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 possible

prices,

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 K

C3 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.

6

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

9

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

90

100

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.

x

yx

y

Knowing that they cross, the expectation does not depend on the initial value (Markov property).

x(T) y(T)

12

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

M

aC M a

M

aM M a M

13

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 0

0

Min

C k s

s k sds

k sK

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.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80

2.00

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

17

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

19

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

20

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

22

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

25

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 K

C K

K K+

26

Tchebitchev

Dominance

•Property:

•In financial terms:

a aP X E X a

Var X

a 2

EurDigPut S a EurDigCall S a

Par S

a0 00

2

S0 S0 + aS0 - a

27

Jensen’s inequality

Dominance

E[X] X

hedge ,buy :])[()price( if

)]([])[(

)(])[('])[(])[(

)]([])[( convex,

fXEff

XfEXEf

XfXEfXEXXEf

XfEXEff

f

28

Applications

Dominance

])[()][( ,)2 KXEKXE(x-K)f(x)

][IX

][][ ,)1

2VIXEV

XEXExf(x)

X

29

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 Y

x

y

:

:

price today of

price today of

2

2

price XYP P P P

P PP P

y x x y

x yx y

2

30

A sight of Cauchy-Schwarz

Dominance

31

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

21

2

1

2

ATM Call ATM Par 1

2

S0

Hedge:

•Short ATM straddle.

•Buy a Par + b.

X XX X X X

X X

P

PX X

x

x

00 0

0

2

0

2

2 2

2

C PX x0

1

2

32

DOOB

Dominance

•Property:

•Hedge at date t with current spot x and current max :

•If x < do nothing.

•If x = -> sell 4 stocks

total short position: 4 () stocks.

E M E X M Max X

T t Tt

2 2

04

,

,

2

x2

x

2 2x

2

2 22 2x x

33

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 a

b a,

UpCrossing

C a

b 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

34

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:

20

2 SSEmSE T

Price Lookback Par S20

2 22

2 2

0 00

0 02 2

20

2

S m S S mS m

S S mS S m

S m S S

T

T T

T T

35

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

40

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

,

0

0 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 k

or 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 dX

t t

t

t

t

T

t

T1 2

1

0

2

0

0

0

,

H H dX H Ht t t

T

1 2

02 10

f H dXt t

T

0

f H dXt t

T

1

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.