International Financial Markets Options

International Financial Markets: Chapter 55-1 Wharton FNCE-219

Professor Amir YaronSpring 2014FNCE-219

International Financial Markets: Options1. Introduction to Currency Options1.1 Call Options, put Options, and Option Premia1.2 Markets for Currency Options1.3 Intrinsic Value and Time Value1.4 graphical Analysis of European Options1.5 Put-Call Parity for European Options

2. Hedging with Options2.1 Hedging without Eliminating Possible Gains2.2 Hedging Positions with Reserve Risk2.3 Hedging non-linear Exposure2.4 Range Forward and Cylinder Options (Collars)

3. Options Pricing Issues3.1 Early Exercise and Relative Interest Pricing3.2 The Black-Scholes Option Pricing Model3.3 Implied Volatility.3.4 Speculating with options


Call Options A call gives the holder the right to buy a stated number of

“underlying” assets at a given price (exercise price or strike price) from the counter party (the writer of the option),

at time T, in the case of a “European” -style option,

at any time until time T, in the case of an “American” -style option.

Example (call) You buy a call on one CHF at USD/CHF 0.50 expiring on June 30th. You

are long the call.

The counter party is the writer of the call; he has the potential obligation to deliver one CHF to you at 50 cts if you want him to (that is if you exercise the option).

If ST = USD/CHF 0.60 you will exercise your right and buy CHF at USD/CHF 0.50, and save USD 0.1.

If ST is less than USD/CHF 0.50, i.e 0.48, you will not exercise the option


Exercise Rules of Call Options A European call option will be exercised, at T,

iff ST-X > 0.

This option allows you to obtain just the "nice" part of the forward purchase: rather than paying X for sure (as in a forward purchase), you pay no more than X for the foreign currency, and possibly less than X.

For early exercise of an American option to be rational, two conditions must be met:

(St - X) > 0, that is, a positive value dead. The option's market value (value "alive") is no higher than

the value dead.

)0,(~

XSMaxC TT


American OptionEXAMPLE

Suppose you have an American option to buy 1 unit of Euro at X=USD/Euro 0.90.

If S t = 0.88, you will not exercise early, you'll wait and see.

If S t = 0.92 but Ct = 0.04, you are better off selling than exercising.

Of course, if the option is still alive at T,


Put OptionsCall Option Put OptionRight to buy at FC at X

Useful to hedge a FC debt:You pay no more than X

CT = Max( ST – X, 0)

Right to Sell FC at X

Useful to Hedge a FC asset:You get no less than X

PT=Max(X- ST, 0)

A put is not the “short” of a call; these are 2 different contracts

A call on Euro in USD is the same as a Put on USD in Euro.Option Premiums A call can be used to “insure” a FC payable against a high ST

A put can be used to “insure” a FC receivable against a low ST

The option buyer pays an (“insurance”) premium (up front) to buy the option The option writer gets paid the (“insurance”) premium, and offers the insurance


Markets for Currency Options Traded on organized exchanges (like futures) OR over the counter (OTC)

(like forwards) Daily transaction volume of $207bn (2010), $212bn (2007), $117bn (2004)

Traded Options Organized exchanges, with a clearing house as a guarantor party. Philadelphia Stock Exchange

Expiration dates: third Wednesday of March/June/September/December + 2 nearest months.

Contract size: 100K EUR, GDP, CHF, CAD, 1M JPY, cash settled Price Quotations: Options with new X’s are introduced around the money

as St changes. Prices quoted in cts/FC, for JPY cts/100JPY. Chicago Mercantile Exchange

Offers options on the foreign currency futures traded at the CME.


http://www.thestreet.com/video/11500363/introducing-phlx-forex-options.html


(Wall Street Journal)


Over the Counter Markets

Majority of currency options, over 90%, are OTC.

Customized options (amount, maturity, exercise price, and type), usually large amounts in excess of USD 1m. Customization does not come for free, spreads are higher than for exchange traded options.

