Tagged Questions
4
votes
1answer
144 views
Non-arbitrage theory and existence of a risk premium
Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
6
votes
1answer
172 views
Upper bound concerning Snell envelope
Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $ \mathbb E \left \{ \bar X \right\}< \infty $ (where $ \bar X =\sup _{0\leq t \leq T} X_t $) and its ...
4
votes
0answers
80 views
Simple question concerning Jump process (Lévy process) model for a risky actif price process [closed]
Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is
$$ \nu \left( dx\right) = A ...
2
votes
1answer
242 views
Regime switching in mean reverting stochastic process
Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an $ARMA(p,q)$.
This time series could be ...
5
votes
2answers
290 views
What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process.
$$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$
Is anyone able to calculate the mean ...
5
votes
2answers
353 views
What is the average stock price under the Bachelier model?
Let's say stock price follows following process:
$$dS(t) = \sigma dW(t)$$
where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price ...
2
votes
0answers
101 views
Measure change in a bond option problem
This is not a homework or assignment exercise.
I'm trying to evaluate $\displaystyle \ \ I := E_\beta \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}}\big]$, where $\beta$ is the ...
3
votes
1answer
173 views
How to measure a non-normal stochastic process?
If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
5
votes
2answers
600 views
How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?
I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
5
votes
3answers
447 views
How to use Itô's formula to deduce that a stochastic process is a martingale?
I'm working through different books about financial mathematics and solving some problems I get stuck.
Suppose you define an arbitrary stochastic process, for example
$ X_t := W_t^8-8t $ where $ W_t ...
