Questions tagged [expected-value]

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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 ...
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 ...
3
votes
1answer
65 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
88 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 ...
3
votes
0answers
612 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
326 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 ...
2
votes
0answers
193 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
vote
1answer
66 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{...
1
vote
1answer
100 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 ...
1
vote
1answer
65 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
123 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
325 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 =\...
0
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1answer
50 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 ...
0
votes
1answer
111 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 ...
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
votes
2answers
77 views

Fourth moment of a itos integral

$I(t)=\int_0^t \sqrt sdW_s$ What is $E(I(t)^4)$