# Questions tagged [expected-value]

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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 ...
107 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 ...
48 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 ...
84 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 ...
98 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 ...
81 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 ...
57 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 ...
112 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 ...
$dY_t=2Y_tdt+2\sqrt{1+Y_t^2}dW_t$ where $W_t$ is $P-$Brownian motion (Wiener process). I have defined a new measure $Q$ where the Kernel density (In Girsanov theorem) is $$\phi_t = \frac{Y_t}{\sqrt{... 0answers 174 views ### Expectation of option value Say we are in a BS world where the (conditional on t) price of a call is given by the usual$$V(S_t)=V(S_t;K,r,\sigma,T|F_t) = \Phi(d_1)S_t - \Phi(d_2)Ke^{-r(T-t)}$$Now, what about the ... 2answers 75 views ### Fourth moment of a itos integral I(t)=\int_0^t \sqrt sdW_s What is E(I(t)^4) 0answers 598 views ### Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process) Let S_t be a process which follows a Geometric Brownian Motion: \frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau By Ito's lemma, we have: S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \... 1answer 295 views ### Intuitive explanation for expectiles I am looking for an intuitive explanation for expectiles. Here is a link to a paper about expectiles: Bellini and Di Bernardino: Risk Management with Expectiles, European Journal of Finance, May ... 1answer 1k views ### Expected Value of Stochastic Process Given the following stochastic process:$$ dX = a(X,t)dt + b(X,t)dz $$where:$$ dz = A \sqrt{dt}$$and A is a random variable with mean zero and variance 1. Is there a way to calculate the ... 1answer 315 views ### Write expectation of brownian motion conditional on filtration as an integral? Let W_t be a Brownian motion, so W_t=z_t \sqrt{t} where z_t \in N(0,1) and the pdf of z is f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}. So$$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz =\...
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 ...