Currency Derivatives

Steven C. MannThe Neeley School of Business at TCUFinance 70420 – Spring 2004.

Currency Exposure

U.S. firm buys Swiss product; invoice: SF 62,500 due 120 days

Spot rate S0 = 0.7032 $/SF

S0+120

$ invoice cost If rate at day 120 (S0+120)

rises, purchase cost rises:

(S0+120) cost.6532 $40,825.7032 43,950.7532 47,075

Forward Hedge of Currency Exposure

Enter into forward contract to buy SF 62,500 in 120 days.Forward exchange rate is f0,120($/SF). Choose forward rate so that initial value of contract is zero.How is forward rate determined?

Need additional information:120 day $ riskless interest rate r$

120 day SF riskless interest rate rSF

Need to find cost today of 1 dollar 120 days from now1 Swiss franc 120 days from now.

E.g. T-bill price = price of dollar to be received at bill maturity

Zero-coupon bond prices

discount rates (id):B(0,T) = 1 - id (T/360); where id is ask (bid) yield

simple interest rates (is):B(0,T) = 1/ ( 1 + is x (T/365))

maturity T-bill Bill Price discount simpleT (days) ask yield B(0,T) rate ( id ) rate ( is )

30 3.96% 0.9967 0.0396 0.040360 4.14% 0.9931 0.0414 0.042390 4.24% 0.9894 0.0424 0.0434120 4.28% 0.9857 0.0428 0.0440180 4.32% 0.9784 0.0432 0.0448

Find B$(0,120) and BSF(0,120)

Need to find cost today of 1 dollar 120 days from now1 Swiss franc 120 days from now.

Given:simple 120 day interest rates:r$ = 3.25%rSF = 4.50%

then B$ (0,120) = ( 1 + .0325 x (120/365)) -1

= $ 0.9894BSF(0,120) = ( 1 + .0450 x (120/365)) -1

= SF 0.9854

Forward rate determination: absence of arbitrage

Strategy One (cost today = 0):Long forward contract to buy SF 62,500 at f0,120 at T=120 value of forward at (T=120) = 62,500 x ( S120 - f0,120) dollars

Strategy Two (cost today depends on forward rate):a) Buy PV(62,500) SF, invest in riskless SF asset for 120 dayscost today = S0 ($/SF) x BSF(0,120) x SF 62,500b) Borrow PV($ forward price of SF 62,500) at dollar riskless rate:borrow today: 62,500 x f0,120 x B$(0,120) dollarspay back loan in 120 days: f0,120 x 62,500 dollars

total cost today of strategy two = cost of (a) + cost of (b)= 62,500 x [0.7032 x BSF(0,120) - f0,120 x B$(0,120) ]

payoff of strategy two at (T=120):a) 62,500 SF x S120 ($/SF); b) repay loan: - f0,120($/SF)x 62,500

net payoff = 62,500 x ( S120 - f0,120) dollars

Forward rate determination: absence of arbitrage

Strategy one:position cost today payoff 120 days laterlong forward 0 62,500 SF x (S120 - f0,120) dollars

Strategy two:position cost today payoff 120 days laterbuy SF bill 62,500 SF x S0BSF(0,120) 62,500 SF x S120 ($/SF) dollarsborrow PV offorward price - 62,500 SF x f0,120 B$(0,120) -62,500 SF x f0,120 dollars

net 62,500 x 62,500 SF x (S120 - f0,120) dollars

(S0BSF(0,120) - f0,120 B$(0,120))

Strategies have same payoff must have same cost:0 = S0BSF(0,120) - f0,120 B$(0,120)

Interest rate parity

Interest rate parity:

f0,120 B$(0,120) = S0 BSF(0,120)

f0,120 ($/SF) = S0($/SF) x

this can be written: f0,120 ($/SF) = S0($/SF) x

if we use continuously compounded interest rates,this can be written:f0,120 ($/SF) = S0($/SF) x

(1 + r$ x (120/365))

(1 + rSF x (120/365))

(1 + r$ ) (120/365)

(1 + rSF) (120/365)

BSF(0,120)

B$ (0,120)

Forward rates via Interest rate parity

Interest rate parity:

f0,120 B$(0,120) = S0 BSF(0,120)

f0,120 ($/SF) = S0($/SF) x = 0.7032

= 0.7032 (.99596)= 0.70035

BSF(0,120) 0.9854

B$ (0,120) 0.9894

Interest rates Forward exchange rates:

r$ > rforeign f 0,T > S0

r$ < rforeign f 0,T < S0

Example

Interest rate parity:f0,180 B$(0,180) = S0BDM(0,180)

f0,180 ($/DM) = S0($/DM) x = 0.6676

= 0.6676 (.98412)= 0.6570

BDM(0,180) 0.964635

B$ (0,180) 0.980199

Data: S0 = 0.6676 ($/DM)180 day T-bill price = $ 98.0199 per $100180 day German bill price = DM 96.4635 per DM100 180 day forward rate = f 0,180 = 0.660 $/DM

Find theoretical forward rate:

Is there arbitrage opportunity?

Exploit arbitrage opportunity

Data: S0 = 0.6676 ($/DM)180 day T-bill price (B$(0,180)) = $ 98.0199 per $100180 day German bill price (BDM(0,180)) = DM 96.4635 per DM100 180 day forward rate = f 0,180 = 0.660 $/DM

Determine that theoretical forward rate is 0.6570 $/DM:

Arb strategy: position cash today payoff 180 days latersell forward 0 - (S180 - f 0,180) x(size)

buy DM bill - S0($/DM)x (BDM(0,180) x (size) DM x (size) x S180 ($/DM)borrow $ costof DM bill + S0($/DM)x (BDM(0,180) x (size) -S0BDM(0,180)x(size)x(B$(0,180))

-1

net 0 (size) x

f 0,180 - S0 BDM(0,180)

B$(0,180)Payoff = (0.660 -0.657) x (size)e.g. 1 million DM gives (.003) x 1,000,000 = $3,000 profit

Currency Options

Example: Buy spot DM call option with strike K = $0.64/ DM option size is 62,500 DM, option life 120 days. Option premium is $0.0062 per mark ( 0.62 cents/DM)

S0+120 ($/DM)

Option payoff

0

.61 .62 .63 .64 .65 .66

Forward vs. Option hedging

U.S. firm buys machinery, cost is DM1 million, due 120 days.Hedge: buy DM 1 million forward at $0.64/DM;or : buy 16 calls, K = $0.64/ DM, @ 0.62 cents/DM) ($6,200)

S120 $ cost of forward net option net

($/DM) equipment value cost value cost

0.60 600,000 -40,000 640,000 0 606,283

0.61 610,000 -30,000 640,000 0 616,283

0.62 620,000 -20,000 640,000 0 626,283

0.63 630,000 -10,000 640,000 0 636,283

0.64 640,000 0 640,000 0 646,283

0.65 650,000 10,000 640,000 10,000 646,283

0.66 660,000 20,000 640,000 20,000 646,283

option hedgeforward hedge

Option cost includes $83 = 6200( 1 + .04 x 120/360) financing cost

Hedge outcomes

S0+120 ($/DM)

Net cost ofequipment

.60 .61 .62 .63 .64 .65 .66

660,000

650,000

640,000

630,000

620,000

610,000

600,000

unhedged

Forward hedge

Option hedge