Similar to the spot/forward markets. Banks trade with corporate customers and hedge their positions in the interbank market.

Interbank trading usually options struck “at-the-money-forward”. Options are traded together with spot positions, this keeps overall positions hedged against exchange rate movements.


Intrinsic Value and Time ValueCall Put

in the money St – X > 0 X – St > 0at the money St – X = 0 X – St = 0out of the money St – X < 0 X – St < 0Intrinsic value Max(St – X,0) Max(X – St,0)Time value Ct – IntVal Pt – IntVal

EXAMPLE: For a call on CHF with strike price X = cts/CHF 43 The intrinsic value is 5 cents if the spot rate is 48 cents. Time value is 1 cent

if the market price is 6 cents. The intrinsic value is 0 if the spot rate is 40 (or any other price equal to or

below 43). Time value is 2 if market price is 2.

EXAMPLE: Put on CHF at X = 43 The intrinsic value is ... cents if the spot rate is 48 cents. Time value is ...

cents if the market price is 2 cents. The intrinsic value is ... if the spot rate is 40. Time value is ... if market price

is 5.


1.4. Lower Bounds on Option Prices

… because

(1) Option have non-negative exercise values (limited liability). (2) American option gives the holder all the rights of the

European put or call plus the right of early exercise.

Ctam > Ct > 0

Ptam > Pt > 0


,0 ,0am amt t t tC Max S X and P Max X S

*,,

*, ,

,011

,01 1

am tt t

t Tt T

am tt t

t T t T

S XC C Max rr

SXP P Max r r

(3) American option can be exercised at any moment

(4) European option is a right, not an obligation, so it is worth at least as much as the comparable forward purchase or sale. (Equality only if exercise is SURE)


Suppose Ft,T= 90cents /FC

Value of forward sale º FST = Ft,T –ST

Value of FC asset = ST

Value FC asset +forward sale = Ft,T

European Options and Graphical Analysis of Forward Contracts

ST


European Options and Graphical Analysis of Forward Contracts

Suppose Ft,T= 90cent $/FCValue of forward purchase FPT = ST - Ft,T

Value of FC debt = - St

Value FC debt+ forward purchase= - Ft,T

-200

-150

-100

-50

0

50

100

0 50 100 150 200

FC Debt

ForwardPurchaseHedged AssetST


Exercise Value of a Call

Call on Euro with strike price X = 35 UScents/Eurovalue to holder:

IF ST = 32 33 34 35 36 37 38

then CT =

value to writer: is just the negative of the value to the holder.

0 0 0 0 1 2 3


Hence: terminology "minus a call" to describe the writer's position:

The payoff from a short (or written) call is minus the payoff from a long position in a call; and

The up-front cash flows also differ by their sign only (the buyer pays the premium, the writer receives it).

-10

-8

-6

-4

-2

0

2

4

6

8

10

30 31 32 33 34 35 36 37 38 39 40

Buy a callSell a Call

ST


Exercise Value of a Put Euro put struck at X = 35 cents:

If ST =32 33 34 35 36 37 38

Then PT=3 2 1 0 0 0 0

-10-8-6-4-202468

10

30 31 32 33 34 35 36 37 38 39 40

buy a putsell a put

ST


Summary:

Put Option Call Option

Long holder Nice part of a forward Sale Nice Part of a Forward Purchase

Short Writer Bad Part of a forward Purchase Bad Part of a forward Sale


Synthetic Forward Sale Buy a Put and Sell a Call You Put “minus” Call at X=35 cents, same T

if ST =32 33 34 35 36 37 38

then PT =3 2 1 0 0 0 0

And - CT =0 0 0 0 -1 -2 -3

Then total: PT-CT =3 2 1 0 -1 -2 -3Equal FST=3 2 1 0 -1 -2 -3


-50-40-30-20-10

01020304050

0 5 10 15 20 2530 35 4045 50 55 6065 70

buy a putsell a callFS

ST


Put-Call Parity on European Options

Buying a put and selling a call is a forward sale

“buy put” “sell call” “forward sale”

What implications does this principle have?

