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
2
votes
1answer
125 views

Dynamical Behavior of Hurst Exponent

I feel that the dynamic of financial market is not really modeled by standard Brownian motion, but fractional Brownian motion or even multifractional Brownian motion. I have read some references on ...
2
votes
4answers
201 views

Basic book on stochastic calculus, Itô and jump processes and Brownian Motion

I was looking for a good book that explains at beginner-level the basic of stochastic calculus, probability and random variables, Itô and jump processes as well as Brownian Motion. At university we ...
2
votes
1answer
269 views

How to get the probability of exercise call option in Black-Scholes model?

From Black-Scholes model, I'm trying to prove: $p(S_t>K) = N(d_2)$ No luck yet! Can anyone suggest a reference showing that how to obtain this equation? All I get is: $S_t = S_0e^{ (\mu-0.5 \...
4
votes
1answer
116 views

Expected payoff at future time

Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
3
votes
1answer
81 views

The conditional mean of a product of standard Brownian motions [closed]

Suppose $\{W_t, t>=0\}$ is a Standard Brownian Motion. How to compute $ \mathbb{E} \left[ W_2 W_3 \vert W_1 =0 \right]$? We know $ W_2 \vert W_1 = 0 \sim N(0,1)$ and $ W_3 \vert W_1 = 0 \sim N(0,...
5
votes
2answers
383 views

Why is the black-scholes model arbitrage free when σ>0?

I want to show that: if $σ$ is positive then there is no arbitrage in the model, even if $r > µ$. Whilst I have satisfied this for $ r > \mu$, I cannot see why the conditioning on $\sigma>0 $ ...
1
vote
1answer
168 views

How to calculate mean and volatility parameters for Geometric Brownian motion?

Say I have a time series $S_K$ for monthly asset prices for the last 30 years. I want to run a monte carlo simulation using geometric brownian motion $$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{...
1
vote
1answer
236 views

negative values in geometric brownian motion

A GBM $ \frac{dx}{x} = \mu dx + \sigma dW $ solves to $x_t = x_o e^{(\mu - \sigma^2)t + \sigma W_t}$ From the solution, it is clear that $x_t$ cannot become negative. However, it is not so clear ...
5
votes
1answer
603 views

GBM probability of hitting barrier

I tried using the brownian bridge approach to determine the probability $$P(S_t<\beta,t\in [0,T]|S_0,S_T)$$ where $S_t$ is a GBM in the usual Black Scholes setup. We know that for a BM $W_t$, $$...
10
votes
3answers
422 views

Finding distribution of $\int_0 ^T uW_u du$

I would like to find the distribution of $\int_0 ^T uW_u du$ where $(W_u)_{u\geq0}$ is the Brownian motion. What I have tried: $$\int_0 ^T uW_u du = \int_0 ^T B_udu - \int_0^T \int_0^tB_sdsdt$$ by ...
3
votes
1answer
121 views

Limits of integration when applying stochastic Fubini theorem to Brownian motion

I'm looking at the solution below from Quantuple, it's a nice, succinct solution but I'm confused about how the limits of the integrals in the second line come from. Could someone please elaborate on ...
4
votes
0answers
90 views

Reference request for research on the maximum drawdown **ratio** (NOT value)

Let's suppose the asset price process follows a Geometric Brownian motion $S_t \sim GBM(\mu, \sigma),\,t\ge 0$, and define the two process: $$ \begin{align} \text{MSF}_t &:= \max_{\tau\in[0,t]} S_\...
3
votes
2answers
314 views

What is the Brownian motion in the model for the return of a stock price trying to capture?

I have read that in the derivation of the Black-Scholes PDE, we assume that the return of a stock $S$ is given by $$\frac{dS}{S}=\mu dt+\sigma dB$$ where $\mu$ is the average growth of $S$, $\sigma$ ...
3
votes
2answers
46 views

What are the underlying events that the random variables map to the real line in the derivation of the Black-Scholes PDE?

When we first try and set up a model for the evolution of S, the value of the underlying stock, I have seen in a lot of textbooks that they model the evolution by the formula $$\frac{dS_t}{S_t}=\mu dt+...
5
votes
1answer
272 views

Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?

Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$? If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
4
votes
1answer
98 views

Discounted asset price is martingale in BS model

I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that $$ \mathrm{d}\mathrm{e}^{-r(T-t)}...
3
votes
4answers
12k views

Difference between ito process, brownian motion and random walk

Can someone explain to a non-math person (myself) what is the difference between these three? If they are so different that a comparison does not even make sense, please point it out. 1.Ito process 2....
3
votes
2answers
107 views

Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?

Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
0
votes
2answers
176 views

Integral of Wiener process over time

This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $W_t$: $$d(\int W_t dt ) = W_t dt$$ I ...
-1
votes
1answer
68 views

Differential product Correlated processes

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...
-1
votes
1answer
171 views

Expectation of the product of two Brownian motions [closed]

Could you please let me know the steps to follow to get to the solution?
1
vote
1answer
110 views

Does GBM stock price model have E[S(t)] unaffected by volatility?

Many an author claims that, if you model stock prices through GBM, $E[S(t)]=e^{\mu t}$, and the expectation is thus not related to volatility. I keep running around in circles on this one. First ...
8
votes
3answers
3k views

Variance of time integral of squared Brownian motion

I want to calculate the variance of $$I = \int_0^t W_s^2 ds$$ I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get $$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
3
votes
1answer
666 views

Calculate drift of Brownian Motion using Euler method

I am working on a project to approximate numerically the solution $X_t$ of a stochastic differential equation (SDE) using the Euler method. I have do to this for the Brownian motion with drift. I am ...
2
votes
1answer
219 views

How to calculate the covariance between two stochastic integrals?

