Questions tagged [expected-value]
The expected-value tag has no usage guidance.
2
votes
1answer
64 views
Expected value of stochastic optimization
I have a optimization problem where the SDE is:
$$
dX(t) = [X(t)(u(t)-\beta(t))+\theta(t)]dt+X(t)u(t)\sigma dW(t), t \in [0,T], X(0) = X_0
$$
where $\beta(t)$ and $\theta(t)$ are deterministic ...
1
vote
0answers
63 views
change of measure expectation
How to find expectation of this stochastic process? Also, to show that the expectation of a stochastic process expression [Xt - St] in one measure is equal to expectation of another expression (of the ...
0
votes
1answer
62 views
Reference material (EV/ betting game questions) for Quant Hedge Funds Interviews [closed]
I need material to practice EV games questions.But I lack practice in betting questions where a set-up of a game is given and one has to respond to the best strategy or best bet to take. A good book ...
1
vote
1answer
44 views
Expectation and variance of standard brownian motion
Assuming that the price of the stock follows the model
$ S(t) = S(0) exp (
mt −
(σ^2/
2)
t + σW(t)
)
,
$
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
1
vote
1answer
67 views
Can the value of a swaption at any time become more negative than the swaption premium?
I am interpolating swaption values as a function of parallel shifts in interest rate and have come across some peculiar shaped options among the data I have at hand.
Here is an example of a simple ...
1
vote
1answer
59 views
How to check if $ E [\exp \{ \int_0^t \frac{Y_u^2}{1+Y_u^2}du \}]< \infty $
$dY_t=2Y_tdt+2\sqrt{1+Y_t^2}dW_t$ where $W_t$ is $P-$Brownian motion (Wiener process).
I have defined a new measure $Q$ where the Kernel density (In Girsanov theorem) is
$$ \phi_t = \frac{Y_t}{\sqrt{...
2
votes
0answers
148 views
Expectation of option value
Say we are in a BS world where the (conditional on t) price of a call is given by the usual
$$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$
Now, what about the ...
-2
votes
2answers
65 views
3
votes
0answers
529 views
Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process)
Let $S_t$ be a process which follows a Geometric Brownian Motion:
$\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$
By Ito's lemma, we have:
$S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \...
2
votes
1answer
241 views
Intuitive explanation for expectiles
I am looking for an intuitive explanation for expectiles.
Here is a link to a paper about expectiles:
Bellini and Di Bernardino: Risk Management with Expectiles, European Journal of Finance, May ...
0
votes
1answer
1k views
Expected Value of Stochastic Process
Given the following stochastic process:
$$ dX = a(X,t)dt + b(X,t)dz $$
where:
$$ dz = A \sqrt{dt}$$
and $A$ is a random variable with mean zero and variance $1$.
Is there a way to calculate the ...
1
vote
1answer
287 views
Write expectation of brownian motion conditional on filtration as an integral?
Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is
$f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\...
9
votes
2answers
439 views
Why is logarithmic mean equal to the arithmetic expectation less one-half its variance?
I've taken it as gospel that the following equality is true:
$$\mathbb{E}[\mu_x] = m_x - \frac{1}{2}\sigma_x^2 $$
where:
$\mathbb{E}[\mu_x]$ is the expected value of the logarithmic mean of some ...
