10
votes
Accepted
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 ...
- 6,260
6
votes
Accepted
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 ...
- 5,949
6
votes
Accepted
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 ...
- 835
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 ...
- 4,968
5
votes
Accepted
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:
$$ P(\tau \leq s | {\cal F}_t) = N\left( -Y_t \sigma^{-1}(s-t)^{-1/2}- \nu(s-t)^{1/2}\...
- 4,968
5
votes
Accepted
I am trying to solve this question about optimal stopping theory. I don't know how to get started. Any hints would be very helpful
The Snell envelope is the smallest super-martingale that is greater than $X$. Since $\tau \le N$, it is obvious that $A_N^{\tau} = A_{N\wedge \tau} = A_{\tau}$.
For part (b), note that, from the ...
- 21k
4
votes
Accepted
What is the probability that a OU process hits an upper barrier U before a lower barrier L?
Assuming $\theta>0$ (take $\tilde{X}=\mu-X$ if it is not the case)
Let us denote $\text{erfi}(x)$ the imaginary error function
Let us denote $\tau_L$,resp.$\tau_U$ the hitting time of $L$resp.$U$ ...
- 2,402
4
votes
On first and last zeros before t in a Brownian Motion
Intuitively speaking, you generally have an event for which you do not know when it occurs (the time of the occurrence of the event is random), but you do know that it will occur at some point in the ...
- 1,706
3
votes
Accepted
is there a dependence between an annotation date of stocks dividend payment and the end fiscal year
What do you mean by annotation date, there is a declaration(announcement) date, ex-date, record date but I've never heard of an annotation date. Dividends are not decided always at the fiscal year end,...
- 2,888
2
votes
Accepted
How to solve one-touch American call
As is often the case, there are at least two solution strategies here.
(Probabilistic) You explicitly solve for the expected discount factor at the first passage time $\nu$ of $S$ to the level $B$ ...
- 5,949
1
vote
American option pricing formulation
In explicitly wording my own question yesterday and naming my doubts, I think I may have stumbled upon the explanation:
On the one hand, indeed we have
$$
{\text{ess}\sup}_{s\in[0,T]}\mathbb{E}\left[...
- 51
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
- 16.2k
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 ...
- 366
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{...
- 21k
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=...
- 2,402
1
vote
Accepted
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 ...
- 14.5k
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