Bruno DUPIRE

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Bruno DUPIRE Bloomberg Quantitative Research
Arbitrage, Symmetry and Dominance
NYU Seminar New York 26 February 2004
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Background
REAL WORLD
MODEL
anything can happen
stringent assumption
1 possible price,
infinite number of possible
1 perfect hedge
prices,
infinite potential loss
Can we say anything about option prices and
hedges when (almost) all assumptions are relaxed?
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Model free properties
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European profilesnecessary sufficient
conditions on call prices
K
K1
K2
K2
K3
K1
0
K
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A conundrum
Call prices as function of strike are positive
decreasing they converge to a positive value a.
Do we necessary have ?

• 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 a gt 0.

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Arbitrage with Infinite trading

N
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Quiz
Strong smile
80
90
S0 100
Put (80) 10, Put (90) 11.Arbitrageable?
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Answer

• At first sight
• P(80) lt P(90), no put spread arbitrage.
• At second sight
• P (90) - (90/80) P (80) is a PF
• with final value gt 0 and premium lt 0.

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Bounds for European claims
If market price lt LB buy f, sell the hedge for
LB
S
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non
Call price monotonicity
Call prices are decreasing with the strikeare
they necessarily increasing with the initial spot?
NO. counter example 1
counter example 2 martingale
120
25
110
100
100
100
90
90
80
75
0
T
0
T
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Call price monotonicity
If model is continuous Markov,Calls are
increasing with the initial spot (Bergman et al)
Take 2 independent paths wx and wy starting from
x and y today.
(1) wx and wy do not cross.
(2) wx and wy cross.
y
y
x
x
Knowing that they cross, the expectation does not
depend on the initial value (Markov property).
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Lookback 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.

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Lookback dominance (2)

• More generally for
• To minimise the price, solve
• thanks to Hardy-Littlewood transform (see Hobson).

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Normal model with no interest rates
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Digitals
1 American Digital 2 European Digitals
Reflected path
From reflection principle, Proba (Max0-T gt K)
2 Proba (ST gt K)
K
Brownian 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.
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Down out call
DOC (K, L) C (K) - P (2L - K)
60.00
50.00
40.00
30.00
20.00
10.00
K
2L-K
0.00
50
60
70
80
90
100
110
120
130
140
150
160
170
-10.00
L
-20.00
-30.00
-40.00
The hedge meets boundary conditions.If S
reaches L, unwind at 0 cost.
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Up out call
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.
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General Pay-off
20.00
15.00
10.00
5.00
0.00
80
90
100
110
120
130
140
150
160
170
180
-5.00
-10.00
-15.00
-20.00
The hedge must meet boundary conditions, i.e.
allow unwind at 0 cost.
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Double knock-out digital
2 symmetry points infinite reflections
0.02
0.9
0.02
0.01
0.4
0.01
0.00
-0.1
90
100
110
130
80
120
-0.01
-0.01
-0.6
-0.02
-1.1
-0.02
Price Hedge infinite series of digitals
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Max option
(Max - K) 2 C (K)
Pricing
Hedge when current Max moves from M to MdM sell
2 call spreads C (M) - C (MdM), that is 2 dM
European Digitals strike M.
K
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Extensions
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Extension to other dynamics
Principle symmetric dynamics w.r.t
L antisymmetric payoff w.r.t L
K
0
L
2L-K
No interpretation in terms of hedging portfolio
but gives numerical pricing method.
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Extension double KO
0
K
0
L
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Martingale inequalities
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Cernov

• Property
• In financial terms

K
Kl

• Hedge
• Buy C (K), sell l AmDig (K l).
• If S reaches K l , short 1 stock.

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Tchebitchev

• Property
• In financial terms

S0
S0 a
S0 - a
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Jensens inequality
f
EX
X
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Applications
X
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Cauchy-Schwarz

• Property
• Let us call
• Which implies

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A sight of Cauchy-Schwarz
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Cauchy-Schwarz (2)

• Call dominated by parabola
• In financial terms

• Hedge
• Short ATM straddle.
• Buy a Par b.

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DOOB

• Property
• Hedge at date t with current spot x and current
  max a
• If x lt a do nothing.
• If x a -gt a da sell 4da stocks
• total short position 4 (a da) stocks.

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Up Crossings

• Product pays U(a,b) number of times the spot
  crosses the band a,b upward.
• Dominance

2
3
1

• 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.

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Lookback squared

• Property ( if S not continuous)
• In financial terms (Parabola
  centered on S0)
• Zero cost strategy when a new minimum is lowered
  by dm, buy 2 dm stocks.
• At maturity long 2 (S0-min) stocks paid in
  average (1/2) (S0min).
• Final wealth

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A simple inequality
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Quadratic variation
Strategy be long 2xi stock at time ti
In continuous time
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Quadratic variation application
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
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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.
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General resultRealise your expectations
Theorem If for any martingale measure
Q Then there exists an adapted process H (the
delta-hedge) such as for any path w That is
any product with a positive expected value
whatever the martingale model (even incomplete)
provides a positive pay-off after hedge.
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Sketch of proof
Lemma If any linear functional positive on
B is positive on f, then f is in B
Proof B is convex so if by
Hahn-Banach Theorem, there is a separating
tangent hyperplane H, a linear functional
and a real a such that
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Sketch of proof (2)
The lemma tells us If for any martingale
measure Q, then
Which concludes the theorem.
stoch. int.
positive
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Equality case
Corollary of theorem If for any martingale
measure Q, Then there exists H adapted such that
Proof apply Theorem to f and -f Adding up
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Bounds for derivatives
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)
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Bounds for European claims1 date
If market price lt LB buy f, sell the hedge for
LB
S
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Example Call spread
100
50
100
200
ST
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Bounds for n dates
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.
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Bounds n dates (2)
Lower bound Apply recursively the operator A
used in the one dimensional case, i.e. define
gives the lower bound
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Bounds for path dependent claimscontinuous time

• 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

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Conclusion

• 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.

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