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© K.Cuthbertson and D.Nitzsche1
Version 1/9/2001
FINANCIAL ENGINEERING:DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)
K. Cuthbertson and D. Nitzsche
LECTURE
Foreign Currency Options
© K.Cuthbertson and D.Nitzsche2
Contracts and Payoffs
Hedging Foreign Currency Receiptsusing Forwards, Options and Futures
Pricing Foreign Currency Options
Topics
© K.Cuthbertson and D.Nitzsche4
Foreign Currency Options Contracts
Table : Foreign Currency Options PHSE
Contract Size K- Increments Min Price Chge
GBP £31,250 $0.0250 $0.0001 = $3.125
DM DM62,500 $0.050 0.0001 = $6.25
JY JY6,250,000 $0.050 0.000001 = $6.25
Can $ CD50,000 $0.050 0.0001 = $5.00Hold, TVS0 = $2m in diversified equity portfolio and ‘
© K.Cuthbertson and D.Nitzsche5
Fig11.2:Long , Foreign Currency Call
ST
Profit
Strike price K = $140
6
C = 4
$150$144K = $1400
0+1
© K.Cuthbertson and D.Nitzsche6
= Max(ST – K, 0) - C = -C if ST < K = ST - K - C if ST > K
Break even spot rate is:ST,BE = K + C
zop = £31,250 at expiryK = 1.40 $/£ C = 4.0 cents/£ = 0.04 $/£ST = 1.50($/£) (see figure 11.2): Gross profit = (ST – K) zop = (1.50 – 1.40) £31,250 = $3,125
Invoice price per contract = zop C = £31,250 (0.04($/£)) = $1,250
Net profit: = (ST - K – C) £31,250 = (1.50($/£) - 1.40($/£) - 0.04($/£)) £31,250 = (0.06($/£))(£31,250) = $1,875
PROFIT FROM A LONG CALL
© K.Cuthbertson and D.Nitzsche7
Fig 11.3 : Long, Foreign Currency Put
Strike price K = $144
ST
Profit
1.50
P = -2.50
$141.50
$140 K = $1440
0-1
© K.Cuthbertson and D.Nitzsche8
Profit from Long Put
If ST < K 140 < 144
Exercise the option (“in-the-money”)
Gross profit = ( K - ST ) z = (1.44-1.40) 31,250 = $1250
Net profit = (K - ST - P) z = 1.44-1.40-0.025 = $468.75 per contract
If ST > K
Do not exercise the option (out-of-the-money)
Loss = (2.5/100) x 31,250 = $781.25
Loss is limited to put premium (insurance)
© K.Cuthbertson and D.Nitzsche9
Buy (long) call on sterling if you expect sterling to appreciate.
Buy long put on sterling if you expect sterling to depreciate
Calls and Puts: Speculation
© K.Cuthbertson and D.Nitzsche10
Hedging Foreign Currency Receiptsusing
Forwards, Optionsand Futures
© K.Cuthbertson and D.Nitzsche11
Hedging Foreign Currency (Intuition)
US firm makes bid for UK contract, outcome of bid is unknownReceipt of f.c. (GBP) is uncertain
FORWARD/FUTURES MARKETBid successful ~ you are hedged
Bid unsuccessful ~ you are not hedged outcome is unfavourable if sterling appreciates ~ have to buy GBP at ‘high’ rate in spot market, to honour delivery in the f.c.
© K.Cuthbertson and D.Nitzsche12
Hedging Foreign Currency (Intuition)
LONG PUT ON GBP
Bid successful and GBP appreciates ~ outcome favourable even though put expires worthless, as you sell GBP at ‘high’ rate
Bid successful and GBP depreciates ~ outcome ‘favourable’ as you exercise the put and receive K
Bid unsuccessful and GBP appreciates ~ loss limited to the put premium, P
Bid unsuccessful and GBP depreciates ~ outcome ‘favourable’ as you exercise the put and receive (K-ST) - P
© K.Cuthbertson and D.Nitzsche13
US firm :bid for sterling contract, V= £12.5mAt F0 = 1.61($/£), USD equivalent of$20.125m.
