Barrier and Lookback options - cuni.czartax.karlin.mff.cuni.cz/~dvorm3bm/1112z/11_Slamova... · 2012. 1. 7. · Pricing of a lookback option Maximum drawdown 4 References 2/40. Reminder

Reminder of notationBarrier options

Lookback options

Barrier and Lookback optionsThe numeraire approach

Lenka Slámová[email protected]

Stochastic modelling in economics and financeDecember 12th, 2011

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Lookback options

1 Reminder of notation

2 Barrier optionsDefinition and examplesPrice of down-and-in call option with constant barrierPricing via power optionsPrice of down-and-in call option with exponential barrier

3 Lookback optionsConnection with Barrier optionsPricing of a lookback optionMaximum drawdown

4 References

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Lookback options

Price of an asset

Consider two assets X and Y .

Price of X is a number representing how many units of assetY are required to obtain a unit of asset X .

Price of the asset X at time t is denoted by XY (t), while byXT we denote the asset itself at time T .

The asset Y serves as a reference asset, or numeraire.

Pairwise relationship

X = XY (t) · Y ⇔ Y =1

XY (t)· X

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Arrow-Debreu security

Arrow-Debreu security is a contract V that pays off one unit ofasset Y at time T when the scenario ω is in A, otherwise it paysnothing.

VT = IA(ω) · YT , or V Y (T ) = IA(ω).

We would like to find its price at time 0. VY is a martingale underthe probability measure P

Y , hence

VY (0) = EY [VY (T )] = E

Y [IA(ω)] = PY (A).

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Price evolution

We assume that the price process XY (t) follows

dXY (t) = σXY (t)dW Y (t),

where PY is such a probability measure that W Y is P

Y -brownianmotion. Under this measure XY (t) is a P

Y -martingale. Thesolution to this SDE is

XY (t) = XY (s) · exp

(σW Y (t − s) − 1

2σ2(t − s)

).

We will assume, that

LYt

(XY (T )

XY (t)

)= LX

t

(YX (T )

YX (t)

).

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Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier




4 References

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Lookback options


Barrier option

contract whose payoff depends on the event that theunderlying price crosses a certain boundary

is cheaper than its corresponding plain vanilla counterpart

often used in foreign exchange markets

Knock–out option

Pays off only if the price stays between the barriers and expiresworthless if the barrier is hit during life of the option.

Knock–in option

Pays off zero if the price stays between the barriers and convertsinto plain vanilla option when the price first hits the barrier.

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Payoffs of barrier options

Barrier knock-out option

Pays off f Y (XY (T )) units of asset Y if fY(t, XY (t)) ≥ 0 for all t.

Hence the price XY (t) must stay in the region at all times.

Barrier knock-in option

Pays off f Y (XY (T )) units of asset Y if fY(t, XY (t)) < 0 for all t.

Hence the price XY (t) must enter the region at least once.

First hitting time of the barrier

τ = inf{t ≥ 0, fY(t, XY (t)) < 0}.

Simple relationship between EC (European call), KO(knock-out) and KI (knock-in) call options

V EC = V KO + V KI .8/40


Lookback options


Knock-out call option

Down-and-out call option

If the price goes down and hits a barrier L < XY (0) then it becomeworthless. The payoff is given by

(XT − K · YT )+ if min

0≤t≤TXY (T ) ≥ L. (1)

Up-and-out call option

If the price goes down and hits a barrier U > XY (0) then itbecome worthless. The payoff is given by


0≤t≤TXY (T ) ≤ U. (2)

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Knock-in call option

Down-and-in call option

If the price goes down and hits a barrier L < XY (0) then it turnsinto a plain vanilla option. The payoff is given by


0≤t≤TXY (T ) < L. (3)

Up-and-in call option

If the price goes down and hits a barrier L < XY (0) then it turnsinto a plain vanilla option. The payoff is given by


0≤t≤TXY (T ) > U. (4)

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Lookback options





4 References

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Lookback options


Price of down–and–in call option with constant barrier

We would like to find a price of a down-and-in call option, in casewhen the interest rate r = 0. The payoff is


0≤t≤TXY (t) ≤ L.

