1 (of 31) IBUS 302: International Finance Topic 11-Options Contracts Lawrence Schrenk, Instructor

2 (of 31) Learning Objectives 1. Explain the basic characteristics of options (using stock options). ▪ 2. Determine the value of a FX option at expiration. 3. Price European FX call options using the Black-Scholes model.▪

3 (of 31) Option Basics

4 (of 31) 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 (of 31) 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 (of 31) 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 (of 31) Call Options 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[S T – E, 0]

8 (of 31) Call Option Payoffs (at expiration) – Stock price ($) Option payoffs ($) Buy a call Exercise price = $50 50

9 (of 31) Put Options 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 – S T, 0]

10 (of 31) Put Option Payoffs (at expiration) – Stock price ($) Option payoffs ($) Buy a put Exercise price = $50 50

11 (of 31) 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 (of 31) Call Option Payoffs (at expiration) – Stock price ($) Option payoffs ($) Sell a call Exercise price = $50 50

13 (of 31) Put Option Payoffs (at expiration) – – –50 Stock price ($) Option payoffs ($) Sell a put Exercise price = $50 50

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. 14 (of 31)

15 (of 31) Valuing FX Options Two questions: 1. What is the value of an option at expiration? 2. What is the value of an option prior to expiration? Two answers: 1. Relatively easy and we have done it. 2. A much more interesting (and difficult) question.

Value at Expiration Call C = Max[S T ($/x) – E, 0] Put P = Max[E – S T ($/x), 0] Note: S T ($/x) is the FX spot rate at expiration, i.e., time T. 16 (of 31)

Value Prior to Expiration Issues: S T ($/x) is not known. E[S T ($/x)] Probability known Assume normal distribution Solution Calculate E [ Max[S T ($/x) – E, 0] ] We use F T ($/x) to estimate E[S T ($/x)] 17 (of 31)

Data F T ($/x) = Forward Rate at T E = Exercise Rate i $ = Dollar Risk Free Rate (Annual)  = Volatility of the Forward Rate (Annual) T = Time to Expiration (Years) 18 (of 31)

19 (of 31) Sensitivities What happens to the call premium if the following increase? Forward Rate Exercise Rate Risk Free Rate Volatility Time to Expiration ▪ ↑ ↓ ↑ ↑ ↑ ▪

20 (of 31) Prince and Time

21 (of 31) The FX Black-Scholes Model C = Call Price (in dollars) F T ($/x) = Forward Rate at T E = Exercise Rate i $ = Dollar Risk Free Rate  = Volatility of the Forward Rate T = Time to Expiration N( ) = Standard Normal Distribution e = the exponential

22 (of 31) The FX Black-Scholes Model: Example Find the value of a three-month call option: F 3 ($/£) = Exercise Rate = Risk free interest rate available in the US (i $ ) = 4% Annual forward rate volatility = 11% Time to expiration = 0.25 (= 3/12 months)

23 (of 31) The Black-Scholes Model First, calculate d 1 and d 2

24 (of 31) The Black-Scholes Model d 1 = N(d 1 ) = N(0.3224) = d 2 = N(d 2 ) = N(0.2674) = The find C

Standard Normal Distribution 25 (of 31) 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.

Black-Scholes Reminders Time is stated in years, so it is normally less than 1. In the formula for d 1, you need variance (  2 ) in the numerator, but standard deviation (  ) in the denominator. In the data, volatility can be given as either variance or standard deviation. d 1 and d 2 can be positive or negative, but C is always positive. 26 (of 31)