Answers to the Chapter Exercises
CHAPTER 2 Markets Exercises
1. If you need to purchase 2,000 ounces of platinum, you can buy it where the marketmaker is willing to sell it—at USD 976 per ounce.
You would respond to the market quote with, “I would like to buy 2,000 ounces of platinum at a spot price of USD 976 please,” or, more simply, “Mine, 2,000 (ounces) at 976.”
In two business days, you will wire USD 1,952,000 to the dealer and receive delivery (in whatever manner has been arranged for this) of 2,000 ounces of platinum.
2. A marketmaker in spot platinum would want to know (among other things): last trade price, market direction (based on fundamental analysis and/or technical analysis), their position, order flow, news coming out, and other relevant information that might move the spot price of platinum.
CHAPTER 3 Interest Rate Exercises
1. The interest on USD 10,000,000 for 3 months (90 days) at r = 4.80% (where this is a simple actual/360 rate):
USD 10,000,000 × (.0480) × (90/360) = USD 120,000
2. The Euros you would receive if you deposited 40 million Euros for 12 years at a quarterly compounded rate of r = 6.00% (treating each quarter as exactly ¼ of a year):
which should make sense if you know the Rule of 72, which says your money should approximately double when the time frame, t (here, 12 years), and the interest rate, r (here, 6.00%), if multiplied together, gives a number around 72; more on this in the answer to Question 4 of this section.
3. The amount of Pounds Sterling today that are equal to 20,000,000 Pounds in 200 days if the relevant U.K. interest rate (quoted annually) is 8.20% depends on how you discount: Either
GBP 20,000,000/[1 + (.0820)(200/365)] = GBP 19,140,010.48
or
GBP 20,000,000 × [1 – (.0820)(200/365)] = GBP 19,101,369.86
The former would be the more likely calculation for market purposes (because that number of Pounds, deposited at r = 8.20% with simple interest would grow to GBP 20,000,000 in 200 days), although U.S. Treasury Bills and commercial paper typically use the latter convention. (Recall: GBP interest rate calculations typically assume that a year consists of 365 days.)
4. How long will it take for your money to double? More concretely, if you are quoted a rate of interest of 6.00% (compounded annually), how long will it take for your money today (PV) to double (i.e., so that FV = 2 × PV)? This answer explains the Rule of 72, mentioned earlier. This is most easily shown using the continuous interest convention: FV = PV ert. In effect, it asks, “What is that time frame, t*, over which your money (PV) will double (turn into 2 × PV) with a given interest rate?” Writing this out, FV = 2 PV = PV ert*. We can cancel the PVs (as the initial amount of money does not matter for solving this math problem), and we get 2 = ert*; we can then take the natural logarithm (ln) of both sides: ln(2) = .69314718 = ln(ert*) = rt*. Rounding off the left-hand side to .70, we get .70 = rt* . . . which says, for example, that, with continuous interest, if r = 5%, then it will take around t* = 14 years for your money to double, whereas, if interest rates are 7%, it will only take around t* = 10 years for your money to double. We said that not many financial institutions give continuous interest, so it may take a little longer; that fact, in conjunction with the observation that 72 seems nicely divisible by a larger number of integers, gives us the Rule of ‘72’: One’s money should approximately double depending on the rate of interest received according to the following general rule of thumb: r × t* = .72.
CHAPTER 5 Spot Exercise #1
1. The name of the exchange rate quoting convention between U.S. Dollars and Canadian Dollars if we quote USD|CAD: a European quote— even though neither currency has anything to do with Europe. [If, on the other hand, someone wanted this currency pair quoted CAD|USD, it would be referred to as an American quote.]
2. If USD|CAD is quoted S = 1.2000, it means that USD 1 (1 U.S. Dollar) will trade for (e.g., can be bought or sold for) CAD 1.2000 (1.20 Canadian Dollars).
3. If USD|CAD S = 1.2000, then CAD|USD S = (1/1.2000) = .83333333 . . . (though we usually only quote spot prices out four decimal places, so, CAD|USD S = .8333).
4. The name of the foreign exchange rate quoting convention between Australian Dollars and U.S. Dollars (“Aussie”) if we quote AUD|USD (as we do in the interbank market) is an American quote (because it reflects how many American (U.S.) Dollars per one Australian Dollar.
