It is therefore necessary to understand how Black-Schools applies to the FAX markets when working with derivatives in these markets. We will also consider assets that are denominated in a foreign currency but whose value we wish to determine in units of a domestic currency. There are two different means of converting the asset’s foreign currency value into domestic currency: (i) using the prevailing exchange rate and (ii) using a fixed predetermined exchange rate. The latter method leads to the concept of a quanta security. We will study both methods in some detail.
Because such securities are exotic, they are generally not priced in reactive using the Black-Schools framework. Nonetheless, we will consider them in this framework for two reasons: (i) given our knowledge of Black- Schools, this is the easiest way of introducing the concepts and (ii) it will afford us additional opportunities to work with martingale pricing using different numerates and Mess. A particular feature of FAX markets are the triangular relationships that exist between currency triples. For example, the USED/SPY exchange rate can be expressed in terms of the SAID/KIER and GERRI/SPY exchange rates.
These implications ought to be borne in mind when working with FAX derivatives. They and other peculiarities of FAX modeling will be studied further in the assignments. Foreign Exchange Modeling Unless otherwise stated we will call one currency the domestic currency and the other the foreign currency. We will then let Ext denote the exchange rate at time t, representing the time t cost in units of the domestic currency of 1 unit Of the foreign currency. For example, taking the usual market convention, if the USED/GERRI exchange rate is 1. 2, then USED is the domestic currency, ERR is the foreign currency and 1 ERR costs $1. 0. Suppose instead we expressed the exchange rate as the cost in Euro of$l LESS. Then USED would be the foreign currency and ERR would be the domestic currency. Note that the designations “domestic” and “foreign” have nothing to do with where you are living or where the transaction takes place. They are a function only of how the exchange rate is quoted. The terms base and quote are also often used in practice. The exchange rate then represents how much of the quote currency is needed to purchase one unit of the base currency. ASS “quote = domestic’ and “base = foreign”.
We will continue to use “domestic” and “foreign” in these lecture notes. Later on we will use “base currency’ to refer to the accounting currency. For example, a US based company would take the USED as its base currency whereas a German company would have the ERR as its base currency. We will let rd and RFC denote the domestic and foreign risk-free rates , respectively. Note that the foreign currency plays the role of the “stock” and RFC is the “dividend yield” of the stock. Note also that holding the 1 In these notes when we write “Curl /Curry” we will always take Curl to be the domestic currency and Curry to be the foreign currency.
Unfortunately, in practice this is not necessarily the case and the domestic and foreign designations will depend n the currency pair convention. 2 But beware: have seen “base” used in place of the domestic currency as well! 3 We will assume that r and r are constant in these notes. When pricing long-dated FAX derivatives, these rates are often d taken to be stochastic. The same applies for long-dated equity derivatives. 2 foreign cash account is only risk-free from the perspective of someone who uses the foreign currency as their unit of account, i. E. Accounting currency.
A domestic investor who invests in the foreign cash account, or any other asset denominated in the foreign currency, is exposed to exchange rate risk. Currency Forwards Let Ft denote the time t price of a forward contract for delivery of the foreign currency at time T . Then since the initial value of the forward contract is O, martingale pricing implies (T) O = CEQ which then implies EXT Assuming that interest rates are constants we obtain t[EXT] -rd (T -t) = red (T -t) CEQ = e(rd a relationship that is sometimes referred to as covered interest parity. Exercise 1 Prove (1) directly using a no-arbitrage argument.
Exercise 2 Compute the time t value of a forward contract that was initiated at time t = O with strike K for delivery at time T > t. The forward currency markets are very liquid and play an important role in currency trading. A forward FAX (T ) rate, Ft , is usually quoted as a premium or discount to the spot rate, Ext , via the forward points. FAX Swaps An FAX swap is a simultaneous purchase and sale of identical amounts of one currency for another with two different value dates. The value dates are the dates upon which delivery of the currencies take place.
In an FAX swap, the first date is usually the spot date and the second date is some forward date, T . An FAX swap is then a regular spot Fixture combined with a forward trade, OTOH of which are executed simultaneously for the same quantity. FAX swaps are often called fore swaps. They should not be confused with currency swaps which are considerably less liquid. A currency swap is typically a long- dated instrument where interest payments and principal in one currency are exchanged for interest payments and principal in another currency.
FAX swaps are regularly used by institutions to fund their FAX balances. Forwards and FAX swaps are typically quoted in terms of forward points which are the difference between the forward price and the spot price. Using (1) we see -Ext- e(rd -RFC )(T -t)- 1 Ext We could also allow interest rates that are (i) deterministic or (ii) stochastic but Centipede of X in our derivation of T (1 But a no-arbitrage argument using zero-coupon bonds could still be used to derive (1) in general. See Exercise 1 . 5 Exchange-traded currency futures also exist. 6 But other value dates are also possible.
It is possible, for example, to trade forward-forward swaps where both value dates are beyond the spot date, or tom-next swaps. 3 When interest rates are identical the forward points are zero. As the interest rate differential gets larger, the absolute value of the forward points increases. Currency Options A European call option on the exchange rate, X, with maturity T and strike K pays Max(D, EXT – K) at time T . If we assume that Ext has KGB dynamics then the same Black-Schools analysis that we applied to options on stocks will also apply here.
In particular the Q-dynamics of the exchange rate are given by (x) txt = (rd – RFC)Ext EDT + ox Ext AWT and the price of the currency option satisfies C(X, t) = e-?RFC (T -?t) Ext ) -? e-rd (T -?t) K˜(do ) where ODL and do Ext + (rd – RFC+ ox – t) ox T – -ODL-to log where ox is the exchange rate volatility. All the Usual Black-Schools Greeks apply. It is worth bearing in mind, however, that foreign exchange markets typically assume a sticky-by-delta implied volatility surface.
