MANAGING THE FOREIGN EXCHANGE RISK Dr. Jean-François VERDIE Professor of Finance Toulouse Business School, France [email protected] [email protected]

MANAGING THE FOREIGN EXCHANGE RISK

Dr. Jean-François VERDIEProfessor of Finance

Toulouse Business School, [email protected]

mailto:[email protected]

FOREIGN EXCHANGE RISK

LESSON CONTENT

• WHAT IS FOREIGN EXCHANGE RISK

• EUR/USD VOLATILITY

• SOLUTIONS TO MANAGE THE FER

The Euro

http://www.economagic.com/chartg/fedny/day-fxus2eu.gif

INDICATIVE TEXTS

• COPELAND « EXCHANGE RATES AND INTERNATIONAL FINANCE » ADDISON WESLEY, 1994.

• BREALEY / MYERS « PRINCIPLES OF CORPORATE FINANCE », MAC GRAW HILL, 7th edition, chapter 28.

• EITEMAN / STONEHILL / MOFFETT « MULTINATIONAL BUSINESS FINANCE » ADDISON WESLEY, 11th edition, chapters 4 to 10.

• WEB SITES : ECB, BIS, IMF, World Bank,…


THE HIDDEN DANGER !


• MEASURING THE FER

• ECONOMIC EXPOSURE OR POSITION

• ACCOUNTING EXPOSURE OR POSITION

• COMMERCIAL/FINANCIAL POSITION = TRANSACTION EXPOSURE


• ECONOMIC POSITION : ANALYSE THE CURRENCIES FLUCTUATIONS IMPACTS ON FIRM ’S VALUE

• ACCOUNTING POSITION : ANALYSE THE CURRENCIES FLUCTUATIONS IMPACTS ON FINANCIAL STATEMENTS


• TRANSACTION POSITION – FOREIGN EXCHANGE EXPOSURE LINKED TO

COMMERCIAL AND FINANCIAL TRANSACTIONS

– USE THE BALANCE SYSTEM TO ESTIMATE THIS POSITION

– CALCULATE THE NET POSITION

– QUALIFY THE NET POSITION • CLOSED / OPENED

• LONG / SHORT

Long and Short Exposures

• A company that is, for example, long the pound, has pound denominated assets that exceed in value their pound denominated liabilities.

• A company that is short the pound, has pound denominated liabilities that exceed in value their pound denominated assets.

Covered Exposure

• If the exposure to foreign exchange risk is completely eliminated through hedging, then the exposure is covered

net position = 0 currency balance is closed

MANAGING FER

• CHOOSING A STRATEGY

• 3 CHOICES DEPENDING ON THE COMPANY POLICY (RISK AVERSE OR NOT)– HEDGING SYSTEMATICALLY– SPECULATION– SELECTIVE STRATEGY

MANAGING FER

• TO IMPLEMENT A SELECTIVE STRATEGY, YOU NEED ANTICIPATIONS !!!

• TO BUILD ANTICIPATIONS, YOU NEED INFORMATIONS (USEFUL !)

FOREIGN EXCHANGE RATES FORECASTING

• FUNDAMENTAL ANALYSIS– INTEREST RATES– EXPECTED INFLATION– BALANCE OF PAYMENTS ANALYSIS– ECONOMIC GROWTH– USD INTERNATIONAL ROLE– OTHER FACTORS : POLITICAL …

• TECHNICAL ANALYSIS eg CHARTISM

MANAGING FER

• EXTERNAL « TOOLS » FOR MANAGING FER ==> ACT ON FOREIGN EXCHANGE MARKETS– OVER THE COUNTER MARKETS (OTC)

• SPOT (CASH)

• CURRENCIES FORWARD / SWAPS

• CURRENCIES OPTIONS (SINGLE, EXOTIC)

– ORGANISED MARKETS (CBOT, LIFFE, …)• CURRENCIES FUTURES CONTRACTS

• CURRENCIES OPTIONS CONTRACTS

MANAGING FER

• FOREIGN EXCHANGE MARKET : A FEW RULES– OBJECTIVES– HUGE TURN OVER– OPERATORS– MOST TRADED CURRENCIES– SPOT RATES

MANAGING FER

• OBJECTIVES – ALLOW INTERNATIONAL TRANSACTIONS

(EXPORTS, IMPORTS, FDI)– INTERNATIONAL DIVERSIFICATION OF

FINANCIAL PORTFOLIOS– HEDGING, SPECULATION, ARBITRAGE

The Foreign Exchange Market is unique not just because of its geographic dispersion, but also because of its extreme liquidity and tremendous volume – around $1.4 T PER DAY !!