Implication 1: You can replicate any of [put, call, FC T-bill, HC T-bill] using the other three instruments

Implication 2: Law of One Price (Put Call Parity for European Options): a portfolio of a put minus call must be the same as a forward sale at X.

TTT SF~ X,0)-S~ (Max - ,0)S~ -Max(X


Replication of Instruments

X (“a home currency T-Bill with face value X”)X = Max(X - ST, 0) - Max(ST - X,0) + ST

With numbersST = 32 33 34 35 36 37 38

Then PT =3 2 1 0 0 0 0

And -CT=0 0 0 0 -1 -2 -3

Total PT -CT

35 35 35 35 35 35 35

“a HC T-Bill with face value X= 35”


(synthetic put = HC T-bill – FC debt + call)

(synthetic call = FC T-bill – HC debt + put)

(synthetic FC T-bill = HC T-bill – put + call)

X,0)-S~(Max S~- X 0)S~-(XMax TT T,

-60-40-20

020406080

100120

0 5 101520 25 30 35 40 45 50 55606570

FC Asset Buy a put Sell a call Bond

0)S~-(XMax X-S~ 0)X-S~(Max T, T ,T

0) X,-S~(Max 0) ,S~-(XMax -X S TTT

ST


Put-Call Parity (European Options)

IMPLICATIONS

1. At the forward puts and calls have the same prices

2. At the money puts and calls usually have different values

3. As soon as we have a Call Option price model, the Put-Call Parity implies the Put option pricing model.

Tt,Tt,Tt,

Tt,tt *r1

Str1

Xr1FX

CP


Put Call Parity for European Options

0r1F-F)F(XC-)F(XP

Tt,

Tt,Tt,Tt,tTt,t

TS~

At the forward means X=Ft,T

The Forward represents the risk adjusted expected future spot rate

The value of a bet that a future ST will be higher than a CEQt( ST) must be equal to the value of a bet that ST will be lower.


At-the-money ( X=St )

If Ft,T < St

“FC” is at a forward discount. Risk adjusted FC expected to depreciate Expect the Put to be more valuable Pt>Ct

Ft,T>St

FC is at a forward premium, and Pt<Ct

A bet on an increase of the spot rate is worth more than a bet on a decrease.

Quick approximate test of Parity for at the money options:

Tt,t

Tt,Tt,

Tt,tt *r1

Sr1

Xr1FX

CP

r*)(1r)(1r*r

SCP


Hedging without Eliminating Possible Gains

EXAMPLEDozier can buy a put struck at USD/GBP 1.42. If the GBP goes down, Dozier is hedged. If the GBP goes up Dozier can throw the option away, and benefits from the rise in the GBP.

0204060

80100120

0 50 100 150

FC Asset Buy a put Hedged Asset-60-40-20

020406080

100120

0 50 100 150

FC Asset Forward Sale Hedged Asset

STST


Hedging a Position with “Reverse” Risk

Assume foreign currency cash flows is conditional on other events

EXAMPLE Dozier, when making its bid in GBP: inflow of GBP if win, no

inflow of GBP if lose.

Risk of forward hedge: you may have bad news on the “other event” and losses on the forward contract.

Advantage of option: avoid having both bad tidings at once.

Example Dozier, submits bid for GBP 1m. F and X are at USD/GBP 1.47


Hedge ST Wins contract No contract

sellforward

> 1.47< 1.47

ST+(1.47- ST)=1.47ST+(1.47- ST)=1.47

0+(1.47 -ST) < 00+(1.47 - ST) > 0

buy put

> 1.47< 1.47

ST +(0) > 1.47ST+(1.47- ST)=1.47

0(1.47- ST) > 0

-60-50-40-30-20-10

01020304050

0 0.5 1 1.5 2 2.5 3 3.5buy a putSell Forward

ST


Barrier Options: Knock-outs What if you want some protection but find the option premium too

high? What if you are willing to deal in the spot market if the exchange

rate has moved sufficiently in favor of your original exposure? Knock-out calls or puts may be appropriate.