How to calculate the covariance between the integral of a Brownian motion at different times: $$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$ I know the answer is: $$\...
1
vote
0answers
44 views

Model of asset substitution/risk shifting in continuous time

Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...
1
vote
1answer
110 views

Fractional Brownian motion references

Does anyone know any good references to understand the fractional Brownian motion and its numerical simulation, preferably applied to finance.Thank you.
-1
votes
2answers
75 views

Fourth moment of a itos integral

$I(t)=\int_0^t \sqrt sdW_s$ What is $E(I(t)^4)$
0
votes
1answer
261 views

Theoretical distribution of (geometric) Brownian motion (with drift)

I am working on a simulation study which focuses on both the Brownian motion with drift (1) and the geometric Brownian motion (2). I denote them by $X_t$. What are the theoretical distributions of ...
4
votes
1answer
182 views

Quadratic variation of an integral of a function of a Brownian motion

I'm asked to find the quadratic variation of the integral $\int_{0}^{t} W_s^2 ds$.
1
vote
0answers
93 views

Correlated GBM and OU processes

I want to model two different stochastic processes, such that: $X_t , V_t$ are correlated with coefficient $\rho$. Where: $\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
4
votes
2answers
656 views

Comparison of Brownian Motion Expected Drawdown and simulated results

Can anyone tell me whether results as predicted by Brownian Motion for a given mean and std, match what you get by measuring actual drawdown from simulated results over a number of iterations?
0
votes
2answers
885 views

Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]

My question is about a stochastic integral of brownian motion w.r.t time. Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this: \begin{eqnarray} X(t)=\int_{0}^t dt' W(t'). \...
3
votes
0answers
605 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) + \...
1
vote
0answers
92 views

Brownian bridge with time varying volatility

I have a question to ask about the Brownian bridge for a process with deterministic volatility varying over time. In other words, we have this dynamic: $dS_t = \sigma_{t} * dW_t$. We want to know the ...
0
votes
1answer
150 views

Asset price simulation under Monte Carlo for option pricing using market data

I am trying to use Monte Carlo to price some exotic options. I have in mind to simulate asset prices under GBM (say S&P prices) using Monte Carlo and price the option accordingly from the payoffs ...
3
votes
1answer
205 views

Conditional Probability - Geometric Brownian Motion

Background I am trying to find a way to price a variant of a gap option by using closed-end expressions. What makes this option a bit tricky is that it can be exercised at four predetermined dates (t=...
15
votes
2answers
8k views

Estimation of Geometric Brownian Motion drift

One can find many papers about estimators of the historical volatility of a geometric Brownian motion (GBM). I'm interested in the estimation of the drift of such a process. Any link on this topic ...
5
votes
2answers
574 views

Does Black Scholes need to assume no arbitrage?

Since Girsanov's theorem guarantees a risk neutral measure for Geometric Brownian motion, by the fundamental theorem of asset pricing there can be no arbitrage. So, why does the model assume no ...
3
votes
1answer
113 views

Using Geometric Brownian Motion for Index Options

As far as I understand, in most of the cases we derive the option valuation assuming that the log-return of the asset is partly driven by its own Brownian motion, and we use Geometric Brownian motion (...
3
votes
2answers
383 views

About SDE of Geometric Brownian Motion

It's known that most of the financial assets are subject to Geometric Brownian Motion, which satisfies the following equations: $\frac{dS}{S}=\mu dt + \sigma dX$ (1) $S_t = S_0 e^{(\mu + \frac{1}{2}...
0
votes
1answer
88 views

Determining the probability of arriving at a price by a time T

A useful calculation for ascertaining the risk of something might be determining the probability of a realization of a set of stock prices $X$ being greater than or equal to some future price $x$. I ...
3
votes
0answers
114 views

Prove the Markov property for the stochastic process $Y^x_t$

Prove the Markov property for the stochastic process $Y^x_t=xe^{at+bW_t}$ Given a function $u(t,x)=\mathbb{E}[f(Y^*_t)]$ with $Y^*_0=x$. For $s<t$ we have $\mathbb{E}[f(Y^*_t)]=u(t-s,Y^*_s)$ by ...
10
votes
3answers
1k views

What's the correct choice for modeling correlated stock prices?

Let's assume we're happy with simulating $n$ stocks as geometric Brownian motion (GBM). But say we also want the prices to be correlated. When I searched around for how to construct correlated paths, ...
0
votes
0answers
42 views

Does it make sense to simulate from the multidimensional GBM?

Suppose I have times series data on 3 assets and I do $N$ simulations (GBM) first for each of assets individually and then from a multidimensional GBM since their log-returns are correlated (I use ...
1
vote
1answer
123 views

GBM in R giving negative numbers?

I was under the impression that simulations involving geometric brownian motion are not supposed to yield negative numbers. However, I was trying the following Monte Carlo simulation in R for a GBM, ...
2
votes
1answer
324 views

Expectation of an Integral of a function of a Brownian Motion

I would really appreciate some guidance on how to calculate the expectation of an integral of a function of a Brownian Motion. Let $B(t)$ be a Brownian motion with drift $\mu$ and standard ...
2
votes
2answers
169 views

Isn't GBM the equivalent of adding infinitessimally small normally distributed returns?

The classic treatment of GBM for asset pricing leads to a point where eventually one gets a solution that is the same as assuming an underlying arithmetic Brownian motion, $X_t$, which has (over unit ...
9
votes
2answers
4k views

Simulation of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + \...
-4
votes
1answer
54 views

Which expression of $S_t$ to use under the Black-Scholes model?

I am currently looking at example exam questions relating to the evolution of a stock price under the Black-Scholes model. However, I am confused by seemingly inconsistent expressions used for the ...