$600B in Spot market Transactions

$800 in Derivative Market

$200B in Forwards

$500B in Swaps

$100B in Options (increasing)

Name % of Volume

Deutsche Bank 17

UBS 12.5

Citigroup 7.5

HSBC 6.4

Barclays 5.9

Merrill Lynch 5.7

JP Morgan Chase 5.3

Goldman Sachs 4.4

ABN Amro 4.2

Morgan Stanley 3.9

The ten most active traders account for 73% of the volume.

USD/JPY20%

USD/EUR31%

EUR/All8%

USD/Other17%

USD/AUD4%

USD/CAD4%USD/CHF

5%USD/GBP

11%

US currency was involved in 89% of transactions, followed by the Euro (37%), the yen (20%) and sterling (17%). ( Note that volume percentages should add up to 200% - 100% for all the sellers, and 100% for all the buyers).

Spot Exchange Rates

• Spot transactions are done immediately. A spot rate is the current domestic currency price of a foreign currency

PAY ATTENTION TO THE UNITS!!!

EUR/USD = 1.3646 USD/EUR = .7328

(Dollars per Euro) (Euros per Dollar)

Arbitrage

• Suppose you see the following currency prices EUR/USD = 1.3656 (SOCGEN, London)

EUR/USD = 1.3500 (HSBC, New York)

Buy Euro in New York and then Sell in London

Profit = $0.0156 per $1 Traded

Cross Rates

Suppose you see the following Currency prices:

JPY/USD = .0097 ($/Y)

EUR/USD = 1.3646 ($/E)

Given the above currency prices, the implied Cross Rate is

.0097/1.3646 = .0071 (E/Y) = JPY/EUR

MANAGING FER

• OPERATORS– PRIVATE PEOPLE (PEANUTS !) – FIRMS (A LITTLE MORE !)– BROKERS– BANKS AND FINANCIAL INSTITUTIONS :

MUTUAL FUNDS, PENSION FUNDS, HEDGE FUNDS

– CENTRAL BANKS

HEDGING STRATEGIES

• TO HEDGE THEIR POSITION, EUROPEAN FIRMS USE :– CURRENCIES EUR/USD FORWARD (OTC)

AND FUTURES (ORGANIZED)– CURRENCIES EUR/USD OPTIONS (OTC AND

ORGANIZED)

EUR/USD FORWARD MARKET

• OBJECTIVE : TO KNOW AT T0 THE PRICE (RATE) FOR AN AMOUNT OF CURRENCY YOU WANT (OR NEED) TO BUY OR SELL LATER.

FORWARD = OBLIGATION TO TRADE(TO DELIVER OR TO RECEIVE THE CURRENCY)

FORWARD MARKET

• EUROPEAN EXPORTER DEALING IN USD (AMOUNT : 1 MILLION USD / MATURITY : 3 MONTHS)

• USD LONG / EURO SHORT POSITION• MARKET SITUATION AT T0 :

– SPOT EUR/USD : 1.0550-70– EURO 3M INTEREST RATES : 3.50%-3.80%– USD 3M INTEREST RATES : 5.50%-5.75%

FORWARD MARKET

• THE BANKER MAKES THE FOLLOWING FORWARD PRICE :

• EUR/USD 3 MONTHS : 1.0640 (USD per unit of EUR)– DO YOU DEAL ?– DO YOU OBTAIN THE FAIR PRICE ?

