Currency Options

1
Currency Options

• Payoffs of foreign currency forward contracts are
  linear in the future spot rate and can have a
  negative expiration value
• Value of a forward purchase at T
• ET,CHF/ - FT,CHF/ gt or lt 0

• We may prefer a non-linear contract that
  precludes negative expiration values. For instance

• Where X is the strike price
• Call option gives the holder the right to buy a
  stated number of foreign currency at the strike
  price from the counterparty (the writer of the
  option)
• At time T, in the case of a European-style option
• At any time ? T in the case of a American-style
  option

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Example

• You buy a call on one at CHF/ 1.6 expiring at
  T October 30, 2007. Hence, X 1.6
• If, on Oct 30 ECHF/ 1.58, you will not
  exercise the option
• If, on Oct 30 ECHF/ 1.62, you will exercise
  the option, buy 1 for 1.6CHF and then sell 1
  for 1.62CHF, making a profit of 0.02CHF
• A European call option will be exercised at T iff
  ECHF/ - X gt 0

• The value of the call option at T is
• CT Max (ET,CHF/ - X, 0)

3
Put options, American options, etc

• Put option gives the holder the right to sell a
  stated number of foreign currency at the strike
  price from the counterparty (the writer of the
  option)
• The value of a put option at T is
• PT Max (X - ECHF/, 0)
• Early exercise at t lt T of an American call
  option is rational if
• (Et,CHF/ - X) gt 0, a positive value dead
• The option market value is no higher than the
  value dead
• Example You buy an american call on one at
  XCHF/ 1.6 expiring at the end of October. If
  today E 1.58, you wait. If E 1.62 but market
  price Ct0.04, you are better off selling than
  exercising
• A call can be used to insure a foreign currency
  debt against a high (depreciated) exchange rate
• You pay an insurance premium (up front) to buy
  the option

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Money Supply

• Money Supply (Ms)
• Ms Currency Checking Deposits
• This is the monetary aggregate called M1
• M1 in Switzerland in July 2007 262348 mil CHF.
  This is roughly 23 of GDP
• An economys money supply is controlled by its
  central bank
• How does the central bank conduct monetary
  policy?
• A look at the SNB
• Hence, most central banks set directly the
  reference nominal interest rate (three-month
  Libor for the SNB, the federal fund rate for the
  FED, etc)
• So, how is money supply controlled?

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Aggregate Money Demand

• The aggregate demand for money can be expressed
  as
• Md P x L(R,Y)
• where
• P is the price level
• Y is real national income
• L(R,Y) is the aggregate real money demand
• We write it as
• Md/P L(R,Y)
• -

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Equilibrium in the money market
Interest rate, R
If the CB raises R
If the CB leaves R unchanged
Real money holdings
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Price stickiness

• In the short run, the price level P is given. It
  does not adjust instantaneously
• Why?
• Some prices adjust instantaneously agricultural
  goods traded in markets, commodities (oil, raw
  materials, etc.)
• Other prices do not adjust instantaneously wages
  are negotiated periodically, prices on catalogues
  (IKEA)
• Wages are a large fraction of the cost of
  producing goods and services

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Equilibrium exchange rate
ECHF/
Return on CHF deposits
Expected return on deposits
0
Rates of return (in CHF terms)
CHF real money holdings
9
Domestic monetary expansion, M2 gt M1
ECHF/
The CHF/ exchange rate depreciates
Return on CHF deposits
1'
Expected return on deposits
E1
0
U.S. real money holdings
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Foreign monetary expansion
The CHF/ exchange rate appreciates
1'
Increase in European money supply
2'
E2
0
1
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The /Yen Exchange Rate
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The / Exchange Rate
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The CHF/ exchange rate
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The Exchange Rate in the Long Run

• Long-run analysis
• The price level is perfectly flexible
• Prices and wages have enough time to adjust to
  their market-clearing levels
• Output is at its full employment level Y Yf
• Price level in the long run

• Long-run neutrality of money An increase in a
  countrys money supply causes a proportional
  increase in its price level
  

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The Law of One Price

• Law of one price (LOP)
• Identical goods sold in different countries must
  sell for the same price when their prices are
  expressed in terms of the same currency.
• This law applies only in competitive markets free
  of transport costs and official barriers to trade
• PiCHF (ECHF/) x (Pi)
• PiCHF is the CHF price of good i when sold in
  Switzerland
• Pi is the corresponding euro price in Europe

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Purchasing Power Parity

• Theory of Purchasing Power Parity (PPP)
• The exchange rate between two counties
  currencies equals the ratio of the countries
  price levels
• ECHF/ PCHF/P
• PCHF is the CHF price of a reference commodity
  basket sold in Switzerland
• P is the euro price of the same basket in Europe

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Purchasing Power Parity

• By rearranging, one can obtain
• PCHF (ECHF/) x (P)
• PPP asserts that all countries price levels are
  equal when measured in terms of the same currency
• The law of one price applies to individual
  commodities, while PPP applies to the general
  price level
• If the law of one price holds true for every
  commodity, PPP must hold automatically for the
  same reference baskets across countries

