Questions tagged [stochastic-processes]

stochastic processes is a collection of random variables representing the evolution of some system of random values over time.

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4
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0answers
44 views

Relating two equations in a jump-diffusion process

I am trying to understand an argument involving the pricing kernel $\xi_t$ in the context of a simple jump diffusion model for the price of an asset $S_t$: \begin{align} \xi_t = \exp \left[ -\theta ...
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100 views

Process Transforms (Fractional Difference)

Let's say I have a process $X_t$ with unknown variance process $V_t$. Then, I write $\mathrm{EMA}[X_t]$ to be the 5 sec exponential moving average of $X_t$. Consider the transformation $$\sum (X_t-\...
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2answers
136 views

Solution to SDE being Evolution of Price Process

I am trying to explain the concept of a solution to SDE being the model for the evolution of a price process. How would you do this to someone who doesn't have a financial engineering background? ...
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1answer
66 views

Are Lévy processes absolutely continuous?

If $X_t$ is a Lévy process, is it absolutely continuous? Meaning, does it have a density?
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1answer
32 views

How to project 1 Year ATM Implied volatility for SPX 500 1Year from now? Final goal is to calculate 1 Year Call prices on SPX 500 1 year from now?

I have the historical data for 1Year ATM Implied Volatility on SPX 500. I want to simulate the 1 year call option prices 1 year from now. What methods and approaches do I need to use? (Heston,GARCH, ...
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2answers
45 views

Negative values in CIR model

I'm having difficulty understanding the well known property of the CIR model that it can't go below zero. Wikipedia says that this is because the random shock on the rate will grow very small as r ...
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1answer
124 views

Expectation of the CIR process

The CIR process follows $$dr_t = (\alpha - \beta r_t)dt + \sigma \sqrt{r_t}dW_t.$$ It can be proved that there exists a unique solution for this SDE but it's not possible to get an expression for ...
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34 views

Why ARIMA is better for shorter term forecasting compared to ECM?

I'm reading up about the Error Correction model and was confused by the statements below, from here: In order to still use the Box–Jenkins approach, one could difference the series and then ...
3
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1answer
105 views

Process with negative quadratic variation

Today seems to be question day for me, sorry. The complex process $$ dX = i\sigma dW $$ where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance ...
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1answer
74 views

Covariance of logarithms of geometric Brownian motion

Suppose I have a Geometric Brownian Motion process, $$dX_t=\mu X_t dt + \sigma X_t dW_t$$ I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
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55 views

Exercise: does Ito integral of a simple stochastic process have normally distributed increments?

I am trying to solve the following problem (exercise 4.3 from Shreve's Stochastic Calculus for Finance, Vol. 2, my adaptation): Let $W(t)$, $0\le t\le T$ be a Brownian motion, and $\mathcal{F}(t)$ ...
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1answer
77 views

Risk Neutral Pricing and the Drift

For risk neutral pricing, why do we want to compute expectation of a martingale? why is this so important? Why do we dislike the drift so much? Avoid math heavy answers please.
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269 views

Solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$

Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation where $a$, $b$, and $c$ are positive constants, so I tried to solve it but I got stuck in ...
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1answer
118 views

Stock price value as a continuous-time stochastic process

I am studying a mathematics textbook on the modelling of stochastic systems. The textbook uses the price of a stock as an example of a continuous-time stochastic process: If $X(t)$ is the value of a ...
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292 views

Variance of a time integral with respect to a Brownian Motion function

Let process $$I_t = \int_0^t f(s) W_s \,\mathrm d s $$ where $W_s$ is standard Brownian motion. My question are the following: We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
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1answer
103 views

OHLC prices after filtering

Assume we have minute-bars of OHLC stock prices. Then, applying Kalman filter to those prices separately, we can remove a measurement noise and obtain the estimates of the states of the price ...
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2answers
111 views

