IEOR E4707: Economical Engineering: Continuous-Time Models
c 2010 by Martin Haugh
Foreign Exchange and " Quantos”
These remarks consider foreign exchange markets plus the pricing of derivative investments in these marketplaces. As is the case with value derivatives, the implied movements surface corresponding to vanilla European FOREIGN EXCHANGE option rates is none flat neither constant. Therefore, it is widely approved that the Black-Scholes GBM model is a poor model pertaining to FX marketplaces. As is the case with value derivatives, nevertheless , vanilla FOREIGN EXCHANGE option prices are cited and their Greeks are worked out using the Black-Scholes framework. Therefore, it is necessary to learn how Black-Scholes is applicable to the FOREIGN EXCHANGE markets whenever using derivatives in these markets. We all will also consider assets which might be denominated in a foreign currency but whose value we desire to determine in units of your domestic forex. There are two different method of converting the asset's foreign currency value in to domestic currency: (i) using the prevailing exchange rate and (ii) using a fixed established exchange rate. The latter approach leads to the concept of a quanto security. We all will analyze both strategies in some depth. Because this sort of securities will be exotic, they are really generally not priced in practice using the Black-Scholes framework. non-etheless, we can consider them in this structure for two reasons: (i) provided our understanding of Black-Scholes, this is the easiest way of introducing the concepts and (ii) it will eventually afford all of us additional in order to work with martingale pricing employing different numeraires and EMMs.
A particular characteristic of FOREX markets are definitely the triangular interactions that exist between currency triples. For example , the USD/JPY exchange rate could be expressed in terms of the USD/EUR and EURO/JPY exchange rates. These ramifications ought to be paid for in mind when working with FX derivatives. They and also other peculiarities of FX building will be researched further inside the assignments.
Foreign Exchange Modeling
Unless normally stated we will call one foreign currency the household currency as well as the other the other currency. We will then allow Xt denote the exchange rate by time capital t, representing the time t cost in devices of the home currency of just one unit from the foreign currency. For example , taking the usual market convention, if the USD/EUR1 exchange level is 1 . 2, in that case USD is the domestic foreign currency, EUR is a foreign currency and 1 EUR costs $1. 20. Imagine instead all of us expressed the exchange rate as the cost in European of $1 USD. Then simply USD would be the foreign currency and EUR is the domestic currency. Note that the designations " domestic” and " foreign” have nothing to do with where you are living or in which the transaction occurs. They are an event only of how the exchange rate is usually quoted.
The terms basic and quote are also frequently used in practice. The exchange rate then signifies how much of the quote forex is needed to purchase one unit with the base forex. So2 " quote = domestic” and " basic = foreign”. We can continue to use " domestic” and " foreign” in these notes from a class. Later on all of us will use " base currency” to refer towards the accounting money. For example , a US based company could take the CHF as its basic currency whereas a A language like german company may have the EUR as its base currency. All of us will let rd and rf denote the domestic and foreign risk-free rates3, respectively. Remember that the foreign forex plays the role from the " stock” and rf is the " dividend yield” of the stock. Note also that holding the 1 In these notes once we write " Curr1/Curr2” all of us will always take Curr1 to be the domestic money and Curr2 to be the money. Unfortunately, in practice this is not actually the case as well as the domestic and foreign designations will depend on the currency set convention.
a couple of But be warned: I have seen " base” used in host to the home currency too! 3 All of us will assume that r and r are constant in these notes. When ever pricing long-dated FX...