FORWARD MARKET

• IF YOU DEAL WITH THE BANKER, YOU ARE SURE, 3 MONTHS LATER, TO BUY EUR AND SELL USD AT THE PRE DETERMINED RATE : EUR/USD 1.0640

• SAME IS TO SAY THAT YOU SELL USD AND BUY EUR AT : 1/1.0640 = USD/EUR 0.9398

FORWARD MARKET

• HOW TO CALCULATE THE FORWARD RATE ?

• USE ARBITRAGE TO SOLVE TO THE PROBLEM

EUR/USD FORWARD PRICING

• C : SPOT RATE EUR/USD

• CT : FORWARD RATE EUR/USD

• i : USD INTEREST RATE

• i* : EUR INTEREST RATE (BASE)

• j : MATURITY (REAL NUMBER OF DAYS)


• ARBITRAGE OPPORTUNITY– YOU ARE EUROPEAN (FRENCH !!!)– YOU HAVE 1 000 000 EUROS TO INVEST– TWO STRATEGIES

• INVEST IN EUROPE

• CONVERT EUR INTO USD/INVEST IN USD MONEY MARKET/CONVERT USD INTO EUR AT THE FUTURE SPOT RATE


• FIRST STRATEGY : YOU INVEST IN EUROLAND

• j DAYS LATER, YOU ’LL RECEIVE :

• ???????????????????????????????????


• SECOND STRATEGY : INVEST IN USA

• j DAYS LATER, YOU ’LL RECEIVE :

• ???????????????????????????????????


• THEORITICAL FORWARD PRICE GIVEN BY AN ARBITRAGE IN EFFICIENT MARKET (LOOK AT ME !!)

• CT = C + e

• e > 0 ==> premium (i* < i)

• e < 0 ==> discount (i* > i)

CURRENCIES FORWARD

• ADVANTAGES– HEDGING TOOL

– RELATIVELY LOW COST : NO PREMIUM TO PAY LIKE FOR OPTIONS

• INCONVENIENTS– THE RATE IS FIXED : OPPORTUNITY LOSSES

– OBLIGATION TO TRADE : DANGEROUS GAME

Hedging With a Currency Future is typically the same !

• To hedge a foreign exchange exposure, the customer assumes a position in the opposite direction of the exposure.

• For example, if the customer is long the pound, they would short the futures market.

• A customer that is long in the futures market is betting on an increase in the value of the currency, whereas with a short position they are betting on a decrease in the value of the currency.

How an Order is Executed (Figure from the CME)

CURRENCIES OPTIONS

• OPTION = DERIVATIVE ASSET =VALUE DEPENDS ON THE VALUE OF THE UNDERLYING ASSET

• OPTION BUYER HAS THE RIGHT NOT THE OBLIGATION TO TRADE

• FORWARD/FUTURES CARRY BOTH THE RIGHT AND THE OBLIGATION

CURRENCIES OPTIONS

• OPTION BUYER HAS A PRIVILEGED POSITION IN THE BILATERAL TRANSACTION

• OPTION SELLER PROVIDES THE PRIVILEGE WHICH OPTION BUYER ENJOYS

• OPTION SELLER IS CALLED THE « WRITER »

CURRENCIES OPTIONS

• OPTION TO BUY OR OPTION TO SELL• CALL = OPTION TO BUY• PUT = OPTION TO SELL• ASYMMETRY 4 POSSIBILITIES

– BUY A CALL– BUY A PUT – SELL A CALL – SELL A PUT

CURRENCIES OPTIONS

• STRIKE PRICE IS THE AGREED PRICE FOR THE PURCHASE OR SALE

• MATURITY DATE IS THE FINAL DATE AT WHICH THE BUYER CAN EXERCISE THE OPTION

• AMERICAN OPTION / EUROPEAN OPTION

CURRENCIES OPTIONS

• UNDERLYING ASSET IS THE ASSET TO BE PURCHASED OR SOLD– STOCKS AND STOCKS INDICES– COMMODITIES (agricultural, energy products)– BONDS / INTEREST RATES– CURRENCIES– OPTIONS !– …

