Questions tagged [expected-value]
The expected-value tag has no usage guidance.
40
questions
1
vote
0answers
29 views
Calculating E^2[σ^2] where σ is a GARCH(1,1) Proces
Given that α =0,113079 β = 0,873884 ω = 0,0000081
Need the calculate a call price using garch volatility I alsa calculated the kurtosis = 235 enter image description here:
https://www.researchgate.net/...
1
vote
2answers
143 views
Question about slides in lecture note: What if we can't assume $\mu=0?$
The question popped up when I was reading these lecture notes online. Consider the MA$(1)$ process given by $X_t=W_t+bW_{t-1}$ where $W_t$ is white noise distributed with constant variance $\sigma_W^2....
1
vote
1answer
83 views
Show that $\text{Cov}[Z_t,Z_{t+h}]=\text{Cov}[Z_s,Z_{s+h}].$
Problem: If $X\sim\text{WN}(\mu,\sigma^2).$ Let then $Z$ be the process defined by \begin{equation}
Z_t=\sum_{i=0}^na_iX_{t-i} \end{equation} for some coefficients $a_1,...,a_n\in\mathbb{R}$ with ...
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votes
0answers
58 views
Is the mean of a stationary timeseries the same everywhere?
Say for example I have the white noise process $Y_t\sim\text{WN}(\mu,\sigma^2)$. Is it true that $\mathbb{E}[Y_t]=\mathbb{E}[Y_{t-h}]$, where $h\in\mathbb{N}?$
0
votes
1answer
100 views
What is the expectation of a change in Brownian motion? [closed]
I know $E[W_T-W_t]=0$ but I have a solution which implies this is wrong.
Question
Answer
1
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1answer
84 views
How to take the expectation of an exponential martingale? And an exponential with a random value?
I am reading Shreve's Stochastic Calculus for Finance II. He states on pages 110 and 111 that,
$$E[exp(\sigma m-\frac{1}{2}\sigma^2 \tau_m)] = 1$$
$$E[exp(-\frac{1}{2}\sigma^2 \tau_m)] = e^{-\sigma m}$...
0
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0answers
85 views
Are these two expectations the same?
I'm studying Markov Processes and Ito diffusion, I'm just at the beginning but I can't understand the different formulation of the expectation formulated in two different books. I'm talking about ...
1
vote
1answer
114 views
Show that $\mathbb{E}[(S+\xi)^2]\rightarrow 0$ as $n\rightarrow\infty$
EDIT: Showing this using Ito's lemma is easy, that's NOT what I want to do. I also realised that $2\mathbb{E}[S\xi]\neq 2\xi\mathbb{E}[S]$ since $\xi$ is also a random variable. Nontheless, if this is ...
2
votes
0answers
81 views
Understanding Bayes Rule of conditional expectation
Let $\mathcal{F}$ be a $\sigma$-algebra, $P$ and $Q$ be equivalent martingale measures and $\frac{dQ}{dP}$ the Radon Nikodym Derivative.
I learned that $\Bbb{E}_Q[X]=\Bbb{E}_P[\frac{dQ}{dP}X] $, which ...
6
votes
1answer
83 views
Esscher Premium: Integral Transform Proof
I have some difficulty understanding the following proof and I hope someone can help me with that.
Claim: I want to show that
$E_\alpha(S)=\frac{d}{dr} \log M_S(r)|_{r=\alpha} $, where $M_S(r)=E(\exp(...
4
votes
1answer
225 views
Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]
I am trying to calculate the expectation of
$$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$
where $(W_t)$ is a Wiener process.
I was told that the value of this expectation is zero. Can someone ...
1
vote
0answers
46 views
Moments of a SDE: a detail on the information set
Very basic questions. Let $(z_t)_{t \geq 0}$ be a standard Brownian motion and let
$$dS_t = \mu S_t dt + \sigma S_t dz_t.$$
When we write $E\left( S_t \right)$, do we mean $E\left( S_t \big| F_0 \...
2
votes
0answers
101 views
Recognizing a Martingale
Under which conditions is the stochastic process $\{X_t\}_{t=0,1,...,T}$ a martingale? Demonstrate and explain clearly for each case below. If it is not necessarily a martingale, provide a ...
0
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1answer
54 views
Log-normal risk-neutral price derivation from binomial trees, not clear about step in derivation process
At page 64 of the book Concepts and practice of mathematical finance, 2nd edition by M. Joshi, paragraph 3.7.2 (Trees and option pricing - A log-normal model - The risk-neutral world behaviour) a ...
3
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0answers
40 views
Infinitesimal Generators and Expectation of First Hitting Time as Solution of Differential Equation
I've been learning about Linear Diffusions and how their infinitesimal generators can be used to relate expectations and deterministic differential equations.
Let $X$ be an one-dimensional diffusion ...
3
votes
0answers
20 views
Limit of conditional expectations (when limit linked to the conditionning)
I am working with conditional expectations and am trying to derive a limit property.
Consider $(Y_n)_{n \in \mathbb{N}}$ a sequence of square integrable random variables, that converge in $L^2$ to a ...
1
vote
1answer
77 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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0answers
38 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 ...
1
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0answers
25 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))]
$$
Under what ...
1
vote
1answer
58 views
How to Evaluate Expected Value powered 4 of a Wiener Process?
Since $X(t_j) - X(t_{j-1})$ is Normally distributed with mean zero and variance $t/n$ we have
$$ \operatorname{E} [(X(t_j) - X(t_{j-1}))^2 ] = \frac{t}{n} \tag{1}$$
and
$$ \operatorname{E} [(X(t_j) - ...
1
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0answers
197 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
133 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.
...
1
vote
1answer
81 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
804 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
105 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
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1answer
124 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
59 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
105 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
137 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
345 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
107 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
410 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 ...
3
votes
1answer
90 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
$$ \phi_t = \frac{Y_t}{\sqrt{...
2
votes
0answers
582 views
Expectation of option value
Say we are in a BS world where the (conditional on t) price of a call is given by the usual
$$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$
Now, what about the ...
-1
votes
2answers
104 views
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0answers
850 views
Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process)
Let $S_t$ be a process which follows a Geometric Brownian Motion:
$\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$
By Ito's lemma, we have:
$S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \...
3
votes
1answer
1k views
Intuitive explanation for expectiles
I am looking for an intuitive explanation for expectiles.
Here is a link to a paper about expectiles:
Bellini and Di Bernardino: Risk Management with Expectiles, European Journal of Finance, May ...
2
votes
1answer
3k views
Expected Value of Stochastic Process
Given the following stochastic process:
$$ dX = a(X,t)dt + b(X,t)dz $$
where:
$$ dz = A \sqrt{dt}$$
and $A$ is a random variable with mean zero and variance $1$.
Is there a way to calculate the ...
1
vote
1answer
454 views
Write expectation of brownian motion conditional on filtration as an integral?
Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is
$f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\...
8
votes
2answers
519 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 ...
