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The option payoff at maturity $T$ is defined by \begin{align*} (S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1}. \end{align*} Let $Q$ be the risk-neutral probability measure and $E$ be the corresponding expectation operator. Let $Q_p$ be a probability measure defined by \begin{align*} \frac{dQ_p}{dQ}\big|_t = \frac{P_t}{e^{rt} P_0}. \end{align*} Moreover, let $E_p$ be the corresponding expectation operator. Then the option value can be computed by \begin{align*} e^{-rT}E\left((S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right) &= e^{-rT}E_p\left(\left(\frac{dQ_p}{dQ}\big|_T\right)^{-1}(S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right)\\ &=P_0 E_p\left(\left(\frac{S_T}{P_T}-1\right)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right), \end{align*} which can be treated as a down-and-out barrier call option, assuming that $S_t/P_t$ is log-normally distributed under the measure $Q_p$.

The option payoff at maturity $T$ is defined by \begin{align*} (S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1}. \end{align*} Let $Q$ be the risk-neutral probability measure and $E$ be the corresponding expectation operator. Let $Q_p$ be a probability measure defined by \begin{align*} \frac{dQ_p}{dQ}\big|_t = \frac{P_t}{e^{rt} P_0}. \end{align*} Moreover, let $E_p$ be the corresponding expectation operator. Then the option value can be computed by \begin{align*} e^{-rT}E\left((S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right) &= e^{-rT}E_p\left(\left(\frac{dQ_p}{dQ}\big|_T\right)^{-1}(S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right)\\ &=P_0 E_p\left(\left(\frac{S_T}{P_T}-1\right)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right), \end{align*} which can be treated as a down-and-out barrier call option, assuming that $S_t/P_t$ is log-normally distributed under the measure $Q_p$.

Note that, under $Q_p$, the process $\{S_t/P_t \mid t \geq 0\}$ is a martingale, that is, we can treat $S_t/P_t$ as an asset process with zero interest and zero dividend. Using the down-and-out barrier call option formula in John Hull, we obtain that \begin{align*} E_p\left(\left(\frac{S_T}{P_T}-1\right)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right) = \frac{S_0}{P_0}-1. \end{align*} That is, the option price is $S_0-P_0$.

The option payoff at maturity $T$ is defined by \begin{align*} (S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1}. \end{align*} Let $Q$ be the risk-neutral probability measure and $E$ be the corresponding expectation operator. Let $Q_p$ be a probability measure defined by \begin{align*} \frac{dQ_p}{dQ}\big|_t = \frac{P_t}{e^{rt} P_0}. \end{align*} Moreover, let $E_p$ be the corresponding expectation operator. Then the option value can be computed by \begin{align*} e^{-rT}E\left((S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right) &= e^{-rT}E_p\left(\left(\frac{dQ_p}{dQ}\big|_T\right)^{-1}(S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right)\\ &=P_0 E_p\left(\left(\frac{S_T}{P_T}-1\right)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right), \end{align*} which can be treated as a usual barrier option, assuming that $S_t/P_t$ is log-normally distributed under the measure $Q_p$.

The option payoff at maturity $T$ is defined by \begin{align*} (S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1}. \end{align*} Let $Q$ be the risk-neutral probability measure and $E$ be the corresponding expectation operator. Let $Q_p$ be a probability measure defined by \begin{align*} \frac{dQ_p}{dQ}\big|_t = \frac{P_t}{e^{rt} P_0}. \end{align*} Moreover, let $E_p$ be the corresponding expectation operator. Then the option value can be computed by \begin{align*} e^{-rT}E\left((S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right) &= e^{-rT}E_p\left(\left(\frac{dQ_p}{dQ}\big|_T\right)^{-1}(S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right)\\ &=P_0 E_p\left(\left(\frac{S_T}{P_T}-1\right)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right), \end{align*} which can be treated as a down-and-out barrier call option, assuming that $S_t/P_t$ is log-normally distributed under the measure $Q_p$.

The option payoff at maturity $T$ is defined by \begin{align*} (S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1}. \end{align*} Let $Q$ be the risk-neutral probability measure and $E$ be the corresponding expectation operator. Let $Q_p$ be a probability measure defined by \begin{align*} \frac{dQ_p}{dQ}\big|_t = \frac{P_t}{e^{rt} P_0}. \end{align*} Moreover, let $E_p$ be the corresponding expectation operator. Then the option value can be computed by \begin{align*} e^{-rT}E\left((S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right) &= e^{-rT}E_p\left(\left(\frac{dQ_p}{dQ}\big|_T\right)^{-1}(S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right)\\ &=P_0 E_p\left(\left(\frac{S_T}{P_T}-1\right)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right), \end{align*} which can be treated as a usual barrier option, assuming that $S_t/P_t$ is log-normally distributed.

The option payoff at maturity $T$ is defined by \begin{align*} (S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1}. \end{align*} Let $Q$ be the risk-neutral probability measure and $E$ be the corresponding expectation operator. Let $Q_p$ be a probability measure defined by \begin{align*} \frac{dQ_p}{dQ}\big|_t = \frac{P_t}{e^{rt} P_0}. \end{align*} Moreover, let $E_p$ be the corresponding expectation operator. Then the option value can be computed by \begin{align*} e^{-rT}E\left((S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right) &= e^{-rT}E_p\left(\left(\frac{dQ_p}{dQ}\big|_T\right)^{-1}(S_T-P_T)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right)\\ &=P_0 E_p\left(\left(\frac{S_T}{P_T}-1\right)1_{\left(\inf_{0 \le t <T}\frac{S_t}{P_t}\right) > 1} \right), \end{align*} which can be treated as a usual barrier option, assuming that $S_t/P_t$ is log-normally distributed under the measure $Q_p$.