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.
312
questions
0
votes
1answer
62 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:
$$...
1
vote
1answer
57 views
Covariance of logarithms of geometric Brownian motion
Suppose I have a Geometric Brownian Motion process,
$$dX_t=\mu X_t dt + \sigma X_t dW_t$$
I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
0
votes
1answer
83 views
Brownian motion Price and Hedge problem
Let $W_t$ be a Brownian Motion and let
$S_t= S_0e^{(rt- \frac{\sigma^2}{3!}t^3 +\int_{0}^{t}\sigma W_s ds )}$
Price and Hedge at time $t=0$ European call with maturity $T$ and strike price $K$, ...
6
votes
2answers
269 views
Variance of a time integral with respect to a Brownian Motion function
Let process
$$I_t = \int_0^t f(s) W_s \,\mathrm d s $$
where $W_s$ is standard Brownian motion. My question are the following:
We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
3
votes
1answer
137 views
Mark Joshi uses forward price to price an option that pays $S_t^2-K$ if $S_t^2>K $ and zero otherwise? Why can we do that?
The following question is taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $6.6$
Suppose a stock follows geometric Brownian motion in a Black-Scholes ...
3
votes
1answer
72 views
Forward rates are martingale under the T-forward measure
Forward rates are martingale under the $T$-forward measure but this derivation is suggesting otherwise. Could anyone please point out the mistake ?
Let $dW_Q$ be a Brownian Motion in the risk ...
3
votes
2answers
98 views
Proof that $\exp(aW(t)-0.5a^2t)$ is a martingale
I'm trying to prove that $Z(t)=\exp(aW(t)-0.5a^2t)$ is a martingale where $W(t)$ is a Wiener process and $a$ is a constant. Here is my attempt:
$$E[Z(t+s)] = E\left[\exp\left(aW(t+s)-0.5a^2(t+s)\...
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, ...
2
votes
1answer
55 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
$$d(tW_t) = ...
2
votes
1answer
108 views
Why it's related to stock price
I am reading a paper High-frequency trading in a limit order book by Avellaneda and Stoikov.
I verified the formula (6) should be correct. However it doesn't make sense to me when I use it for ...
4
votes
1answer
57 views
Three proofs regarding brownian motions and martingales
1. Let $(B_t)_{t \geq 0}$ and $(W_t)_{t \geq 0}$ be two standard Brownian motions and let $X_t := B_t W_t$. Is $(X_t)_{t \geq 0}$ a martingale?
The easiest way to proceed seems to be to apply Ito's ...
7
votes
1answer
139 views
Option pricing with Brownian Bridge
Say I have an asset following arithmetic Brownian motion
$$
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) = ...
7
votes
2answers
286 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 = ...
0
votes
0answers
34 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
253 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
145 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)...
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vote
3answers
212 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 $...
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 $ ...
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
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
1answer
84 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
2answers
148 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
37 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 (...
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
88 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
34 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
$$dS_t^i = \mu^i(t) S_t^i dt + \sigma^i(K, t) ...
1
vote
2answers
287 views
Probability that the price of stock following a brownian motion goes under a certain value
The price of the stock XYZ follows a brownian motion pattern with
starting price = 10, μ = 0 and σ = 20 (on annual basis). What's the probability that in 6 months the price is less or equal to 8?
...
2
votes
1answer
75 views
Valuation of Cash-Or-Nothing option
Studying options pricing, I'm stuck with the following problem:
The price of a stock is described by the dynamic:
$$dS_t = \mu\, dt + \sigma\,dW_t$$
Compute the fair price of a Cash or Nothing ...
1
vote
1answer
45 views
In search of double barrier out option on a BM
We have a BM $X_t$ with $dX_t=\sigma dB_t$ ($X_0$ not necessarily zero!) under the risk neutral measure $\Bbb Q$. Given upper barrier $U$, lower barrier $L$, "strike" $K$ such that $L<X_0<U, L&...
1
vote
1answer
86 views
Proof standard Brownian Motion under change of measure
Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
1
vote
0answers
80 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 ...
1
vote
1answer
112 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, ...
3
votes
1answer
154 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
0
votes
0answers
103 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 ...
1
vote
0answers
48 views
Brownian motion from price-series, what is the time step?
If I assume a given empirical price-series is a brownian motion, I can estimate the drift and standard deviation as long as I know what the time step was when the process was 'generated'.
But since ...
4
votes
1answer
473 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}
2
votes
2answers
204 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 ...
1
vote
0answers
94 views
The conditional expectation of a geometric brownian motion
In this question it states that
$$\mathbb{E}[e^{\sigma(W_t-W_s)}|\mathcal{F}_s] = \mathbb{E}[e^{\sigma(W_t-W_s)}],$$
and I assume that $0 \leq s \leq t$.
The accepted answer states that this step is ...
4
votes
1answer
133 views
Geometric Brownian Motion unable to model / predict jumps
In my finance course, we were talking about the flaws of modelling Stock Prices with the geometric Brownian Motion. According to my professor:
"Since the geometric Brownian Motion has continous time ...
2
votes
0answers
58 views
Novikov condition for Vasicek process
Suppose that we have a money account $S^{(0)}$ with dynamics
\begin{align}
dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt,
\end{align}
where
\begin{align}
dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}.
\...
1
vote
1answer
135 views
Bitcoin dynamics - C++ Simulation
I would like perform a simulation of Bitcoin future prices given a sample of the 4 past years (2014-2018). My problem is that I do not know what model to use! For common stocks I used the geometric ...
5
votes
0answers
80 views
Distribution of portfolio values with constant spending rate
If your portfolio is invested in an asset that follows a geometric Brownian motion, and you withdraw a constant dollar amount at the beginning of each year, is there an approximate analytical ...
3
votes
2answers
133 views
Find the brownian motion associated to a linear combination of dependant brownian motions
I have $N$ correlated standard one-dimensional Brownian motions $W_1,\ldots,W_N$ with correlation matrix $\rho$ and I consider the process $Z_t \equiv \sum_{i=1}^N \mu_i (t) W_t$ where the $\mu_i$ are ...
1
vote
0answers
65 views
Brownian motion for modelling future asset values
Assume that an asset price $S$ is given by a Brownian motion. Argue from the definition why it is not possible to predict future values of the asset based on the past values of $S$.
I am not sure ...
1
vote
0answers
236 views
Geometric Brownian Motion with Dividends
I am working on a problem and had a quick question. I understand that for Geometric Brownian Motion we use the formula:
$$X_{t_n} = X_{t_{n-1}} + \mu X_{t_{n-1}} \Delta t + \sigma X_{t_{n-1}} \...
2
votes
0answers
129 views
For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$
In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
1
vote
1answer
37 views
CDF&density of stock price modeled by standard brownian motion
Assume that the price of the stock follows the model
$S(t) = S(0) exp (
mt −
((σ^2)/2 )
t + σW(t)
)$
, (1)
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some constants.
Derive the ...
1
vote
1answer
69 views
Expectation and variance of standard brownian motion
Assuming that the price of the stock follows the model
$ S(t) = S(0) exp (
mt −
(σ^2/
2)
t + σW(t)
)
,
$
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
2
votes
1answer
86 views
If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?
Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say
$$
\begin{equation}
dS_t = \mu S_t dt + \sigma S_tdW_t
\end{equation}
$$
My question is what is the distribution of $...
2
votes
2answers
159 views
How to numerically simulate exponential stochastic integral
For example given an integral
$$
\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')\,...
