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

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

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

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))

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) = ( x (120/365)) -1 = $ B SF (0,120)= ( x (120/365)) -1 = SF

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 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 [ 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 f 0,120 ) dollars

Forward rate determination: absence of arbitrage Strategy one: positioncost todaypayoff 120 days later long forward 0 62,500 SF x (S 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 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)

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)

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 = = (.99596) = B SF (0,120) B $ (0,120) Interest ratesForward exchange rates: r $ > r foreign f 0,T > S 0 r $ < r foreign f 0,T < S 0

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

Exploit arbitrage opportunity Data: S 0 = ($/DM) 180 day T-bill price (B $ (0,180)) = $ per $ day German bill price (B DM (0,180)) = DM per DM day forward rate = f 0,180 = $/DM Determine that theoretical forward rate is $/DM: Arb strategy: positioncash todaypayoff 180 days later sell forward 0- ( S f 0,180 ) x (size) buy DM bill - S 0 ($/ 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 = ( ) 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 $ per mark ( 0.62 cents/DM) S ($/DM) Option payoff

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/ 0.62 cents/DM) ($6,200) Option cost includes $83 = 6200( x 120/360) financing cost

Hedge outcomes S ($/DM) Net cost of equipment , , , , , , ,000 unhedged Forward hedge Option hedge