CURRENCIES OPTIONS

• OPTION BUYER ’S ASYMMETRIC PRIVILEGE MUST HAVE A PRICE

• THIS PRICE IS CALLED THE PREMIUM

• OPTIONS CAN BE STANDARDISED AND EXCHANGE TRADED OR BILATERAL AND TAILORMADE CONTRACTS (OTC)

ProfitProfit

SPOT RATESPOT RATE

0

Call WriterCall Writer

Call HolderCall Holder

Profit Profiles for CallsProfit Profiles for Calls

Profit Profiles for PutsProfit Profiles for Puts

0

Profits

SPOT RATE

Put Writer

Put Holder

How to know if you sell or buy a currency ?

• Use the « + » and « -) system

• + : buy, option to buy, call

• - : sell, option to sell, put

EUR/USD OPTIONS HEDGING

• ASSUMPTIONS– SPOT PRICE = FORWARD = STRIKE = 1.00

EUR/USD– EUROPEAN EXPORTER DEALING IN USD– PAYMENT MATURITY : 3 MONTHS

EUR/USD OPTIONS

• 3 STRATEGIES TO MANAGE THE SITUATION– NO HEDGING– HEDGING WITH THE FORWARD MARKET :

PRICE = 1.00 EUR/USD– HEDGING WITH A CALL EUR / PUT USD

• MATURITY : 3 MONTHS (EUROPEAN)• STRIKE = 1.00 EUR/USD• PREMIUM = 2%

EUR/USD OPTIONS

• 2 CASES TO ANALYSE– EXPORTER IS PAID : NO CREDIT RISK OR

COUNTRY RISK • IN THIS CASE, YOU RECEIVE THE USD. YOU

WANT TO GET THE BEST RATE

– EXPORTER IS NOT PAID : CREDIT RISK OR COUNTRY RISK

• IN THIS CASE, YOU DON ’T RECEIVE THE USD. SO, YOU HAVE NOTHING TO SELL.

EUR/USD OPTIONS

• SEE GRAPHS TO HELP YOU TO UNDERSTAND THE ADVANTAGES OF CURRENCIES OPTIONS COMPARED TO OTHER STRATEGIES

EUR/USD OPTIONS

• RESULT = WITH CURRENCIES OPTIONS, YOU ARE ABLE :– TO MANAGE THE CURRENCY VOLATILITY– TO MANAGE CONDITIONAL OPERATIONS

(CREDIT OR COUNTRY RISKS)

EUR/USD OPTIONS

• ONLY INCONVENIENT = THE PREMIUM THE BUYER HAS TO PAY TO THE WRITER

• ==> ZPO : ZERO PREMIUM OPTION– OBJECTIVE IS TO REDUCE THE COST OF

HEDGING– CREATED BY CITYBANK IN 1985

EUR/USD OPTIONS

• How to choose the premium ? The objective is to link the premium to pay to chosen strike.

• At The Money options (ATM) : strike equal to the market conditions (spot or forward).

• In The Money options (ITM) : strike is better than the market conditions.

• Out The Money options (OTM) : strike is defavourable compared to the market conditions

EUR/USD OPTIONS

• EUROPEAN EXPORTER CASE (currency forward rate = 1.00)

• STRATEGY – BUY CALL EUR/PUT USD STRIKE = 1.10– SELL PUT EUR/CALL USD STRIKE = 0.90

• 2 CASES TO ANALYSE– EXPORTER IS PAID– EXPORTER IS NOT PAID

EUR/USD OPTIONS

• TENDER ZPO FOR LONG AND CONDITIONAL POSITION

• STRATEGY TO IMPLEMENT– BUY CALL EUR/PUT USD STRIKE = 1.20– SELL PUT EUR/CALL USD STRIKE = 1.00– BUY PUT EUR/CALL USD STRIKE = 0.90

EUR/USD OPTIONS

• 2 CASES TO ANALYSE– EXPORTER IS PAID ==> LONG POSITION IN

USD / SHORT POSITION IN EUR– EXPORTER IS NOT PAID ==> OBJECTIVE =

NOT TO GO TO THE « DEEPEST HELL »

