Exotic Options - Computational Methods for Quantitative Finance - page 77

Fig. 6.1 Down-and-out,
down-and-in barrier put and
plain vanilla put option

Example 6.1.2 Consider a down-and-out and down-and-in put option. Set K = 100,

T 0 . 01. We plot the price of both options and the

corresponding plain vanilla put price in Fig. 6.1 . Here, we computed the down-
and-out barrier option using finite elements and applied formula ( 6.1 ) to obtain the
corresponding down-and-in contract.

= 1, B = 80, σ = 0 . 3 and r =

6.2 Asian Options

Asian options are path-dependent options where the payoff depends on the price
history of the underlying, in particular on the arithmetic average price at maturity,

T

1

S T := S(τ) d τ. (6.3)

T − t 0

t 0

The term T − t 0 denotes the length of the averaging period. Applying Itô's formula

we set t 0 = 0. There are different types of o pt ions. The fixed strike call has payoff

(S T − K) + and the floating strike call (S − S T ) + . To derive the partial differential

equation, we introduce the new variable

t

=

Y(t) S(τ) d τ.

0

Since this history of the asset price is independent of the current price S(t) , we
may treat S , Y and t as independent variables . The value of an Asian can then be
obtained in the form V(t,S,Y) . We need a stochastic differential equation for the
vector process (S, Y ) . Applying Itô formula, we obtain that (S, Y )

is a diffusion

process,