PUT CONTRACTStrike Price K = 1.60 ($/£)Size of Contract, zp = £31,250Put Premium P = 0.025 ($/£)Cost, one Put contract (= zpP) = $781.25
Number of Put Contracts NP = (V/zp) = (£12.5m / £31,250) = 400 contracts
Cost of Np puts = Np (zp P) = VP = $312,500 (Note that V = Np zp )
Hedging Foreign Currency Receipts: Detail
© K.Cuthbertson and D.Nitzsche14
Possible outcomes
ST = 1.65($/£) or ST = 1.50($/£)
Bid Successful or Unsuccessful
Hedging Foreign Currency Receipts: Detail
© K.Cuthbertson and D.Nitzsche15
A: Bid Successful (appreciation £)
ST= 1.65
No Hedge
= V.ST =(12.5m)1.65= $20.625m
Forward Market at F0=1.62
= V.F0=(12.5m)(1.61) = $20.125m
Put Option
ST >K, puts not exercised convert £’s at 1.65:
= Spot revenue - Cost of Put
V.ST – Np (zp.P)= V ( ST – P )
= 12.5 (1.65 – 0.025) =$20,312,500Equivalent to Long put + long spot = long call
© K.Cuthbertson and D.Nitzsche16
B: Bid Successful(depreciation £)
ST = 1.50
No Hedge
= V.ST =(£12.5)1.50 = $18.75m
Forward Market at F0=1.62
= V. F0 =£12.5(1.61) = $20.125m
Put Option : Exercise Puts (locked in K= 160):
Payoff from puts+long spot - cost of puts
= = [(K - ST) + ST] .V – Np (zp P)
= K .V – Np (zp P) = 1.60 (12.5m) - $312,500
= $19,687,500 Had you chosen put with K = 161 then the put would have a gross payoff
equal to that of the forward.
© K.Cuthbertson and D.Nitzsche17
C: Bid Unsuccessful (appreciation £)
ST = 1.65
No Hedge: No Cash Flow
Forward Market at F0=1.62
Purchase £12.5m at a cost of ST = 1.65 and receive F0 =1.61
=(Fo – ST).V == (1.61 – 1.65) £12.5= - $500,000
(Equivalent to open short position in the F.C. and you are exposed to potential large losses as S increases)
Put Option :Not Exercised:(equiv to naked put)
Lost put premium = Np (zp P) = V. P = $312,500
© K.Cuthbertson and D.Nitzsche18
D: Bid Unsuccessful (depreciation £)
ST = 1.50
No Hedge: No Cash Flow
Forward Market at F0=1.61
Purchase £12.5m at a cost of ST = 1.50 and
receive F0 =1.62 on (£12.5m)
=(Fo – ST)V == (1.61 – 1.50) £12.5= $1.375m
(Equivalent to open short position in the F.C. and you have potential large gains as S increases)
Put Option : Exercise Puts:(equiv to naked put)
Purchase, at ST= 1.50 and exercise puts K = 1.60
= (K - ST – P ) V=(1.60– 1.50 –0.025) 12.5 = $937,500
© K.Cuthbertson and D.Nitzsche19
Bid Successful = V S1 + V (Fo – F1) = V [F0 + (S1 – F1)]
Bid Unsuccessful = V (Fo – F1)
The outcomes are the same as for the forward market if the futures are held to maturity, F1 = S1 (and ‘close’, if futures are closed out and basis is small)
Hedging: Using Futures
© K.Cuthbertson and D.Nitzsche21
Pricing
Replace q= dividend yield by rf
[11.13] C = S N(d1) - K N(d2) [11.14] P = K N(-d2) - S N(-d1)
d1 =
d2 = = d1 -
S is measured as $ per £ (or cents per £),
T
TrrKS fd
)2/()/ln( 2
T
TrrKS fd
)2/()/ln( 2
T
© K.Cuthbertson and D.Nitzsche22
Pricing: Alternative Representation
[11.16] S = F
Substituting [11.16] in [11.13] and [11.14]::
[11.17] C = [F N(d1) - K N(d2)]
[11.18] P = [K N(-d2) – F N(-d1)]
d1 =
d2 =
Trr fde )(
T
TKF
)2/()/ln( 2
T
TKF
)2/()/ln( 2
© K.Cuthbertson and D.Nitzsche23
Table 11.5: Put-Call Parity: Currency Options
Case : ST > K Case : ST < K
Portfolio A(1): Cash ST ST
Long Put 0 K-ST
Total A ST---------------------------- K
Portfolio B(2): Long Call ST - K 0
US T-bond K K
Total A ST------------------------- K
Portfolio A = One long put, plus cash of $A = S0
invested in a foreign bond
Portfolio B = One long call, plus domestic (US) bond of ($) K
Tfre
Tdre
Returns from Two Portfolios :