Questions tagged [expected-value]
The expected-value tag has no usage guidance.
46
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SDE linear combination of stock price
Assume that $X_t$ is a process with dynamics $dX_t = \sigma X_t dW_t$ is where $W_t$ is a standard Brownian motion. Given two deterministic functions $p(t)$ and $q(t)$, compute $\mathbb{E}[p(t)X(t)+q(...
- 11
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Valuation of companies, which belong to each other
There are three companies: A, B and C.
A fully belongs to B, B fully belongs to C and C fully belongs to A.
Company A has USD 1 mln of cash and no debt.
Company B and C each have USD 10 mln of cash ...
- 101
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Expectation of Product of two European Option when vol smile exist
Currently I'm thinking about how to calculate the expectation of the product of two euro option, which is
$E[(S_T-K_1)^+(S_T-K_2)^+]$
I can fit some parametric vol model from the market listed option ...
- 51
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2
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Use Half-Normal to estimate Expected Loss
Say a stock return follows a normal distribution with 0% mean and 50% volatility. If I want to calculate the Expected Loss (ie. only the expected value of the negative returns), does it make sense to ...
- 21
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Probablity distributions of zero crossings in 1D random-walk
Consider a simple 1D random walk that starts at position zero, and each second changes position by either +1 or -1 with 50-50 probabalities.
I know it is proven to cross zero infinitely many times, ...
- 101
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117
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Kelly Criterion — maximize expected value and minimize the variance in card game with $x$ red and $y$ black cards
You have $x$ red cards and $y$ black cards. I flip them over one at a time. The probability of flipping a particular colour is proportional to the amount of those coloured cards left. You start with $...
- 155
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How to compute $E[W(T)\exp(W(T)]$
I have got this interview question twice. Does anyone know from which interview question book or another source this question comes from? It may be some well known source as two different interviewers ...
- 785
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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/...
- 11
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2
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155
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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....
- 221
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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 ...
- 221
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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 $h\in\mathbb{N}?$
- 221
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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.
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Answer
- 113
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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,
$$E[exp(\sigma m-\frac{1}{2}\sigma^2 \tau_m)] = 1$$
$$E[exp(-\frac{1}{2}\sigma^2 \tau_m)] = e^{-\sigma m}$...
- 113
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Show that $\mathbb{E}[(S+\xi)^2]\rightarrow 0$ as $n\rightarrow\infty$
EDIT: Showing this using Ito's lemma is easy, that's NOT what I want to do. I also realised that $2\mathbb{E}[S\xi]\neq 2\xi\mathbb{E}[S]$ since $\xi$ is also a random variable. Nontheless, if this is ...
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Understanding Bayes Rule of conditional expectation
Let $\mathcal{F}$ be a $\sigma$-algebra, $P$ and $Q$ be equivalent martingale measures and $\frac{dQ}{dP}$ the Radon Nikodym Derivative.
I learned that $\Bbb{E}_Q[X]=\Bbb{E}_P[\frac{dQ}{dP}X] $, which ...
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Esscher Premium: Integral Transform Proof
I have some difficulty understanding the following proof and I hope someone can help me with that.
Claim: I want to show that
$E_\alpha(S)=\frac{d}{dr} \log M_S(r)|_{r=\alpha} $, where $M_S(r)=E(\exp(...
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Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]
I am trying to calculate the expectation of
$$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$
where $(W_t)$ is a Wiener process.
I was told that the value of this expectation is zero. Can someone ...
- 347
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Moments of a SDE: a detail on the information set
Very basic questions. Let $(z_t)_{t \geq 0}$ be a standard Brownian motion and let
$$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 \...
- 2,356
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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 ...
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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 ...
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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 ...
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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 ...
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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/...
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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 ...
- 157
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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))]
$$
Under what ...
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How to Evaluate Expected Value powered 4 of a Wiener Process?
Since $X(t_j) - X(t_{j-1})$ is Normally distributed with mean zero and variance $t/n$ we have
$$ \operatorname{E} [(X(t_j) - X(t_{j-1}))^2 ] = \frac{t}{n} \tag{1}$$
and
$$ \operatorname{E} [(X(t_j) - ...
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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 ...
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161
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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.
...
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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 ...
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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}+\...
- 83
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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 ...
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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
...
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70
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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 ...
- 221
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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 ...
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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 ...
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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 ...
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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 ...
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555
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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 ...
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1
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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
$$ \phi_t = \frac{Y_t}{\sqrt{...
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Expectation of option value
Say we are in a BS world where the (conditional on t) price of a call is given by the usual
$$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$
Now, what about the ...
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Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process)
Let $S_t$ be a process which follows a Geometric Brownian Motion:
$\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$
By Ito's lemma, we have:
$S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \...
- 2,845
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Intuitive explanation for expectiles
I am looking for an intuitive explanation for expectiles.
Here is a link to a paper about expectiles:
Bellini and Di Bernardino: Risk Management with Expectiles, European Journal of Finance, May ...
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1
answer
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Expected Value of Stochastic Process
Given the following stochastic process:
$$ dX = a(X,t)dt + b(X,t)dz $$
where:
$$ dz = A \sqrt{dt}$$
and $A$ is a random variable with mean zero and variance $1$.
Is there a way to calculate the ...
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Write expectation of brownian motion conditional on filtration as an integral?
Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is
$f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So
$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz
=\...
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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 ...
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