IBUS 302: International Finance

1. IBUS 302: International Finance Topic 11-Options Contracts Lawrence Schrenk, Instructor

2. Learning Objectives • Explain the basic characteristics of options (using stock options). ▪ • Determine the value of a FX option at expiration. • Price European FX call options using the Black-Scholes model.▪

3. Option Basics

4. Option Basics • An option gives the holder the right, but not the obligation, to buy or sell a given quantity of an asset on (or before) a given date, at prices agreed upon today. • The buyer has the long position; the seller has the short position. • Roughly analogous to a forward contract with optional exercise by the buyer.

5. Option Basics (cont’d) • Exercising the Option • The act of buying or selling the underlying asset • Strike Price or Exercise Price • Refers to the fixed price in the option contract at which the holder can buy or sell the underlying asset. • Expiry (Expiration Date) • The maturity date of the option

6. Option Basics (cont’d) • European versus American options • European options exercised only at expiry. • American options exercised at any time up to expiry. • In-the-Money • Exercising the option would result in a positive payoff. • At-the-Money • Exercising the option would result in a zero payoff. • Out-of-the-Money • Exercising the option would result in a negative payoff. • Premium • The Price paid for the option.

7. Call options gives the holder the right, but not the obligation, to buy a given quantity of some asset on or before some time in the future, at prices agreed upon today. Payoff: C = Max[ST – E, 0] Call Options

8. Call Option Payoffs (at expiration) 60 40 Option payoffs ($) Buy a call 20 80 20 40 60 100 120 50 Stock price ($) –20 Exercise price = $50

9. Put options gives the holder the right, but not the obligation, to sell a given quantity of an asset on or before some time in the future, at prices agreed upon today. Payoff: P = Max[E – ST, 0] Put Options

10. Put Option Payoffs (at expiration) 60 50 40 Option payoffs ($) 20 Buy a put 0 80 0 20 40 60 100 50 Stock price ($) –20 Exercise price = $50

11. Selling Options • The seller receives the option premium in exchange. • The seller of an option accepts a liability, i.e., the obligation if buyer exercises the option. • Unlike forward contracts, option contracts are not symmetric between buyer and seller.

12. Call Option Payoffs (at expiration) 60 40 Option payoffs ($) 20 80 20 40 60 100 120 50 Stock price ($) Sell a call Exercise price = $50 –20

13. Put Option Payoffs (at expiration) 40 20 Option payoffs ($) Sell a put 0 80 0 20 40 60 100 50 Stock price ($) –20 Exercise price = $50 –40 –50

14. FX Options (versus Stock Options) • Underlying Asset: The Forward Rate • Each option gives you the right to exchange a certain amount of one currency for another. • Exercise Price is an FX Rate • Risk Free Rate • Rate for the domestic currency • Premium is priced in the domestic currency. • The FX Black-Scholes formula is slightly different from the one used for stock options.

15. Two questions: What is the value of an option at expiration? What is the value of an option prior to expiration? Two answers: Relatively easy and we have done it. A much more interesting (and difficult) question. Valuing FX Options

16. Value at Expiration • Call • C = Max[ST($/x) – E, 0] • Put • P = Max[E – ST($/x), 0] Note: ST($/x) is the FX spot rate at expiration, i.e., time T.

17. Value Prior to Expiration • Issues: • ST($/x) is not known. • E[ST($/x)] • Probability known • Assume normal distribution • Solution • Calculate E[Max[ST($/x) – E, 0]] • We use FT($/x) to estimate E[ST($/x)]

18. Data • FT($/x) = Forward Rate at T • E = Exercise Rate • i$ = Dollar Risk Free Rate (Annual) • s = Volatility of the Forward Rate (Annual) • T = Time to Expiration (Years)

19. Sensitivities • What happens to the call premium if the following increase? • Forward Rate • Exercise Rate • Risk Free Rate • Volatility • Time to Expiration ▪ • ↑ ↓ • ↑ • ↑ • ↑ ▪

20. Prince and Time

21. The FX Black-Scholes Model C = Call Price (in dollars) FT($/x) = Forward Rate at T E = Exercise Rate i$= Dollar Risk Free Rate s = Volatility of the Forward Rate T = Time to Expiration N( ) = Standard Normal Distribution e = the exponential

22. The FX Black-Scholes Model: Example Find the value of a three-month call option: • F3($/£) = 1.7278 • Exercise Rate = 1.7.00 • Risk free interest rate available in the US (i$) = 4% • Annual forward rate volatility = 11% • Time to expiration = 0.25 (= 3/12 months)

23. The Black-Scholes Model First, calculate d1 and d2

24. The Black-Scholes Model The find C d1 =0.3224 N(d1) = N(0.3224) = 0.6264 d2 =0.2674 N(d2) = N(0.2674) = 0.6054

25. Standard Normal Distribution • Find x in the bold row and column, N(x) is the value at the intersection. • This is a partial table. There is also a table for x < 0. • Table values are only approximate.

26. Black-Scholes Reminders • Time is stated in years, so it is normally less than 1. • In the formula for d1, you need variance (s2) in the numerator, but standard deviation (s) in the denominator. • In the data, volatility can be given as either variance or standard deviation. • d1 and d2 can be positive or negative, but C is always positive.