Nonlinear Option Pricing

Euler discretization error 37

deltas, Delta

i

= ∂

e

i

v

E

Q

[φ(S

v

t

)]; and the gammas, Gamma

i,j

= ∂

e

i

+e

j

v

E

Q

[φ(S

v

t

)]

of the European option of maturity t and payoff φ ((e

1

, . . . , e

d

) is the canonical

base of R

d

). For convenience, let us set x = ln v (i.e., x

i

= ln v

i

) and X

x,i

t

=

ln(S

v,i

t

). X is then the solution to (2.4) with b = µ − kσk

2

/2 ∈ C

∞

b

(R

d

),

where kσk

2

i

(x) =

P

r

j=1

σ

2

i,j

(x). If we set exp(x) = (exp(x

1

), . . . , exp(x

d

)) and

f(x) = φ(exp(x)), we define a function f ∈ E

1

and, since Price = E

Q

[f(X

x

t

)],

(2.13) leads to

Price

n

− Price = C

Price

t

φ(v)/n + O



n

−2

t

−2

exp (c

2

kln vk)



where Price

n

stands for the approximated price E

Q

[f(X

n,x

t

)] and

C

Price

t

φ(v) =

Z

(R

∗

+

)

d

φ(u