This means that as the exchange rate moves, the volatility of an option with a given strike is also assumed to move in such a way that the volatility skew, as a function of delta, does not move. The notional of the option is the number of foreign currency units that the option holder has the right to buy or sell at maturity. So for example, if the notional of a call option is N , then the time t value of the option is N x C(X, t). Delta and the Option When trading a currency option, the price of the option may be paid in units of the domestic currency or it may be paid in units of the foreign currency.
This situation never arises when trading stock options. For example, if you purchase an option on IBM then you will pay for that option in dollars, i. E. The domestic currency, and not in IBM stock which plays the role of the foreign currency. Note that this is a matter of practice as there is nothing in theory to stop me paying for the IBM option in IBM stock. In currency markets, however, and depending on the currency pair, it might be quite natural to pay or the currency option in the foreign currency. When computing your delta it is important to know what currency was used to pay for the currency option.
Returning to the stock analogy, suppose you paid for an IBM call option in IBM stock that you borrowed in the stock-borrow market. Then I would inherit a long delta position from the option and a short delta position position from the option payment. My overall net delta position will still be long (why? ), but less long that it would have been if had paid for it in dollars. The same is true if you pay for a currency option in units of foreign currency. When an option remit is paid in units of the foreign currency and the delta is adjusted to reflect this, we sometimes refer to it as the premium-adjusted delta.
As is probably clear by now, currencies can be quite confusing! And it takes time working in the FAX markets before most people can get completely comfortable with the various market conventions. The one advantage that currencies have over stocks from a modeling perspective is that currencies do not pay discrete dividends and so the (often) ad-hoc methods that have been developed for handling discrete dividends are not needed when modeling cue irenics. Other Deltas There are other definitions of “delta” that are commonly used in FAX-space.
These alternative definitions arise because (i) the value of currency derivatives can be expressed naturally in domestic or foreign currency units and (ii) forwards play such an important role in FAX markets. In addition to the regular Black-Schools delta (which is the usual delta sensitivity) and the premium- adjusted delta we also have: 4 ; The forward delta. This is the sensitivity of the option price with respect to changes in the value of the underlying forward contact with the same strike and maturity as the option. The premium-adjusted forward delta. This is the same as the forward delta but you need to adjust for the fact that the option premium was paid in units of the foreign currency. Note that all of the delta definitions we have discussed thus far have implicitly assumed that we want to hedge the domestic value of the option. However, we might also want to hedge the foreign value of the option. Example 1 (Hedging Domestic Value v Foreign Value) Suppose an FAX trader in Australia buys a SPY/ADD call option from an FAX trader in Japan.
If the Australian trader hedges the ADD value of the option and the SPY trader hedges the SPY value of the option, then they will not put n equal and opposite hedges. For example, suppose the value of the option is 1 00 ADD and the current ADD/SPY exchange rate is 65 SPY per ADD. If the Australian trader has hedged the LAID value of the option and the exchange rate moves from 65 to 67, say, his position will still be worth 100 ADD. However, the SPY value of his position will have changed from 100 -k 65 6, 500 SPY to 67 * 100 = 6, 700 SPY.
So hedging the ADD value of his position clearly does not hedge the SPY value of his position. As a result, we have another set of deltas that correspond to hedging the foreign value of the option. There are also “sticky-by-delta” versions of all these deltas. Clearly then, one must be very careful in specifying and interpreting a delta. The default delta that is quoted in the market place Will, as always, depend on the currency pair and market conventions. Trading systems will generally be cap bled Of reporting these different deltas.
What Does At-the-Money (AT M) Mean? The usual definition of “at-the-money’ in equity-space is that the strike of the option under consideration is equal to the current spot price. But there are other alternative definitions that commonly occur in r-X-space. They are: 1 . At-the-money spot which is our usual definition. 2. At-the-money forward. 3. At-the-money value neutral. This is the strike, K, such that the call value with strike K equals the put value with the same strike, K. 4. At-the-money delta-neutral.
This is the strike, K, where the delta of the call option with strike K is equal to minus the delta of the put option with the same strike K. But as there many different definitions Of delta, we have many possibilities here. Building an FAX Volatility Surface The construction of a volatility surface in PIX space is unfortunately much more difficult than in equity-space. The simple reason for this is that the liquid options in the FAX markets are quoted for a given delta and not for a given strike.
In particular, FAX markets typically quote volatility prices for 1 . ATM options. However, the particular definition of ATM will depend on the cue reentry-pair and market conventions. 2. 25-delta and 1 0-delta strangles. A strangle is a long call option together with a long put option. 3. And 25-delta and 1 a-delta risk-reversals. A risk reversal is a long call option together with a short put option. In each of the strangles and risk reversals, the strike Of the put option will be rower than the strike of the call option.
These strikes then need to be determined numerically using combinations of the quoted volatilities and the designated deltas. Once the various strikes and their implied volatilities have been calculated, curve fitting 5 techniques are then used to build the entire volatility surface. So building an FAX implied volatility surface is considerably more complicated than building an equity implied volatility surface. This appears to be a legacy of how FAX options markets developed and in particular, of how risk reversals and strangles became the liquid securities in the market.
Until recently there have been very few (if any) detailed descriptions of the various conventions in FAX markets and how FAX volatility surfaces are built. More recently, however, some papers have been published on this topic. They can be consulted for further (tedious but necessary) details. The Triangle Relationships The fact that all currency pair combinations can be traded imposes restrictions on the dynamics of the various exchange rates. To see this we will introduce some new notation. In particular, let a/b # Of units Of currency A required to purchase 1 unit Of currency B.