Questions tagged [expected-value]

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8
votes
2answers
485 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
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1answer
689 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 ...
3
votes
1answer
115 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
254 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
81 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
92 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
711 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
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0answers
394 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
76 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 ...
1
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1answer
78 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
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1answer
64 views

Properties of integrated GBM

(I asked this question on MSE but I think it might have more success here) Good day, I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
1
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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
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1answer
41 views

How to Evaluate Expected Value powered 4 of a Wiener Process?

Since $X(t_j) - X(t_{j-1})$ is Normally distributed with mean zero and variance $t/n$ we have $$ \operatorname{E} [(X(t_j) - X(t_{j-1}))^2 ] = \frac{t}{n} \tag{1}$$ and $$ \operatorname{E} [(X(t_j) - ...
1
vote
1answer
113 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
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1answer
77 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
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1answer
232 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
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1answer
376 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 =\...
1
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0answers
66 views

Expected value and variance of the stock log-returns under Local Volatility framework

I want to calculate the expected value and the variance of the stock process log-returns in the Local Volatility setting (and the realized/terminal correlation but let us begin in the one-dimentional ...
0
votes
1answer
109 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
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1answer
53 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
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1answer
190 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
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0answers
28 views

Intuitive explanation for - Shreve Vol 1 Ch 3 Lemma 3.2.6

I need help in getting a more intuitive understanding of this lemma 3.2.6 in Vol 1 of Shreve's Stochastic Calculus for Finance: Here, in the context of multi-period binomial pricing model, Y is a ...
0
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0answers
23 views

Order of expectation versus expectation of order (error terms in Taylor expansion)

Given a payoff function $F(X)$ of a random variable $X$, and a Taylor expansion of $F(X)$ around $X=a$, then the expecation of $F(X)$ can be written as $$ E[F(X)] = F(a) + E[ O((X-a))] $$ Under what ...
0
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0answers
28 views

How to approximate expectation and variance of an integral from a discrete Time series financial dataset?

I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this. ...
-1
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2answers
80 views

Fourth moment of a itos integral

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