©2001, Mark A. Cassano Exotic Options Futures and Options Mark Cassano University of Calgary

©2001, Mark A. Cassano

Exotic Options

Futures and OptionsMark Cassano

University of Calgary

©2001, Mark A. Cassano

Asset Price

• Assume the asset has volatility of 30%, the price is 100, the time intervals are one month and options expire in three months. Also assume no dividends (although we could easily adjust it by altering the risk neutral probabilities using the “dividend yield” adjustments). r = 5%.

• Going to the tree..............

©2001, Mark A. Cassano

The Tree

Asset PriceTime

0 0.0833 0.1667 0.25129.6681

118.911109.0463 109.0463

100 10091.70415 91.70415

84.0965177.11999

©2001, Mark A. Cassano

European Call OptionTime

0 0.0833 0.1667 0.2529.66806

19.3267911.91292 9.046318

7.082795 4.5263262.264748 0

00

European Put OptionTime

0 0.0833 0.1667 0.250

02.036732 0

5.840575 4.1105269.730726 8.295849

15.4876922.88001

At -The-Money Options

©2001, Mark A. Cassano

Exotic Options

• Exotic options alter some of the characteristics of a standard (plain vanilla) options:– Time to Maturity (e.g. Barrier)– Exercise Price (e.g. Lookback)– Position (e.g. chooser, Swing)– Underlying Asset (e.g. Quantos,

Asian)– Payoff Structure (e.g. Binary)

©2001, Mark A. Cassano

Path Dependent Options

• Note that for regular options the option payoff does not depend on how it got there. For example the above call option paid $9.046 since S in three months is 109.046. The payoff depended on S1/4 with no dependence on how it got to the final value (i.e. uud or udu or duu).

• Mathematically:),....,,( NOT )( TttT SSSCSC 2

©2001, Mark A. Cassano

Path Dependent Options

• Path Dependent Options have payoffs that depend on previous values the asset takes.

• Mathematically, the value of the option at expiration is:

• To make the notation easier we will use subscripts for the step in the tree (as opposed to calendar time in years).

nnTttT CSSSCSSSCC ),....,,(),....,,( 212

©2001, Mark A. Cassano

Lookback Options

• Our first path dependent option are lookback options.

• Lookback (Call) Options on the Minimum– Strike Price at Expiration:

• Example: 6 month lookback call on the Japanese Yen. Importer can buy a lookback option that allows her to purchase yen at the lowest price that occurs in 6 months.

],0max[

},...,,min{ 21

nnn

nn

XSC

SSSX

©2001, Mark A. Cassano

Lookback Options

• Lookback (Put) Options on the Maximum– Strike Price at Expiration:

• Lookback Options on the Average

],0max[

},...,,max{ 21

nnn

nn

SXP

SSSX

)0,max(

)0,max(

1

1

nnn

nnn

n

iin

SXP

XSC

Sn

X

©2001, Mark A. Cassano

Pricing: Analytical Results

• There are BSM type results for these. Often there are clever static hedges (use a portfolio of standard options).

• Most of these formulas assume the stock is observed continuously. In practice they often are based on closing prices. (see p. 466 for reference).

• The following binomial trees do a poor job of approximating the formulas but I think it gives the student a better grasp of pricing. (Monte Carlo would be the easiest way for most of these).

©2001, Mark A. Cassano

Look Back on the MinimumCall Option 29.6681

19.326819.0463

11.912939.0463

4.5263180

9.04640717.34215

8.6771470

6.227694

7.607643.806484

0

©2001, Mark A. Cassano

Exercise

• Price a Lookback on Maximum and Lookback on Average.

• Analytical Formulas: – Call(Min) 11.98, Put (Max) 11.86

• Answers:– Maximum: European, $8.18; American, $8.39– Average Call: European $3.57; American

$4.06– Average Put: European $2.95; American

$3.50

Verify This!Need a Binomial Tree for American.

©2001, Mark A. Cassano

Average Rate Options

• These are also known as Asian options although I find this an unfortunate name since these Asian options can be American or European style options.

• The lookback options had the exercise price being random and changing with time. Average Rate Options have a fixed exercise price but a random underlying asset value.

©2001, Mark A. Cassano

Average Rate Options

• The Call will pay off, at expiration,

• Similarly for the put,

n

iinnn S

nSXSC

1

10 where ,max

,max 0nn SXP

©2001, Mark A. Cassano

Analytical Approximations

• If the averages are geometric (i.e. the nth root of the product) we can get an exact formula.

• There are several approximations you can use for arithmetic averages. See pp. 468-469 if you are curious (you will not need to know this analysis).

©2001, Mark A. Cassano

Average Rate At-the-Money European Call

Option

Average Price 114.3984109.3085

109.243

104.5073104.5152

103.0049100.1797

100

100.187697.23472

95.8520895.85208

91.876291.93355

88.23016

Call Option 14.3984311.78408

9.242983

7.0596994.515233

2.3482350.179695

3.5555940.187613

0.0938720

0.046969

00

0

Worth More or Less Than Plain Vanilla Call Option?

©2001, Mark A. Cassano

Barrier Options

• An immediate rebate barrier option pays a given rebate as soon as a barrier is reached.

• Example: Up & Out Call Option has a payout of Max(0, Sn - X) only if the stock price fails to reach a certain level, call it H. If the stock reaches this barrier, the call option will be exercised, paying H - X.

• IMPORTANT: The text (and the formulas) assume no rebate; i.e. Up and Out terminates with no payment of H - X. We can adjust this by adding (H-X)P*(S>H).

©2001, Mark A. Cassano

More “Knockout” Options• Down & Out Call Option: If the stock price

falls below a threshold, H, the option immediately expires worthless.

• Up & Out Put Option: If the stock price rises above a threshold, H, the option immediately expires worthless.

