Barrier Options and Lumpy Dividends

price
2.5
2
1.5
1
0.5
0
94 95 96 97 98 99 100 101 102 103
Figure 3: The price of the down-and-out call option for dividends D1 = D2 = 1 paid
at times τ1 = T /3 and τ2 = 2T/3, computed by using the approximative numerical
integration method (solid) and the Crank-Nicolson scheme (dashed) with very small step
sizes.
and so it only remains to compute P(t,St;K,σ,τ1,τ2) by evaluating one-dimensional
integrals of the function P(τ1,g1(·);K adj
1 ,σ adj
1 ).
For N > 2, we just continue this procedure recursively: at the kth step, k =
S
1,... ,N, compute an approximation of the time-τN−k price P(τN−k,St;K,σ,τN−k+1,... ,τN)
by integrating P(τN−(k−1),g N−(k−1)(·);K adj
k−1 ,σadj
k−1 ). By convention, Kadj 0
σ adj
0 = σ, and τ0 = 0. Adjust the strike to
K adj
k := K +
N�
j=k+1
and choose the adjusted volatility σ adj
k such that
and iterate.
e r(T −τj) Dj = K adj
k−1 + er(T −τk+1) Dk+1
P(τN−k,St;K,σ,τN−k+1,... ,τN) = P(τN−k,St;K adj
k ,σadj
k ),
= K,
Figure 3 compares the price computed with the described approximative method
to the result from the Crank-Nicolson scheme with very small step sizes. We consider
the case N = 2 with dividends D1 = D2 = 1 paid at times τ1 = T/3 and τ2 = 2T/3.
The approximation works nicely — relative errors are in the order of 0.1%-1%,
10