10 votes
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Expected value and Variance of a stopped random process

Although Math SE might be a bit more suited for this one, I wanted to give it a try. The answer relies on the law of total expectation, the law of total variance, and the relationship between Euler's ...
Kermittfrog's user avatar
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6 votes
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Figure of Stopping and Continuation Region

The exercise boundary $B_t$ for a finite maturity American put option is not a constant function of time as in your plot. As mentioned in the excerpt, $B_T = K$ at maturity. But for $t < T$, we ...
LocalVolatility's user avatar
6 votes
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Why do we need to split market and default information into 2 separate filtrations?

I think you are absolutely correct if the hazard rate is deterministic, although I think you are forgetting a discounting factor in your example. But sometimes the hazard rate cannot be assumed to be ...
mmencke's user avatar
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5 votes

Why do we need to split market and default information into 2 separate filtrations?

Your ${\cal F}$ is actually ${\cal G}$, that is the already enlarged filtration/probability space. So, the claim here seems to be that we do not have to consider the smaller, market filtration, ${\cal ...
ir7's user avatar
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5 votes
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Default intensity in Black-Cox model

As shown in Credit Risk Modeling Notes (Bielecki, Jeanblanc, Rutkowski), Corollary 1.3.1, for $t < s$, we have: $$ P(\tau \leq s | {\cal F}_t) = N\left( -Y_t \sigma^{-1}(s-t)^{-1/2}- \nu(s-t)^{1/2}\...
ir7's user avatar
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5 votes
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I am trying to solve this question about optimal stopping theory. I don't know how to get started. Any hints would be very helpful

The Snell envelope is the smallest super-martingale that is greater than $X$. Since $\tau \le N$, it is obvious that $A_N^{\tau} = A_{N\wedge \tau} = A_{\tau}$. For part (b), note that, from the ...
Gordon's user avatar
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4 votes
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What is the probability that a OU process hits an upper barrier U before a lower barrier L?

Assuming $\theta>0$ (take $\tilde{X}=\mu-X$ if it is not the case) Let us denote $\text{erfi}(x)$ the imaginary error function Let us denote $\tau_L$,resp.$\tau_U$ the hitting time of $L$resp.$U$ ...
M. Jeunesse's user avatar
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4 votes

On first and last zeros before t in a Brownian Motion

Intuitively speaking, you generally have an event for which you do not know when it occurs (the time of the occurrence of the event is random), but you do know that it will occur at some point in the ...
Hans-Peter Schrei's user avatar
3 votes
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is there a dependence between an annotation date of stocks dividend payment and the end fiscal year

What do you mean by annotation date, there is a declaration(announcement) date, ex-date, record date but I've never heard of an annotation date. Dividends are not decided always at the fiscal year end,...
Lliane's user avatar
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2 votes
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How to solve one-touch American call

As is often the case, there are at least two solution strategies here. (Probabilistic) You explicitly solve for the expected discount factor at the first passage time $\nu$ of $S$ to the level $B$ ...
LocalVolatility's user avatar
1 vote
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American option pricing formulation

In explicitly wording my own question yesterday and naming my doubts, I think I may have stumbled upon the explanation: On the one hand, indeed we have $$ {\text{ess}\sup}_{s\in[0,T]}\mathbb{E}\left[...
Martin K's user avatar
1 vote

Convergence rate of Bermudan to American option

Is this paper useful? Discussed usage of Richardson extrapolation for such purposes http://www.fin.ntu.edu.tw/~conference2002/proceding/5-4.pdf
dm63's user avatar
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1 vote

Regression techniques for bermudan Monte-Carlo

Nowadays there are a lot of methods related to the machine learning. Most of them are based on Gaussian Process Regression and they are particulary good if you would like to price high dimensional ...
ltrd's user avatar
  • 501
1 vote

Proof of optimal exercise time theorem for American derivative security in N-period binomial asset-pricing model

I think the proof has already been provided at the end of the proof in Shreve's Theorem 4.4.5. Specifically, note that, since \begin{align*} \frac{1}{(1+r)^{n \wedge \tau^*}}V_{n \wedge \tau^*}. \end{...
Gordon's user avatar
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1 vote

What is the probability that a Brownian Bridge hits an upper barrier $U$ before a lower barrier $L$?

Idea Let $B$ be a standard brownian motion starting from $x_0=0$, $m_T = \inf_{u\leq T}B_u$ and $M_T =\sup_{u\leq T}B_u$. Let's define if it exists for $A\in\sigma(B_u,u\leq T)$, $\mathbb{P}(A | B_T=...
M. Jeunesse's user avatar
  • 2,422
1 vote
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Probability that return exceeds a certain level before a certain time (Black-Scholes)

In that case, the problem becomes a non-trivial stopping time problem. Consider a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P})$ equipped with the natural filtration of a standard ...
Quantuple's user avatar
  • 14.6k

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