Hence the option activates if the barrier L is hit. Let

A =

{min

0≤t≤TXY (t) ≤ L, XY (T ) ≥ K

}.

The down–and–in call option corresponds to a combination of 2Arrow-Debreu securities V and U:

1 V pays off 1 unit of Y on the event A (and we take −K

contracts)

2 U pays off 1 unit of X on the event A

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Price of Arrow-Debreu security V

Denote VT = IA(ω) · YT . Assume that XY (t) = L, i.e. the barrierwas hit at time t. Then the price of V at time t is

Vt = PYt (XY (T ) ≥ K ) · Yt

= PYt

(XY (t) exp

(σW Y (T − t) − 1

2σ2(T − t)

)≥ K

)· Yt

= PXt

(XY (t) exp

(σW X (T − t) − 1

2σ2(T − t)

)≥ K

)· 1

L· Xt

= PXt

(XY (t)

YX (T )

YX (t)≥ K

)· 1

L· Xt = P

Xt

(1

K≥ XY (T )

(XY (t))2

)· 1

L· Xt

= PXt

(L2

K≥ XY (T )

)· 1

L· Xt

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Price of Arrow-Debreu security V - continued

Hence the Arrow–Debreu security V corresponds to 1/L units ofArrow–Debreu security U with payoff UT = IB(ω) · XT , where theevent B is

B =

{XY (T ) ≤ L2

K

}.

Which is a plain vanilla option. Denote the first hitting time of theboundary L by τL. Then

VτL∧T =1

LUτL∧T .

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Price of Arrow-Debreu security U

Denote UT = IA(ω) · XT . Assume that XY (t) = L, i.e. the barrierwas hit at time t. Then the price of U at time t is (following thesame arguments as before)

Ut = PXt (XY (T ) ≥ K ) · Xt = P

Yt

(L2

K≥ XY (t)

)· L · Yt .

Hence the Arrow–Debreu security U has the same price as L unitsof Arrow–Debreu security V with payoff VT = IB(ω) · YT .

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Price of down-and-in with constant barrier – conclusion

Hence we have shown that the down-and-in call option with payoff


0≤t≤TXY (t) ≤ L.

corresponds to a plain vanilla option with a payoff

(LYT − K

L· XT

)+

up to the first time τL when the barrier is hit.

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Distribution of the first hitting time

We can also determine the distribution of the first hitting time

τL = inf{t ≥ 0 : XY (t) ≤ L}

using the prices of two Arrow-Debreu securities. Consider acontract V that pays off a unit of Y at time τL, i.e. VτL

= YτL, or

VT = I(τL ≤ T ) · YT . Define two Arrow-Debreu securities V 1 andV 2 by

V 1T = I

(min

t∈[0,T ]XY (T ) ≤ L, XY (T ) > L

)· YT

V 2T = I

(min

t∈[0,T ]XY (T ) ≤ L, XY (T ) ≤ L

)· YT = I (XY (T ) ≤ L) · YT

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Distribution of the first hitting time – continued

V 2 is a plain vanilla option with priceV 2 = P

Y (XY (T ) ≤ L) · Y .

V 1 is a knock-in Arrow-Debreu security with barrier L andstrike K = L, hence its price is V 1 = P

X (XY (T ) ≤ L) · 1L

· X

up to time τL

Since V = V 1 + V 2, we have

V = PY (τL ≤ T ) · Y =

= PX (XY (T ) ≤ L) · 1

L· X + P

Y (XY (T ) ≤ L) · Y .

And so

PY (τL ≤ T ) = P

X (XY (T ) ≤ L)XY (0)

L+ P

Y (XY (T ) ≤ L).

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Lookback options





4 References

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Lookback options


Pricing via power options

When the interest rate r is positive then the barrier takesexponential form. We can show that when XY (t) hits theexponential barrier, there is a corresponding power option Rα

whose price is hitting constant barrier.

Power option

Power option Rα pays off [XY (T )]α units of asset Y at time T .We have R0 = Y and R1 = X . Its price is given by

Rα

Y (t) = exp

{1

2α(α − 1)σ2(T − t)

}· [XY (t)]α .

