1
Bruno DUPIRE
Bloomberg
Quantitative Research
Arbitrage, Symmetry and Arbitrage, Symmetry and DominanceDominance
NYU SeminarNYU SeminarNew York 26 February 2004 New York 26 February 2004
Background
Dominance
Can we say anything about option prices and hedges when (almost) all assumptions are
relaxed?
REAL WORLD:
anything can happen
infinite number of possibleprices,
infinite potential loss
MODEL:stringent assumption
1 possible price,1 perfect hedge
3Dominance
Model free properties
4
European profilesnecessary & sufficient conditions on call prices
Dominance
S K C K 0 0
S K S K C K C K
1 2 1 20
K K S K K K S K K K S KC
3 2 1 3 1 2 2 1 3 0
convex
S S K
KC
1 0 1'
K
K1 K2
K2 K3K1
0K
5
A conundrum
Dominance
Do we necessary have ? limK
C K
0
Call prices as function of strike are positive decreasing: they converge to a positive value .
It depends which strategies are admissible!
•If all strikes can be traded simultaneously, C has to converge to 0.
•If not, no sure gain can be made if > 0.
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Arbitrage with Infinite trading
Dominance
N
payoff ofequality : Tat t2
gain : 0at t
) premium (receive Sell
)2
(cost N) (i theallBuy :Arbitrage
2/
price its be and be Let
Ni0i0
n
0i
1
N
i
iii
iiii
C
CS
yNyCCSn
y -CCCS
7
Quiz
Dominance
Strong smile
Put (80) = 10, Put (90) = 11.Arbitrageable?
80 90 S0 = 100
8
Answer
Dominance
•At first sight:
P(80) < P(90), no put spread arbitrage.
•At second sight:
P (90) - (90/80) P (80) is a PF
with final value > 0 and premium < 0.
80 90
90
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Bounds for European claims
Dominance
European pay-off f(S). What are the non-arbitrageable prices for f?Answer: intersection of convex hull with vertical line S S 0
SS0
UB
LB
f
arbitrageable price
arbitrage hedge
If market price < LB : buy f, sell the hedge for LB:
0 initial cost
>0 pay-off
{
arbitrage
10
Call price monotonicity
Dominance
Call prices are decreasing with the strike:are they necessarily increasing with the initial spot?
non
NO.
counter example 1:
0 T
90
110
100
0 T
90100
80
12025%
75%
counter example 2:
martingale
100
11
Call price monotonicity
Dominance
If model is continuous Markov,Calls are increasing with the initial spot
(Bergman et al)Take 2 independent paths x and y starting from x and y today.
(1) x and y do not cross. (2) x and y cross.
xy
x
y
Knowing that they cross, the expectation does not depend on the initial value (Markov property).
x(T) y(T)
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Lookback dominance
Dominance
•Domination of
•Portfolio:
•Strategy: when a new maximum is reached, i.e.
sell
The IV of the call matches the increment of IV of the product.
Max K
0
1
0a
C K a dKK
M M M
Ma
C M aMa
M M a M
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Lookback dominance (2)
Dominance
•More generally for
•To minimise the price, solve
thanks to Hardy-Littlewood transform (see Hobson).
k s s
C k s
s k sds Max K
K
dominates 00
Min
C k ss k s
dsk s
K
0
14Dominance
Normal model with no interest rates
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Digitals
Dominance
1 American Digital = 2 European Digitals
From reflection principle,
Proba (Max0-T > K) = 2 Proba (ST > K) K
Brownian path
Reflected path
As a hedge, 2 European Digitals meet boundary conditions for the
American Digital.
If S reaches K, the European digital is worth 0.50.
0.000.200.400.600.801.001.201.401.601.802.00
50 70 90 110 130 150
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Down & out call
Dominance
DOC (K, L) = C (K) - P (2L - K)
The hedge meets boundary conditions.
If S reaches L, unwind at 0 cost.
-40.00
-30.00
-20.00
-10.00
0.00
10.00
20.00
30.00
40.00
50.00
60.00
50 60 70 80 90 100 110 120 130 140 150 160 170
K2L-K
L
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Up & out call
Dominance
UOC (K, L) = C (K) - C (2L - K) - 2 (L - K) Dig (L)
The hedge meets boundary conditions for the American Digital.
