We now have the machinery to price European style options under models that provide a fuller description of the FX spot process than Black–Scholes. While European options are especially liquid, they are often seen as particularly expensive, and clients are often interested in cheaper alternatives.
One particularly common way to cheapen a European option in FX is to add a path-dependent barrier feature to it, so that the option can only be exercised if it expires in the money and has remained within a specified region of interest over its lifetime. Part of the reason these barrier options are popular in FX is that currency spot rates are often thought to be more range-bound or mean-reverting than stock prices, and clients sometimes feel that the price is unnecessarily high as a result of purchasing option protection for large but unlikely moves in the FX spot rate.
The price of a European option is just the discounted domestic risk-neutral expectation of the payout function at expiry. For a call option,
V0=exp(−rdT ) ∞
K
fST(u)(u−K )+du (8.1) Neglecting the discount factor for now, this is just the inner product of the PDF (for ST) and the option payout. Though the option payout for a European call increases without bound beyond the strike K , the product of the PDF and the payout profile decays to zero beyond K , as the e−x2eventually dominates.
The price paid for a European option (8.1) therefore takes into account the payout at all possible future levels of ST, including some potential future scenarios that the client may view as quite unlikely. From a client’s point of view, therefore, if they have a view that the spot rate is unlikely to trade beyond a certain range, then they may be happy to consider an option that has a value at expiry function which goes abruptly from (ST −K )+to zero, beyond a certain level H . This product is known as a European barrier option, and exists for both calls and puts.
The European up-and-out call pays (ST−K )+if K <ST <H (and zero otherwise), and its counterpart, the European down-and-out put (with barrier L), pays (K−ST)+if L <ST <K (and zero otherwise). These products will be covered in Section 8.2.1.
Clearly the knock-out barrier is sampled only at time T ; the asset price process can make any excursion whatsoever up to that time, without affecting the value at expiry. One can make the product even cheaper still by extending the barrier monitoring over the entire path history of the option.
In such a case, we have a continuously monitored barrier option, which depends on either the continuously monitored maximum MT, the continuously monitored minimum mT or potentially both (we introduce these in Section 8.3 below).
Even more simply, it is quite commonplace in FX to strip out the option payout (ST−K )+ entirely, and just have a path-dependent option, which pays one unit of either domestic or foreign currency at time T depending on whether Mtand/or mthave remained entirely within
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Table 8.1 American digitals/binaries
Product code Name Payout function VT
OT1C Foreign upside one-touch ST1{MT≥U}
OT1P Foreign downside one-touch ST1{mT≤L}
OT2C Domestic upside one-touch 1{MT≥U}
OT2P Domestic downside one-touch 1{mT≤L}
DOT1 Foreign double-touch ST1{mT≤L∪MT≥U}
DOT2 Domestic double-touch 1{mT≤L∪MT≥U}
NT1C Foreign upside no-touch ST1{MT<U}
NT1P Foreign downside no-touch ST1{mT>L}
NT2C Domestic upside no-touch 1{MT<U}
NT2P Domestic downside no-touch 1{mT>L}
DNT1 Foreign double-no-touch ST1{mT>L}1{MT<U}
DNT2 Domestic double-no-touch 1{mT>L}1{MT<U}
particular ranges over the entire lifetime of the option. These are referred to as American digitals (or binaries) (see Table 8.1), and are basically just the path-dependent counterpart of the European digitals introduced in Section 2.12. Domestic corresponds to ccy2 and foreign corresponds to ccy1, in the manner introduced in Chapter 3.
Suppose the initial spot rate is 1.0, volatility is a flat 10 % and domestic and foreign interest rates are zero. In this example, we have a 1 year American cash-or-nothing digital that pays
$1 in one year if an up-and-in barrier level of 1.1 is touched at any time during that year and a 1 year European digital that pays $1 if spot ST at expiry exceeds the trigger level of 1.1 on the expiry date.
Clearly if the European condition is met, then the American option will have touched too – at time T (if not before). So the American digital must be worth more than the European digital. How much more? About twice as much – but capped at the one year discount factor.
We see this in Figure 8.1, where the TV prices of European and American style digitals are graphed as a function of the barrier level, with down-and-in digitals and up-and-in digitals
0%
20%
40%
60%
80%
100%
1.2 1.1
1 0.9
0.8
European American
Figure 8.1 Digitals
H′
T W~T
W~T
2H′−
Figure 8.2 The reflection principle
separated either side of unity. Note the absence of symmetry, due to the presence of−12σ2 in N (d2). For our particular example, the European digital is priced at 15.8 % whereas the American digital is worth about twice as much, at 32.4 %.
The rest of this chapter attempts to explain why, using the reflection principle, and to introduce the techniques required to price these first generation exotic options. It’s never clear exactly what products fall in the remit of first generation products in FX, though binary and barrier options are always taken to fall into this category. I shall basically follow Wilmott (1998) and talk about the so-called weakly path-dependent products in this chapter, i.e. products that can be priced on one finite difference grid (in one tradeable factor, though possibly including other untradeable factors such as stochastic volatility). Strongly path-dependent options will be covered in Chapter 9.
8.1 THE REFLECTION PRINCIPLE
The reflection principle, as mentioned in Section 2.6 of Karatzas and Shreve (1991) basically states that for any driftless Brownian motion that hits a particular level H , we can construct a ‘shadow path’ that, upon arrival of the original trajectory at H, is thereafter just the mirror image of the original trajectory reflected about that level.
We have two possible paths: ˜Wt, and ˜Wt∗, where the second is defined by W˜t∗=
⎧⎨
⎩
W˜t, t< τH, 2H−W˜t, otherwise, whereτH=inf{t : ˜Wt =H}is the hitting time.1
Somewhat heuristically, if we identify the trajectories shown in Figure 8.2 with possible asset price paths, and suppose that Hcorresponds to the barrier level of U =1.10, then both trajectories will cause the American one-touch to expire with value of unity, whereas only one
1Note that this is not the time-reversedτ, which is used in the PDE formulation of option pricing; it is a physical hitting time.