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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
43 views

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
1 vote
0 answers
48 views

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
  • 37
0 votes
1 answer
87 views

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 ...
Mango's user avatar
  • 31
0 votes
1 answer
70 views

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 ...
Lanazo's user avatar
  • 31
2 votes
1 answer
310 views

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
2 votes
0 answers
42 views

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 ...
Luca Gi's user avatar
  • 327
4 votes
1 answer
141 views

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. ...
Niko's user avatar
  • 43
2 votes
0 answers
95 views

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 - ...
yrual's user avatar
  • 151
1 vote
1 answer
117 views

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
  • 151
0 votes
0 answers
64 views

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
1 vote
1 answer
208 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
0 answers
102 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
2 votes
1 answer
234 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
0 votes
0 answers
74 views

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
148 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
  • 893
1 vote
1 answer
253 views

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
1 vote
0 answers
87 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
0 votes
0 answers
81 views

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
  • 13
0 votes
1 answer
104 views

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
0 votes
1 answer
167 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
274 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
106 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
  • 327
1 vote
0 answers
33 views

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
169 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
156 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
62 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
0 votes
2 answers
122 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
  • 11
4 votes
1 answer
266 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
178 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
  • 221
0 votes
0 answers
178 views

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
0 votes
0 answers
55 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
90 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
142 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
  • 103
1 vote
0 answers
83 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
438 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. ...
Landscape's user avatar
  • 548
-1 votes
1 answer
205 views

Integration of exponential raised with Brownian Motion wrt the Brownian Motion

I have to derive several things for my thesis, however, I have the following expression: $$ \int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t} $$ Does anyone know what the solution for this is? Kind regards.
cem's user avatar
  • 5
7 votes
3 answers
803 views

Why does the diffusion term remain the same when we change pricing measure?

Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion In plenty of interest rate examples, I have ...
user9078057's user avatar
0 votes
2 answers
193 views

Expectation of functions with Brownian Motion embedded

Trying to solve a problem set with: Let $W_t$ be a Brownian Motion and $X_t = e^{izW_t}$ where $z$ is real, $i = \sqrt{-1}$. I need to find $\mathbb{E}\left(X_t\right)$... I am a bit stuck. I have ...
LondonGuest's user avatar
2 votes
0 answers
168 views

In the Black-Scholes model with stochastic interest rates, what are the 3 assets used to compute measures?

Suppose I have a model with 2 primary assets, a stock $S$ and a short rate. The stock will be driven by a Brownian motion $W_1$. The short rate will be random and will be driven by a Brownian motion $...
user60304's user avatar
2 votes
3 answers
295 views

Integral of brownian increments

I'm stuck at a problem and I'm not sure on how to proceed. My question is how would one go about and integrate the following $$\sigma\int_{t}^{T}\mathrm{e}^{a\cdot u}\cdot (W_{u}-W_{t})du.$$ I've been ...
Marc Allan's user avatar
0 votes
1 answer
326 views

GBM - How to make make annualized dividend reflected in one quarter

I want to simulate the price path of a stock for one quarter using geometric Brownian motion. The stock has a continuous dividend yield of 5% based on the annual dividend yield. However, historically ...
Chell's user avatar
  • 3
0 votes
1 answer
418 views

Drift rate in Geometric Brownian Motion

I have two questions regarding the drift term in the geometric Brownian motion that I cannot find any clear answers to online. When would we use risk-free rate as drift and when would we use the ...
Chell's user avatar
  • 3
0 votes
0 answers
85 views

Brownian Bridge from timestep 1 to timestep @ expiration, proper mathematical way to generate

When I was learning finance, we didn't cover the subject of Brownian Bridges. So I am trying to learn the proper way of generating paths when you have an arithmetic Asian option which has an ...
Matt's user avatar
  • 139
0 votes
1 answer
99 views

Lemma (maybe) to imply the sign of the sensitivity to correlation

Can anybody please help me to understaind if this result is true ? Let $\pi=\mathbb{E}\left(f(X_{T})g(Y_{T})\right)$ where $f$ and $g$ are increasing functions. Hence, $\pi$ is increasing with respect ...
DeepInTheQF's user avatar
2 votes
2 answers
977 views

Solving SDE using integration factor and Ito's lemma [closed]

I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential ...
Bohdan_'s user avatar
  • 21
1 vote
2 answers
480 views

How can there be Brownian motions under different measures?

According to the definition, for a Brownian motion it holds that $W_0 = 0$, and $W_t - W_s \in N(0, t-s), \quad t > s$. This implies that $W_t \in N(0, t)$, for all $t \geq 0$. Hence, the ...
user3221037's user avatar
4 votes
0 answers
139 views

How to integrate Itô integral w.r.t time?

Let $W_t$ be a Brownian motion. How to calculate the following integral $$ I:=\int_0^t\left( \int_0^u(u-s)dW_s\right) du? $$ My attempt so far is: First note that $$ \int_0 ^u (u-s)dW_s = \int_0^u ...
user9312's user avatar
-2 votes
2 answers
186 views

Python - Problem of random numbers in MC simulation

I am interested in estimating the price of a European Call Option using the Montecarlo simulation, to get a good approximation of the analytical Black Scholes formula, so a very simple task. ...
John_maddon's user avatar
1 vote
1 answer
172 views

standard/brownian market with different brownian motion

Consider for simplicity the following brownian market: $$dS^0_t= r S^0_tdt$$ $$dS^1_t= S^1_t(r dt + dW^1_t + dW^2_t) $$ where the filtration is generated by $W^1,W^2$ Consider now $W_t:= \frac{1}{2}(W^...
Steven Hunt's user avatar
5 votes
0 answers
132 views

Covariance of (fractional) Brownian motions with different Hurst parameters

I'd like to calculate the covariance function for fractional Brownian motions $$ E_t \left[ dW^H(t) dW^{H'}(t) \right] $$ but where the Hurst parameters are not equal: $H \neq H'$. My first idea would ...
user avatar

1
2 3 4 5
10