5. If AUD|USD S = .7500, then one Australian Dollar will trade for .75 U.S. Dollars.
Spot Exercise #2
One might guess that the exchange rate between Japanese Yen and Swiss Francs (knowing that Yen are small) would be quoted CHF|JPY or Swiss-Yen or Japanese Yen per one Swiss Franc (as, indeed, it typically is).
One might further guess that the exchange rate between Euros and Swiss Francs is quoted EUR|CHF or Euro-Swiss or Swiss Francs per one Euro.
Spot Exercise #3
1. The price of a Dollar could fall against the Swiss Franc and rise against the Japanese Yen; this would imply USD|CHF went down (the U.S. Dollar weakened against the Swiss Franc) and USD|JPY went up (the U.S. Dollar strengthened against the Japanese Yen). If this is the case, there is no doubt that the Swiss Franc just strengthened against the Japanese Yen (i.e., that CHF|JPY went up).
2. Can you buy U.S. Dollars with Pounds Sterling? Yes, and no, and yes! Of course you can buy Dollars with British Pounds, but, in the interbank market, a professional dealer would never say that; since the quoting convention between U.S. Dollars and Great Britain Pounds in the marketplace involves GBP|USD (i.e., Dollars per Pound), what you trade (that is, the underlying asset) is Pounds. If you want to buy Dollars with Pounds, one would say, “I sell Sterling-Dollar,” or “I sell Cable.” Confusing, isn’t it?
If you wanted to sell Japanese Yen in exchange for Dollars, what would you say?
Spot Exercise #4
Complete the following table of foreign exchange (cross) rates (bold numbers given): 
(Hint: Look back at Figure 1.9. To get oriented, start with USD and JPY.) Start by “inverting” all the quotes you are given (e.g., EUR|USD = 1.2500 so USD|EUR = (1/1.2500) = .8000). Then start looking for other pairs; knowing USD|EUR = .8000 and USD|JPY = 110.00, then EUR|JPY = 110.00/.8000 = 137.50. Similarly, knowing it takes 2.40 CHF per GBP 1 and 96 JPY per CHF 1, then it should take 230.40 JPY per GBP 1. (Answers are rounded to two, four, or six decimal places.)
Spot Triangular Arbitrage Exercises #5
1. Given the following spot prices:
USD|JPY 110.00
USD|CHF 1.2500
CHF|JPY 85.00
triangular arbitrage starts by identifying where one currency pair is (i.e., too high or too low) in relation to the other two. Given the first quote USD 1 = JPY 110.00 and the second quote USD 1 = CHF 1.2500, then the exchange rate between CHF and JPY should follow from CHF 1.2500 = JPY 110.00 or, to put it in market terms, CHF|JPY should = 88.00. If CHF|JPY S = 85.00 in the market, it is too low (relative to where it should be trading). More precisely, the price of one Swiss Franc is lower (in terms of Japanese Yen) than it should be; one would want to BUY CHF and SELL JPY (as a single trade). If you start with CHF, you don’t want to sell them for JPY, so sell them for USD, then sell the USD for JPY, and finally, sell the JPY for CHF. The cash flows would look like this:
Start: CHF 40,000,000 to USD 32,000,000 to JPY 3,520,000,000 back to CHF 41,411,764.71. Your profit in Swiss Francs is CHF 1,411,764.71 (or in percentage terms, you make 3.529412%, which, coincidentally, is exactly
[.8800/.8500 – 1].
See Figure A.1.
FIGURE A.1 Triangular Arbitrage
2. Given the following spot quotes:
EUR|GBP .6850
GBP|USD 1.8420
EUR|USD 1.2500
a triangular arbitrage starting with EUR 100,000,000 and realizing your profit in USD can be done two ways. One could simply do a regular triangular arbitrage starting with EUR and ending with EUR and then converting your EUR profit back into Dollars, or you can “take” your profit along the way.
Let’s try to consider the latter: First, looking at the first two quotes, EUR|USD should = 1.26177 (so the market price is too low; therefore, BUY EUR = SELL USD). Since we start with Euro, this would mean we go from EUR to GBP to USD to EUR. Starting with EUR 100,000,000 gives GBP 68,500,000, which subsequently gives USD 126,177,000.
