Generally in thinking about options, one assumes that one is buying an asset: for instance, you can have a call option on oil, which allows you to buy oil at a given price. One can consider this situation more symmetrically in FX, where one exchanges: a put on USD/GBP allows one to exchange GBP for USD: it is at once a put on GBP and a call on USD.
As a vivid example: people usually consider that in a fast food restaurant, one buys hamburgers and pays in dollars, but one can instead say that the restaurant buys dollars and pays in hamburgers.
There are a number of subtleties that follow from this symmetry.
Ratio of notionals
The ratio of the notionals in an FX option is the strike, not the current spot or forward. Notably, when constructing an option strategy from FX options, one must be careful to match the foreign currency notionals, not the local currency notionals, else the foreign currencies received and delivered don't offset and one is left with residual risk.
Non-linear payoff
The payoff for a vanilla option is linear in the underlying, when one denominates the payout in a given numéraire. In the case of an FX option on a rate, one must be careful of which currency is the underlying and which is the numéraire: in the above example, an option on USD/GBP gives a USD value that is linear in USD/GBP (a move from 2.0000 to 1.9000 yields a .10 * $2,000,000 / 2.0000 = $100,000 profit), but has a non-linear GBP value in USD/GBP. Conversely, the GBP value is linear in the GBP/USD rate, while the USD value is non-linear in the GBP/USD rate. This is because inverting a rate has the effect of , which is non-linear.
Change of numéraire
The implied volatility of an FX option depends on the numéraire of the purchaser, again because of the non-linearity of .
Tuesday, March 31, 2009
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