Limit of product

Suppose $g(X, \delta_t)$ approaches a constant $J$ as $\delta_t$ approaches $0$, where $X$ is a random variable, and suppose $Y^2/\delta_t$ approaches some constant $K$ as $\delta_t$ approaches $0$, ...
3
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1answer
77 views

Forward rates are martingale under the T-forward measure

Forward rates are martingale under the $T$-forward measure but this derivation is suggesting otherwise. Could anyone please point out the mistake ? Let $dW_Q$ be a Brownian Motion in the risk ...
3
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2answers
126 views

Proof that $\exp(aW(t)-0.5a^2t)$ is a martingale

I'm trying to prove that $Z(t)=\exp(aW(t)-0.5a^2t)$ is a martingale where $W(t)$ is a Wiener process and $a$ is a constant. Here is my attempt: $$E[Z(t+s)] = E\left[\exp\left(aW(t+s)-0.5a^2(t+s)\...
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2answers
60 views

Heston Model and antithetic variables

I was implementing some variance reduction techniques for the heston model and came up with a question when implementing the antithetic variable technique. Namely, I was not sure if I had to implement ...
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29 views

Realized Variance as an approximation of the Integrated Variance

Realized Variance is written as $RV_{[0,T]}^{n} = \sum_{j = 1}^{n} r_{j,n}^2$, where $r_{j,n}$ is the log return for the $j$th increment, and $n$ is the total number of sample points in the time ...
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1answer
72 views

Ito isometry and the covariance of an Ito process

Let $(B_t)_{t \geq 0}$ et $(W_t)_{t \geq 0}$ be two independent Brownian motions and let $f: \mathbb{R} \rightarrow \mathbb{R}$ a deterministic function of time. We define the following process: \...
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1answer
61 views

Boundaries for Call Spread

I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold: Question: What are the ...
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1answer
95 views

Dynamic Programming: Dynamic Card Game

I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts about the solution provided by the book, so I really appreciate your advice if my doubt ...
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1answer
21 views

What's the relationship between the risk-neutral probability in HJM and the risk-neural probability under domestic money market?

In shreve's book, we model the stock price dynamics as: $$S_i(t) = \alpha(t)S_i(t)dt +S_i(t)\sum ^d_{j=1}\sigma _{ij}(t)dW_j(t)$$ and the forward rate can be written as : $$df(t,T) = \gamma(t,T)dt + \...
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1answer
146 views

Option pricing with Brownian Bridge

Say I have an asset following arithmetic Brownian motion $$ dX(t) = \sigma dW^\bot (t) $$ with $\sigma$ constant, and I have prices of vanilla options on $X$. I introduce a Brownian bridge $$ dY(t) = ...
3
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1answer
108 views

Computing Itô differential of conditional expectation process (Heston SDE)

Going through this article on Heston's model, where the variance evolves following the SDE \begin{equation} \label{sd1} d\sigma^2_t = \kappa \bigg( m - \color{red}{\sigma^2_t} \bigg)dt + \nu \sqrt {\...
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1answer
213 views

Discretization of Wiener process

The Wiener process $(W_t)$ is a continuous stochastic process that satisfies the following there conditions: $W_0 = 0$, the increments $\mathrm{d}W_t = W_{t + \mathrm{d}t} - W_t$ are normally ...
3
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1answer
145 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}+\...
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1answer
65 views

Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $

Let $X_t$ be a stochastic process such that $$X_{t} =\frac{1}{t}\int_0^t u dW_u $$ I know that for $$Y_{t} =\int_0^t u dW_u$$ $Y_t-Y_s$ is independent of $Y_s$ where $t>s$. But is this also true ...
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2answers
140 views

Probability distribution of the stochastic process $\int_{0} ^{t}\frac{u}{t}dW_{u}$

I am wondering about the probability distribution of the stochastic process $$X_t=\int_0^t \frac{u} {t} dW_{u}$$ I thought of using the Kolmogorov equation but after converting this into An SDE $$...
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1answer
41 views