A knock-out is an option that ceases to exist if the spot price touches or goes through a predetermined barrier level (out-strike)

Compared to a vanilla option, the premium of a knock-out is lower, because there are paths of the exchange rate for which the knock-out does not pay, but the vanilla option does

If you agree to give up protection along these paths, then the premium reduction may be worthwhile


EXAMPLEDozier, an American exporter has an accounts receivable in GBP the current USD/GBP is at 1.45. They can buy a European style put struck at USD/GBP 1.42, with a knock-out barrier at USD/GBP 1.50.

There are three types of possible outcomes:1) The spot rate stays below 1.50 and at maturity it is below 1.42:

Dozier exercises the put

1) The spot rate stays below 1.50 and at maturity it is at or above 1.42: the option expires out of the money

2) The spot rate hits 1.50: the option immediately ceases to exist and Dozier has no more cover

Dozier can now decide whether to:

- forgo cover, and deal at the spot at maturity- get new cover in the forward market or with a new option

For a 1 year option, the premium would be lowered roughly by one third in this case


Hedging Non-Linear Exposure

A company’s financial situation may be a non-linear function of the exchange rate. Options, may then be better suited for hedging than forwards because of their nonlinear /asymmetric payoff profiles.

Companies are subject to economic exposure, that is, their competitive situation and thus their sales depend on the exchange rate.


EXAMPLEKodak can sell 10m films in the domestic market for USD 3. If the JPY depreciated below USD/JPY 0.0075 (JPY/USD 133), Fuji will reduce the price of its films in the American market to SJPY 400. Kodak will then have to lower its price in step to stay competitive. Hedge with put

“strong yen” Value if S > 0.0075

10m films (USD) 30mPut 4000m JPY X= .0075 0Total Hedged Position 30m

“Weak yen”Value if S<0.007510m films (USD) 10m( S*400) <30m Put 4000m JPY X= .0075 4000m(.0075-S)Total Hedged Position 30m


0

5

10

15

20

25

30

35

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175

Sale Receipt Buy a put Hedged Position

-45

-30

-15

0

15

30

45

0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175

Sale Receipt Forward Sale Hedged Position

What would have happened if the hedge had been with forward contracts?

ST

ST


The Use of Options in Risk ManagementIt is Friday, 10/1/10: Pfimerc has a receivable of £500,000 on Friday, 3/19/11.

Spot (U.S. cents per £): 158.34 Strike Call Prices Put Prices170-day forward rate (U.S. cents per £): 158.05 158 5 4.81U.S. dollar 170-day interest rate: 0.20% p.a. 159 4.52 5.33British pound 34-day interest rate: 0.40% p.a. 160 4.08 5.89Option data for March contracts in ¢/£:

How should Pfimerc hedge?£ Put Option: gives them the right (but not the obligation) to sell pounds at a specific price if the £'s value falls

Because Pfimerc wants to sell £500,000, it must pay:£500,000 * ($0.0481/£) = $24,050

They will exercise if the £ falls below $1.58/£500,000 * $1.58/£ = $790,000 if S(t+170) ≤ $1.58/£

They will sell £'s in the spot market if the £ is worth more than $1.58500,000 * S(t+170) > $790,000 if S(t+170) > $1.58/£

Either way, the cost of the put = [$24,050*(1+(0.002*170/360))]=$24,073

The minimum revenue is therefore: $790,000-$24,073=$765,927


Hedging Pound Revenues—Profit vs. Revenue


Combinations of Options and Exotic Options

Exotic options - options with different payoff patterns than the basic options Range forward contract – allows a company to

specify a range of future spot rates over which the firm can sell or buy forex at the future spot rate No money up front

