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Questions tagged [brownian-motion]

In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.

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Probability distribution function for boundaries on brownian motion

What would be the probability distribution function for brownian motion with 2 boundaries ie a stop loss and take profit. The process is a standard brownian motion. However at values a and b any ...
oxheron's user avatar
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Is there a "standard" "textbook" model for making re-financing decisions?

You have a loan with an x% interest rate. Rates fall to y%. Should you pay a fee to refinance? Presumably not if the NPV of the saved interest is less than the fee. However, if you always refinance ...
Thomas Redding's user avatar
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Sample Wiener process constrained to open (initial), high (max), low (min), close (final)

With a Brownian bridge, one can sample a Wiener process constrained to a specified initial value and a final value. Can the same be done when the process is constrained also to have a specified ...
JoseOrtiz3's user avatar
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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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Weak stationarity of continuous ARMA process from Brockwell

I am currently working on Brockwell "Levy-driven CARMA processes" (2001) and I am stuck in the introduction. So we have a continuous AR process (CAR(p)) \begin{align*} X_t=e^{At}X_0+\...
Valentin's user avatar
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Find expected rate of return without drift based on ito process

I would like to know how to solve question (ii), I know it is a cash-or-nothing option but I have no idea how to get the expected rate of return even I use put-call parity. Could anybody guide me I ...
Jun's user avatar
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Orthogonalizing brownian path

I want to improve the stability of my SDE sample (statistical properties do not change much when using a different seed). I am using a sobol brownian bridge to generate the brownian path increments dw....
Madhuresh's user avatar
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Option pricing boundary condition

I am currently working on this paper "https://arxiv.org/abs/2305.02523" about travel time options and I am stuck at Theorem 14 page 20. The proof is similar to Theorem 7.5.1, "...
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Pricing PDE of Asian option by Shreve

I am currently working on "Stochastic Calculus for finance II, continuous time model" from Shreve. In chapter 7.5 Theo 7.5.1 he derives a pricing PDE with boundary conditions for an Asian ...
Valentin's user avatar
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Volatility of a stochastic Process given by an SDE

I am currently working on this thesis: http://arks.princeton.edu/ark:/88435/dsp01vd66w212h and i am stuck on page 199. There we have a portfolio $P=\alpha F+\beta G $ with $\alpha +\beta =1$ and ...
Valentin's user avatar
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Moments of the integral of the exponential of Brownian motion/Normal random variable

I'm studying arithmetic Asian options and there is integral of the following form: $$X_T=\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$ where $W_t$ is a Brownian motion/Wiener process....
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The conditionnal law of a brownian motion

Please, I have a question about the conditionnal law of a brownian motion. Here is the statement: We have $\mathcal{B}_{h}$ the $\sigma$-field generated by the $\left(S_{t_{k}}, k=0, \ldots, N\right)$...
Raphael Morel's user avatar
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Deriving probability of hitting stop loss given annual return and Sharpe

Suppose I have a strategy with a mean return and defined Sharpe. Given a preset stop loss, I want to calculate the probability of the stop being hit. In the example below I use the following ...
insomniac's user avatar
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Change of numeraire : quotient

Let's consider $X_1(t)$ a geometric brownian motion (with variable volatility) and $X_2(t)$ a Brownian bridge : $dX_1(t) = \mu X_1(t) dt + \sigma_1(t) X_1(t) dW(t)$ $dX_2(t) = \frac{b - X_2(t)}{T - t} ...
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Brownian Motion as a Limit of Simpler Models

Let $Δ$ be a small increment of time, and consider a process such that every $Δ$ time units the value of the process either increases by the amount $σ \cdot sqrt(Δ)$ with probability $p$ or decreases ...
Sir Fart-A-Lot's user avatar
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Ito Process: How to calculate expected return?

On page 300 of Hull's Options, Futures and Other Derivatives It is tempting to suggest that a stock price follows a generalized Wiener process; that is, that it has a constant expected drift rate and ...
user546106's user avatar
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modelling time series using semi-martingale process

During this week lecture my professor said that the semimartingale( brownian motion contamined by noise) is a model in reduced form because we do not specify the dynamic which leads to price ...
XY0's user avatar
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Return distribution with stop loss

Assume there is a investment with payoff going like a brownian motion, i.e. $dS=\mu dt+\sigma dW$, for simplicity, setting $\mu= \sigma=1$. At $t=1$, the payoff distribution is $P=Normal(1,1)$. If we ...
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How to apply CLT on scaled symmetric random walk--Shreve unclear

