The previous post about parity relations between installment calls and American puts has not taken into account dividends. This is due to the fact that \[ \lim_{q_n \rightarrow 0} c(t, S_t; t_n, q_n) \neq S_t \] whenever \( \delta \neq 0 \).

However, numerical experiments show that

Hypothesis

\[ C(t, S_t; \chi(t, S_t)) + K = P(t, S_t) + S_t \] where the variable installment rate \[ \chi(t, S_t) = rK – \delta S_t \] is path-dependent upon the price process \( S_t \). Similarly, for American calls: \[ P(t, S_t; -\chi(t, S_t)) + S_t = C(t, S_t) + K. \]