Nonlinear Option Pricing

The Monge-Kantorovich distance and its financial interpretation 269

Eventually, we compute the infimum over M

+

. If at some point, say (X

0

, Y

0

) ∈

R

n

× R

n

, u

1

(X

0

) + u

2

(Y

0

) > d(X

0

, Y

0

)

p

, then the infimum is −∞ and is

“attained” by P = λδ

X

0

δ

Y

0

with λ −→ +∞. This case can be disregarded

as we compute a supremum over u

1

, u

2

in a second step. If for all (X, Y ),

u

1

(X) + u

2

(Y ) ≤ d(X, Y )

p

, then the infimum is zero. Therefore, we conclude

that

d

MK

(P

1

, P

2

)

p

= sup

u

1

,u

2



E

P

1

[u

1

(X)] + E

P

2

[u

2

(Y )]



subject to the constraints

∀(x, y) ∈ R

n

× R

n

, u

1

(x) + u

2

(y) ≤ d(x, y)

p

This dual problem has a clear economic interpretation: the objective func-

tion u

1

(X) + u

2

(Y ) represents the value of the portfolio composed of two

options written respectively on X and Y with market prices