"Theorem 3.2.1 (Central limit)" in the book "Stochastic Calculus for Finance II Continuous-Time Models" by Steven Shreve says: Theorem. Fix $t\geq0$. As $n\to \infty$, the ...
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Integrated Brownian motion

I occasionally see a post here: Integral of brownian motion wrt. time over [t;T]. This post has the conclusion that $\int_t^T W_s ds = \int_t^T (T-s)dB_s$. However, here is my derivation which is ...
Wang Jing's user avatar
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Pricing equation with two correlated states

Consider the following asset pricing setting for a perpetual defaultable coupon bond with price $P(V,c)$, where $V$ is the value of the underlying asset and $c$ is a poisson payment that occurs with ...
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Quadratic Variation Of Mixed Brownian Motion and Poisson Process

I am trying to solve this problem where we're asked to compute the quadratic variation of a process. I assume that it is necessary to apply Ito's formula but not sure how to get the right solution. ...
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multivariate geometric brownian motion equivalent martingale measure

Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by \begin{equation} dS(t) = S(t)\big((\mu - ...
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Confusion about the formula for gain process in a financial market

In this wikipedia page, we consider the following financial market The formulas for the stocks are given here And the gain process of a portfolio $\pi$ is defined such that From what I understand, ...
yrual's user avatar
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Equivalent definition of brownian motion

I'm having a question about this characterization of Brownian Motion : Theorem : If a process : $\big( X_t \big)_{t\geq 0}$ satisfies these conditions, $\big( X_t \big)_{t\geq 0}$ is a Gaussian ...
Ahmed EL YOUSEFI's user avatar
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314 views

If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns? [closed]

Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why? I tried to ask ...
nearhome's user avatar
3 votes
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156 views

Feynman-Kac formula: Ito's lemma for exponentiated integrals $e^{-\int b dr}$

Consider the stochastic process $$ dy = f(y,s)ds + g(y,s)dw $$ where, $w$ is Brownian motion. Now consider the following exponentiated integral $$ z_1(s) = \exp \left[ - \int_t^s b(y(r),r) dr \right] $...
TheTwistedSector's user avatar
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261 views

How do your solve for trader's optimal demand in market similar to Kyle's model?

Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a standard probability space and $Z_t=(Z_t^1,Z_t^2)$ is a two dimensional Brownian motion with the filtration $\mathcal{F}^Z_{t}$ and $Z_t^1$, $Z_t^2$ ...
Oliver Queen's user avatar
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Maximum likelihood estimation of system of correlated SDEs

I have the following system of SDEs (which you can think of as 3 different stocks) $$dX_t^1 = \mu_t X_t^1 dt + \sigma_t X_t^1 dW_t^1$$ $$dX_t^2 = \mu_2 dt + \sigma_2 dW_t^2$$ $$dX_t^3 = \mu_3 dt + \...
Spandaver's user avatar
4 votes
1 answer
193 views

Estimating the knockout probability of a discretely observed autocall note

For simplicity, let's suppose the underlier follows a Geometric Brownian Motion $S_t\sim\text{GBM}(\mu, \sigma), t\ge 0$ with $S_0=1$. A discretely-observed binary autocall note is a derivative ...
Vim's user avatar
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On first and last zeros before t in a Brownian Motion

Suppose we have the following random variables, given a fixed $t$ we define the last zero before $t$ and the first zero after $t$: \begin{align*} \alpha_t &= \sup\left\{ s\leq t: B(s) = 0 \...
Eduardo Contreras's user avatar
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97 views

How to derive this HJB equation?

I'm reading the paper by J.Gatheral and A.Schied (2012) - "Optimal Trade Execution under Geometric Brownian Motion in the Almgren and Chriss Framework". On page 6, the authors provide a ...
matvey kormushkin's user avatar
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How to simulate from instantaneously correlated Brownian motions?