Cylinder options – allow buyers to specify a desired trading range and either pay money or potentially receive money up front for entering into the contracts

Both can be synthesized buying a call and selling a put (at a lower X) For range forward – X must be set such that P(Xp) =

C(Xc)


Combinations of Options and Exotic Options

Average-rate options (or “Asian” option) - most common exotic option; payoff is max[0,Ŝ – X] where Ŝ defines the average forex rate between the initiation of the contract and the expiration date (source and time interval are agreed upon)

Barrier options – regular option with additional requirement that either activates or extinguishes the option if a barrier forex rate is reached

Lookback options – option that allows you to buy/sell at least/most expensive prices over a year (more expensive than regular options)

Digital options (“binary” options) – pays off principal if X is reached and 0 otherwise (think lottery)


Range Forwards and Cylinder Options (Collars) What if you want insurance, but find the option premium too high? What if you want to insure against large movements in the exchange rate?

Example: American company has accounts payable in 3 months to Australian firm. Contract to buy forward only when big exchange rate changes from current

exchange rate St = USD/AUD 0.75.

In 3 months: if St+1 > 0.765, buy AUD at USD/AUD 0.765

if $/AUD 0.725 < St+1 < $/AUD 0.765, buy AUD at St+1if St+1 < 0.725, buy AUD at USD/AUD 0.725

To create this synthetic instrument, buy a call at XC = USD/AUD 0.765, and sell put at XP = USD/AUD 0.725.


Range Forward: Find call and put with same premium. Chose a ceiling: call with XC = USD/AUD 0.765, the premium is Ct = $ 0.006.Now, find a put that has the same premium Pt = $ 0.006, the strike is XP = USD/AUD 0.725.The customer locks a maximum exchange rate without having to pay for it.Cylinder Option: Customer chooses call and put, the premiums do not need to offset each other, and then either pays or receives the net difference.

-0.2

-0.15-0.1

-0.05

0

0.050.1

0.15

0.2

0.6 0.63 0.65 0.68 0.7 0.73 0.75 0.78 0.8 0.83 0.85

sell a put Buy a call

ST


ST

30m

X= .0075

Note: Average price options (“Asian options”) may be particularly appropriate for this purpose


Exchange-listed currency warrants Longer-maturity foreign currency options (>1 year) Issued by major corporations Actively traded on exchanges such as the American

Stock Exchange, London Stock Exchange, and Australian Stock Exchange

American-style option contracts Issuers include AT&T, Ford, Goldman Sachs, General

Electric, etc. Allow retail investors and small corporations too small to

participate in OTC market to purchase L/T currency options

Additional Foreign Currency Option Contracts


Example: Macquarie Put Warrant:

AUD put warrant against $ - maturity date of December 15, 2010X = $0.90/AUD; multiplier of AUD10Payoff of put warrant specified in contract as:Max [0,(X-S)/S]*Multiplier

Suppose spot at maturity is $0.85/AUD, the payoff then is:([($0.90/AUD) – ($0.85/AUD)]/ $0.85/AUD) * 10 = $0.59

Since the actual spot exchange rate at maturity was $1.0233/AUD, the holder of the warrant at maturity received no payoff.

https://www.macquarie.com.au/mgl/au/personal/investments/specialised/listed/trading-warrants

Example: Foreign Currency Option Warrant


Early Exercise and Relative Interest Rates What is the value of early exercise on American options?

How likely is early exercise? depends on 2 conditions(1) Early exercise becomes more likely when options are deep in the money.(2) Relative interest rates:

Call: Opportunity cost of not exercising the call, is the foreign return, r*t,T. So, when r*t,T is high, early exercise becomes more likely. Conversely, if r*t,T = 0, there is no cost to waiting, and there would be no early exercise. Rule-of-thumb: If r* more than 3% p.a. lower than r, then early

exercise for a short-lived call (<9 months) is not likely.

