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of the option is significantly different from the average premium paid to own the option.2

      That this is not always the case may be surprising. But that options can be systematically ‘cheap’ or ‘expensive’ throughout the history of the market, depending on their precise nature, is even more surprising, and should be of significant interest to many different areas of the finance community.

      Why is this not widely known? In part it is simply the focus of the market participants. Most trading desks will operate on a daily mark to market P/L with drawdown and stop-loss limits.3 Another way of putting this is that they will want to make money all or most days, with limited risk. So the timescale and nature of a trading desk dictates that the price of a contract ‘now’ is the focus of the market. Further, depending on the hedging strategy and how the option is traded, different end results can be seen. So pricing the contract ‘now’ is in many ways simpler than trying to model an option's performance. Later, we will discuss in detail how a desk manages its portfolio of options to make money, but we may summarise it now by saying that, ideally, deals are done and hedged so that a small but almost riskless profit is locked in almost immediately. After that, the combination of the deal and its offsetting hedges should be almost immune to market movements – so a systematic tendency for deals to be cheap or expensive over time may well not be noticed on a trading desk, as long as they can be hedged at a profit. The situation is complicated by the fact that a perfect hedge is rarely available, combined with the fact that a trading desk may want to have a ‘position’ – a sensitivity to market movements – when they believe that certain moves are likely to occur.

      But the other reason that the long-term mispricing of parts of the FX option markets is not well known is that FX options are a young market! Before one can say that a contract is generally cheap or expensive, one needs to observe it under a variety of circumstances. To say that 12M options bought in 2006, when market confidence was high and volatility low, were cheap because they paid out large sums in 2007, when confidence was greatly shaken and volatilities had begun a very sharp rise, would be to look at a particular case which does not represent the generality of market conditions. It is only really now, with widely available option data available going back to the 1990s, that we can say we have information available for a wide variety of market regimes, and importantly, the transitions between these regimes. We will discuss exactly what data are needed and available in the next chapter, but for now we may say that for most liquid currencies there will be perhaps 20 years of daily data available, with longer time series or higher frequencies available in some cases.

      So, we are now in a position to say whether FX options have performed well or badly for their buyers and sellers. We can take a day in the past, collect all the data needed to calculate the cost of the option and look ahead to the payoff of the option at expiry to compare the two. We can tell, on average and for different time periods, whether the options have had the correct price.

      If they have not had the correct price – and the fact that there is a book being written on the subject implies that this has been the case at least some of the time! – then the situation becomes much more interesting. Why did the market appear to be inefficient? Was there a good reason? Is it connected to the way options are used, the way they are hedged, differences in demand and supply? We will show that indeed, in different ways, the payoff and the cost of the options have differed significantly throughout the history of the market, and moreover these differences have been systematic, repeated in different currency pairs and market regimes.4

      1.2 THIS BOOK

      The book is laid out in increasing order of complexity. We give a brief history of the market and describe how options are valued – this will cover simple widely used valuation techniques; it is not our intention to go deeply into the details of exotic option pricing. Then we set the scene by introducing the available dataset and discussing the way that the market operates. We next introduce the first set of comparisons, looking at payoff vs cost or premium for options of different tenors.5 We then move on to look at different types of option: puts, calls, options which pay out at different levels or strikes, and options on emerging market currencies, which present particular features and may have less data available. Finally we examine whether some of the anomalies we see are predictable and whether it is possible to use some market indicators to buy and sell options in a dynamic fashion to improve the protection they provide or to deliver value.

      Perhaps we need to say at this point – before the reader gets too far – that there will be no magical profit-making trading strategy found in these pages. Though the market can consistently show features which seem to indicate that it lacks efficiency, inevitably they are not those which lead to a fast buck and early retirement for those who happen upon them. That is not to say that the information here may not be useful to those looking for trading strategies. At the very least it could prevent them from reinventing the wheel, show them where opportunity may lie and where they may be wasting their time. But the authors confess freely that they have not yet discovered the Holy Grail of risk-free yet profitable trading. And if they do, they may not be publishing it in a book…

      1.3 WHAT IS AN FX OPTION?

      Before we discuss which market participants can use this information, we should define more precisely what kind of contract we are talking about. Foreign Exchange (FX) options are contracts whose payoff depends upon the values of FX rates, and they are widely used financial instruments.

      Let's look at a definition from a popular website…6

      A foreign-exchange option is a derivative financial instrument that gives the owner the right but not the obligation to exchange money denominated in one currency into another currency at a pre-agreed exchange rate on a specified future date.

The price or cost of this right is called the premium, by analogy with the insurance market, and it is usually (depending on the tenor and the market at the time) a few percent of the insured amount (notional amount). The specified future date is called the expiry or expiry date.7 The payoff profile at expiry of the simplest type of option is shown schematically in Figure 1.1.

FIGURE 1.1 Payoff profile at expiry for a call option

      The figure shows the payoff received by the holder of an at-the-money-forward (ATMF) call option on an FX rate. This means that the strike of the option is the forward rate, and the option is the right to buy the base currency, or, in other words, an option to buy the FX rate.8 In other markets such as commodities and equities it is obvious what the call or put is applied to but in FX more clarity is needed. For instance a call option associated with the currency pair USDJPY could be a call on USD (and thereby a put on JPY) or a call on JPY (and therefore a put on USD). As different currency pairs have different conventions it is always best to clarify the exact details before trading. A put option would be the right to sell the base currency, or FX rate. We will discuss forward rates and their relationship with options more completely in later chapters but in essence the forward rate is the current FX rate adjusted for interest rate effects. If the interest rates for the period of the option were identical in both currencies involved in the FX rate, then the forward rate would be identical to today's FX rate. Because they usually are not the same, the rate which one may lock in an exchange without risk for a future date will be somewhat different from today's rate.

      The figure shows the premium cost of the option. At all FX rates at expiry which are less than the forward rate, this will be what the option holder loses, meaning that he or she paid a premium to buy the option and will make no money from it. The net result is the loss of the premium. At the forward rate, the payoff begins to rise, at first reducing the overall cost and then taking the owner of the option into profitable territory for higher FX rates at expiry. We have also shown the payoff from a forward contract, which is simply when the owner of the contract

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