FIGURE A.2 Triangular Arbitrage
At this stage, you might recognize that with EUR|USD S = 1.2500, all you should have to give up is USD 125,000,000 to get EUR 100,000,000. This would mean that you could take your profit in USD. That amount is USD 1,177,000.
If instead one had gone through the entire triangular arbitrage: EUR to GBP to USD to EUR, the resultant Euro position would have been EUR 100,941,600.00, which, net of the original EUR 100,000,000, gives a profit (in Euro) of EUR 941,600, which, translated at EUR|USD S = 1.2500 gives (as expected) USD 1,177,000. Reality check: 1.26177/1.2500 – 1 = .009416%, which is obvious enough in Euro—but that’s also the ratio of the USD profit to the magnitude of the USD trade: USD 1,177,000/125,000,000. (See Figure A.2.)
CHAPTER 6 Forward Exercise #1
If a stock is trading at S = 100.00, interest rates are r = 4.00 percent, we are thinking about a one-year time frame (t = 1), and, over the course of the year, this stock will pay a quarterly dividend of Div = 0.25, where would the one-year forward price be quoted?
Using our forward relationship:
we would have
F = S + Srt – Div
F = 100 + 100(.04)(1) – (.25) × 4 = 103
Forward Exercise #2
If USD|JPY is trading in the spot market at S = 110.00, Japanese interest rates are rJ = 1.00%, U.S. rates are rUS = 5%, then where would you expect to see the one-year USD|JPY forward price to be quoted?
Forward Pricing/Forward Point Exercises #3
1. The one-year forward outright market in Cable (GBP|USD):
| Spot Price | 1.8021 | — | 1.8025 | (4 pips wide) |
| Forward Points | 118 | — | 115 | (3 pips wide) |
| Forward Price | 1.7903 | — | 1.7910 | (7 pips wide) |
2. The six-month forward market in USD|JPY:
| Spot Price | 112.49 | — | 112.57 | (8 pips wide) |
| Forward Points | 310 | — | 308 | (2 pips wide) |
| Forward Price | 109.39 | — | 109.49 | (10 pips wide) |
3. The one-month forward in USD|CHF:
| Spot Price | 1.2498 | — | 1.2504 | (6 pips wide) |
| Forward Points | 27.25 | — | 26.75 | (½ pip wide) |
| Forward Price | 1.247075 | — | 1.247725 | (6½ pips wide) |
4. If you wanted to sell USD 10,000,000 in one month, how many CHF you would get? Selling USD 10,000,000 with F = 1.247075 (note: You sell on the marketmaker’s bid) means you will get CHF 12,470,750.
Foreign Exchange Forward Exercises #4
1. Undertake a spot-forward arbitrage in the following circumstances:
Do this on a spot notional of EUR 100,000,000 and check your answer.
The one-year forward should be F = 1.2940 (1.03)/(1.04) = 1.28155769
so, BUY EURO FORWARD.
What this means:
BUY EUR FORWARD = SELL USD FORWARD
and SELL EUR SPOT = BUY USD SPOT.
Let’s look at the cash flows.
In the spot market
Borrow and sell EURO spot = buy and deposit USD spot
Sell EUR 100,000,000 and Buy USD 129,400,000
In one year, you’ll owe EUR 104,000,000 and own USD 133,282,000.
Now, if you sell all the USD you have at a forward price of F = 1.2750,
it will translate into EUR 104,534,902 or a Euro profit of EUR 534,902.
BUY EURO FORWARD = SELL USD FORWARD
If you had chosen to take your profit in Dollars, you’d get USD 682,000.
(Reality check: You made a little more than .5%; 1.28156/1.2750 – 1 = .005+%.)
2. What could you say about Turkish interest rates versus U.S. interest rates if the forward points in USD|TRY are positive?
If the USD|TRY forward points are positive, then F > S, which means (given the way this currency pair is quoted) that Turkish interest rates are higher than U.S. interest rates (for the time horizon under consideration).
