individual stock parameters. Contrast this with a contract that pays off the difference between stocks, where you would need to model each stock very carefully. Averaging is your friend when parameter uncertainty is high.

Knockout Options

Double Knockout

A double knockout on an exchange rate (Mexican peso vs. US dollar). The option has two barriers, upper and lower. If the exchange rate hits either barrier, the option dies.

This is low-dimensional, well understood, and best solved with finite-difference methods. Monte Carlo would be slower and less flexible for this type of problem. Wilmott provides explicit finite-difference code for it.

The holder wants the exchange rate to stay within the corridor. They get a call payoff at expiry if the barriers are never breached. The setup includes domestic and foreign interest rates, volatility, and barrier levels. Standard barrier option math applies.

Instalment Knockout

Now it gets more interesting. This is an up-and-out put option, but instead of paying the full premium upfront, the holder pays in installments over the life of the contract. Each installment is optional: if the holder decides the option is not worth it anymore, they stop paying and walk away.

This creates an American-style optimization problem at each installment date. As you solve backwards in time, at each payment date you compare: is the option worth more than the installment I am about to pay? If yes, pay and continue. If no, throw the option away.

Mathematically, this means:

V(just before payment) = max(V(just after payment) - installment, 0)

This is exactly like the early exercise condition in American options, but reversed. Instead of deciding whether to exercise, you decide whether to continue paying.

Wilmott shows a 3D plot of the contract value against stock price and time, and you can clearly see the regular jumps at each payment date. The code is straightforward: at each payment date in the backward solve, subtract the installment and floor at zero.

Range Notes

A range note pays off at expiry based on how long the underlying stayed within a specified range. A simple version pays an amount proportional to the fraction of time spent in the range.

The simple range note is only weakly path-dependent. You can add a source term to the Black-Scholes equation that accumulates credit whenever the asset is inside the range, and present-value it to expiry. Or you can treat it as an infinite number of binary options. Both approaches work.

But the term sheet Wilmott examines has a clause saying “subject to a minimum of zero.” This makes the contract strongly path-dependent. The total accumulated time can go negative (for time outside the range), but the payoff floors at zero. To handle this correctly, you need an extra state variable (a clock) that tracks the net in-range versus out-of-range time, similar to Parisian options.

The code for the simple version is clean. A function F(S) returns 1 when S is in the range and 0 otherwise. This function acts as a source term in the finite-difference scheme, and at each timestep you present-value the accumulated credit.

Lookback with Discrete Sampling

Two lookback term sheets appear. One has continuous sampling (unusual in practice) and one has discrete sampling (more common). For the discretely sampled version, the option value depends on S, the realized maximum M, and time.

Using the similarity variable x = S/M reduces the problem by one dimension. You solve in x and time, with a jump condition at each sampling date that updates the maximum. The code implements this with an explicit finite-difference scheme and linear interpolation for the jump condition.

Cliquet Options

Cliquets are popular in equity derivatives. The contract in this term sheet is “globally floored, locally capped.” At each periodic fixing date, the return of the stock is computed, capped at some maximum. These capped returns are accumulated, and at the end, the total payoff is floored at zero.

The appeal to investors: you get participation in the upside of the stock with protection against the total being negative. The cap ensures the product is not too expensive for the seller.

The path dependency requires two state variables beyond S and time: the last fixing price S’ and the running total Q. This makes the problem four-dimensional. With a similarity reduction using S/S’ as a variable, it drops to three dimensions, which is manageable for finite differences.

But here is Wilmott’s big warning, which he devotes an entire later chapter to: the cliquet is extremely sensitive to the volatility model. In a constant-volatility world, the pricing is straightforward. But in any realistic model where volatility can change, the price swings dramatically. The individual building blocks may look like simple call spreads, but the optionality from the floor and cap interactions creates subtle dependencies that blow up when you change the volatility model.

The lesson: never try to approximately decompose a cliquet into simpler instruments. You might get the price very wrong.

Passport Options

Two passport option term sheets appear. The first is standard: the holder trades one underlying with a limit on trades per day. The second is a dual passport where the holder can trade two underlyings but only hold a position in one at a time. Not both simultaneously.

The dual passport is significantly more complex because the holder now has a three-way choice at each moment: long asset 1, long asset 2, or flat. This expands the control problem from Chapter 27 but follows the same framework.

Decomposition Warning

Wilmott closes the chapter with important advice. If a contract can be cleanly decomposed into simpler vanilla products, absolutely do that for pricing and hedging. The contingent premium put was a good example.

But if it cannot be cleanly decomposed, do not try to force an approximate decomposition. The cliquet is the poster child for this danger. The individual pieces look like call spreads, but the interactions between the floor, the cap, and the periodic resetting create effects that no simple combination of vanillas can replicate.

As the number of exotic structures keeps growing, it is better to think about new products in mathematical terms (what are the state variables? what is the PDE? what are the boundary conditions?) rather than trying to map them onto combinations of familiar instruments. The classification framework from earlier chapters is more useful than decomposition.

Key Takeaways

Real term sheets are messy, incomplete, and full of small details that matter. The difference between weakly and strongly path-dependent can be a single clause like “subject to a minimum of zero.” The choice between Monte Carlo and finite differences depends heavily on the dimensionality of the problem. And the volatility model can matter much more than you think, especially for contracts with periodic resetting features like cliquets.

The practical skill this chapter builds is the ability to look at a term sheet and immediately identify: what are the state variables, is this weakly or strongly path-dependent, can it be decomposed, and what numerical method should I use?

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