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State Space model with heteroskedastic disturbance - approximation of error term

We have a state-space model with a heteroskedastic disturbance term modelled according to some time-varying process (e.g. ARCH, GARCH etc.). The disturbance term is modelled according to e.g. $h_{t+1}=...
Energy Media's user avatar
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1 answer
56 views

Possibility of obtaining a positive mathematical expectation in a quoted currency

There is a currency pair C/USD = 1. C - currency in which I want to invest in order to make a profit in USD. Suppose its price changes discretely: 50% - increases by 20%, 50% - decreases by 20%. This ...
Randomuser's user avatar
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64 views

How to simulate a conditional expectation given a filtration

I had a question regarding how to simulate a certain conditional expectation. I am given two processes $X_1(t), X_2(t)$ which both follow their own SDE, but both are of the form \begin{equation*} dX_i(...
Tipeg's user avatar
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0 answers
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Margin period of risk and scaling (MPoR)

I'm analyzing the formula to approximate the Margin Period of Risk (MPoR) for linearly linearly decreasing to zero exposure. Given the MPoR at $\tau$ one can evaluate the continious total exposure at $...
bag_dush's user avatar
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1 answer
301 views

Approximation of an Itô integral with python

Exercise 3.11 (Approximation of an Itô Integral). In this example, the stochastic integral $\int^t_0tW(t)dW(t)$ is considered. The expected value of the integral and the expected value of the square ...
Jessie's user avatar
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2 votes
2 answers
139 views

Conditional expectation of increments of stochastic process [closed]

I have come across the following result in my book on stochastic finance and I have trouble understanding the proof. On a filtered probability space with filtration $(\mathcal{F}_t)_{t \in \mathbb{R}^+...
Michaël's user avatar
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0 answers
23 views

Determining an Appropriate Locate Fee Threshold for Short Selling Based on Expected Return

I'm working on an automated stock trading program and often consider short selling as part of my strategy. For each potential short sale, there's an associated "locate fee" that I have to ...
David's user avatar
  • 33
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1 answer
111 views

Analytical formula for discounted exposure of a European Put on a stock in Real-World measure

Is there an analytical formula to approximate the discounted exposure for a European Put on a Stock in the Real-World measure? This is just an initial phase to be able to assess the accuracy of using ...
Rhoyourway's user avatar
2 votes
0 answers
55 views

Black's formula derivation: expectation of a indicator times a random variable

In this derivation of Black's formula for puts, we have that $\mathbb{E}[e^X 1_{e^X \leq K/S_0}]$ somehow equals $S_0 e^{\mu + 0.5 \sigma^2} N$ (as above in the formula). I tried breaking apart the ...
Jerry Qu's user avatar
2 votes
0 answers
100 views

Method of conditional expectations for basket

I am reading paper "An analysis of pricing methods for baskets options". Unfortunatly, I can not find the working paper "Beisser, J. (1999): Another Way to Value Basket Options, Working ...
Nick's user avatar
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1 answer
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What is the risk neutral expectiation of an option price given a move in spot?

Lets say we have a volatility surface for the SPX at time t with spot S. We consequently know the price of some call option at maturity T with strike K. What is the risk neutral expectation of the ...
Rodrigo's user avatar
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210 views

What is the dynamic of the forward price process under $\mathbf{Q}$?

Let me define the Spot price process of an underlying as follows: $$dS_{t}=\mu_{S}S_{t}dt+\sigma_{S}S_{t}dW_{t},$$ where $\left(W_{t}\right)_{t\geq0}$ is an appropriate Wiener-process, so $\left(S_{t}\...
Kapes Mate's user avatar
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Why don’t methods focus on constructing expected distributions and solving the integrals

Everyone knows assumptions of normality etc are bad and that the expected distributions of financial quantities (such as returns) change depending on the circumstances. We know that we can compute the ...
thankfulperson's user avatar
8 votes
1 answer
261 views

Propagation of Errors of Sharpe Ratio

Looking at Opdyke, J.D., Comparing Sharpe Ratios: So Where are the P-Values?, page 22 (Appendix A) an application is given for the Propagation of Errors formula on a ratio of two random variables: $$\...
oronimbus's user avatar
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0 answers
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Expected value of option spreads

I'm attempting to write a tool that will automatically calculate the expected value of arbitrary options positions, and I need to clarify my understanding. I am neither a statistician nor a ...
nathanvy's user avatar
1 vote
1 answer
74 views

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(...
Siron's user avatar
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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 ...
Ilya Tegmark's user avatar
2 votes
0 answers
58 views

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 ...
OneDayMemo's user avatar
1 vote
2 answers
95 views

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 ...
user1016491's user avatar
3 votes
2 answers
557 views

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 $...
Mining's user avatar
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10 votes
5 answers
1k views

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 ...
Joanna's user avatar
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2 votes
0 answers
58 views

Calculating E^2[σ^2] where σ is a GARCH(1,1) Proces [closed]

Given that α = 0,113079 β = 0,873884 ω = 0,0000081 I need to calculate a call price using garch volatility; I also calculated the kurtosis = 235: https://www.researchgate.net/publication/...
Andrian's user avatar
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2 answers
161 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....
Parseval's user avatar
  • 221
1 vote
1 answer
115 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 ...
Parseval's user avatar
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0 answers
67 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 $h\in\mathbb{N}?$
Parseval's user avatar
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0 votes
1 answer
229 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
cona's user avatar
  • 113
1 vote
1 answer
372 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}$...
cona's user avatar
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1 vote
1 answer
137 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 ...
Parseval's user avatar
  • 221
2 votes
0 answers
249 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 ...
user619755's user avatar
6 votes
1 answer
115 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(...
Wombat's user avatar
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4 votes
1 answer
272 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 ...
Daniel's user avatar
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1 vote
0 answers
55 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 \...
Stéphane's user avatar
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2 votes
0 answers
159 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 ...
John Paris's user avatar
0 votes
1 answer
134 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 ...
Giogre's user avatar
  • 366
3 votes
0 answers
145 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 ...
Victor Felipe's user avatar
3 votes
0 answers
25 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 ...
potato's user avatar
  • 31
1 vote
1 answer
94 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/...
ʎpoqou's user avatar
  • 157
1 vote
0 answers
72 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 ...
qwerty_uiop's user avatar
1 vote
0 answers
30 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 ...
user avatar
1 vote
1 answer
364 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) - ...
Syed Hadi's user avatar
1 vote
0 answers
636 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 ...
Pxx's user avatar
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0 votes
1 answer
284 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. ...
Chp's user avatar
  • 1
1 vote
1 answer
101 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 ...
Dreason94's user avatar
  • 311
3 votes
1 answer
1k 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}+\...
Cloud's user avatar
  • 83
3 votes
1 answer
437 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 ...
Dhruv Gupta's user avatar
3 votes
1 answer
155 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 ...
A. Henderson's user avatar
0 votes
1 answer
83 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 ...
Dmitriy's user avatar
  • 243
3 votes
1 answer
154 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 ...
Ranu Castaneda's user avatar
1 vote
1 answer
169 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 ...
happyGiraffe's user avatar
0 votes
1 answer
993 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 ...
divyanshu kumar's user avatar