If S reaches L, unwind at 0 cost.
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
20.00
80 90 100 110 120 130 140 150 160
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General Pay-off
Dominance
The hedge must meet boundary conditions, i.e. allow unwind at 0 cost.
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
20.00
80 90 100 110 120 130 140 150 160 170 180
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Double knock-out digital
Dominance
2 symmetry points: infinite reflections
Price & Hedge: infinite series of digitals
-1.1
-0.6
-0.1
0.4
0.9
90 100 110 130
-0.02
-0.02
-0.01
-0.01
0.00
0.01
0.01
0.02
0.02
80 120
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Max option
Dominance
(Max - K)+ = 2 C (K)
Hedge: when current Max moves from M to M+M sell 2 call spreads C (M) - C (M+M), that is 2 M European Digitals strike M.
Pricing: Max K AmDig K dK EurDig K dK C KK K
2 2
K
21Dominance
Extensions
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Extension to other dynamics
Dominance
No interpretation in terms of hedging portfolio but gives numerical pricing method.
Principle: symmetric dynamics w.r.t L
antisymmetric payoff w.r.t L
0 value at L
L
K
2L-K
0
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Extension: double KO
Dominance
L
K
00
24Dominance
Martingale inequalities
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Cernov
Dominance
•Property:
•In financial terms:
Hedge:
•Buy C (K), sell AmDig (K+ ).
•If S reaches K+ , short 1 stock.
P M K E S KT T0,
AmDig KC K
K K+
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Tchebitchev
Dominance
•Property:
•In financial terms:
a aP X E X a
Var Xa 2
EurDigPut S a EurDigCall S a
Par Sa0 0
02
S0 S0 + aS0 - a
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Jensen’s inequality
Dominance
E[X] X
hedge ,buy :])[()price( if
)]([])[()(])[('])[(])[(
)]([])[( convex,
fXEff
XfEXEfXfXEfXEXXEf
XfEXEff
f
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Applications
Dominance
])[()][( ,)2 KXEKXE(x-K)f(x)
][IX
][][ ,)12VIXEV
XEXExf(x)
X
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Cauchy-Schwarz
Dominance
•Property:
Let us call:
Which implies:
E XY E X E Y 2 2
For all and
so XY is dominated by the Portfolio:
X Y P X P Y
XYP X P Y
P P
y x
x y
x y
, ,
2
2 2
0
2
P X
P Yx
y
:
:
price today of
price today of
2
2
price XYP P P P
P PP Py x x y
x yx y
2
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A sight of Cauchy-Schwarz
Dominance
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Cauchy-Schwarz (2)
Dominance
• Call dominated by parabola:
•In financial terms:
E XY E X E Y E S S E S S E S ST T 2 2
0 0 0
212
12
ATM Call ATM Par 12
S0
Hedge:
•Short ATM straddle.
•Buy a Par + b.
X XX X X X
X XP
PX X
x
x
00 0
0
2
0
22 2
2
C PX x0
12
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DOOB
Dominance
•Property: •Hedge at date t with current spot x and current max :
•If x < do nothing.•If x = -> sell 4 stockstotal short position: 4 () stocks.
E M E X M Max X
T t T t2 2
04
,
,
2
x2
x
2 2x
2
2 22 2x x
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Up Crossings
Dominance
•Product: pays U(a,b) number of times the spot crosses the band [a,b] upward.
•Dominance: E U a b
E S ab a
,
UpCrossing
C ab a
Hedge:
•Buy 1/(b-a) calls strike a.
•First time b is reached, short 1/(b-a) stocks.
•Then first time a is reached, buy 1/(b-a) stocks.
•etc.
12
3
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Lookback squared
Dominance
•Property: ( if S not continuous)
•In financial terms: (Parabola centered on S0)
•Zero cost strategy: when a new minimum is lowered by m, buy 2 m stocks.
•At maturity: long 2 (S0-min) stocks paid in average (1/2) (S0+min).
