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Hedging with Foreign Exchange Derivatives. Alex Russell Ben Davidson. Mona Lisa. History.

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  • August 21,1911 Louis Béroud, a painter, walked into the Louvre and went to the Salon Carré where the Mona Lisa had been on display for five years. However, where the Mona Lisa should have stood, he found four iron pegs.

  • Béroud contacted the section head of the guards, who thought the painting was being photographed. A few hours later, Béroud checked back with the section head of the museum, and it was confirmed that the Mona Lisa was not with the photographers. The Louvre was closed for an entire week to aid in the investigation of the theft.

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History Continued

  • At the time, the painting was believed lost forever. It turned out that on August 20,1911 Louvre employee Vincenzo Peruggia stole it by simply by entering the building during regular hours, hiding in a broom closet, and walking out with it hidden under his coat after the museum had closed.

  • Con-man Eduardo de Valfierno master-minded the theft, and had commissioned the French art forger Yves Chaudron to make copies of the painting so he could sell them as the missing original. Because he didn't need the original for his con, he never contacted Peruggia again after the crime. After keeping the painting in his apartment for two years, Peruggia grew impatient and was caught when he attempted to sell it to a Florence art dealer; it was exhibited all over Italy and returned to the Louvre in 1913.

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Present Day

  • A short time after the theft Yves Chaudron escaped to the United States where he hid from the international art community. Two weeks ago, his great grandson Frances contacted us with a business proposal.

  • Frances Chaudron, grandson of the great art forger Yves Chaudron is currently one of the best painters in the world. As an artist, he has made a small fortune painting old masterpieces. Recently, in his studio in Salem, Oregon, Francis presented a profitable scheme inspired by his great grandfather.

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The Plan

  • Francis wants us to steal the Mona Lisa so that he can make precise forgeries to sell on the black market. Current technology requires that he actually have the original painting in his possession so that the borders match that of the original. Although the borders are not seen by the public, and are never photographed, a few collectors know their exact composition.

  • Acting in our capacity as an art broker, we were intrigued by the idea and contacted our associate in Rome, Italy. She believes that she can steal the painting and transport it to the United States, but requires payment of € 15,000,000 upon delivery.

  • After consulting our thief, we decided to take the job on the condition that Francis pays us for our services up front. He agrees, but says that we will only deal with dollars since foreign exchange rates confuse him. Because of this complication, he offers us $ 19,000,000 for the painting.

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The Problem

  • As an intermediary, how do hedge our positions to make the most money on the transaction?

    Long $19,000,000

    Short € 15,000,000

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  • Used to manage against risk

  • Delta Hedge—attempts to offset the change in the value of the exposure with an opposite change in the value of the hedge position.

  • Matching ‘longs’ and ‘shorts’.

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Hedging With a Currency Future

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

  • For example, if the customer is short the Euro, they would go long in the futures market (which we will do).

  • 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.

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Futures Contracts

  • Four fixed Dates a year; 3rd Wednesday of March, June, September and December.

  • Exchange traded, price determined through market trading, Liquidity.

  • Standard sizes (by Date).

  • Available only in a few currencies (Cross-hedge).

  • Daily settlement and Margin Requirements.

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Forward Contracts

  • Written by Banks

  • Tailor Size and Date (large and precise, respectively).

  • Traded Inter-Bank

  • No Settlement, only on Date.

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Our Homework Applied

  • Remember the last assignment?

  • We buy 120 (E15 mil / E125,000) June Euro Futures contracts at ($1.2247 * E125,000) * 120 costing us $18,370,500.

  • Break even (just on the contract) at $2.4494.

  • Current Profit = $629,500

  • Change in Account; (Buy Price – Settle) * Contracted Amount.

  • Maintain Margin Levels.

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Futures Time Line

When hedging with a foreign currency there are three important dates that you must consider.


Inception liquidation Maturity

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Future Equation (To Maturity)

Gain(Loss)= X[(St+n- bSt) - (Zt+n,T-Zt,T)]

X = Amount of Exposure

b= [1+it,t+n]1/a^/ [1+ i*t,t+n]1/a^ (The interest rate ratio)

a^= annualized factor for the interval from t to t+n

(S t+n - bSt) = is the deviation of the future spot exchange rate (or forward rate) at inception of the position

(Z t+n,T-Z t,T) = is the change the price of a futures contract maturing time T, over the time period from ,t inception of the position, to t+n, liquidation of the position

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Futures Equation 2

  • Previous equation was for a contract held to maturity.

