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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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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50 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:
$$Rate(1)=P(t,...
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Summary of Pricing Options of Log-Normal Claims Using Black's Formula
Cross posted from here.
Let $B$ be a $Q$-Brownian motion and $X^{s,x}$ given by
$$dX_t = X_t(\mu_t dt + \sigma_t dB_t),\quad X_s = x$$
for $\mu, \sigma$ deterministic. Let $\mu_{s,t}=\int_s^t \mu_u du$...
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How to fix my Ornstein-Uhlenbeck parameter MLE in Python?
I am trying to fit time-series data into an Ornstein-Uhlenbeck process.
Here is my code so far:
...
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4answers
372 views
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Develop a formula for the price of a derivative paying
$$\max(S_T(S_T-K))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
3
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4answers
190 views
Price of Call Option with or without jumps
Suppose two assets in the Black Scholes world have the same volatility, but different drifts and that one has downward jumps at random times. How does this affect the option prices?
I would have ...
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0answers
56 views
How to price a down-and-out leveraged barrier call option using Brownian motion?
I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below. I am not sure how to correctly model the increasing ...
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1answer
130 views
Simulate stock prices with Geometric Brownian Motion motion with mu and signa based on 'normal' or continuous compounding?
I have written a simple script for modelling stock prices using Geometric Brownian Motion. The time series I am downloading are daily adjusted closing prices. My aim is to be able to change the ...
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mixing fractional Brownian motions
Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by
$$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$
where $W_t^{2}$ and $Z_t$ are independent of each other.
My question then: is there ...
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1answer
80 views
Convert option inputs to standard Brownian motion
I want to know the probability that the strike price of an option is touched.
My input values are:
P = price
S = strike
v = vol
t = time to expiration
According ...
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1answer
63 views
Sampling from SDE
In the case of the classic Geometric Brownian motion
$$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] $$
and ...
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How to expand lognormal approximation of Brownian motion
How can we expand this sum? $\sum_{i=1}^n (e^{rt_i-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}})^2$ where: $w_{t_i}$ is a standard Brownian motion.
If we let $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
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1answer
60 views
Ito's lemma for a Forward
I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$
$dS_{t} = S_{t}(\...
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1answer
78 views
true or false: the risk-neutral measure is useless in this situation
Example 2 of this Wiki article on the risk-measure describes how a stock price $S_t$ that is modeled with Geometric Brownian motion with drift $\mu$
$$
dS_t = \mu S_t dt + \sigma S_t dW_t
$$
can be ...
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0answers
26 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
65 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
71 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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0answers
22 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
56 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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43 views
Compo/Quanto Adjustment & Multivariate Ito
Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:
By exploring StackExchange, I noticed the ...
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1answer
76 views
Conditional distribution of $X_t = \int_0^t W_s \mathrm{d}s$
What is the conditional distribution of $$X_t = \int_0^t W_s \mathrm{d}s$$with respect to $W_t = x$?
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1answer
55 views
Exact solution stock price with Vasicek interest rate model
Define two correlated stock price- and interest rate (Vasicek) processes, governed by the Wiener processes $W^{S}(t)$ and $W^{r}(t)$
$$dS(t)=r(t)S(t)dt+\sigma S(t)dW^{S}(t)$$
$$dr(t)=\kappa(\theta-r(...
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38 views
Simulate correlated Brownian motions conditioned on future state(s)
Consider a model defined by 2 geometric Brownian motions
$$dY_{1}(t) = \sigma_{2} Y_{1}(t)dW_{1}(t)$$
$$dY_{2}(t) = \sigma_{2} Y_{2}(t)dW_{2}(t)$$
with $Y_{1}(0) = y_{1}$, $Y_{2}=y_{2}$ and $dW_{1}(...
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1answer
111 views
Difference between $W_t$ and $X_t= \sqrt{t}Z$
$W_t$ is a brownian motion and $X_t= \sqrt{t}Z$, where: $Z\sim N(0,1)$.
How to show that for a bounded continuous $f$ process, $$U_t = \int_0^t (f(W_s))ds$$ and $$V_t = \int_0^t (f(X_s))ds$$ have the ...
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31 views
Distribution of the random variable $X (t) = ā« s*B_sds$ [duplicate]
What is the distribution of the random variable $$X (t) = ā« s*B_sds ?$$ The integral is taken over [0,t].
$$B_s$$ is a brownian motion.
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2answers
181 views
Why is $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for prices?
Why is the Geometric Brownian Motion defined as $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for stock prices? $S(t)$ has a lognormal distribution which is right skewed. Another problem ...
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1answer
64 views
Intuition behind prices modeled by Geometric Brownian Motion
Suppose that we model a price $P_t$ to evolve per
$$\frac{dP_t}{P_t}=\mu dt+\sigma dW_t$$
for $\mu\in\mathbb{R}$ and $\sigma>0$. The unique strong solution to this diffusion is
$$P_t=P_0e^{(\mu-\...
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1answer
80 views
Integration of a deterministic function w.r.t. a Brownian motion
Help me solve this problem:
Let $W_t$ be a Brownian motion and suppose
$X_t = \int_{0}^{t}\delta _{s}dW_{s}$
where $\delta _{s}$ is a deterministic function. Then show that $X_t$ is a Gaussian ...
