# Questions tagged [stopping-time]

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12 questions
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### American put option. Exercise time is a random variable, calculation of expected payoff

I got an American put option, where the payoff is $V_\tau = \max(K - X_{\tau}, 0)$ and $X_{\tau}$ is the price of an underlying at the stopping time $\tau < T$. The underlying follows a standard ...
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I want to solve the one-touch American call at $t = 0$ with level $B,$ maturity $T$ under the following assumption: $$d S= rSd t + \sigma SdW,\quad S_0<B.$$ We have following formula: $$V(S_0,0) = \... 1answer 37 views ### is there a dependence between an annotation date of stocks dividend payment and the end fiscal year I know that the fiscal year in USA from 1 October till 30 September. I'd like to know: is whether there a dependence between a declaration date and an end of fiscal year? I think this dependence ... 1answer 297 views ### Figure of Stopping and Continuation Region I am reading Alternative Characterizations of American Put Options by Carr et al. It is stated there that: Consider an American put option on the stock with strike price K and maturity date T. ... 3answers 229 views ### Proof of optimal exercise time theorem for American derivative security in N-period binomial asset-pricing model At least two textbooks (Shreve's Stochastic Calculus for Finance - I, theorem 4.4.5 or Campolieti & Makarov's Financial Mathematics, proposition 7.8) prove the optimal exercise theorem that says ... 1answer 534 views ### What is the probability that a Brownian Bridge hits an upper barrier U before a lower barrier L? The probability that an arithmetic Brownian motion process dt = \mu dt + \sigma dW hits an upper Barrier U before it hits a lower barrier L is given by$$ \mathbb{P}(\tau_U\leq \tau_L) = \frac{\...
What is the probability that the arithmetic OU process $dx_t= \theta(\mu-x_t)dt+\sigma dW_t$ hits barrier $U$ before hitting barrier $L$ when $L<x_0<U$ ?