A passport option gives the holder a zero-strike call option on the value of a trading account. The option holder trades in and out of a credit-default swap contract up to a notional limit as frequently as desired over a pre-agreed time period. At maturity of the passport, if the trades have realized a profit then the holder receives the proceeds of the trading account; if the account is in deficit then the buyer walks away and the seller bears the costs.

The more careful reader will have spotted that crucial phrase, "for an upfront fee". The old adage about the scarcity of free lunches is appropriate here. This product is an option; an investor pays to be given the opportunity to trade the market with no downside. The investor's end trading account has to have a bigger profit than the upfront fee. In other words, trading strategies have to be successful enough to break even with the initial premium.

Walking Through A Passport Trade

To clarify, let's walk through a simplified example of an investor who decides to buy a three-month passport option on the iTraxx IG index. The investor pays 40 cents for the right to go either long or short up to EUR40 million nominal of five-year iTraxx IG4 CDS.

To begin with, the investor thinks the market is going wider and informs the market maker that he wants to buy EUR40 million of protection at the current market rate of 36 basis points. After 10 days the market widens five basis points to 41 bps. At this point the investor thinks the market looks overblown and decides to sell protection, but this time only EUR5 million. The index tightens three basis points to 38 bps and the investor decides to reverse the position and to leave himself long EUR10 million of protection. The index, however, moves against him by one basis point to 37 bps. After the first 20 days the investor has the following P&L:

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Default

Passports have different rules on default dependent on whether the underlying is a single name or an index. The rules are the same as for vanilla default swaptions.

In the case of the single name, a credit event will result in the passport terminated with neither loss nor gain for the investor.

In the case of the index, a default event results in a P&L change which is passed on to the investor depending on whether they are long or short. This will be in two parts: a payout due to the credit default and any resultant P&L change due to change in the index as a result of the defaulted credit being taken out. For example, an index with 125 names with an average of 40 bps and a single credit at 400 bps would have its average spread fall by 2.9 bps once the high spread name was removed.

*So what are the advantages of trading passports over trading CDS on margin with a firm stop-loss point?*

Trading a passport is actually quite similar to trading CDS on margin with a firm stop-loss point. There are some crucial differences however. The first is that CDS are prone to gap and a stop-loss order is far from guaranteed to provide an accurate limited downside. Therefore it is possible to suffer losses beyond that specified in the stop-loss order and there is potential for these to be large.

The second is that investors speculating on short-term market moves are subject to default risk while trading CDS but are not subject to default risk with passports. In contrast, investors trading passports don't even have to have the credit lines required to trade CDS as all the risks associated with the products are taken by the market-maker and the investor cannot lose more than the upfront premium.

*But couldn't investors achieve this one sided directional strategy by trading in and out of options?*

Behind the scenes, passport market makers are hedging their client's trading strategies using a series of complex vanilla option trades. Armchair theorists might argue that investors could replicate their own passport options using the same tactics as the market makers.

This is much more complicated in practice as replicating strategies require the frequent trading of deep in- and out-of-the-money transactions.

* *

So how are passport options priced?

There are several schools of thought with passport options. The first attempts to replicate the decision making made by investors in tree format, the second attempts to assess the replication cost using options.

The tree approach is quite complex but has the advantage the bid/offer and smile curves can be incorporated. A tree is created with the underlying price at every node of the tree. The investor's trade decisions are then incorporated. These sorts of models tend to suffer from a lack of transparency and are not as popular as the replication approach.

Pricing Passport Options Using The Black Model: Replication Approach

It is possible to demonstrate that the theoretical value of a passport option **Q(t)** at time t is:

Mathematical Justification

To justify this price we have to make two assumptions. The first is that forward credit spread is symmetric; which is relatively accurate for short-maturity options. The second is that the passport holder's trades are always profitable and that the full notional amount is always traded; clearly a conservative assumption.

The option payoff is dependent on the path **P** followed by the forward spread over the life of the option. Suppose that the holder executes *n* trades over the life of the option at times *t*_{k}.

We have constructed a path **P*** that increases when **P** increases, and also increases when **P** decreases.

This is where our symmetry assumption comes in useful, the probability of the spread process following the path **P*** is the same as that for **P** (equal size movements in either direction are equally likely).

Therefore, the path **P*** travels the same distance **d** as the original path **P**--thus a call struck at the forward (=**P**(0)) makes **d** basis points.

By construction:

Hedging Replication Strategy

By leaning on the assumption that the distribution of credit spreads is symmetric, it is theoretically possible to perfectly replicate the payoff of the passport option using a combination of vanilla options and CDS transactions.

Let us say that USD10 million notional of passport is traded. At start of the trade the option seller receives the value of the option given by the formula **(1)**. If the holder goes long USD10 million protection, then the option seller hedges by buying an at-the-money payer on the same notional.

By time T1, the holder has made a profit of say 3 bps and decides to reverse the trade and sell protection. To hedge, the market-maker sells the payer which is now in-the-money by 3 bps and buys an in-the-money receiver with the proceeds from the sale of the payer swaption.

Theory Vs. Practice

Even the only modestly sceptical will pick up that a strategy that relies upon perfect liquidity for ITM and OTM default swaptions in potentially non-standard maturities is going to result in market-makers being left out-of-pocket. Therefore, adjustments have to be made for liquidity of both the underlying and default swaptions markets. Market-makers have better access to liquidity but are unlikely to follow the theoretically perfect replication technique outlined above; instead opting for combinations of ATM default swaptions and CDS trades which are both easier to source.

Although it is possible to model these frictional costs--using historical simulation or Monte Carlo--many passport option market-makers tend to prefer to make these adjustments qualitatively.

*This week's Learning Curve was written by* *Michael Hampden-Turner**, structured credit strategist at* *The Royal Bank of Scotland**in London.*