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Passport or vacation options are not options which give you the opportunity to go on holiday, but rather, these options are options on the balance of a trading account. By this, we can consider a simple example where Alpha bank has a passport call on The trading account sold to them by Beta bank. If the trading account makes a profit, Alpha bank takes the trading gain, whereas Beta bank will bear the loss.
Introduction || Exotic Passport Options || Pricing || Binomial Method || Finite Differences || Other Known Names & Variants || References
Passport or vacation options are not options which give you the opportunity to go on holiday, but rather, these options are options on the balance of a trading account. By this, we can consider a simple example where Alpha bank has a passport call on The trading account sold to them by Beta bank. If the trading account makes a profit, Alpha bank takes the trading gain, whereas Beta bank will bear the loss.
Before we continue, we should make the distinction between passport and vacation options; the difference between them is analogous to the difference between a European and American options, with passports being the European type. For discrete passport options, we can consider a time horizon (0, T) with an H+1 number of discrete "settlement" dates
The payoff of a European passport option is then given as:
Where k takes on a value of -1 or 1 depending on investor position at each date S(t) is the stock price at a particular date t and H is the above defined number of settlement dates.
The payoff is therefore dependent on the investor strategy u, and the number of measurement dates H there are during the life of the option. At each settlement / measurement date, the only changes being made is that the investor (holder of the option) has the right to 'tell' the writer of the option what position he or she wishes to take - in other words, the investor can change his or her trading strategy without physically buying or selling any assets; this is left up to the writer of the option to handle. The investor effectively pays for the right to adjust a portfolio held by the option writer.
However, continuous passport options where the number of settlement dates is infinite (in that the measurement point can be any time during the life of the option can also be priced, but in a slightly different manner.
Just when you thought things couldn't get more 'exotic', you'll be happy to find out that within the passport option family, there are a host of other so-called exotic passports. These exotic passports give investors the ability to explore various alternatives based on market perception and level of risk adversity - we highlight some of these exotic passport options, and point readers to Ahn, Penaud & Wilmott (1999) for further insight into these extreme exotics:
Barrier Passport Options Some of you might already be accustomed to standard barrier options, options which become knocked in or knocked out upon reaching a prescribed barrier. In the case of passport barriers, the same structure is present as in standard barriers in that if the trading account a hits a certain barrier level, the option expires worthless and the investore recieves a prescribed amount. These options are useful for investors who expect large movements in the underlying asset price and does not see the use of holding the option if and when the asset price is too large. Similar to standard chooser options, passport choosers allow the holder to select between one of two payoffs at each settlement date prior to maturity:
At each settlement date, assuming the investor is not near zero and a is positive, he or she is likely to choose the payoff which gives a positive a, and the opposite is true if a is negative. The usefulness of passport choosers lies in the market perception that there will be a large uncertain move between settlement dates and chooser passports give a degree of flexibility to the investor. Casino gamblers often double up on their bets if they strongly believe that their hand is a sure win in an attempt to maximise their gain in the event that they do win. As we know, option trading is also a bet, and one can also "double up" on passport options in order to maximise gains. In general, standard vanila passports will enable the investor to take on positions of -1, 0, or 1 representing the position they are holding; in the case of double stake passports, investors can take on positions of -2 or 2 or even more depending on the individual contract. If the trading account moves in the perceived direction, investors can double his or her gain, but will take on double the loss of their perception is wrong. Beginning to sound like a computer game, we come across to magic potion passports - which enables investors to drink the magic potion and make part of his or her trading history disappear. For example, after 6 settlement periods, the investor thinks that erasing the history between periods 5 and 6 will improve their overall gain. These options can be useful for investors who do not have a tuned perception of the market, or beginners who are prone to mistakes. This type of passport is generally the most expensive due to its flexibility. It is worth seeing the section on smooth trader passport options first as that gives way for this type of passport option. For the investors who feel that the smoot trader passports are too restrictive, they can take up a type of passport which lies somewhere in between a smooth trader passport and a vanilla passport option. The investor in this case takes on a similar strategy as if he or she were to hold a smooth trader passport, but the piecewise structure enables the investor to take on a number of predetermined jumps if he or she decides to change their mind on the direction of the underlying. Tacking more exotics together, we can consider the case of reset options, or options which have a function where the investor can reset some aspect of it during the life of the option, whether it be time to maturity or strike price. The intuition behind reset passports is the same in that the investor, instead of reseting the strike or time to maturity, can reset the 'memory' of the positions he or she has taken. This is generally constrained to reseting of the past N trading account positions. In other words, the investor can erase his or her mistakes if there is a belief that the underlying is going to move in the direction that is believed in the future. Increasing the number of reset times will increase the price of the option, beit at a decreasing rate. These types of passport options are not that much different from standard passport options. In the following 'pricing' section, we consider the Hamilton-Jacobi-Bellman equation which governs standard vanilla passport options, and we see that the number of assets held is constrained by a value C. In the case of smooth trader passports, the investor cannot adjust the number of assets from, lets say -1 to 1 instantaneously. Although an adjustment is allowed, the constraint C now represents the speed at which one can adjust the assets held. These are useful for investors who believe the underlying asset price will not fluctuate alot, and because of it's reduced flexibility, these are cheaper than vanilla passports. The final, on our list of exotic passports is the so-called switch passport. These options enable the investor to switch between 2 assets at any time, but both assets cannot be held at once and furthermore, there is a limit to the number of times the investor can switch assets. These can be useful in two main ways:
Investors selecting two assets A and B which are subject to many of the same factors, but one or two major factors which are distinct and there have a relatively low correlation between them. At any point, the investor can consider these factors and decide which will be more profitable in the future.
Alternatively, assets A and B could be in completely different sectors and have strong negative correlation, giving investors the flexibility based on general market trends.
Closed form valuation of these types of options only takes place in special circumstances, which we shall detail in a subsequent section, but in general, lattice or finite grid methods are the best way of solving these options.
The pricing of these types of options, although not entirely straight-forward; by making use of some stochastic control theory such as application and solution of the Hamilton-Jacobi-Bellman equation, will help us to determine the valuation governing the option. Under our standard Black & Scholes framework one assumes a continuous pricing framework; and although we do not go into detail the analysis of the differential equation, one can give it as:
Which is our standard equation with
and claiming that:
Penaud, Wilmott & Ahn (1999) give the price of a vanilla passport option as:
Whereby C is the so-called constraint level of the portfolio, in that the investor can only adjust the portfolio within a range of (-C, C), a is the traded account, r is the risk free rate of interest, q is the number of assets held (in the range of (-C, C), S represents the price of the traded asset and The special case in which closed form valuation exists is where the risk free rate of interest is equal to the constant dividend yield r = D In cases where the risk free rate does not equal the constant dividend yield, one has to resort to other techniques such as a lattice or the use of finite differences methods.
Account Options
Additional/Useful List of resources Papers: Ahn, H., Penaud, A., & Wilmott, P., "Various Passport Options and Their Valuation" Oxford University OCIAM Working Paper, 1998 |
, where n takes on the values (0, 1, 2, 3, 4, ..., H-1, H).
equal to the drift term (risk free - dividend yield) and z is our Wiener process. By building a risk free portfolio
is the volatility. It is also pointed out that the risk free rate in fact does not matter in determining the price of the option - see Ahn, Penaud & Wilmott (1998). The mathematics involved in solving the HJB (Hamilton-Jacobi-Bellman) equation is not in the scope of this text and we recommend the readers to see further readings in Ahn et al (1998, 1999).