EUR/USD OPTIONS

• OPTIONS CAN ALSO BE USED FOR SPECULATIVE STRATEGIES

• EXAMPLE : STRADDLE WHICH CONSISTS IN – BUYING A CALL EUR/PUT USD– BUYING A PUT EUR/CALL USD– SAME STRIKE / SHORT MATURITY– AMERICAN OPTIONS

EUR/USD OPTIONS

• OTHER STRATEGIES : SPREAD– VERTICAL SPREADS

• OPTIONS WITH DIFFERENT EXERCISE PRICES BUT OTHERWISE IDENTICAL

– HORIZONTAL SPREADS• OPTIONS WITH DIFFERENT TIMES TO MATURITY BUT

OTHERWISE IDENTICAL

– DIAGONAL SPREADS• OPTIONS WITH DIFFERENT TIMES TO MATURITY AND

DIFFERENT EXERCISE PRICES

EUR/USD OPTIONS

• EXAMPLES – BULL SPREADS : SPREADS WHICH BENEFIT

FROM PRICE INCREASES• BULL CALL SPREAD

– BUY CALL EUR/USD AT PE1

– SELL CALL EUR/USD AT PE2 (PE1 < PE2)

– BEAR SPREADS : SPREADS WHICH BENEFIT FROM PRICE DECREASES

EUR/USD OPTIONS

• OTHER SPECULATIVE STRATEGIES : SYNTHETIC OPTIONS– SYNTHETIC LONG FUTURES

• BUY CALL EUR/USD AND SELL PUT EUR/USD

• SAME STRIKE

• BULLISH ANTICIPATIONS

– SYNTHETIC SHORT FUTURES• BUY PUT EUR/USD AND SELL CALL EUR/USD

• SAME STRIKE

• BEARISH ANTICIPATIONS

EUR/USD OPTIONS

• 2nd GENERATION OF CURRENCIES OPTIONS : EXOTIC (SEE HULL, OPTIONS FUTURES AND OTHER DERIVATIVES))– AVERAGE RATE OPTIONS (ASIATIC)– LOOKBACK OPTIONS– COMPOUND OPTIONS (OPTION ON

OPTION)– KNOCK IN AND KNOCK OUT OPTIONS ….

EUR/USD OPTIONS

• CURRENCIES OPTIONS PRICING : SOME ELEMENTS

• PREMIUM (W) = INTRINSIC VALUE + TIME VALUE

• INTRINSIC VALUE• IT ONLY EXISTS FOR ITM OPTIONS

• LINEAR FUNCTION DEPENDING ON THE DIFFERENCE BETWEEN THE STRIKE PRICE OF THE OPTION AND THE LEVEL OF THE UNDERLYING FORWARD OR SPOT

EUR/USD OPTIONS

• TIME VALUE– DIFFERENCE BETWEEN THE PREMIUM OF

THE OPTION AND THE INTRINSIC VALUE– FUNCTION OF

• THE IMPLIED VOLATILITY OF THE CURRENCIES

• THE MATURITY OF THE OPTION : THE LONGER IT IS THE MORE UNCERTAINTY WE HAVE

Option Pricing and Valuation

• The total value (premium) of an option is equal to the intrinsic value plus time value.

• Intrinsic value is the financial gain if the option is exercised immediately.

– For a call option, intrinsic value is zero when the strike price is above the market price

– When the spot price rises above the strike price, the intrinsic value become positive

– Put options behave in the opposite manner

– On the date of maturity, an option will have a value equal to its intrinsic value (zero time remaining means zero time value)

• The time value of an option exists because the price of the underlying currency, the spot rate, can potentially move further and further into the money between the present time and the option’s expiration date.

