1. Lec 15B: Options on Foreign Currencies (Hull, Ch. 12.9, 15)
Call Option Example:
Buy one Dec Au$ 6400 Call (at PHLX), Call Premium of $/Au$. (Say What?)
• Size of Contract: Au$100,000
• Call Premium: C0 = $/Au$
→ (0.012 $/Au$) * 100,000 Au$ = $1,200
• Exercise Price (K) = $/Au$
→ (0.64$/Au$) * Au$100,000 = $64,000
• Buyer has the right to buy Au$100,000 at the K = $64,000
Lec 15B Currency Options

2. Cash Flow Analysis: At t = 0, you (Long) pay $1,200 to seller of call
At t = Expiration (December). Possible scenarios:
Spot FX
Rate
$Price of Au$100,000
Exercise?
Value of
{ +CT }
➀ $/Au$
$67,600
Yes
$3,600
➁ $/Au$
66,400
$2,400 ➂
65,200
$1,200 ➃ No ➄
Lec 15B Currency Options

3. Hedging an A/P with a Call Option Example:
▸ UTC has an A/P = Au$100,000 due in December.
▸ To hedge risk, UTC buys one Dec Au$ 6400 Call.
Call Premium is $/Au$ → $/Au$*100,000 Au$=$1,200
▸ Cost of hedge is C0 = $1,200
Cash Flow Analysis. At t = Expiration (December). Possible scenarios
Spot FX Rate
$ Value of A/P
Value of { +CT }
A/P plus{ +CT }
➀ $/Au$
$66,400
$2,400
$64,000
➁ $/Au$
65,200
1,200
64,000 ➂ ➃
62,800 ➄
60,800
Lec 15B Currency Options

4. Example: Buy one Dec Au$ 6400 Put. Put Premium = 0.0160 $/Au$
Put Options (Long Put)
Example: Buy one Dec Au$ 6400 Put. Put Premium = $/Au$
▸ Size of Contract: Au$100,000
▸ Price: P0 = $/Au$, → $/Au$ * 100,000 Au$ = $1,600
▸ Exercise Price (K) = $/Au$ * Au$100,000 = $64,000
▸ Long Put is the right (but not the obligation)
to sell Au$100,000 at K = $64,000
Cash Flow Analysis. At t = Expiration (December). Possible scenarios
Spot FX
Rate
$Price of Au$100,000
Exercise?
Value of
{ +PT }
➀ $/Au$
$66,400 No
➀ $/Au$
64,000
➁0.6280
62,800
Yes
$1,200
➂0.6080
60,800
3,200
Lec 15B Currency Options

5. Hedging an A/R with a Put Example:
▸ UTC has an A/R = Au$100,000 due in December.
▸ To hedge risk, UTC buys one Dec Au$ 6400 Put.
Put Premium is $/Au$ → $/Au$*100,000 Au$=$1,600
▸ Cost of hedge is P0 = $1,600
Cash Flow Analysis. At t = Expiration (December). Possible scenarios
Spot FX Rate
$ Value of A/R
Value of { +PT }
A/R plus{ +PT }
➀ $/Au$
$66,400 $0
➁ $/Au$
65,200
$65,200 ➂
64,000
$64,000 ➃
62,800
1,200 ➄
60,800
3,200
Lec 15B Currency Options

6. BOPM for Currencies (p. 5) Consider a 1-year CALL option on the ¥.
▸ Size of contract Z¥ = ¥1,000
▸ K = 0.02 $/¥ ➟ K = $20.
▸ Assume r¥ = 7.51% and r$ = 11.2% (both c.c.)
FX rate movements Call Price Tree (in$)
t= T= t= T=1
0.04 $/¥ (SU) ➁ CU =$20 ➁
➀ S0 = 0.02 $/¥ ➀ C0 =
0.01 $/¥ (SD) ➂ CD = $0 ➂
Suppose the ¥ ↑ .
At ➁ (T =1) the spot rate is SU = 0.04 $/¥, will call be exercised?
Intuition: buy ¥1,000 in the spot market for ¥1,000(0.04$/¥) =$40.
Or, You can buy ¥1,000 thru the call for ¥1,000(0.02 $/¥ ) = $ 20. Which is better?
Thus, at point ➁ the call value (in $) is CU = ?
Lec 15B Currency Options

7. FX rate movements Call Price Tree (in$) t=0 T=1 t=0 T=1
0.04 $/¥ (SU) ➁ CU =$20 ➁ ➀ S0 = 0.02 $/¥ ➀ C0 =
0.01 $/¥ (SD) ➂ CD = $0 ➂
Suppose ¥ ↓ . At ➂ the spot rate is SD = $/¥, will call be exercised? Yes No
Intuition:
You can buy ¥1,000 in the spot market for ¥1,000(0.01 $/¥ ) = $10.
Or, You can buy ¥1,000 thru the call for ¥1,000(0.02 $/¥ ) = $20. Which is better?
Thus, at point ➂ CD = ?
Lec 15B Currency Options

8. FX rate movements Call Price Tree (in$) t=0 T=1 t=0 T=1
0.04 $/¥ (SU) ➁ CU =$20 ➁ ➀ S0 = 0.02 $/¥ ➀ C0 =$6.408
0.01 $/¥ (SD) ➂ CD = $0 ➂
Last, at t=0 ➀. To replicate the call
let Δ = amount of ¥ Bonds, B = amount of $ Bonds.
Δ er¥ (0.04 $/¥) +B er$ =$20
Δ er¥ (0.01 $/¥) +B er$ =$0
Solution: Δ = ¥ , and B = -$5.96
Therefore, the Call Value is
➀ C0 = Δ S0 +B = ¥ ( 0.02 $/¥ ) - $5.96 = $6.408
And
Synthetic Call = { Long a Yen bond with PV= ¥ , and
short a $ bond with PV = -$5.96 }
Lec 15B Currency Options

9. ▸ Invest $6.408 and borrow $5.96 in Storrs at r$ = 11.2%.
Proof: (p. 5)
At t=0 (now)
▸ Invest $6.408 and borrow $5.96 in Storrs at r$ = 11.2%.
▸ Buy ¥ =$ [1/0.02] =¥ and invest it in Tokyo at r¥ =7.51%.
▸ Net CF0 = $6.408 (same as the call)
At T=1 there are two possible outcomes:
➁ ¥ ↑ SU = 0.04 $/¥.
Yen investment becomes = ¥618.40( e ) = ¥
Sell these Yen and buy $ = ¥ ( 0.04$/¥ ) = $26.665
Pay back US $ loan = $5.96 ( e0.112 ) = $6.666
Net CF = = $20 ➟ same as the Call
➂ ¥ ↓ SD = 0.01 $/¥.
Sell these Yen and buy $ = ¥ ( 0.01$/¥ ) = $6.666
Net CF = = $0 ➟ same as the Call. It works!!
Lec 15B Currency Options

10. Thank you
(A Favara)