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1answer
25 views
Find Arithmetic Brownian Motion's transition density
Consider the following stochastic differential equation, an Arithmetic Brownian Motion: šš(š”) = š šš” + š šš(š”) . Find its solution, integrating from t to T, then find its transition density. ...
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1answer
65 views
Prove that $d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t$ gives a Brownian motion under forward measure
Let $N_t$ be a numeraire and $(W_t)$ be the standard Brownian motion under the risk-neutral probability measure $P$.
Recall that forward measure $\hat{P}$ is defined as the Radon-Nikodym derivative:
$...
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1answer
100 views
Calculation of a process's drift
Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift.
The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint):
I ...
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0answers
51 views
Two- (multi) dimensional geometric Brownian Motion
I am trying to calculate the value of a Basket Option with two stocks and the following information:
S1 = 100, S2 = 120, r = 0.06 L = Volatilitymatrix = ((0.3, 0.1), (0.0, 0.2)), weight of Stock 1 = 1/...
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1answer
101 views
Advantages of pathwise calculus over stochastic calculus in continuous self-financing trading models
I am new to stochastic calculus but the statement below confuses me:
Beside the issue of the impossible consensus on a probability measure,
the representation of the gain from trading lacks a ...
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1answer
311 views
What is a Brownian motion “under the risk-neutral” measure?
I understand that the risk-neutral measure is one under which the discounted price (acc. to the risk-free rate) of any asset is a martingale. But we also see notation like $\mathbb{W}^Q_t$ to denote a ...
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0answers
48 views
Are the increments of a stochastic process driven by fractional Brownian motion independent?
I'm studying the following equation
$$\tag1
dX_t = \mu X_t dt + \sigma X_t dB^H_t
$$
where $B^H$ is the fractional Brownian motion (fBm) of Hurst parameter $H\in(0,1)$, that is a continuous ...
3
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1answer
157 views
Solving Stochastic Differential Equation for Geometric Brownian Motion with time-dependent drift
Given the stochastic differential equation:
$$dZ_t = -Z_t \theta_t dB_t, \quad Z_0 = 1.$$
for an adapted process $\theta_t$ and Brownian motion $B_t$, how exactly do I apply ItƓ's Lemma to obtain:
...
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2answers
172 views
Solution to SDE being Evolution of Price Process
I am trying to explain the concept of a solution to SDE being the model for the evolution of a price process. How would you do this to someone who doesn't have a financial engineering background?
...
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1answer
75 views
Showing BM $W(s)$ is independent of $W(t)-W(s)$ [closed]
Consider $0\leq s<t$ where $t,s$ represent time index.
I want to show a Brownian motion $W(s)$ is independent of $W(t)-W(s)$.
Specifically, show that $E[W(s)(W(t)-W(s))]=0$
Proof:
Writing $W(s)$...
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38 views
Risk-neutral Simple Return Moment Log-return Moment
I am trying to find a way to link Risk-neutral moment of simple return to risk-neutral moment of log-returns.
Specifically, by making the same standard assumptions of the Black-Scholes model with the ...
3
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1answer
129 views
Process with negative quadratic variation
Today seems to be question day for me, sorry.
The complex process
$$
dX = i\sigma dW
$$
where $i = \sqrt{-1}$ and $dW$ is a standard (real-valued) Brownian motion will have a negative variance ...
2
votes
1answer
182 views
Steven Shreve: Stochastic Calculus and Finance
The lecture notes have the following theorem:
Let $\theta\in \mathbb{R}$ be given and $B(t)$ stands for the Brownian motion which is a martingale, then $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
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1answer
81 views
Why do I get this difference when simulating geometric Brownian motion?
I tried simulating GBM using both the SDE definition and the closed form solution. The paths I get through these methods are very different. Can someone help me figure my mistake?
...
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1answer
104 views
Simulation Heston Model, markovianity
I am trying to simulate the instanteneous volatility of a Heston process.
My equations are the following :
wealth process:
$$dX_t = r_t X_t + \theta \sqrt {V_t} u_t dt + u_t dW_{1t}$$
Volatility:
$$...
2
votes
1answer
116 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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1answer
91 views
Brownian motion Price and Hedge problem
Let $W_t$ be a Brownian Motion and let
$S_t= S_0e^{(rt- \frac{\sigma^2}{3!}t^3 +\int_{0}^{t}\sigma W_s ds )}$
Price and Hedge at time $t=0$ European call with maturity $T$ and strike price $K$, ...
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2answers
342 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 ...
3
votes
1answer
174 views
Mark Joshi uses forward price to price an option that pays $S_t^2-K$ if $S_t^2>K $ and zero otherwise? Why can we do that?
The following question is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $6.6$
Suppose a stock follows geometric Brownian motion in a Black-Scholes ...
3
votes
1answer
101 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
votes
2answers
156 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
4k views
Why is Brownian motion useful in finance?
The following is an interview question from Mark Joshi et al. Quant Job Interview.
Question: Why is Brownian motion useful in finance?
I am from a Pure Maths PhD background (functional analysis, ...