Put: Opportunity cost of not exercising the put, is the domestic return, rt,T. So, when rt,T is high, early exercise becomes more likely.Conversely, if rt,T = 0, there is no cost to waiting, and there would be no early exercise. Rule-of-thumb: If r is more than 3% p.a. lower than r*, then

early exercise for a short-lived put (<9 months) is not likely.

Option Pricing Issues


Example: r = 9% p.a. r* = 12% p.a.St = USD/GBP 1.40 T‐t = .25 (90 days to maturity)

S:

• In‐the‐money calls: no more time value below 1.30, with intrinsic value S‐X = 1.40‐1.30 = 0.10

• In‐the‐money puts: must go much deeper in the money until no more time value: X‐S = 1.95‐1.40 = 0.55.

Why? because r* > r



The Black-Scholes Option Pricing Model

Black-Scholes model is most widely used option pricing model. It gives pricing formulas for European options.

Other approaches: Binomial model, numerical approaches.

Assumptions of the Black-Scholes model:

The distribution of percentage changes in the exchange rate is lognormal (the continuously compounded change in the exchange rate is normal)

Constant variance of percentage changes of the spot rate over the option’s life.

Constant risk free rate(s) over the option’s life


Price of European Call:

Tt

Tt

TtTt

TtTt

tt

dd

XF

d

dNr

XdNrSC

,12

,

,2,

1

2,

1,

*

21ln

11

where N(d) denotes the cumulative standard normal probability; t,T is the standard deviation of ln( ).

Standard Notation

Ct = St e‐r*(T‐t) N(d1) ‐ X e‐r(T‐t)N(d2)

tTdd

tT

tTtTrrXS

dt

12

2*

1

21ln

TS~


Analyzing the Black-Scholes Formula

When valuing the call, we have in fact computed the value of a portfolio containing a certain amount of FC and HC T‐bills.

The first term says that one buys N(d1) FC T‐bills, by paying N(d1)/(1+r*t,T) in FC.

The second term corresponds to a sale of X N(d2) in domestic T‐bills, at a cost of ‐X N(d2)/(1+rt,T), that is, one takes out a loan for X N(d2)/(1+rt,T).

The idea behind the formula is that the replicating portfolio of domestic borrowing and foreign investment is continuously updated.

TtTt

tt rdNX

rdNSC

,

2

,*1

1

1


N(d1)/(1+r*t,T ) is the call’s delta, because:

that is, if St changes by 1, then Ct changes by (for small change in St).Note: 0 1

1

,

*1

Ttt

trdN

SC

Call premium and intrinsic value

‘delta’ = slope

Option traders can hedge options with cash positions in FC. Example: Buy call, for GBP 10m, delta = 0.5. Can hedge call by selling 0.5×10m= GBP 5m spot; have a “delta neutral” position.


Standard notational convention for the Black and Scholes call pricing model:The convention in the literature and among practitioners is to quote all data on an annualized basis.The p.a. variance is typically denoted by the (non-subscripted) symbol 2. Thus 2

t,T = 2 (T-t).

The risk-free rate is typically denoted by the continuously compounded p.a. interest rate, denoted by the (non-subscripted) symbol r (HC) and r* for the (FC). Thus: (1+rt,T) = er(T-t) &(1+r*t,T) = er*(T-t).Example

life is 201 days T‐t = 201/365 = 0.55

: volatility 14.14% Variance .14142 = 0.02 p.a.2

t,T = 2(T‐t) = .02 .55 = .011r (p.a., cc): 9.7347% 1+rt,T = e.097347×0.55 = 1.055

r*(p.a., cc): 5.9031% 1+r*t,T = e.059031×0.55 = 1.033


Example: Price a CallS = USD/CHF 0.45, or 45 cents, and X = USD/CHF 0.43, or 43 cents.

Use exchange rates expressed in cents, likewise the Black-Scholes formula will give an option premium in cents.

