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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45 views

Expectation of incremental brownian motion with drift [closed]

Given $B(s)$ with drift mu, what would the expected value and variance be for $B(t)$ such that $0<s<t$? I know that for simple Brownian motion, the expected value would be $B(s)$ itself, since $...
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How to calculate expectation and variance of smooth function applied to brownian motion

I applied a smoothing function to a Brownian equation and obtained a stochastic differential equation by using Ito's lemma. The smoothing function is exp(Bt). How do I get the expected value and ...
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Non attainable claim - Incomplete market

I am wondering whether there is a standard procedure to find a non attainable (i.e. non replicable) asset in an incomplete market. As an example, let us have the following market ($B = (B^1, B^2, B^3)$...
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88 views

How to show that a Brownian motion is normally distributed and that the covariance is zero? [closed]

I need help under standing this question. So i have the following given the logarithm of the price of a share of stock is given by \begin{align*} p(t)=p(0)+\mu t+\sigma W(t), \quad t \in[0, T] \end{...
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Summary of Stochastic Derivatives, Integrals, Expectations, and Variances

I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
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1answer
112 views

Black Scholes price of exotic claim

Given a time horizon N, I want to know the time-$t$ Black-Scholes fair price of $$\int_0^T S_u du$$ where $S_u$ denotes the time-$u$ stock price. I have used the formula I have been given as follows: $...
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1answer
87 views

Using geometric brownian motion for stock price forecasting [closed]

I am doing a dissertation in finance on a maths degree. I wanted to forecast stock prices using artifcial neural networks but none of my tutors are able to supervise so I'm having to do something else....
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630 views

Expectation of exponential of 3 correlated Brownian Motion

Consider, are correlated Brownian motions with a given I want to calculate the, , I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
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1answer
204 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 ...
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1answer
39 views

Multiple underlying brownian motions

I'm trying to find a way to price a triple product forward with payoff XYZ at time T using risk-neutral pricing. But I don't really have a math background and I have trouble finding a way to account ...
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Probability of Hitting time of Brownian motion

Let $B =\{ B(t); t \ge 0\}$ be Brownian motion. What is the probability that $B$ hits state one and then state minus one before time one? My take: Let $T_x = \inf \{ t\ge 0 : B(t) = x\}$, the first ...
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177 views

Integral of the square of Brownian motion using definition of variance

Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
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Continuous option pricing: Brownian Bridge

I have a question on the proof of the formula of Sup(S) between 2 simulation points. Do you know how the prove the following formula? Thanks
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38 views

Rate of return of a bond price

Assume $B(t, T)$ is a zero coupon bond price and assume that it has dynamics $dB(t, T) = B(t, T)[\mu(t, T)dt + \sigma(t, T)dW_t]$, where $W_t$ is a Brownian motion under $(\Omega ;F; P)$ and $P$ is an ...
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1answer
69 views

Let $W_t$ denote a standard Brownian motion. Evaluate this integral [closed]

$$ \int_{0}^{t}d(W_{u}^2) $$ How can I deal with this kind of problem? If there is no function given to apply Itô's formula.
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1answer
110 views

Long-Term Energy Price Modelling: Log Returns, Distributions, Time-Weighting

I wish to forecast energy prices in the long-term (ca. 20 years) for energy-efficiency investments. While I understand that the energy carriers are particularly sensitive to external (geo-political) ...
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2answers
140 views

Proof of Feller condition for CIR square root process. Any reference?

Could you please give me some reference for the proof of the so-called Feller condition as to a stochastic differential equation of the form: $$dr_t=a(b-r_t)dt+\sigma\sqrt{r_t}dB_t\tag{1}$$ with $\...
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Relationship between risk and return for GBM and riskless bond

Suppose we have $S$, a stock following geometric Brownian motion ($dS_t = S_t (\mu dt + \sigma dZ_t)$ for $Z =$ Brownian motion) and $B$, a zero coupon bond with rate $r$, i.e. $dB_t = rB_t dt$. In ...
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95 views

Estimating constant and local volatility based on passage times

Consider a Brownian motion B_t with constant instantaneous volatility σ and zero drift where ...
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1answer
318 views

If $W_t$ is standard Brownian motion, what is $\int_0^T W_t \ln(W_t) dW_t$?

If $W_t$ is standard Brownian motion, what is meant by $\int_0^T W_t dW_t$ in finance? Furthermore, what then is the meaning of $\int_0^T W_t \ln(W_t) dW_t$?
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Simulating artificial asset prices: Random walk vs Brownian motion?