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Pricing via power options – continued

Let us consider an Arrow-Debreu security V that pays off a unit ofY at time T on the event

A =

{min

t∈[0,T ]

(e−r(T−t)XY (t)

)≤ L, XY (T ) ≥ K

}.

The first time of hitting the exponential boundary isτL = inf{t ≥ 0 : XY (t) ≤ Ler(T−t)}.

Theorem

The Arrow-Debreu security V specified above has the same price

up to the first hitting time τL as(

1L

)α

units of a plain vanilla

Arrow-Debreu security V α that pays off

V α

T = IB(ω)·Rα

T , with B = {XY (T ) ≤ L2/K} and α = 1− 2r

σ2.

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Lookback options





4 References

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Lookback options


Price of down-and-in call option with exponential barrier

The payoff of down-and-in call option in case of positive interestrate r > 0 is


0≤t≤T

(e−r(T−t)XY (t)

)≤ L.

By the previous theorem we know, that the price up to time τL isequivalent to a plain vanilla European call option V with a payoff

VT = L2r

σ2

(LR

− 2r

σ2

T − K

LR

(1− 2r

σ2

)

T

)+

The option is a combination of two Arrow-Debreu securities

first pays off a unit of X and has a plain vanilla counterpart

that is settled in units of R(− 2r

σ2 ).

second pays off a unit of Y and has a plain vanilla counterpart

that is settled in units of R(1− 2r

σ2 ).

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Lookback options

Connection with Barrier optionsPricing of a lookback optionMaximum drawdown




4 References

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Lookback options


Lookback option

The payoff of lookback option depends on either the maximumprice of XY (t) or the minimum price of XY (t).Consider the maximal asset

M∗t =

[max

0≤s≤tXY (s)

]· Yt .

It is also known as the High watermark. M∗ is an arbitrage asset,but the contract to deliver M∗ at a future time is a no-arbitrageasset.

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Lookback options


Connection with barrier options

Consider a contract V to deliver the maximal asset M∗ at time T .Let X be a stock S and Y a dollar $. Then the payoff of V is

VT = M∗T =

[max

0≤t≤TS$(t)

]· $T =

[max

0≤t≤TS$(t)

]· BT

T

or

VT =

[∫ ∞

0I(τL ≤ T )dL

]· BT

T .

When the maximal price if m, then the price process S$ must havecrossed all levels L ≤ m by time T and all levels L > m were notreached by time T .The hitting time τL is defined as the first time the price S$ reacheslevel L from below,

τL = inf{t ≥ 0 : S$(t) ≥ L}.

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Lookback options


Connection with barrier options – continued

A contract that pays off I(τL ≤ T ) units of BT

I(τL ≤ T ) · BTT = I(τL ≤ T , S$(T ) ≥ L) · BT

T +

I(τL ≤ T , S$(T ) < L) · BTT

= I(S$(T ) ≥ L) · BTT + I(τL ≤ T , S$(T ) < L) · BT

T .

First is a plain vanilla option and second is a knock–inArrow-Debreu security, a barrier up-and-in with payoff on the eventS$(T ) < L. We have already shown how to price this barrieroptions in the previous part. First note that

S$(t) ≥ L ⇔ SBT (t) ≥ Ler(T−t).

Hence the second contract has the same price as(

1L

)α

units of

plain vanilla Arrow-Debreu security that pays off a power optionRα

T if S$(T ) ≥ L, for α = 1 − 2rσ2 .

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Lookback options





4 References

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Lookback options


Pricing of a lookback option

Denote by m = M∗$ (t) the current running maximum of the stock

price. The payoff of the lookback option is

VT = m · BTT +

∫ ∞

mI(S$(T ) ≥ L)dL · BT

T +

∫ ∞

m

(1

L

)α

I(S$(T ) ≥ T )dL · Rα

T

= m · BTT + V 1

T + V 2T .

Again we will find forms of the two contracts V 1 and V 2.

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Lookback options


Price of the contract V1

Since

V 1T =

∫ ∞

mI(S$(T ) ≥ L)dL · BT

T =

∫ S$∨m

mdL · BT

T

= (S$(T ) − m)+ · BTT = (ST − mBT

T )+,

the contract V 1 is a plain vanilla call option with strike m.

