Discounted perpetual game call options

Pricing cancellable American put options on the finite time horizon

“…For each node of this grid we use a procedure with three steps to derive the boundaries' approximations. First, by the use of Zaevski (2020b) we derive the exercise boundaries for the related perpetual option. We have that the writer's optimal boundary is B = 15.7208 (the holder's one is A = 5.2520).…”

“…If we denote the dividend rate by $$\delta $$ and the corresponding model by the $$(r,\lambda ,\delta )$$‐model, then it is equivalent to the $$(r-\delta ,\lambda +\delta ,0)$$‐model. The proof of this relation is based on the fact that after discounting with rate $$r-\delta $$ the underlying asset has to be a martingale under the $$(r,\lambda ,\delta )$$‐model—for more details see proposition 2.3 from Zaevski (2020a). All this allows us to work with the $$(r,\lambda ,0)$$‐model without loss of generality.…”

“…If we read carefully the proofs of propositions 2.2-2.4 from Zaevski (2020b), related to the perpetual options, we can see that they are true when the maturity is finite too. We summarize them in the following proposition.…”

“…It is well known that the option valuation problem is easier to solve when there are no maturity constraints, because the exercise boundaries are flat due to the absence of a coercive exercise at the maturity date. Emmerling (2012) and Zaevski (2020a) examine such call‐style options, whereas Kyprianou (2004) and Suzuki and Sawaki (2007) obtain the corresponding results for the put options. Kunita and Seko (2004) and later Yam et al (2014) explore the call versions when the maturity is finite.…”

“…Together with the introduced time dependence in the payment structures, this additional discounting has another outstanding importance-a model based on a continuously paying dividend underlying asset can be expressed in this framework. The perpetual case is examined in Zaevski (2020b).…”