Questions tagged [expected-value]
The expected-value tag has no usage guidance.
20
questions
0
votes
0answers
24 views
St. Petersburg Paradox [closed]
I am currently trying to solve the St. Petersburg paradox from the lecture notes, but somethings bugs, me. Bernoulli shows that we can solve it using marginal utility theory. This is how it is derived:...
0
votes
1answer
91 views
What's the expected value of a repeated game with 50% chance to win 0.5 and 50% to lose 0.5?
Assume we start with 1.
In the first bet the expected value of remained balance is 1.5 * 0.5 + 0.5 * 0.5 = 1
For N times, is it still 1 according to E(XYZ)=E(X)E(Y)E(Z)?
But 1.5^50 * 0.5^50 is not 1.
...
0
votes
1answer
70 views
Determining the No Arbitrage price of max[B(T), S(T)]
Following is given,
$dB(t)=rB(t)dt$
$dS(t)= (r-\delta)S(t)dt+\sigma S(t)dW(t)$
where, $r$ is the risk-free interest rate, $\delta$ the continous dividend yield $\sigma$ is the stock asset ...
3
votes
1answer
126 views
How to calculate the mean and variance of this Ito integral?
I tried to calculate this integral use Ito's lemma, $W_{t}$ is the Wiener Process.
$$I_{T}=\int_{0}^{T}\sqrt{|W_{t}|}dW_{t}$$
We have
$d f\left(W_{t}\right)=f^{\prime}\left(W_{t}\right) d W_{t}+\...
3
votes
1answer
72 views
Proof that we can price any derivative as the discounted value of its expected return under the risk neutral measure
I am reading a paper which tries to convey the intuition behind the Black-Scholes pricing formula. In that paper, the author states the following two things without proof, and I would like to know why ...
3
votes
1answer
108 views
How to calculate the expected stock returns for an individual stock?
I know about CAPM. My question is if this method is also viable:
Calculate monthly logReturns
...
0
votes
1answer
52 views
Condition expectation calculation examples and theory [closed]
I want to ask you an advice about reading theory and examples of conditional expectation and conditional variance. I want to have my understanding deeper, because sometimes I can't understand ...
3
votes
1answer
89 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
1answer
106 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
140 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
69 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
144 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
71 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
$$ \phi_t = \frac{Y_t}{\sqrt{...
2
votes
0answers
241 views
Expectation of option value
Say we are in a BS world where the (conditional on t) price of a call is given by the usual
$$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$
Now, what about the ...
-1
votes
2answers
77 views
3
votes
0answers
641 views
Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process)
Let $S_t$ be a process which follows a Geometric Brownian Motion:
$\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$
By Ito's lemma, we have:
$S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \...
2
votes
1answer
353 views
Intuitive explanation for expectiles
I am looking for an intuitive explanation for expectiles.
Here is a link to a paper about expectiles:
Bellini and Di Bernardino: Risk Management with Expectiles, European Journal of Finance, May ...
0
votes
1answer
2k views
Expected Value of Stochastic Process
Given the following stochastic process:
$$ dX = a(X,t)dt + b(X,t)dz $$
where:
$$ dz = A \sqrt{dt}$$
and $A$ is a random variable with mean zero and variance $1$.
Is there a way to calculate the ...
1
vote
1answer
338 views
Write expectation of brownian motion conditional on filtration as an integral?
Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is
$f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\...
8
votes
2answers
461 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 ...