  • If a futures contract is held to maturity, it is the same as a forward contract.

  • Why?

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Futures Equation (Not held to Maturity)

  • Gain (loss) =

    X((St+n – bSt) – c (St+n –bSt)

    C is

    C = (1+it+n,T)^1/a / (1+i*t+n,T)^1/a

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Futures Contracts (Not held to Maturity) 2

  • Remaining time to maturity prices using Covered Interest Parity.

  • Interest rates at liquidation are not known in advance, and subject to change during the interval.

  • Futures price has not had the full interval to converge to spot price.

  • Optimal hedge involves 1/c units of foreign currency futures, since c(1/c) = 1.

  • C > 1 smaller amount, C < 1 larger amount.

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  • Using a hedge ratio of 1/c sets the expected gain (loss) to zero, but does not create a perfect hedge.

  • C is stochastic

  • Interest rates, etc. are unpredictable.

  • Frequently adjusted.

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How much risk can we Hedge?

% of un-hedged risk eliminated by futures contracts;

(1- (Var(St+n – bSt) – (Zt+n,T – Zt,T)) / (Var(St+n – bSt))) * 100

Or as a % of open risk;

SQRT (Var(St+n – bSt) – (Zt+n,T – Zt,T)) / (Var(St+n – bSt))) * 100

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  • Futures contracts, unlike forward contracts, nearly always fail to be a complete hedge.

  • Futures are used when the transaction date is not definite.

  • Futures are used when the transaction amount is not precise.

  • Futures are used if the transaction is too small for the forward market.

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  • Foreign Currency options: are financial contracts that give the holder the right, but not the obligation, to buy or sell a specified amount of foreign currency on or before a specified maturity date.

  • Types of Options

    • American

    • European

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Calls and Puts

  • Calls

    • Gives the owner of the option the right but not the obligation to buy currency at a specified exercise price (X)

      • This is a long position

  • Puts

    • Gives the owner of the option the right but not the obligation to sell currency at a specified exercise price

      • This is a short position

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Why use an Option?

  • Your exposure to foreign exchange rate risk is uncertain

    • We are not sure that our thief will succeed

      • If our thief fails we no longer have exchange exposure

      • With Futures and Forwards we now will have an open position

  • You are uncertain about exchange rates in the future and want to capture the upside

    • Allows you still to hedge against risk

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Black Scholes Pricing

  • Developed by Fischer Black and Myron Scholes in 1973

  • Assumptions

    • European exercise terms are used

    • Markets are efficient

    • No commissions

    • Interest rates are known and remain constant

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The Model

  • C = SN(d1) – Ke(-rt)N(d2)

    • S = Current spot price

    • t = Time until option is exercised

    • K = Option strike price

    • r = Risk free interest rate

    • N = Cumulative standard deviation

    • s = standard deviation of returns

  • d1 = ln(S/K) + (r+s2/2)t st

  • d2 = d1 - st

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Results of the Model

  • Spot price = 1.2178

  • Risk free rate = 4.75%

    62,500 Euro – European style

    Strike Date Call Put

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How do we Hedge our Position?

  • We are short € 15,000,000

    • We need to take a long position to offset this position

    • This requires 240 contracts

      • € 62,500 each

  • The price of each 1200 contract is $1,731.25

    • Equals the number of Euros in the contract times price per Euro

    • Total price of the position = $415,500

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The Resulting Position

  • The 1200 contract gives us an effective rate of:

    • 1.2477 $/€

  • Why is this so much more than the Forward contracts?

    • Protects against the appreciation of the Euro

    • Allows for capture of falling Euro prices

    • Locks in the maximum loss

    • No maximum gain

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  • The foreign exchange options market is very liquid

  • Time of payment is not dependant upon the contract date

    • Execution of the options are not required

      • Buy € 15,000,000 at the spot rate

      • Sell 240, 1200 Euro Call options

        Final cost = $ 18,115,500 to $18,715,500

    • Compare to $ 17,700,000 to $19,500,000 at original spot rate

      • 1.18 $/€ to 1.30 $/€

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Methods of Offsetting the Price of Options

  • Straddle

    • Requires selling puts offsetting the gain when the price of the Euro falls against the dollar

      • The premium from the sale (price) offsets the cost of the call options

      • Generally used when both the call and the put are bought and sold out of the money

    • Straddle using the same strike price for both calls and puts

      • Effectively creates a forward contract