All Questions
Tagged with finance brownian-motion
88
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Boundary hitting and arbitrage in Reflecting Brownian Motion approximation
I am facing the following quantitative finance problem.
Suppose that $z$ follows a Brownian Motion centered at $0$. There are two boundaries that define a no-arbitrage region: $[-b,b]$. Trades take ...
- 21
1
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1
answer
63
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Conditional distribution of Brownian motion given the passage time?
I am stuck with this question. Suppose a stock price follow Brownian motion starting at price $p$. Denote by $\tau_r$ the passage time reaching level $r$ with $r<p$. What would be the distribution ...
- 197
2
votes
1
answer
88
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Heaviside under Geometric Brownian Motion
I'm new to using Geometric Brownian Motion, so I'm not sure if what I've done is correct.
Be the Geometric Brownian Motion $dS_t = \mu S_tdt + \sigma S_t dW_t$, $H$ a Heaviside, and $p_r, r_k$ ...
- 43
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372
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Expected value of Ornstein-Uhlenbeck process
In the paper "The Impact of Jumps in Volatility and Returns" by Nicholas Polson, Bjorn Eraker, and Michael Johannes (2003), the authors state in footnote 6 on page 1273 that, given an ...
- 392
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37
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How do Brownian/Wiener processes involve randomness?
My financial mathematics course notes have
A Brownian motion is a family of random variables $\{B_t|t\geq0\}$ on some probability space $(\Omega,\mathcal{F},P)$ such that: \begin{align}
(1) \; & ...
- 2,281
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41
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Returns of an asset in risk-neutral measure and its PDE
I am a bit confused regarding how an asset returns in a risk-neutral measure (say $\mathbb{Q}$), and subsequently its Black-Scholes-esque PDE.
In class, we learned about the approach to take when ...
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30
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Notation for modeling the price process of risky asset in d-dimensional Black-Scholes model
I'm learning the Black-Scholes model as beginner. In our lecture notes, we have following setting:
Let $W=(W_t^1,\dots,W_t^d)_{t\in[0,T]}$ be a d-dim standard Brownian motion. The financial market ...
- 168
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106
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Expectation of stock price that follows a stochastic differential equation
Let the price $S_t$ of an asset satisfy $dS_t = \alpha (\mu - \ln{S_t})S_tdt + \sigma S_t dW_t$, where $W_t$ is a Brownian motion.
I managed to show that $x_T = x_te^{-b(T-t)} + \frac{a}{b}(1 - e^{-b(...
- 167
1
vote
1
answer
85
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Whose statement about Brownian Motion correct?
My classmate and I are disagreeing about a point about $X(t)$, a BM.
His statement is that $X(t)$ is normal, my statement is that only increments of $X(t)$ are normal (thus (thus $X(t) - X(0)$ is ...
- 1,173
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1
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203
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Analysing Geometric Brownian Motion
For uni I'm doing this exercise on brownian motion, specifically geometric brownian motion. For this part of the exercise I'm required to compute certain values: sample mean, sample variance, and the ...
- 255
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176
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Expected value of powers of Brownian Motion
At the moment I am following a uni course on Financial mathematics, the current subject is Brownian Motion. A subject I have now encountered a couple of times which I don't really understand is the ...
- 255
1
vote
1
answer
39
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Expectation of Brownian function.
$(W_t)t≥0$ be a standard one-dimensional Brownian motion
Let $f : R → R$ be a given function and $0 ≤ s ≤ t$. Write down an
expression for $E(f(W_t)|W_s = x)$ in terms of $ϕ(x) = Φ′(x)$
Where $Φ(x)$ ...
- 95
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165
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Variance of Stock Price - St - given by GBM
I am working on a problem, where I'm interested in computing the variance of the stock price in the next two years.
Using the GBM notation for the stock price,
I can write St as
$
S_t = S_0e^{(\mu - \...
- 145
1
vote
1
answer
132
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Convergence of semi-implicit Euler scheme for SDE
I am confronted with the following problem: For $W$ being a one-dimensional brownian motion and $\alpha\in[0,1]$, what are the conditions for the numerical scheme
$X_{n+1}=X_n+(1-\alpha)\mu X_n\Delta ...
- 13
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33
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Stochastic differential notation in Standard Brownian Market Models
I am trying to become familiar with stochastic integration and stochastic differential notation. I tried to do the following little exercise. In my lecture notes the risky Asset is defined in the ...
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