3. What can you say about the forward points for USD|CHF if the respective (zero) interest rate curves in the respective countries currencies are as pictured in Figure 6.8 (reproduced here as Figure A.3)?
At the short end of the curve, rUS > rCH, so F will be < S and forward points will be negative. Where the interest rate curves first cross, forward points will be zero. In that range in which rCH > rUS, F will be > S and forward points will be positive. They will go to zero once again where the interest rate curves cross a second time, and for longer-dated forwards, the forward points will again be negative (where rUS > r CH).
FIGURE A.3 Interest Rate Curves
4. If GBP|USD three-month forward is F = 1.7568 , U.S. interest rates are 4.00%, U.K. interest rates 7.00%, and we assume three months involves 91 days, then
• What is the GBP|USD spot price? Working our spot-forward relationship backward:
so
F = 1.7568 = S(1 + (.04)(91/360))/(1 + (.07)(91/365))
GBP|USD S = 1.76956748 or probably S = 1.7696
Note: For U.S. rates, we used an “actual/360” day count convention and for U.K. interest rates, we used an “actual/365” day count convention.
• What do you think would happen to this forward price if the Bank of England raises interest rates? It depends on whether the rate hike was anticipated, and whether the magnitude of the rate hike was in line with expectations. If the rate hike was either unanticipated or greater than expected, S will likely go up; on the other hand, if the interest rate hike was anticipated, but the actual increase in U.K. rates was less than expected, S will likely fall. If the spot price is unchanged, though, then we can say that F will go down.
• How much would you have to pay (today) for a three-month off-market forward of F = 1.7200 on a notional of GBP 10,000,000? If paying in USD, it will be the PV(USD.0368 per GBP 1) × GBP 10,000,000 (where this is discounted using rUS) = USD 364,316.36.
CHAPTER 7 Currency Futures Exercises
1. If a U.S. corporation had a firm commitment of two billion in Japanese Yen receivables arriving in December of that year
What futures contracts could they use to “hedge”?
Would they buy or sell them?
How many would they want to trade?
The corporation should SELL 160 (JPY 2,000,000,000/12,500,000) December Japanese Yen futures contracts.
2. In what ways do FX futures contracts differ from FX forward contracts? Exchange-traded versus OTC, standardized versus tailored (with respect to contract size, maturity, delivery dates, and processes, etc.), margined or marked-to-market daily versus (possibly) no cash flows, and the majority of futures contracts do not result in delivery, whereas many forwards do.
CHAPTER 8 Cross-Currency Swaps Exercise
Using the numbers in our earlier example: USD|CHF S = 1.2500, rUS = 5.00% and rCH = 3.00% (for all maturities):
1. Show, in two different ways, that if the U.S. corporation issues a 3.00% annual coupon bond in CHF in Switzerland and does a cross-currency swap, then, effectively, it will be borrowing at 5.00% in USD.
PVing the CHF cash flows (scaling for a notional of CHF 100):
PVing the USD cash flows on an equivalent amount of Dollars = USD 80:
(Note: The USD coupon of “4” is 5% of the notional of USD 80.)
We can alternatively use the FX forward rates (calculated earlier in Chapter 8), turn all the cash flows into one currency (say, CHF here), PV (using Swiss interest rates), and sum. If the result = 0, then there is no “savings” from issuing abroad.
In Year 1 – CHF 3 + USD 4 × F1 (1.2262) = + CHF 1.9048
In Year 2 – CHF 3 + USD 4 × F2 (1.2028) = + CHF 1.8112
In Year 3 – CHF 3 + USD 4 × F3 (1.1799) = + CHF 1.7196
In Year 4 – CHF 3 + USD 4 × F4 (1.1574) = + CHF 1.6296
In Year 5 – CHF 103 + USD 84 × F5 (1.1354) = – CHF 7.6264
PVing and summing
2. Determine the fair USD coupon for a five-year coupon-only FX swap where one counterparty receives the 3.00% CHF on a notional of CHF 125,000,000.
Here, USD c = 2.53875%
CHAPTER 9 Put-Call Parity Exercise #1
If a nondividend-paying stock is trading S = 78.50, the time frame is one year; interest rates are 4.00%, and the one-year 80 strike European call is trading at 80C = 3.75, where should the one-year 80 strike European put be trading?
