Nonlinear Option Pricing

Stochastic representation of solutions of linear PDEs 21

Example 1.6 Digital Neumann options and Az´ema-Yor martingales

Let us consider Φ(x, M) = 1

x≥ζ

(with ζ > 0) and D

0

= {(x, M) | x ≥ g(M)}

with g an increasing function such that g(M ) < M. Equations (1.30)–(1.32)

read

∂

2

x

u(x, M) = 0 g(M) < x < M (1.35)

∂

M

u(x, M)|

x=M

= 0 (1.36)

u(g(M), M) = 1

g(M)≥ζ

(1.37)

Using Equations (1.35) and (1.37) we get

u(x, M) = a(M )(x − g(M)) + 1

g(M)≥ζ

u(X

t

, M

t

) is an example of an Az´ema-Yor’s martingale (see [42] and Exercise

3). Using the remaining equation (1.36), we get

a(M) = αe

R

M

0

g

0

(y)

y−g(y)

dy

−

1

g(M)≥ζ

g

−1

(ζ) − ζ

e

R

M

g

−1

(ζ)

g

0

(y)

y−g(y)

dy

The integration constant α is fixed