Lookbacks and Barriers A lookback option has terms that depend on - - PowerPoint PPT Presentation

Lookbacks and Barriers

• A lookback option has terms that depend on the path.
• For example, a lookback call with a floating strike allows the

holder to choose the strike from any value taken by the asset: K = Sτ, where the holder chooses τ ∈ [0, T].

• The optimal choice is

K = min

0≤t≤T St

and hence the payoff is ST − min

0≤t≤T St.

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• A lookback call with a fixed strike K allows the holder to

choose the time at which the option is exercised: the payout is (Sτ − K)+, where the holder chooses τ ∈ [0, T].

• The optimal choice is

τ = arg max St and hence the payoff is

• max

0≤t≤T St − K

• +

.

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• The price of a lookback option is, as always, the expected

value of the discounted payoff under the risk-neutral distri- bution. – This requires the joint distribution of ST and either min0≤t≤T St

• r max0≤t≤T St ;

– use the reflection principle.

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• A barrier option is an option (European or American) with

the additional condition that it is worthless if the path crosses (knock-out) or fails to cross (knock-in) a barrier level.

• Knock-in:

– up-and-in; – down-and-in.

• Knock-out:

– up-and-out; – down-and-out.

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• For a European barrier option, the payoff is a function of ST

and either min0≤t≤T St or max0≤t≤T St.

• Calculating the expected value similarly requires their joint

distribution.

• More complex barrier options may have barriers that are ef-

fective in only the early part of the option’s life, or in only the later part.

• One exchange-traded barrier put had a knock-out upper bar-

rier and automatic exercise at a lower barrier.

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Asian options

• Payoff depends on the average price.
• E.g. Asian call with fixed strike K: payoff is

CT =

• 1

T

T

0 St dt − K

• +
• CT is FT-measurable, so the value at time t is again

EQ e−r(T−t)CT

• Ft
• But note that this is not a function of only St.

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• Asian call with floating strike: payoff is

CT =

• ST − 1

T

T

0 St dt

• +
• Again, the value at time t is

EQ e−r(T−t)CT

• Ft
• and again this is not a function of only St.

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