This paper deals with pricing formulae for a European call option and an exchange option in the case where underlying asset price processes are represented by stochastic delay differential equations with jumps (hereafter “SDDEJ”). We introduce a new model in which Poisson jumps are added in stochastic delay differential equations to capture behaviors of an underlying asset process more precisely. We derive explicit pricing formulae for the European call option and the exchange option by proving a Lemma on the conditional expectation. Finally, we show that our “SDDEJ” model is meaningful through some numerical experiments and discussions.

中文翻译：

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基于跳变的随机延迟微分方程的交易期权定价公式

本文讨论了在标的资产价格过程由带有跳跃的随机延迟微分方程（以下简称“SDDEJ”）表示的情况下，欧式看涨期权和交换期权的定价公式。我们引入了一种新模型，其中在随机延迟微分方程中添加了泊松跳跃，以更精确地捕捉基础资产过程的行为。我们通过证明条件期望的引理，推导出欧式看涨期权和交换期权的明确定价公式。最后，我们通过一些数值实验和讨论表明我们的“SDDEJ”模型是有意义的。