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 0.0120 $/Au$. (Say What?) • Size of Contract: Au$100,000 • Call Premium: C0 = 0.012 $/Au$ → (0.012 $/Au$) * 100,000 Au$ = $1,200 • Exercise Price (K) = 0.6400$/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

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 0.6760 $/Au$ $67,600 Yes $3,600 ➁ 0.6640 $/Au$ 66,400 $2,400 ➂ 0.6520 65,200 $1,200 ➃ 0.6400 No ➄ 0.6280 Lec 15B Currency Options

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 0.0120 $/Au$ → 0.012 $/Au$*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 0.6640 $/Au$ $66,400 $2,400 $64,000 ➁ 0.6520 $/Au$ 65,200 1,200 64,000 ➂ 0.6400 ➃ 0.6280 62,800 ➄ 0.6080 60,800 Lec 15B Currency Options

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 = 0.0160 $/Au$ ▸ Size of Contract: Au$100,000 ▸ Price: P0 = 0.016 $/Au$, → 0.016 $/Au$ * 100,000 Au$ = $1,600 ▸ Exercise Price (K) = 0.6400$/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 0.6640 $/Au$ $66,400 No ➀0.6400 $/Au$ 64,000 ➁0.6280 62,800 Yes $1,200 ➂0.6080 60,800 3,200 Lec 15B Currency Options

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 0.0160 $/Au$ → 0.016 $/Au$*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 0.6640 $/Au$ $66,400 $0 ➁ 0.6520 $/Au$ 65,200 $65,200 ➂ 0.6400 64,000 $64,000 ➃ 0.6280 62,800 1,200 ➄ 0.6080 60,800 3,200 Lec 15B Currency Options

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=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 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

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 = 0.01 $/¥, 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

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 = ¥618.4338, and B = -$5.96 Therefore, the Call Value is ➀ C0 = Δ S0 +B = ¥618.4338( 0.02 $/¥ ) - $5.96 = $6.408 And Synthetic Call = { Long a Yen bond with PV= ¥618.4338, and short a $ bond with PV = -$5.96 } Lec 15B Currency Options

▸ 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 ¥ =$12.368 [1/0.02] =¥618.40 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( e0.0751 ) = ¥666.63. Sell these Yen and buy $ = ¥666.63 ( 0.04$/¥ ) = $26.665 Pay back US $ loan = $5.96 ( e0.112 ) = $6.666 Net CF = + 26.665 - 6.666 = $20 ➟ same as the Call ➂ ¥ ↓ SD = 0.01 $/¥. Sell these Yen and buy $ = ¥666.63 ( 0.01$/¥ ) = $6.666 Net CF = + 6.666 - 6.666 = $0 ➟ same as the Call. It works!! Lec 15B Currency Options

Thank you (A Favara)