Say I have obtained a distribution for different forward rates F_k such that: $$ dF_k (t) = \sigma (t) * F_k (t) * dW_k(t) $$ with $$ dW_k(t) * dW_l(t) = \rho_{k,l} (t) dt. $$ From this I want to ...
Stann98's user avatar
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1 answer
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Standard Brownian Motion and Exponential Martingale calculation [closed]

Let $W(t)$ be a standard brownian motion and let $Z(t) = \exp (\lambda W(t) - \frac{1}{2}\lambda^2 t).$ In Xinfeng Zhou's Green Book in the section on Brownian Motion (p.130) he writes as part of the ...
phhhlpfk's user avatar
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1 answer
253 views

Expectation of Bt^4 given BS [closed]

What is the expectation of Bt^4 and Bt^3 given Bs? Given t>s. I understand that the expectation of Bt given Bs is Bs and that the expectation of Bt^2 given Bs is something like Bs - s + t.
Lawrence Chun's user avatar
1 vote
1 answer
389 views

4th Order Brownian Motion Martingale [closed]

I understand the first order MG of brownian motion is Bt.. the second order is Bt^2 - t and the third order is bt^3 - 3tBt. How can I find the fourth and beyond order of a Brownian Motion Martingale?
Lawrence Chun's user avatar
4 votes
1 answer
111 views

Discounted expectation of generic $\mathbb{C}^2$ function

Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$. It holds that the discounted expectation is $$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right]...
Luca Gi's user avatar
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Mean of diffusion term not zero using NORMINV? [closed]

maybe this is a question considered too basic for all of you but im new so please excuse: I wanted to buid a simulation in excel using the usual suspect(STANDNORMINV(RAND()) and i tried to calculate ...
BussiHasi's user avatar
2 votes
1 answer
231 views

Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach

I am looking to compute the following using Ito's formula. $$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$ Knowing the properties of brownian motion, it is rather easy to show that the ...
ilikemath3.14's user avatar
3 votes
1 answer
161 views

Integral of Function of Brownian Motion w.r.t Time (Context: Computing Quadratic Variation)

I am looking to compute the quadratic variation of $$S_t = S_0e^{\sigma B_t}$$ where $B_t$ is Brownian Motion. Applying Itô's lemma, I having the following $$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...
ilikemath3.14's user avatar
1 vote
0 answers
67 views

Quantile function for fractional Brownian motion (fBm)

If anyone could help me to understand if it is possible calculate the quantile function for fBm? I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised ...
Serg Gini's user avatar
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2 answers
133 views

Construction of Itos integral

I am trying to understand the below: Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] =some positive number be a profit or loss? In this calculation the purchase price is not taken ...
KD007's user avatar
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4 votes
1 answer
273 views

Simulating Iterated Brownian Motions

I was going through an interesting article (https://arxiv.org/pdf/1112.3776.pdf) while I was trying to read about subordinated processes. I wanted to simulate subordinated processes (in R or python) ...
Rishabh Kumar's user avatar
2 votes
0 answers
211 views

How did Bachelier characterize the Brownian motion?

The model for a stock price $$ dS_t=\mu dt + \sigma dB_t $$ where $B_t$ is a Brownian motion on $(\Omega, \mathcal{F},P)$, is commonly attributed to the work that Bachelier has carried out in his PhD ...
Mr Frog's user avatar
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Hitting time of Brownian motion with drift using Feynman-Kac

I was studying this question from "A Practical Guide to Quantitative Finance Interviews" and was having some trouble understanding one solution. Please advise if misunderstood anything or if ...
Richardhxw's user avatar
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60 views

Does the Lévy characterization imply that the price process of any asset is a Brownian motion?

While studying Brownian motion applied to mathematical finance, I came across these lecture notes by prof Steve Lalley. In the prologue, he gives this explanation for the occurrence of Brownian motion ...
sound wave's user avatar
2 votes
0 answers
99 views

NFT Floor Price

I'm interested in modeling NFT Floor Price. Specifically, I'm trying to answer the question: Given current bid-ask info on an NFT collection, what is the probability distribution of the lowest ask ...
Kalev Maricq's user avatar
0 votes
1 answer
172 views

Implication of unique risk neutral measure

I'm reading Shreve Stochastic Calculus II, theorem 5.4.9 (Second fundamental theorem of asset pricing), This is the part that confuses me : suppose there is only one risk-neutral measure. This ...
C.C.'s user avatar
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1 vote
0 answers
91 views

Dividend Dynamics under Q Measure / Using Girsanov Theorem with Covariance

I want to find the value of a dividend stream. I can do it under the P-measure, but now I would also like to do it under the Q-measure but cant figure out how to derive the dynamics of the dividend ...
Hedgehog's user avatar
1 vote
1 answer
535 views

Integral of brownian motion wrt. time over [t;T]

From the post Integral of Brownian motion w.r.t. time we have an argument for $$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$ However, how does this generalise for the interval $[t;T]$? I.e. ...
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