Derivatives and Stochastic Control: Passport Options

Most options we have seen so far give the holder a choice at one specific moment. With a European option, you decide at expiry. With an American option, you pick the best time to exercise. But what if the option let you actively trade during its entire life, and then insured you against losses? That is the idea behind the passport option, and Chapter 27 of Wilmott’s book uses it to introduce stochastic control.

The Perfect Trader Fantasy

Imagine you trade a stock over some time period. You buy, you sell, you follow your gut or your favorite chart pattern. At the end, you look at your trading account. If you made money, great, you keep it. If you lost money, someone covers your losses. You walk away with at least zero.

That is exactly what a passport option does. The holder gets to make hypothetical trades in an underlying asset. The option then pays out the positive part of the trading account at expiry. Losses are written off. It is like having a trading license with built-in insurance.

The name “perfect trader option” comes from the idea that we need to figure out what the optimal trading strategy would be. Not because anyone actually follows it, but because the seller of the contract needs to assume the worst case when pricing it.

The Math Setup

To price this thing, we need a new variable: the trading account, usually called pi. This tracks the total value of the stocks you hold plus any accumulated cash. We also introduce q, the quantity of stock held at any time. This q is the control variable, the thing the holder gets to choose.

The key equation for the trading account looks like a combination of two effects. First, whatever cash you are not using to hold stock earns interest. Second, your stock position changes in value as the stock price moves. The option value V depends on three variables: the stock price S, the trading account pi, and time t.

Here is where it gets interesting. The pricing equation looks similar to our usual Black-Scholes equation, but it has this extra variable pi. And because S and pi are perfectly correlated (the trading account moves because of the stock), the equation is not a full two-dimensional diffusion problem. It has one space-like variable and two time-like variables.

Choosing the Optimal Strategy

The stochastic control part comes in choosing q. If you are selling this contract, you assume the holder trades optimally to make the option as expensive as possible. The holder might not actually follow this strategy in practice, but that is irrelevant for pricing. You hedge against the worst case.

Finding the optimal q turns out to be surprisingly clean. You look at the terms in the pricing equation that contain q, and you maximize them. With a constraint that the position size stays between -1 and +1 (you cannot go too crazy), the optimal strategy depends on the sign of a particular derivative of the option value.

For the simplest version of the passport option, where the payoff is just max(pi, 0) at expiry, there is a nice similarity solution. The option value can be written as the stock price times some function of pi/S and time. This reduces the problem to a simpler equation in fewer variables.

The result? The optimal strategy is just q = +1 or q = -1. You are always fully long or fully short. No half positions. The decision is binary, and it depends on the sign of a derivative. This makes intuitive sense: if you are trying to maximize the trading account, you go all in one direction or the other.

Limiting the Number of Trades

The basic passport option lets the holder trade as often as they want. But in practice, trading has costs, and the contract might restrict how often you can flip your position.

What happens when you limit the number of trades to, say, n? The problem becomes more complex because now you need to track two functions: the option value when you are currently long with n trades remaining, and the option value when you are currently short with n trades remaining.

The math looks like the American option problem. At each moment, the holder decides whether to flip from long to short (or vice versa), using up one trade. The decision to flip is optimal when the option value with n trades in one position equals the value with n-1 trades in the opposite position. Just like the early exercise boundary in American options, you get inequalities instead of equalities.

Here is the surprising part from Wilmott’s numerical examples: even with only three or four trades allowed, the option value is close to the unlimited case. The ability to trade freely is not worth much more than the ability to trade a few times. This is a useful practical result because it means the simpler, restricted version is almost as valuable.

Limiting Time Between Trades

Instead of capping the total number of trades, you could require a minimum waiting period between trades. This introduces a “clock” variable that resets to zero after each trade. You cannot trade again until the clock reaches some threshold omega.

The structure is very similar to Parisian options, which we will see in the next chapter. The option value now depends on S, pi, time t, and the clock tau. The results look similar to the limited-trades case: requiring the holder to wait between trades does not dramatically reduce the option value.

When the Holder Does Not Trade Optimally

This section is where Wilmott gets philosophical, and I think it is one of the more interesting points in the chapter.

We defined “optimal” as the strategy that maximizes the option value for someone who is delta hedging. But the passport option holder is probably not delta hedging. They bought this contract because they think they are a hot trader. They have views about the market. They are going to trade based on their own opinions, not based on what makes the option expensive.

Is the holder wrong to do this? Not necessarily. Their definition of “optimal” is different from the hedger’s definition. The holder is trying to make money on direction. The hedger is trying to neutralize risk.

Here is the key insight: the seller of the passport option should be happy when the holder trades non-optimally. The seller priced the contract assuming the worst case. If the holder deviates from that worst case, the seller effectively overprice the contract and pockets the difference. The holder might still do great (if their directional bets pay off), but the seller’s risk is reduced.

This is a common theme in derivatives pricing. You price for the worst case, and then reality is usually better than the worst case. The seller collects a small premium for bearing a risk that never fully materializes.

Real Term Sheets

Wilmott includes a real term sheet for a passport option on the USD/DEM exchange rate. The contract allows two trades per day with a maximum position size. The holder makes hypothetical trades, and at the end, they receive the positive part of their trading account balance. If the balance is negative, it is written off.

These contracts are not just theoretical curiosities. They have been traded in the FX markets, and the mathematical framework for pricing them works well in practice. The main challenge is the hedging: the option writer must hedge at least as often as the holder trades, and frequent hedging means higher transaction costs.

Key Takeaways

Passport options are one of the most creative derivatives structures out there. The holder gets to play trader with a safety net. The seller gets paid for providing that insurance. The pricing requires stochastic control theory, but the results are surprisingly clean.

Three things to remember from this chapter. First, the optimal strategy is always all-in: fully long or fully short, no middle ground. Second, limiting trades or requiring waiting periods does not reduce the option value by much. Third, the seller benefits whenever the holder deviates from the theoretical optimal strategy. That last point is a useful lens for thinking about many types of exotic options, not just passport options.

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