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Price of the contract V2 – case of r = 0

The value of the second contract depends on the value of α orequivalently r . In case of α = 1, or r = 0, we can write

V 2T =

∫ ∞

m

1

LI(S$(T ) ≥ L)dL · ST =

∫ S$∨m

m

1

LdL · ST

= log

(S$(T )

m

)+

· ST = log

(SBT (T )

m

)+

· ST .

Hence it is again a plain vanilla option.

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Price of contract V – case of r = 0

Hence V is a combination of two plain vanilla options

VT = m · BTT + (ST − m · BT

T )+ +

[log

(SBT (T )

m

)+]

· ST .

Let uT (t, x , y) = ET[VBT (T )|SBT (t) = x , M∗

BT (t) = y]. Then

using Black-Scholes formula and a simple computation,

uT (t, x , y) = y+xN(d+)−yN(d−)+xσ√

T − t[d+N(d+)+φ(d+)].

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Price of the contract V2 – case of r > 0

In case of α < 1, or r > 0, we can write

V 2T =

∫ ∞

m

(1

L

)α

I(S$(T ) ≥ L)dL · Rα

T =

∫ S$∨m

m

(1

L

)α

dL · Rα

T

=1

1 − α

((S$(T ))1−α − m1−α

)+· Rα

T

=1

1 − α

((SBT (T ))1−α − m1−α

)+· Rα

T

=1

1 − α

(ST − m1−αRα

T

)+,

where we used the fact that

[SBT (T )]1−α·Rα

T = [SBT (T )]1−α·[SBT (T )]α·BTT = SBT (T )·BT

T = ST .

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Price of contract V – case of r > 0

Hence V is once again a combination of two plain vanilla options

VT = m · BTT + (ST − m · BT

T )+ +σ2

2r

[ST − m

2r

σ2 R

(1− 2r

σ2

)

T

]+.

The price of the lookback option u(t, x , y) at time t can be onceagain computed using Black-Scholes formula, but it is a bit morecomplicated because of the positive interest rate.

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Hedging

Once we know the price of the lookback option uT (t, x , y), we cancompute the hedging portfolio. In the case r = 0 we have

∆S(t) = uTx (t, x , y) = 2N(d+) + σ

√T − t[d+N(d+) + φ(d+)],

∆T (t) = uTy (t, x , y) = yN(−d−) − xN(d+).

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4 References

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Lookback options


Maximum drawdown

popular portfolio performance measuredrawdown asset = difference between the maximal asset M∗

and the stocka large value of the drawdown indicates that the price is farfrom its running maximum

Example

Hedge fund charges 20 % of the returns above the high watermark.Initial price of the fund is 100, at the end of year 1 the price of thefund 110, the return over the high watermark is 10, hence thehedge fund manager gets compensation of 2. At the end of year 2price drops back to 100, he gets nothing. If the price gets to 115at the end of year 3, the manager receives compensation only forthe return above the running maximum 110, so 1. Hence themanager gets a compensation only if the drawdown is zero.

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Maximum drawdown – continued

Consider two basic assets: Y plays role of the reference asset inthe economy (Dow Jones index), X is the hedge fund portfolio.Drawdown asset is defined by

M∗T − XT .

Now we have three natural numeraires to consider - Y , X and M∗.The maximal drawdown can be thus defined in three ways

DYT =

[max

0≤t≤T(M∗

Y (t) − XY (t))

]· YT , absolute maximum drawdown,

DXT =

[max

0≤t≤T(M∗

X (t) − 1)

]· XT ,

DM∗

T =

[max

0≤t≤T(1 − XM∗(t))

]· M∗

T , relative maximum drawdown,

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References

Večeř, JanStochastic calculus – A numeraire approach

Chapman & Hall, 2011

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Thank you for your attention.

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Barrier and Lookback options - cuni.czartax.karlin.mff.cuni.cz/~dvorm3bm/1112z/11_Slamova... · 2012. 1. 7. · Pricing of a lookback option Maximum drawdown 4 References 2/40. Reminder

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