Barrier Options and Lumpy Dividends

Quantity Symbol Value
call strike K 100
knock-out barrier L 95
initial spot S0 K
barrier option expiry T 0.2
dividend date τ T/2
interest rate r 0.05
stock volatility σ 0.1
Table 1: Parameters in the numerical experiments.
Consider the price at time 0 of the down-and-out call option when the underlying
stock pays a dividend d = min{Sτ−,D} at time τ, where D ≥ 0 and 0 < τ < T.
Figure 1 shows the price as a function of D and τ. The dividend makes barrier
crossings more likely since the contract is down-and-out and makes the option more
out-of-the-money. So, the price is a decreasing function of D for each fixed τ. For
D = 0, we obtain the usual no-dividend Black-Scholes price of 2.2981. Figure 2
shows the price as a function of τ for D = 2,5 and 8. We see that the timing of the
dividend is more important for large dividends and that the contract is cheaper the
earlier the dividend is paid.
4.1 Comparison with the Crank-Nicolson scheme
Since the underlying has geometric Brownian motion dynamics everywhere except
when the dividend is paid, the price at time t of the contract is given by P(t,St),
where P(t,s) solves
∂P
∂t
1
+
2 σ2s 2∂2P + rs∂P − rP = 0
∂s2 ∂s
for t ∈ [0,T], t �= τ, and s > L. The boundary and terminal conditions are given by
P(t,L) = 0, for t ∈ [0,T],
P(T,s) = max{s − K,0}, for s > L,
and the behavior at the dividend payment date τ is given by the following jump
condition:
P(τ−,s) = P(τ,s − d(s)).
6
(7)