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Relative PPP

• Take percent deviations of PPP
• (ECHF/,t - ECHF/, t 1)/ECHF/, t 1 ?CHF, t
  - ?, t
• where ?t inflation rate
• It states that the percentage change in the
  exchange rate between two currencies over any
  period equals the inflation differential
• If ?CHF, t gt ?,t the CHF/ exchange rate should
  depreciate in the long run

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Long-Run Effects of a Permanent Change in Money
Supply

• A permanent increase in a countrys money supply
  causes a proportional long-run depreciation of
  its currency against foreign currencies
• In the long run

• A permanent increase in Ms raises PCHF which
  raises ECHF/ in the long run

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Short-Run Effects
Long-Run Effects of a Permanent Change in Money
Supply

• Take P and Y as given
• Suppose MCHF rises. Then
• RCHF falls while R unchanged
• By the IPC
• RCHF R (EeCHF/ - ECHF/)/ECHF/
• it follows that (EeCHF/ - ECHF/)/ECHF/ falls
• EeCHF/ depreciates proportionally to MCHF
• This implies that the spot exchange rate ECHF/
  depreciates more than the expected future
  exchange rate EeCHF/
• This is the Exchange-Rate Overshooting the
  short-run response is greater than the long-run
  response

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Exchange Rate Volatility
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Subsequent Adjustment Process

• In the long run, PCHF rises, MCHF/PCHF goes back
  to its initial level
• RCHF increases over time
• EeCHF/ is unchanged
• By the IPC
• RCHF R (EeCHF/ - ECHF/)/ECHF/
• it follows that (EeCHF/ - ECHF/)/ECHF/
  increases, which implies that ECHF/ falls
  (appreciates)
• In the long run, the exchange rate has
  depreciated proportionally to the increase in the
  money supply

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Short- and Long-Run
CHF return
E2
E4
E1
1'
M2CHF P1CHF
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A Long-Run Exchange Rate Model Based on PPP

• IPC
• RCHF R (EeCHF/ - EeCHF/ ) / EeCHF/
• Relative PPP
• (EeCHF/ - ECHF/)/ECHF/ ?eCHF - ?e
• This implies the Fisher relationship
• RCHF - R (EeCHF/ - ECHF/)/ECHF/ ?eCHF
  - ?e
• The international interest rate differential is
  the difference between expected national
  inflation rates

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Empirical Evidence on the Fisher Relationship
Switzerland
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Empirical Evidence on the Fisher Relationship
The United States
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Empirical Evidence on the Fisher Relationship
Switzerland and the United States
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Empirical Evidence on the Fisher Relationship
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Empirical Evidence on PPP and the Law of One
Price

• Big Mac Currencies, published by The Economist
• Started in 1986 it is a survey of Big Mac
  hamburger prices at McDonalds restaurants around
  the world
• It is a homogeneous good. Hence the Law of One
  Price would apply
• Evidence The Economist

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Big Mac

• Column 1 Big Mac price in local currency Px
• Column 2 Px multiplied by E/x, the U.S.
  dollar/local currency price exchange rate. This
  is the dollar price of the Big Mac in country X
• Column 3 Px/ PUS. This is the implied PPP of
  the U.S. dollar, EPPP/X
• Column 3 (EPPP/X - E/X)100 / E/X
• This is the under(-)/over() valuation of
  currency X against the U.S. dollar
• The CHF is 53 overvalued relative to the US
  Dollar

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Big Mac (cont.)

• China most undervalued, Iceland the most
  overvalued
• If you can keep the Big Mac fresh, buy it in
  China for the equivalent of 1.45 and sell it in
  Iceland for 7.6!
• Trade barriers, transport costs and differences
  in taxes
• Use of non-traded goods and services, like labor
  and rent contribute about 60 of the price of the
  Big Mac
• Most expensive Big Mac Iceland, Norway,
  Switzerland, Denmark
• Least expensive Big Mac Sri Lanka, Indonesia,
  Hong Kong

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Explaining the Problems with PPP

• The failure of the empirical evidence to support
  the PPP and the law of one price is related to
• Trade barriers and transport costs
• Nontradable goods
• Departures from free competition
• International differences in price level
  measurement

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Trade Barriers and Transport Costs

• Transport costs and governmental trade
  restrictions make trade expensive
• Equivalent to 170 tariff in advanced economies

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Non-tradable goods

• The domestic prices of non-tradable goods can be
  very different when expressed in the same
  currency
• Example housing price in Lausanne and in New
  Delhi
• No arbitrage
• Non-tradable goods account for about 50 of GNP
  and hence 50 of the CPI
• Non-tradable goods are more expensive in richer
  countries

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Balassa-Samuelson Model

• In poorer countries, productivity in tradable
  goods and therefore wages are lower
• Prices of non-tradable goods are lower
• PN gt E PN
• Where PN is the foreign price of non-tradable
  goods
• Because non-tradable goods account for almost 50
  of the CPI (consumer price index)
• P gt E P

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Explaining the Problems with PPP