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The most common closed-form solution for valuing currency options is usually
attributed to Garman Kohlhagen (1983). This model is equivalent to an appropriately
configured version of the generalized Black Scholes model, where the net cost
of carry parameter for all models is calculated as
g = r - rf
Where,
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g = net cost of carry.
r = 'domestic' riskless interest rate.
rf = 'foreign' riskless interest rate.
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Both the Garman Kohlhagen and generalized Black Scholes models are derived
under a fairly restrictive set of assumptions, including:
1. |
The stochastic behaviour of the underlying exchange rate is assumed to be well represented by a Geometric Brownian Motion process. In crude terms this means that rates follow a smooth random walk through time, without any 'extreme' price changes or price spikes.
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2.
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The option can only be exercised at maturity
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All of the currency option pricing models supported in Vanilla Options retain
the first assumption. Some of the models are, however, designed to deal with
options for which the second assumption is relaxed. The following table indicates
the option pricing models that are appropriate for each of the various option
types supported by this component.
| Exercise Style
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Appropriate Model
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| European
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GK, GBS, BLACK, BIN, BIN2
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| American
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BAW, BIN, BIN2
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| Bermudan
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BIN2
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| Where,
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GK = Garman Kohlhagen.
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GBS = Generalized Black Scholes.
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BLACK = Black model for futures options.
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BAW = Barone-Adesi Whaley.
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BIN = Binomial Option Pricing Model with constant timesteps.
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BIN2 = Binomial Option Pricing Model with variable timesteps.
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Note that while we list all of the pricing models that can be applied to each
type of option, some models are more appropriate than others. For example, although
both the Garman Kohlhagen model and the binomial option pricing model can be
used to value a European currency option, the Garman Kohlhagen model would be
preferred in this case. As the binomial model involves a numerical approximation
to the 'true' value (given by Garman Kohlhagen), it is slower to compute and
less precise than the Garman Kohlhagen model.
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