Intrinsic Value, Time Value & Total Value for a Call Option on British Pounds with a Strike Price of $1.70/£

1.69 1.70 1.71 1.72 1.731.681.671.660.0

1.0

2.0

3.0

4.0

5.0

Spot rate ($/£)

Option Premium(US cents/£)

3.30

5.67

4.00

6.0

1.74

1.67

Total value

Intrinsic value

Time value

-- Valuation on first day of 90-day maturity --

EUR/USD OPTIONS

• FACTORS INFLUENCING THE PRICE OF AN OPTION– PRICE OF THE UNDERLYING ASSET

• POSITIVE ON THE CALL PREMIUM

• NEGATIVE ON THE PUT PREMIUM

– STRIKE PRICE• NEGATIVE ON THE CALL PREMIUM

• POSITIVE ON THE PUT PREMIUM

EUR/USD OPTIONS

• FACTORS INFLUENCING THE PRICE OF AN OPTION– TIME TO MATURITY

• POSITIVE ON CALL PREMIUM

• POSITIVE ON PUT PREMIUM

– IMPLIED VOLATILITY• POSITIVE ON CALL PREMIUM

• POSITIVE ON PUT PREMIUM

EUR/USD OPTIONS

• FACTORS INFLUENCING THE PRICE OF AN OPTION– INTEREST RATES

• POSITIVE ON CALL PREMIUM : WHY ?– THE SELLER OF A CALL MUST PURCHASE A CERTAIN

QUANTITY OF THE UNDERLYING ASSET TO COVER HIMSELF IN CASE OF EXERCISE. THE PURCHASE OF THE UNDERLYING ASSET MUST BE FINANCED

• NEGATIVE ON PUT PREMIUM

EUR/USD OPTIONS

• EXAMPLE : CALL USD / PUT EUR WRITER (SELL USD / BUY EUR IF EXERCISED)– TO HEDGE YOUR POSITION, YOU NEED TO BUY A CERTAIN

AMOUNT OF USD (how much ????????????)– TO DO THIS, YOU BORROW A CERTAIN AMOUNT OF EUR– YOU SELL EUR/BUY USD ON THE SPOT.– USD ARE INVESTED ON THE MONEY MARKET;SO (EVERYTHING BEING EQUAL) :

=> IF INTEREST RATES ON USD INCREASE, PREMIUMS FOR CALL USD/EUR DECREASE=> IF INTEREST RATES ON EUR INCREASE, PREMIUMS FOR CALL USD/EUR INCREASE

EUR/USD OPTIONS

• CURRENCIES OPTIONS PRICING IN PRACTICE– MARKET PARAMETERS : INTEREST RATES,

VOLATILITY, …– CUSTOMER PARAMETERS : TIME TO

MATURITY, STRIKE– ==> STATISTICAL MODEL– ==> RESULT = PREMIUM TO PAY/RECEIVE

EUR/USD OPTIONS

• MODELS– COX / ROSS / RUBINSTEIN : BINOMIAL

MODEL– GARMANN / KOHLHAGEN 1982 :

DERIVATED FROM BLACK AND SCHOLES FORMULA

Currency Option ValuationOption valuation involves the mathematics of

stochastic processes. The term stochastic stochastic means random ; stochastic processes model

randomness.

Myron Scholes and Fischer Black

Binomial Option PayoffsValuing options prior to expiration

Given : You are a resident of Japan. You want to buy a European Call on 1 USD.

The current spot rate S is 100 USD/JPY

The contract has an exercise price K = to the expected future spot exchange rate (E[S]), which is also 100 USD/JPY,


Now, assume two equally likely possible payoffs: 90 USD/JPY or 110 USD/JPY, at the expiration of the contract 90¥//$

110¥//$

100¥//$

.5

.5


What do you do if the yen price of $ is 90?

90¥//$

110¥//$

.5

.5


Right! You don’t exercise your option. Hence:

0 ¥//$

10¥//$

.5

.5

5¥//$

Buy a $, Borrow ¥

Next, let’s replicate the call option payoffs with money market instruments and then find its value.

How do you do that?