Alternatively, If S = 78.50, the one-year F = 81.64 (F = S(1 + rt) = 78.50(1.04)), so the 80 call has value or is in-the-money with respect to the forward and has an intrinsic value of 1.64. If you present value the 1.64 (= 1.58) (recalling equation [9.8]) and take that away from the 80 call, you get 2.17.
If it’s trading for .50 more than you think that it should, what trades would you do?
Sell high. Sell the 80 put (+2.67), buy the 80 call (-3.75), and sell the spot (+78.50). This results in a net cash inflow today of 77.42. Deposited in the bank (at 4.00%), it will grow into 80.52. At expiration, you would have to buy back the underlying at the strike price of X = 80.00, so your profit is .52. Why did you make more than .50 (your original “edge”)? Because you waited to “capture your profits,” you would realize an amount equal to the future value of your original “edge”: .50 × (1.04) = .52.
FX Put-Call Parity Exercise #2
USD|JPY S = 110.00, rJ = 1.00 percent, rUS = 3.00 percent, t = 3 months (t = ¼ = .25)
1. For the 107.50 strike (X = 107.50) three-month European options, by how much will the call and put differ and which will be more valuable?
Using equation [9.11]:
Since the call is in-the-money (forward), the 107.50 C is more valuable than the 107.50 put.
Alternatively, using equation [9.12]:
and
(F – X)/(1 + r1t) = 1.954/1.0025 = 1.95
2. If the 107.50 call is trading at JPY 2.62 per USD 1, where should the put be trading? The 107.50 call is in-the-money forward, so it should be trading over the put by 1.95, or the 107.50 put should be trading for JPY .67.
FX Option Premium Exercise #3
You telephone an FX option dealer/bank and ask for a three-month (
= .25) at-the-money European option on EUR|JPY on a notional of EUR 40,000,000. Spot is currently trading at S = 140.00, Japanese interest rates are 1.00%, and Euro rates are 5.00%.
= .25) at-the-money European option on EUR|JPY on a notional of EUR 40,000,000. Spot is currently trading at S = 140.00, Japanese interest rates are 1.00%, and Euro rates are 5.00%.The dealer/bank quotes the premium in Yen pips: 263.
1. What is the strike price, X?
Using simple interest: X = F = 138.62 (rounded) (more precisely 138.617284)
2. What is this option (i.e., call or put, on what currency)?
This is either a Euro call or a Euro put; for valuation, it doesn’t matter; at-the-money (forward) European calls and puts have the same value.
This is an option to exchange EUR 40,000,000 for JPY 5,544,800,000.
3. Calculate the option premium in terms of
Total EUR: EUR 751,428.57; total JPY: JPY 105,200,000.
Percent of Euro face: 1.88% (really 1.87857%); percent of Yen face: 1.90% (really 1.89727%).
Euro pips per Yen: (remember, Euro-Yen pips go out 6 decimal places): 135.52.
FX Option Breakeven Graph Exercise #4
1. Sketch the P/L (profit/loss) or breakeven graph of a long three-month 125.00 put with a premium of 6.58. Identify the breakeven spot price, S*. Does it matter whether this is a three-month option or a one-year option? Briefly explain. See Figure A.4.
Breakeven graphs are typically drawn independent of the expiration date (so it doesn’t matter, for our graph, whether expiration is in three months or one year), but it will impact the cash flows and, therefore, our actual profit/loss.
FIGURE A.4 Solution to FX Option Breakeven Graph Exercise #4 Profit/Loss Graph for the 125 Put.
CHAPTER 10 Exotic Option Exercise
What would you label a put option with spot at S = 22.50, strike price at X = 20.00, and an out barrier at 21.00? What if, instead, it was a call?
I would label this option “Stupid.” Think about it. If spot trades down to or past 21.00, the option disappears, and this put would only have value if the spot price ends up below the strike price of X = 20.00. Nevertheless, we would label such an option, which goes out as the spot price moves in the in-the-money direction a “kick out” put. With these same parameters (S = 22.50, X = 20.00, B = 21.00), a call option would be referred to, among other things, as a “knock out call.”
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