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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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Reminder of notationBarrier options
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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Reminder of notationBarrier options
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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Reminder of notationBarrier options
Lookback options
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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Reminder of notationBarrier options
Lookback options
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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
Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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
(XT − K · YT )+ if min
0≤t≤TXY (T ) ≤ U. (2)
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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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
(XT − K · YT )+ if min
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
(XT − K · YT )+ if min
0≤t≤TXY (T ) > U. (4)
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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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
(XT − K · YT )+ if min
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
Price of down-and-in with constant barrier – conclusion
Hence we have shown that the down-and-in call option with payoff
(XT − K · YT )+ if min
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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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Reminder of notationBarrier options
Lookback options
Definition and examplesDown-and-in with constant barrierPricing via power optionsDown-and-in with exponential barrier
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
(XT − K · YT )+ if min
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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$(t) ≥ L}.
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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
Connection with barrier options – continued
A contract that pays off I(τL ≤ T ) units of BT
I(τL ≤ T ) · BTT = I(τL ≤ T , S$(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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
Price of the contract V1
Since
V 1T =
∫ ∞
mI(S$(T ) ≥ L)dL · BT
T =
∫ S$∨m
mdL · BT
T
= (S$(T ) − m)+ · BTT = (ST − mBT
T )+,
the contract V 1 is a plain vanilla call option with strike m.
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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
Price of the contract V2 – case of r = 0
The value of the second contract depends on the value of α orequivalently r . In case of α = 1, or r = 0, we can write
V 2T =
∫ ∞
m
1
LI(S$(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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
Connection with Barrier optionsPricing of a lookback optionMaximum drawdown
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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Reminder of notationBarrier options
Lookback options
References
Večeř, JanStochastic calculus – A numeraire approach
Chapman & Hall, 2011
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Reminder of notationBarrier options
Lookback options
Thank you for your attention.
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