Time to maturity: T-t = 201/365 = 0.55Volatility: 2(T-t) = 0.02 × 0.55= .011, √(T-t) = .1414 × √0.55 = .10488.Returns: e.097347×0.55 = 1+rt,T = 1.055, and e.059031×0.55 = 1+r*t,T = 1.033.Intermediate calculations:

582038.010488.0

686908.010488.0

011.021066536.0

066536.43958.45lnln

958.45033.1055.145

*11

12

1

,

,

,*,

dd

d

XF

ctsrr

SeSF

Tt

Tt

Ttt

tTrrtTt


N(d1) = 0.753930, N(d2) = 0.719729 Get values from probability table, calculator, or computer.

St e-r*(T-t) = St / (1+r*t,T) = 45/1.033 = 43.5624 (cts)Xt e-r(T-t) = X / (1+rt,T) = 43/1.055 = 40.7583 (cts)Ct = (43.5624×0.753930)-(40.7583×0.719729) = 3.50807 cts.

Note:Practically, use calculator with option function or software package.For currency options use the standard Black-Scholes option pricing model for “option on stock paying known dividend yield” (also called Garman-Kohlhagen model). Use foreign interest rate, r*, as the dividend yield.


The value of a European Put Option

Now that we have a pricing equation for a Call option we can use Put‐Call parity to find the price of a put:


Implied Volatility

Volatility is an important factor determining option premia; the more volatile the exchange rate, the more valuable the option.

Option buyer can never lose more than the premium paid, yet can realize big gains on the upside.

The more volatile the asset, the larger the expected gain on the option, so the larger the premium.


An option pricing model can be used to uncover the market’s “implied volatility”.

In the Black-Scholes model, everything is observed but . Can use the formula together with the actual price of the option to uncover the “implied volatility”:

Ct = C(, S, X, r, r*, T-t)

We can reverse this to give: (need numerical methods to find ) =G(c,s,x,r,r*,T-t)

Implied Volatility serves as a benchmark for evaluating whether an option is cheap or expensive

Maps premia into common units In the interbank markets option prices are mainly quoted in terms of

implied volatility “vols”, with bid and ask.


Different Probability Distributions of Future USD/EUR

k=x=strike price


Different Probability Distributions of Future USD/EUR





EXAMPLE Have a March call and a June put option on JPY, trading at

wildly different prices. Are they both correctly priced? Is one overpriced relative to the other? Compare their implied volatilities.

EXAMPLEIf think implied < “true ”, what to do?

Options are undervalued Straddle Strategy—Payoff when exchange rate moves

significantly up or down Straddle: buy a put and call

Has to pay premium but since market thinks movement inS are going to be small, the price will be small.

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.6 0.63 0.65 0.68 0.7 0.73 0.75 0.78 0.8 0.83 0.85

Buy a putBuy a call

ST


Butterfly

Buy a call at Xc=0.725 Buy a call at Xc=0.775 Sell 2 calls at Xc=0.750

-0.12

-0.07

-0.02

0.03

0.08

0.675 0.7 0.725 0.75 0.775 0.8 0.825 0.85

Buy a callBuy a callsell 2callsButterfly

This strategy allows to bet that market perception of volatility is too large with a limited liability. Loses are kept to initial investment.

Compare to shorting a straddle.

ST


Bullish Spread

Suppose investor thinks call at Xc=0.775 is overvalued

Buy a call at Xc=0.725 Sell a call at Xc=0.775

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.68 0.705 0.73 0.755 0.78 0.805 0.83

Buy a callSell a CallBull Spread

The investor is making the bet that?

ST


Bear Spread

Suppose investor thinks Xc=0.725 is overvalued

Sell a call at Xc=0.725 Buy a call at Xc=0.775

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.68 0.705 0.73 0.755 0.78 0.805 0.83

Buy a callSell a CallBear

The investor is making the bet that?

ST

Documents

International Financial Markets Options