How well can each simulate the real-life behavior of stock prices, and what considerations or (dis-)advantages must we be aware of when deciding to use each: Random walk with drift Random walk ...
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1answer
68 views

General Dynamics of a Tradable Asset under the Risk Neutral Measure

Is it true that every tradable asset must have a log-normal dynamics under the risk neutral measure where the drift term is the short rate $r$? I.e., is it true that if $X$ is a tradable asset then $$\...
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Expectation of number of hits by a brownian motion

If we denote $\tau_i$ the sequence of stopping times defined by: $\tau_i = \inf(t>\tau_{i-1} : |B(t)-B(\tau_{i-1})| > a)$, $\tau_0=0$. If we denote N the number of stopping times below T. What ...
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68 views

Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown

I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2]. I do not ...
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181 views

Geometric Brownian Motion and Energy-Efficiency Investments

Suppose the payoff $X$ on an investment follows a Geometric Brownian Motion: $$ dX/X = \mu dt + \sigma dz\ , $$ for $dz$ an increment of a Wiener process. I wish to compute the expected present value ...
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83 views

law of absolute of max of brownian motion

What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion? Any references for properties of this process?
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70 views

Brownian function and Clark's formula

I was reading a paper (link) from Richard Bass about Brownian functionals, and came across the following passage : Let $X_t$ be a Brownian motion, $F_t$ its filtration and $g$ a real-valued function. ...
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67 views

Alternatives to Lognormality for negative Prices

If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a ...
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63 views

Modelling Power Prices

Since electricity prices involve strong seasonality, jump components as well as negative prices and can not be modelled by the GBM, what models/distributions exist, which would allow for modelling ...
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1answer
80 views

Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process

We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
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4answers
663 views

Ito Integral of functions of Brownian motion

How does one show that: $$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$ For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
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Simulating correlated stock paths to calculate VaR

So I wanted to generate a Monte Carlo simulation for two correlated assets to derive then the VaR as a quantile of the generated distributions. My code is the following, where the input parameters are ...
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1answer
104 views

Instantaneous correlation in the 2 factor Hull White model

I'm trying to understand which parameter controls the instantaneous correlation in the 2 F HW model. As in, correlation b/w 2 rates observed at the same time. My thinking is as follows: $$Rate(1)=P(t,...
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59 views

Summary of Pricing Options of Log-Normal Claims Using Black's Formula

Cross posted from here. Let $B$ be a $Q$-Brownian motion and $X^{s,x}$ given by $$dX_t = X_t(\mu_t dt + \sigma_t dB_t),\quad X_s = x$$ for $\mu, \sigma$ deterministic. Let $\mu_{s,t}=\int_s^t \mu_u du$...
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2answers
684 views

How to fix my Ornstein-Uhlenbeck parameter MLE in Python?

I am trying to fit time-series data into an Ornstein-Uhlenbeck process. Here is my code so far: ...
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4answers
551 views

Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$

Develop a formula for the price of a derivative paying $$\max(S_T(S_T-K))$$ in the Black Scholes model. Apparently the trick to this question is to compute the expectation under the stock measure. So,...
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4answers
294 views

Price of Call Option with or without jumps

Suppose two assets in the Black Scholes world have the same volatility, but different drifts and that one has downward jumps at random times. How does this affect the option prices? I would have ...
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59 views

How to price a down-and-out leveraged barrier call option using Brownian motion?

I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion. My python script is below. I am not sure how to correctly model the increasing ...
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1answer
511 views

Simulate stock prices with Geometric Brownian Motion motion with mu and signa based on 'normal' or continuous compounding?

I have written a simple script for modelling stock prices using Geometric Brownian Motion. The time series I am downloading are daily adjusted closing prices. My aim is to be able to change the ...
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2answers
157 views

Itos Lemma Derivation notation

So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the ...
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86 views

mixing fractional Brownian motions

Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by $$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$ where $W_t^{2}$ and $Z_t$ are independent of each other. My question then: is there ...
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1answer
97 views

Sampling from SDE

In the case of the classic Geometric Brownian motion $$dS_t = \mu S_t dt + \sigma S_tdW_t$$ we solve it as $$ S_t = S_0 \exp\left[ \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma dW_t\right] $$ and ...
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How to expand lognormal approximation of Brownian motion

How can we expand this sum? $\sum_{i=1}^n (e^{rt_i-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}})^2$ where: $w_{t_i}$ is a standard Brownian motion. If we let $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
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1answer
85 views

Ito's lemma for a Forward

I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$ $dS_{t} = S_{t}(\...
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1answer
92 views

true or false: the risk-neutral measure is useless in this situation

Example 2 of this Wiki article on the risk-measure describes how a stock price $S_t$ that is modeled with Geometric Brownian motion with drift $\mu$ $$ dS_t = \mu S_t dt + \sigma S_t dW_t $$ can be ...
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0answers
48 views

How to find the derivative for a multi-factor geometric brownian motion model

Does anyone know how to find the derivative for a multi-factor geometric brownian motion model $ \frac { dS_{i}}{S_{i}} $. I have seen solutions for the standard GBM model however I suspect that the ...
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1answer
71 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/...
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1answer
99 views

Brownian motion and Stochastic Integration

I have two questions relating stochastic integration which perhaps could be answered together. First question: First of all, I don't really understand why we can't use Riemann-Stieltjes integration ...
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24 views

Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)

For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could ...
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1answer
66 views

Sample path simulation using two random variables

I was wondering if there is a way of generating a sample path of a Geometric Brownian Motion using two independent standard normal random variables instead of just one. The exact scheme that uses ...

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