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.
318
questions
4
votes
2answers
125 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?
...
1
vote
1answer
58 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 $W(s)$...
0
votes
0answers
36 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
votes
1answer
102 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
138 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 $Z(t)=exp\{-\theta B(t)-\dfrac{1}{2}\theta^2t\}$...
1
vote
1answer
72 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?
...
0
votes
1answer
83 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:
$$...
2
votes
1answer
67 views
Covariance of logarithms of geometric Brownian motion
Suppose I have a Geometric Brownian Motion process,
$$dX_t=\mu X_t dt + \sigma X_t dW_t$$
I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
0
votes
1answer
87 views
Brownian motion Price and Hedge problem
Let $W_t$ be a Brownian Motion and let
$S_t= S_0e^{(rt- \frac{\sigma^2}{3!}t^3 +\int_{0}^{t}\sigma W_s ds )}$
Price and Hedge at time $t=0$ European call with maturity $T$ and strike price $K$, ...
6
votes
2answers
290 views
Variance of a time integral with respect to a Brownian Motion function
Let process
$$I_t = \int_0^t f(s) W_s \,\mathrm d s $$
where $W_s$ is standard Brownian motion. My question are the following:
We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
8
votes
7answers
3k views
Consensus on Cauchy distribution for stock prices
What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded.
My motivation is to find ...
3
votes
1answer
140 views
Mark Joshi uses forward price to price an option that pays $S_t^2-K$ if $S_t^2>K $ and zero otherwise? Why can we do that?
The following question is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $6.6$
Suppose a stock follows geometric Brownian motion in a Black-Scholes ...
3
votes
1answer
76 views
Forward rates are martingale under the T-forward measure
Forward rates are martingale under the $T$-forward measure but this derivation is suggesting otherwise. Could anyone please point out the mistake ?
Let $dW_Q$ be a Brownian Motion in the risk ...
3
votes
2answers
123 views
Proof that $\exp(aW(t)-0.5a^2t)$ is a martingale
I'm trying to prove that $Z(t)=\exp(aW(t)-0.5a^2t)$ is a martingale where $W(t)$ is a Wiener process and $a$ is a constant. Here is my attempt:
$$E[Z(t+s)] = E\left[\exp\left(aW(t+s)-0.5a^2(t+s)\...
2
votes
2answers
162 views
How to numerically simulate exponential stochastic integral
For example given an integral
$$
\int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+
$$
where $W(t')$ is a standard Wiener process.
I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
2
votes
1answer
114 views
Why it's related to stock price
I am reading a paper High-frequency trading in a limit order book by Avellaneda and Stoikov.
I verified the formula (6) should be correct. However it doesn't make sense to me when I use it for ...
12
votes
2answers
3k 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, ...
30
votes
5answers
22k views
Integral of Brownian motion w.r.t. time
Let
$$X_t = \int_0^t W_s \,\mathrm d s$$
where $W_s$ is our usual Brownian motion. My questions are the following:
Expectation?
Variance?
Is it a martingale?
Is it an Ito process or a Riemann ...
7
votes
2answers
864 views
Risk neutrality correction for Monte Carlo Bootstrapping according to PRIIP regulation for products of category III
The PRIIP (packaged products) regulation prescribes Monte Carlo bootstrapping simulation for calculation of VaR for products of category III (non-linearly leveraged products). The idea is based on ...
2
votes
1answer
67 views
Expectation and variance of $\int_0^t (W_s)^n ds$ for any positive integer $n$?
It is well known that the integral
$$\int_0^t W_s ds,$$
where $(W_s)_s$ is a Brownian motion, can be derived using Ito's Lemma. More precisely, Ito's lemma on $d(tW_t)$ implies that
$$d(tW_t) = ...
4
votes
1answer
57 views
Three proofs regarding brownian motions and martingales
1. Let $(B_t)_{t \geq 0}$ and $(W_t)_{t \geq 0}$ be two standard Brownian motions and let $X_t := B_t W_t$. Is $(X_t)_{t \geq 0}$ a martingale?
