Nonlinear Option Pricing

Conclusion 123

for instance [139, 32, 182, 126] and the references therein. A passport option

is an option on the balance of a trading account: each day, or week, or month,

the holder decides to be long (∆

t

i

= +1) or short (∆

t

i

= −1) some index X

and receives at maturity T = t

N

some function F

T

= g(π

T

) of the portfolio

value π

T

, where

π

T

=

N

X

i=1

∆

t

i−1

(X

t

i

− X

t

i−1

)

By considering the continuous time limit, the fair value of this option can be

written as a stochastic control problem:

u(t, x, π) = sup

∆∈∆

t,T

E

Q

[g(π

T

)|X

t

= x, π

t

= π]

with dπ

t

= ∆

t

dX

t

, and where ∆

t,T

is the set of all adapted processes

(∆

s

)

t≤s≤T

such that for all s ∈ [t, T ], ∆

s

∈ {−1, +1}. Assuming for