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IBUS 302: International Finance

Topic 11-Options Contracts

Lawrence Schrenk, Instructor

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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.▪

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Option Basics

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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.

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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

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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.

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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[ST – E, 0]

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Call Option Payoffs (at expiration)

–20

12020 40 60 80 100

20

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Buy a

cal

l

Exercise price = $50

50

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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 – ST, 0]

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Put Option Payoffs (at expiration)

–20

0 20 40 60 80 100

20

0

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Buy a put

Exercise price = $50

50

50

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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.

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Call Option Payoffs (at expiration)

–20

12020 40 60 80 100

20

40

60

Stock price ($)

Op

tio

n p

ayo

ffs

($)

Sell a call

Exercise price = $50

50

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Put Option Payoffs (at expiration)

–20

0 20 40 60 80 100

–40

20

0

40

–50

Stock price ($)

Op

tio

n p

ayo

ffs

($)

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)

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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[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.

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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)]

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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)

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Sensitivities What happens to the call premium

if the following increase?

Forward Rate

Exercise Rate

Risk Free Rate

Volatility

Time to Expiration ▪

↑

↓

↑

↑

↑ ▪

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Prince and Time

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The FX Black-Scholes Model $-i T

T 1 2C = F $/x N(d ) - EN(d ) e

C = Call Price (in dollars)FT($/x) = Forward Rate at TE = Exercise Ratei$ = Dollar Risk Free Rates = Volatility of the Forward RateT = Time to ExpirationN( ) = Standard Normal Distributione = the exponential

T 2

1

F $/x 1ln + σ T

E 2d =

σ T

2 1d = d - σ T

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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)

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The Black-Scholes Model

First, calculate d1 and d2

T 2

1

F $/x 1ln + σ T

E 2d =

σ T

2 1d = d - σ T = 0.3224 - 0.11 .25 = 0.2674

2

1

1.7278 1ln + 0.11 .25

1.7000 2d = = 0.3224

0.11 .25

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The Black-Scholes Model

d1 =0.3224 N(d1) = N(0.3224) = 0.6264

d2 =0.2674 N(d2) = N(0.2674) = 0.6054

$-i TT 1 2C = F $/x N(d ) - EN(d ) e

-.04 .25C = $1.7278 0.6264 - 1.7000 0.6054 e

C = $0.0526

The find C

Standard Normal Distribution

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x 0.00 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 x 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.00 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 0.00 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

-0.10 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 0.10 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753-0.20 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 0.20 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141-0.30 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 0.30 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517-0.40 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 0.40 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879-0.50 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 0.50 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224-0.60 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 0.60 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549-0.70 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 0.70 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

Standard Normal Distribution Table, N(x), for x > 0

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 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.

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