Accumulation Rate of Variance in Random Walk

I am slightly confused with the terminology Shreve (2008), he states: "The variance of the symmetric random walk accumulates at rate one per unit time, so that the variance of the increment over ...
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1answer
86 views

Justify a backward differential equation

Regards of 4.5.1, how we get 4.5.5?
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1answer
110 views

Statistics related question about ruin theory

I am trying to solve the following problem: 'An insurance company has an initial surplus of 150 and premium loading factor of 15%. Assume that claims arrive according to a compound Poisson process $(...
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1answer
74 views

some doubts about answers to ticket line question from interview book

I'm reading an interview book called A Practical Guide to Quantitative Finance Interviews (nickname: Greenbook) and cannot understand the answer to the following question: Question: From Chapter 5/5....
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1answer
72 views

Cannot Understand The Ticket Line Question From Interview Book

I'm reading an interview book called A Practical Guide to Quantitative Finance Interviews (nickname: Greenbook) and cannot understand the following question(the question itself instead of its answers):...
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0answers
45 views

Let $dp=\mu(t)p(t)dt − k\,p(t)h(t)dt.$ Why $E[dp]=0$?

Assuming for simplicity that the price falls during a crash by a fixed percentage $k \in (0, 1)$, the asset price dynamics is given by $$dp=\mu(t)p(t)dt − k\,p(t)h(t)dt.$$ In a paper I read: The no-...
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1answer
125 views

stochastic dominance displaced diffusions

Suppose I have two processes both satisfying a displace lognormal diffusion: $$ dX(t) = \alpha(t)[X(t) - a] dW(t) $$ $$ dY(t) = \beta(t)[Y(t) - b] dW(t) $$ Note that the processes are perfectly ...
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2answers
91 views

Instantaneous change in value of portfolio

I am trying to figure out an intuitive explanation for the instantaneous change for the value of a portfolio (essentially I'm creating a self-financing portfolio to replicate a derivative payoff). ...
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64 views

What is the relevant application of mathematics?

I want to model an asset (like a currency) that is sensitive to relative economic performance between two countries, which can be measured by GDP (for example). This is a very simple case with many ...
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32 views

Hedging a long position-one period from Steven Shreve Stochastic Calculus for Finance

The following question is taken from Steven Shreve Volume 1, Chapter 1, Exercise $1.6$ (Hedging a long position-one period) Consider a one period binomial stock model with $S_0=4$, $S_1(H)=8$ and $...
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28 views

Hedged portfolio dynamics under T-forward measure

I'm looking to find the hedging PDE for a multi-currency derivative $u(F_d, F_f, X,t, T)$ under the T-forward measure, using the delta-hedging argument (F - forward rate, X - forward FX rate). ...
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84 views

Alternative derivation of Black Scholes by Merton

I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I ...
4
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1answer
109 views

What the expectation of S^2 is from GBM? [closed]

I was at an interview and was asked to write down the SDE for GBM. $$ dS = S\mu dt + S\sigma dX $$ Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
4
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1answer
174 views

Evaluating the SDE $dX_t = t\,dS_t$

The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
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142 views

Estimating Market Price of Risk

I need help with estimating market price of risk. Assume money market account and two risky assets which exposed to same two sources of risks follow process: $dM(t)=rM(t)dt$ $dS_1(t)=S_1(t)(\mu_1dt+\...
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0answers
62 views

Interchange Expectation and Supremum in Snell Envelope/American Options

I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options. I know that in general,...
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56 views

Pricing exchange options

I am really puzzled about the mechanism of pricing of exchange options using a change in numeraire: Suppose that $S^{(1)}$ and $S^{(2)}$ are stocks satisfying SDEs $$dS^{(1)}_t = \mu_1 S^{(1)}_t \,...
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1answer
93 views

Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?

In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma: Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying $$dX_t = \mu(X_t,t)dt + \sigma(...
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1answer
126 views

Invariance Scaling of Brownian Motion

Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...

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