Buy USD, Borrow JPYYou BUY 1 USD at a cost of 100 JPY (S) and you

borrow present value of 90 JPY at 5% (90/1.05 = 85.71 JPY)

The yen value of the JPY at the end of the year will be either 90 or 110, but you have a liability of precisely 90 JPY. Hence, your expected payoff is 10 JPY. Which, as you have probably noted, is a multiple of your option payoff (10¥/2=5¥)

Buy a USD, Borrow JPYYou BUY one USD at a cost of 100 JPY (S) and you

borrow 90 JPY at 5% (90/1.05 = 85.71 JPY)

What you probably overlooked is the present value of buying one USD at a cost of 100 JPY (S) and you borrow 90 JPY at 5%

100 JPY - 85.71 JPY cost of the bank loan= 14.29 JPY

Buy a $, Borrow ¥So, how do you scale down the “buy a dollar, borrow

yen strategy until it is the same as the payoff on a call?

Of course, if you can do that, you can also value the call option.

Using the Hedge Ratio to Value Currency Options

(called the option delta)

The Hedge Ratio indicates the number of call options required to replicate one unit (in this case, one USD) of the underlying asset.

Hedge Ratio = spread of option prices/ spread of possible

underlying asset valuesHence 0-10/0-20 = 10/20 = 0.5 (Delta call option

is at the money !)


What next?


What next?

You buy 0.5 of one USD at a cost of 50 JPY and you borrow 0.5 of 90 JPY or 45 JPY at 5% or 42.86 JPY

The difference between 50 JPY and 42.86 JPY is 7.14 JPY

Hence, the yen value of a one-dollar call option is 7.14 JPY.

The General Case of the Binomial Model

The General Case of the Binomial Model

At the limit, the distribution of continuously compounded exchange rates approaches the normal distribution (which is described in terms of a mean (expected value, in this case E[S]) and a distribution (variance or standard deviation)

This makes it equivalent to Black-Scholes model.

The Black-Scholes Option Pricing Model

Call = [S*N(d1)] - [e-iT*K* N(d2)]Where:

Call = the value of the call optionS = The spot market price

K = the exercise price of the optioni = risk free instantaneous rate of interest = instantaneous standard deviation of S

T = time to expiration of the optionN(.) = f(the standard normal cumulative P distribution)


Call = [S*N(d1)] - [e-iT*K* N(d2)]

d1 = [ln(S/K) + (i + ((d2 = d1 -

e-iT = 1/(1+i) T

Discounts the exercise or strike price to the present at the risk-free rate of interest


At expiration, time value is equal to zero and there is no uncertainty about S (call option value is composed entirely of intrinsic value).

CallT = Max [0, ST - K]Prior to expiration, the actual exchange rate remains

a random variable. Hence, we need the expected value of ST - K, given that it expires in the money.


In Black-Scholes, N(d1) is the probability that the call option will expire in the money


S* N(d1) is the expected value of the currency at expiration, given S>K.

K* N(d2) is the expected value of the exercise price at expiration

e-iT discounts the exercise price to PVOption Price

The Foreign Interest Rate

• We denote the foreign interest rate by rf

• When a U.S. company buys one unit of the foreign currency it has an investment of S0 dollars

• The return from investing at the foreign rate is rf

S0 dollars

• This shows that the foreign currency provides a “dividend yield” at rate rf

Valuing European Currency Options

• A foreign currency is an asset that provides a continuous “dividend yield” equal to rf

• We can use the formula for an option on a stock paying a continuous dividend yield :

Set S0 = current exchange rate (spot)

SSEE HULL «EE HULL « OPTIONS AND DERIVATIVES »

Formulas for European Currency Options (Equations 13.9 and 13.10, page 297)

T

Tf

rrKSd

T

Tf

rrKSd

dNeSdNKep

dNKedNeScTrrT

rTTr

f

f

)2/2()/ln(

)2/2()/ln(

)()(

)()(

0

2

0

1

102

210

where

Alternative Formulas(Equations 13.11 and 13.12, page 298)

F S e r r Tf

0 0 ( )Using

Tdd

T

TKFd

dNFdKNep

dKNdNFecrT

rT

12

20

1

102

210

2/)/ln(

)]()([

)]()([

Currency Option Pricing Sensitivity :the GREEKS (EITEMAN/STONEHILL/MOFFETT CHAPTER 7)

• If currency options are to be used effectively, either for the purposes of speculation or risk management, the individual trader needs to know how option values – premiums – react to their various components.

