1. Currency Derivatives Steven C. Mann The Neeley School of Business at TCU Finance 70420 – Spring 2004.

2. Currency Exposure U.S. firm buys Swiss product; invoice: SF 62,500 due120 days Spot rate S 0 = 0.7032 $/SF S 0+120 $ invoice cost If rate at day 120 (S 0+120 ) rises, purchase cost rises: (S 0+120 ) cost.6532$40,825.7032 43,950.7532 47,075

3. Forward Hedge of Currency Exposure Enter into forward contract to buy SF 62,500 in 120 days. Forward exchange rate is f 0,120 ($/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 r SF Need to find cost today of 1 dollar 120 days from now 1 Swiss franc 120 days from now. E.g. T-bill price = price of dollar to be received at bill maturity

4. Zero-coupon bond prices discount rates (i d ): B (0,T) = 1 - i d (T/360); where i d is ask (bid) yield simple interest rates (i s ): B (0,T) = 1/ ( 1 + i s x (T/365))

5. Find B $ (0,120) and B SF (0,120) Need to find cost today of 1 dollar 120 days from now 1 Swiss franc 120 days from now. Given: simple 120 day interest rates: r $ = 3.25% r SF = 4.50% then B $ (0,120) = ( 1 +.0325 x (120/365)) -1 = $ 0.9894 B SF (0,120)= ( 1 +.0450 x (120/365)) -1 = SF 0.9854

6. Forward rate determination: absence of arbitrage Strategy One (cost today = 0): Long forward contract to buy SF 62,500 at f 0,120 at T=120 value of forward at (T=120) = 62,500 x ( S 120 - f 0,120 ) dollars Strategy Two (cost today depends on forward rate): a) Buy PV(62,500) SF, invest in riskless SF asset for 120 days cost today = S 0 ($/SF) x B SF (0,120) x SF 62,500 b) Borrow PV($ forward price of SF 62,500) at dollar riskless rate: borrow today: 62,500 x f 0,120 x B $ (0,120) dollars pay back loan in 120 days: f 0,120 x 62,500 dollars total cost today of strategy two = cost of (a) + cost of (b) = 62,500 x [0.7032 x B SF (0,120) - f 0,120 x B $ (0,120) ] payoff of strategy two at (T=120): a) 62,500 SF x S 120 ($/SF); b) repay loan: - f 0,120 ($/SF)x 62,500 net payoff = 62,500 x ( S 120 - f 0,120 ) dollars

7. Forward rate determination: absence of arbitrage Strategy one: positioncost todaypayoff 120 days later long forward 0 62,500 SF x (S 120 - f 0,120 ) dollars Strategy two: positioncost today payoff 120 days later buy SF bill62,500 SF x S 0 B SF (0,120) 62,500 SF x S 120 ($/ SF ) dollars borrow PV of forward price - 62,500 SF x f 0,120 B $ (0,120) -62,500 SF x f 0,120 dollars net62,500 x 62,500 SF x (S 120 - f 0,120 ) dollars (S 0 B SF (0,120) - f 0,120 B $ (0,120)) Strategies have same payoff must have same cost: 0 = S 0 B SF (0,120) - f 0,120 B $ (0,120)

8. Interest rate parity Interest rate parity: f 0,120 B $ (0,120) = S 0 B SF (0,120) f 0,120 ($/SF) = S 0 ($/SF) x this can be written: f 0,120 ($/SF) = S 0 ($/SF) x if we use continuously compounded interest rates, this can be written: f 0,120 ($/SF) = S 0 ($/SF) x (1 + r $ x (120/365)) (1 + r SF x (120/365)) (1 + r $ ) (120/365) (1 + r SF ) (120/365) B SF (0,120) B $ (0,120)

9. Forward rates via Interest rate parity Interest rate parity: f 0,120 B $ (0,120) = S 0 B SF (0,120) f 0,120 ($/SF) = S 0 ($/SF) x = 0.7032 = 0.7032 (.99596) = 0.70035 B SF (0,120) 0.9854 B $ (0,120) 0.9894 Interest ratesForward exchange rates: r $ > r foreign f 0,T > S 0 r $ < r foreign f 0,T < S 0

10. Example Interest rate parity: f 0,180 B $ (0,180) = S 0 B DM (0,180) f 0,180 ($/ DM ) = S 0 ($/ DM ) x = 0.6676 = 0.6676 (.98412) = 0.6570 B DM (0,180) 0.964635 B $ (0,180) 0.980199 Data: S 0 = 0.6676 ($/ DM ) 180 day T-bill price = $ 98.0199 per $100 180 day German bill price = DM 96.4635 per DM 100 180 day forward rate = f 0,180 = 0.660 $/ DM Find theoretical forward rate: Is there arbitrage opportunity?

11. Exploit arbitrage opportunity Data: S 0 = 0.6676 ($/DM) 180 day T-bill price (B $ (0,180)) = $ 98.0199 per $100 180 day German bill price (B DM (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 $/DM: Arb strategy: positioncash todaypayoff 180 days later sell forward 0- ( S 180 - f 0,180 ) x (size) buy DM bill - S 0 ($/ DM ) x (B DM (0,180) x (size)DM x (size) x S 180 ($/ DM ) borrow $ cost of DM bill+ S 0 ($/ DM ) x (B DM (0,180) x (size)-S 0 B DM (0,180)x(size) x(B $ (0,180) ) -1 net0 (size) x f 0,180 - S 0 B DM (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

12. 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 $0.0062 per mark ( 0.62 cents/DM) S 0+120 ($/DM) Option payoff 0.61.62.63.64.65.66

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

14. Hedge outcomes S 0+120 ($/DM) Net cost of equipment.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