• Down & Out Put Option: If the stock price falls below a threshold, H, the option is immediately is exercised: Value X - H. (Again text assumes no rebate).

©2001, Mark A. Cassano

Pricing: Analytical Results

• There are BSM type formulas for these options.

• We can use the fact that a regular call is a down and out plus a down and in; also, a regular call is an up and out plus an up and in (“& in” to be defined shortly). Similar for puts.

• These formulas assume the stock is observed continuously. In practice they often are based on closing prices. (see p. 464)

©2001, Mark A. Cassano

Tree Exercises

• Consider a Down & Out Call with a barrier of $95; verify its price is $5.96. (BSM = 4.21)

• Consider a Down & Out Call with a barrier of $90; verify its price is $7.08. (BSM = 6.00)

• A Up and Out Call with a barrier at 105. $3.12. Look at the tree.... All options assume X = 100 (at-the money)

Note these prices differ a great deal, see pp. 478-481 for adjustments.

©2001, Mark A. Cassano

Up and Out Call

129.6681118.911 109.0463

109.0463 100 109.046391.70415

100109.0463

91.70415 100 91.70415

84.09651 91.7041577.11999

$5

$5

©2001, Mark A. Cassano

“Knock-in” Options

• These will become exercisable only if they attain a barrier H (with a particular direction).

• A Down and In Call will become exercisable (essentially it will “exist”) only if the stock price falls below a threshold H<X.

• Up and In Put: Exist only if S rises above a threshold H>X.

©2001, Mark A. Cassano

Example

• Down & In Call with Barrier 95129.6681

118.911 109.0463

109.0463 100 109.046391.70415

100109.0463

91.70415 100 91.70415

84.09651 91.7041577.11999

Valid

Never Valid

$9.05$1.12

©2001, Mark A. Cassano

Complex Threshold Rules

• The terms of knocking in and out may not be as simple. For example a baseball option is a regular call option that gets knocked out (becomes worthless) if closing price falls below the threshold on three different days (prior to expiration).

©2001, Mark A. Cassano

Cliques & Ladders & Shouts

• (Oh My!) We can classify a set of options with payoffs

• H is determined by some rule. If H = X we have an ordinary option.

),,0max(

,,0max

HXSXP

XHXSC

nn

nn

In general, H is random.

©2001, Mark A. Cassano

Cliques & Ladders & Shouts

• If H is the maximum price then the call has a lookback feature that gives the right to buy for X and sell it not at the spot price @ expiration (Sn ) but the highest price during the life of the contract.

• If H is the stock price at a single pre-determined date then it is a one-click option.

©2001, Mark A. Cassano

Ladders• If H is a predetermined level only

if it is reached by the stock, otherwise it is X, then the option is a one-rung-ladder. Can you guess what a two-rung-ladder is?

H H

Left Figure, Option will pay at least H - X. Right figure is a regular call option (so far).

©2001, Mark A. Cassano

Shouts

• H is the (contemporaneous) price at any moment the buyer chooses. (Could have multi-shout contracts also).

• Let us try to price an at-the-money shout option.

• Same Idea: Work backwards asking each step: “to shout or not to shout”?

©2001, Mark A. Cassano

Pricing a Shout Option

129.6681118.911 109.0463

109.0463 100 109.046391.70415

100109.0463

91.70415 100 91.70415

84.09651 91.7041577.11999Never Shout

$9.0463

Shout Now$18.911

$9.0463$0

$29.668

$8.30

©2001, Mark A. Cassano

Pricing a Shout Option

129.6681118.911 109.0463

109.0463 100 109.046391.70415

100109.0463

91.70415 100 91.70415

84.09651 91.7041577.11999Never Shout

$9.0463

Shout Now$9.0463

$9.0463$9.0463

$29.668

$8.19

Better not to shout in one month; Value is $8.30

©2001, Mark A. Cassano

Others: Forward Start • Typical executive compensation

contracts. You will receive an at the money option some date in the future, T1. Note that at-the-money options are proportional to the stock and exercise price. Let c be the value of a current at the money option with the same duration. An at-the-money call at T1 will be worth

1

0TS

Sc

©2001, Mark A. Cassano

Forward Start Options

• What is the current value? Well the constant is known today hence we need the present value of the future stock price. We’ve done this enough:

DPVSSc

ceeSSc

SSc

PV qTqTT

0

00

00

or 11

1

©2001, Mark A. Cassano

As You Like It

• These chooser options allow the holder to choose, during a specified time period, whether the option is a call or a put.

• Assume the choice must be made at T1 then the value of this option at that time is max(c,p). If the options are both European and have the same strike price we can use put-call-parity for valuation.

©2001, Mark A. Cassano

As You Like It

• Let T2 be the expiration date of the options on which the chooser is based. PCP at time T1 yields:

• Result, Cost today is identical to the cost of this standard option strategy:

),0max

,max),max(

1

1212

12

1

12

TTTrqTTq

TTqT

TTr

SXeec

eSXeccpc

),(),( 121212 TXepeTXcChooser TTrqTTq

©2001, Mark A. Cassano

Rainbow Options

• These are options that depend on more than one risky asset.

• Exchange Options: A call on asset A with exercise price being the price of asset B. This pays max( SA - SB , 0) at expiration.

• Option on the Best• Call on the Minimum

XSSTC BA ),max(,0max

XSSTC BA ),min(,0max

©2001, Mark A. Cassano

Other Options

• Spread Options: Based on the difference between two prices, e.g. Max(0,SA - SB )

• Digital/Binary Options (pp. 464-465)– Cash or Nothing Options will pay a fixed

amount if the option is in the money. Can you price a European digital option using Black-Scholes?

– Asset of Nothing Options will pay the stock value if in the money.