Currency Option Pricing Sensitivity

• Spot rate sensitivity (delta):– The sensitivity of the option premium to a small

change in the spot exchange rate is called the delta

delta = Δ premium

– The higher the delta, the greater the probability of the option expiring in-the-money

Δ spot rate


• Time to maturity – value and deterioration (theta):– Option values increase with the length of time to

maturitytheta = Δ premium

– A trader will normally find longer-maturity option better values, giving the trader the ability to alter an option position without suffering significant time value deterioration

Δ time

Theta: Option Premium Time Value Deterioration

Days remaining to maturity

Option Premium(US cents/£) A Call Option on British Pounds: Spot Rate = $1.70/£

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

90 80 70 60 50 40 30 20 10 0

In-the-money (ITM) call ($1.65 strike price)

At-the-money (ATM) call ($1.70 strike price)

Out-of-the-money (OTM) call ($1.75 strike price)


• Sensitivity to volatility (lambda):– Option volatility is defined as the standard deviation of daily

percentage changes in the underlying exchange rate

– Volatility is important to option value because of an exchange rate’s perceived likelihood to move either into or out of the range in which the option will be exercised

lambda = Δ premiumΔ volatility

Currency Option Pricing Sensitivity• Volatility is viewed in three ways:

– Historic

– Forward-looking

– Implied

• Because volatilities are the only judgmental component that the option writer contributes, they play a critical role in the pricing of options.

• All currency pairs have historical series that contribute to the formation of the expectations of option writers.

• In the end, the truly talented option writers are those with the intuition and insight to price the future effectively.

• Traders who believe that volatilities will fall significantly in the near-term will sell (write) options now, hoping to buy them back for a profit immediately volatilities fall, causing option premiums to fall (cf. : AIB case).


• Sensitivity to changing interest rate differentials (rho and phi):

– Currency option prices and values are focused on the forward rate

– The forward rate is in turn based on the Theory of Interest Rate Parity

– Interest rate changes in either currency will alter the forward rate, which in turn will alter the option’s premium or value

• A trader who is purchasing a call option on foreign currency should do so before the domestic interest rate rises. This timing will allow the trader to purchase the option before its price increases.


• The expected change in the option premium from a small change in the domestic interest rate (home currency) is the term rho.

rho = Δ premium

• The expected change in the option premium from a small change in the foreign interest rate (foreign currency) is termed phi.

phi = Δ premium

Δ US $ interest rate

Δ foreign interest rate


• The sixth and final element that is important to option valuation is the selection of the actual strike price.

• A firm must make a choice as per the strike price it wishes to use in constructing an option (OTC market).

• Consideration must be given to the tradeoff between strike prices and premiums.

Option Premiums for Alternative Strike Rates

Call strike price (U.S. dollars/£)

Option Premium (US cents/£)

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75

Current spot rate = $1.70/£

OTM Strike rates

ITM Strike rates

Currencies derivatives

• Objectives of the AIB case :– Analyse the reasons that led to loss at ALLFIRST.– Discuss the importance of proper supervision and

control systems to mitigate risk.– Understand the use of derivatives as a risk management

tool in foreign exchange trading.

Value at risk• Since 1996, the Basel Committee endorsed VaR as a

measure for market risk.• 1999+ credit risk• Basel II: VaR is the landmark for banking risk

measurement.• Example: portfolio with a 1 million 90 % daily VaR: this

means that there is 10 % chance that the daily loss on this portfolio is more that 1 million.

Value at Risk

• Normal distribution• 50% of the datas are below the average and

50% are upper• z fonction is used (0,1).• Portfolio : 8 500 000 USD (10 000 000 EUR)• Exchange rate = 0.85 EUR/USD 1.1764

USD/EUR• Annual standard deviation on the spot = 4%

Documents

MANAGING THE FOREIGN EXCHANGE RISK Dr. Jean-François VERDIE Professor of Finance Toulouse Business School, France [email protected] [email protected]