•Final wealth:
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2 SSEmSE T
Price Lookback Par S20
2 22
2 2
0 00
0 02 2
20
2
S m S S m S m
S S mS S m
S m S S
T
T T
T T
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A simple inequality
Dominance
][2][][ and,
][][
][][ lly,symmetrica
])[(][][back)price(look
])[(])[(][
page,last From ).continuousy necessaril(not martingale
,0,0,0
0,0
,0
20,0
20
2,0
2,0
TTTT
TT
TTT
TTTT
TTTTT
SSTDmMERangeE
SSTDSME
SSTDSME
SSESSTDmSE
SSEmSEmSE
S
36
Quadratic variation
Dominance
E QV E X XT T02
0 0,
Strategy: be long 2xi stock at time ti
P L x x x x x x x
P L x x x
i i i i i i i i
N i ii
N
N
&
&
1 1 1
2 21
2
21
2
0
1
2
In continuous time:
P L x QVT T& , 20
37
Quadratic variation: application
Dominance
Volatility swap:
to lock (historical volatility)2 ~ QV (normal convention)
1) Buy calls and puts of all strikes to create the profile ST 2
2) Delta hedge (independently of any volatility assumption) by holding at any time -2St stocks
38
Dominance
Dominance
We have quite a few examples of the situation for any martingale measure, which can be interpreted financially as a portfolio dominance result.
Is it a general result? ; i.e. if you sell A, can you cover yourself whatever happens by buying B and delta-hedge?
The answer is YES.
E A E B
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General result:“Realise your expectations”
Dominance
Theorem: If for any martingale measure Q
Then there exists an adapted process H (the delta-hedge) such as for any path :
That is: any product with a positive expected value whatever the martingale model (even incomplete) provides a positive pay-off after hedge.
E f XQ 0
f H dXt t
T
0
0
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Sketch of proof
Dominance
A attainable claims K positive claims B A K
Lemma: If any linear functional positive on B is positive on f, then f is in B
B f f B 0 0
Proof: B is convex so if by Hahn-Banach Theorem, there is a separating tangent hyperplane H, a linear functional and a real such that:
f B
As B is a cone
B and f,
00 0
B f
0 0
0
H
41
Sketch of proof (2)
Dominance
B
A
K
Q E ggQ
00
01
defined by
is a martingale measure
The lemma tells us:If for any martingale measure Q,
then E fQ 0
f B
f B a A k K f a kor f a
,0
stoch. int. positiveWhich concludes the theorem.
42
Equality case
Dominance
Corollary of theorem:If for any martingale measure Q, Then there exists H adapted such that
E f XQ 0
Proof:apply Theorem to f and -f:
Adding up:
H H
f H dX
f H dXt t
t
t
t
T
t
T1 2
1
0
2
0
0
0,
H H dX H Ht t t
T1 2
02 10
f H dXt t
T
0
f H dXt t
T1
0
43
Bounds for derivatives
Dominance
The theorem does not give a constructive procedure:
In incomplete markets, some claims do not have a unique price.
What are the admissible prices, under the mere assumption of 0 rates (martingale assumption)
44
Bounds for European claims1 date
Dominance
European pay-off f(S). What are the non-arbitrageable prices for f?Answer: intersection of convex hull with vertical line
S S 0
SS0
UB
LB
f
arbitrageable price
arbitrage hedge
If market price < LB : buy f, sell the hedge for LB:
0 initial cost
>0 pay-off
{
arbitrage
45
Example: Call spread
Dominance
100
50
200
Arbitrage bound for C100 - C200 ( S0=100, ST>0)
100
ST
46
Bounds for n dates
Dominance
Natural idea: intersection of convex hull of g with (0,…,0) vertical line
This corresponds to a time deterministic hedge: decide today the hedge at each date independently from spot.
Define g by g y y f y yn i1 1,..., ,...,
47
Bounds n dates (2)
Dominance
Lower bound:
Apply recursively the operator A used in the one dimensional case, i.e. define
x x A g
where
g x g x x x
p x x
x x p
p
p
1
1
1
1
,...,
... ,
...
...
0 gives the lower bound
48
Bounds for path dependent claimscontinuous time
Dominance
•Brownian case: El Karoui-Quenez (95)
•Analogous to American option pricing
American: sup on stopping times
Upper bound: sup on martingale measures
In both cases, dynamic programming
For upper bound: Bellman equation
49
Conclusion
Dominance
• It is possible to obtain financial proofs / interpretation of many mathematical results
• If claim A has a lesser price than claim B under any martingale model, then there is a hedge which allows B to dominate A for each scenario
• If a mathematical relationship is violated by the market, there is an arbitrage opportunity.