The easiest way to proceed seems to be to apply Ito's ...
7
votes
1answer
144 views
Option pricing with Brownian Bridge
Say I have an asset following arithmetic Brownian motion
$$
dX(t) = \sigma dW^\bot (t)
$$
with $\sigma$ constant, and I have prices of vanilla options on $X$.
I introduce a Brownian bridge
$$
dY(t) = ...
4
votes
2answers
631 views
Asymptotic behavior property of geometric Brownian Motion proof
Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as:
If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \...
1
vote
3answers
272 views
How to calculate standard deviation of continuously compounded four-year stock returns?
Currently I am preparing for quant interview and I encounter the following question in Heard on the street.
Question: If the standard deviation of continuously compounded annual stock returns is $...
2
votes
2answers
210 views
Geometric Brownian Motion - Price Probabilities
I am modeling a stock price that follows Geometric Brownian Motion and have the following:
$E(X)$ = .16 (16%)
$\sigma$ = .24 (24%)
$X_0$ = 95
$T$ = 1 (12 months)
I am trying to find the ...
7
votes
2answers
294 views
Stochastic Integral Graph
As we can represent the integration of $f(x)$ on $[a,b]$ with the graph below,
I was wondering how to represent the following integral with $X(t)$ a Brownian motion, $f(t)$ any function and $t_j = ...
3
votes
2answers
865 views
probability question about brownian motion
Assume $W_{t}$ is a standard Brownian Motion, calculate the the probability that $W_{t}*W_{2t}$ is negative, i.e., $P(W_{t}*W_{2t}<0)$. I find it tricky to calculate the probability.Thank you.
0
votes
0answers
39 views
Swap rate in the annuity measure and Martingale Representation Theorem
As we know, swap rate evolves as a martingale in the appropriate annuity measure. Martingale representation theorem says if I can find a Brownian motion in the annuity measure and the swap rate is ...
5
votes
2answers
280 views
More questions about integral of Brownian Motion w.r.t time
A similar question have been posted earlier but one part has remained unanswered. Let us define:
$$X_t = \int_0^t W_s ds,$$
where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
1
vote
2answers
147 views
What is the stock price expectation?
The Hull textbook (and accompanying technical note) says that the expected stock price $\mathbb{E}[S_T]=S_0 \exp(\mu T)$. However, the answers to a British actuarial examination (Q4 for September 2018)...
4
votes
1answer
126 views
Invariance Scaling of Brownian Motion
Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
1
vote
1answer
73 views
Mathematical proof of $g = \mu - \frac{\sigma^2}{2}$ relationship between CAGR and average returns
I found in a paper the relation between the CAGR and the arithmetic average of returns to be
$$g \sim \mu - \frac{\sigma^2}{2}$$
where g is the geometric average, $\mu$ the arithmetic average and $ ...
1
vote
1answer
95 views
Integrating Brownian Motion [closed]
I just wonder how to integrate standard Brownian motion on time interval $(t, T)$.
Let $Z$ be a standard Brownian motion with mean $0$ and standard deviation $1$, with $dZ^2 = dt$. How to derive the ...
1
vote
1answer
38 views
Moments of discrete Asset Price Model
Say if B is standard Brownian motion then:
$S(t) = S0e^{((𝜇- σ^2)/2)t+σB(t)}$
The mean of this SDE would be
$𝐄[𝑆(𝑡)]=𝑆_0𝑒^{𝜇𝑡}$
I know to do this you use the density function and ...
1
vote
2answers
149 views
Understanding $N(d_1)$ and $N(d_2)$
Firstly, if the solution to geometric Brownian motion is $S_t = S_0 \exp((r-\sigma^2)t + \sigma W_t$ then if I have a payment that is not necessarily a full call option e.g. if the exercise price $K$ ...
0
votes
0answers
39 views
Risk neutral measure in the binomial approximation of geometric Brownian motion
Suppose an asset is described by geometric Brownian motion with a drift, i.e.
$$dS_t = S_t\mu dt + S_t \sigma dW_t$$
for a Wiener process $W_t$ and $S_0=1$. By some consequence of Girsanov's theorem (...
5
votes
1answer
87 views
Autocovariance of increments of a semimartingale
Say that $X_t$ is an Itō process with
\begin{equation}
dX_t = \mu_t dt + \sigma_t dW_t
\end{equation}
where $\mu_t$ and $\sigma_t$ are adapted processes.
Is it always true that
\begin{equation}
E[...
2
votes
0answers
41 views
How to calculate the multiple integrals where the integral domain is based on the sum of normal distribution random variables?
The integral is shown below:
And how to use python to calculate pi (better if we don't need to code for each pi)?
2
votes
1answer
91 views
What is the annualized realized volatility of simulated Brownian motion paths?
I saw this following question in an exam.
Take a Brownian motion simulation with drift 5% and annualized volatility of 20% for a period of 1 year. Then the annualized realized volatility of the ...
0
votes
0answers
35 views
Negative drift when calibrating GBM parameters
Setup for question:
Consider a basket of $N$ stocks $\{S^1, S^2, \dots, S^N\}$. For fixed strike $K$, each stock in the basket, $S^i$, follows the SDE
$$dS_t^i = \mu^i(t) S_t^i dt + \sigma^i(K, t) ...
2
votes
1answer
77 views
Valuation of Cash-Or-Nothing option
Studying options pricing, I'm stuck with the following problem:
The price of a stock is described by the dynamic:
$$dS_t = \mu\, dt + \sigma\,dW_t$$
Compute the fair price of a Cash or Nothing ...
1
vote
2answers
288 views
Probability that the price of stock following a brownian motion goes under a certain value
The price of the stock XYZ follows a brownian motion pattern with
starting price = 10, μ = 0 and σ = 20 (on annual basis). What's the probability that in 6 months the price is less or equal to 8?
...
1
vote
1answer
45 views
In search of double barrier out option on a BM
We have a BM $X_t$ with $dX_t=\sigma dB_t$ ($X_0$ not necessarily zero!) under the risk neutral measure $\Bbb Q$. Given upper barrier $U$, lower barrier $L$, "strike" $K$ such that $L<X_0<U, L&...
1
vote
1answer
87 views
Proof standard Brownian Motion under change of measure
Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
2
votes
2answers
262 views
What are some examples of non-solvable SDE where Monte Carlo discretization is necessary
Reading Glasserman - "Monte Carlo Methods in Finance" it says in the introduction to Chapter 6 - Discretization Methods, that moste models arising in derivatives pricing can be simulated only ...
1
vote
0answers
81 views
How to solve these SDE Problems
Quuestion1.
I make a solution $r(t)$ used by Ito's lemma
$r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$
Is this right?
and I try to make ...
3
votes
1answer
158 views
Correlation between stock prices given correlation between returns
assume I have two stocks with known volatilities and a known correlation coefficient of returns - does anyone know how to determine the correlation between the prices and NOT THE RETURNS
1
vote
1answer
114 views
Brownian Motions theorems
I know that if $W$ and $W′$ are two independent brownian motions, then $dWt \ dWt′$ = 0.
How can I prove/demonstrate this theorem?
Additionaly, how can we prove that if $W$ and $W′$ are dependent, ...
4
votes
1answer
479 views
Ito`s Lemma problem
Can someone help me with calculus for this problem.
I have these 3 equations and with Ito`s Lemma I have to find $dXt$.
\begin{cases} dY= μYdt+σYdB
\\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
1
vote
0answers
108 views
Predicting time series using Jump Diffusion model and Neural Networks
I am trying to understand the difference between using Jump diffusion model and Neural Networks or more precisely LSTM to predict time series data regardless what that data contains for example a ...
