# 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.

485 questions
Filter by
Sorted by
Tagged with
52 views

### Complete market price and incomplete market price specification

We know that if a liquid market of an asset exists, then the standard derivative pricing theorem implies an equivalent martingale measure exists, not necessarily unique, under which the discounted ...
• 419
66 views

### ABM Crossing Times

Suppose I have a process that follows an arithmetic brownian motion $dX_t = \sigma dW_t$ How do I calculate, within a certain interval $\Delta t$ , the expected number of times that the process will &...
87 views

### Volatility in simulated paths different to monte carlo parameters

I am trying to convince myself that I have set up my monte carlo simulation correctly by looking at the results and trying to get them to agree with the model parameters. Please help me understand ...
1 vote
53 views

### How is Itô's Lemma connected to Messmore's Variance Drain?

How does Itô's Lemma explain the concept of volatility drain in investment returns, and how do the associated equations illustrate this effect? I did the following considerations so far: In financial ...
613 views

### Geometric Brownian Motion as the limit of a Binomial Tree?

Consider the price of a stock whose drift and volatility parameters are $\mu, \sigma$ respectively, over the time interval $[0, t]$. Suppose we use an $n$-stage binomial tree to model the price ...
125 views

### What are $\mu$ and $a$ in $\mu = a + \frac{\sigma^2}{2}$

Considering GBM: $$S(t_i) = S_0 \exp(a \cdot t_i + \sigma \cdot W(t_i)) = S_0 \exp\left((\mu - \frac{\sigma^2}{2}) \cdot t_i + \sigma \cdot W(t_i)\right)$$ I am interested ...
1 vote
103 views

### Identifying stochastic process from data

Suppose I am given the values of a stochastic process $S_t$ satisfying some unknown SDE from say 2000 to 2024 so I have a lot a data. How do I identify, model this stochastic process ? First I thought ...
• 121
332 views

### Differentiating Wiener process

I have come across an expression as below $d\left({W_t}^4\right) = 4 {W_t}^3 d\left({W_t}\right) + 6{W_t}^2 dt$ where $W_t$ is standard Wiener process. While I understand the first part of the RHS, I ...
• 858
33 views

### Probability distribution function for boundaries on brownian motion

What would be the probability distribution function for brownian motion with 2 boundaries ie a stop loss and take profit. The process is a standard brownian motion. However at values a and b any ...
1 vote
99 views

### Is there a "standard" "textbook" model for making re-financing decisions?

You have a loan with an x% interest rate. Rates fall to y%. Should you pay a fee to refinance? Presumably not if the NPV of the saved interest is less than the fee. However, if you always refinance ...
124 views

### Sample Wiener process constrained to open (initial), high (max), low (min), close (final)

With a Brownian bridge, one can sample a Wiener process constrained to a specified initial value and a final value. Can the same be done when the process is constrained also to have a specified ...
• 422
58 views

### Possibility of obtaining a positive mathematical expectation in a quoted currency

There is a currency pair C/USD = 1. C - currency in which I want to invest in order to make a profit in USD. Suppose its price changes discretely: 50% - increases by 20%, 50% - decreases by 20%. This ...
38 views

### Weak stationarity of continuous ARMA process from Brockwell

I am currently working on Brockwell "Levy-driven CARMA processes" (2001) and I am stuck in the introduction. So we have a continuous AR process (CAR(p)) \begin{align*} X_t=e^{At}X_0+\...
• 135
24 views

### Find expected rate of return without drift based on ito process

I would like to know how to solve question (ii), I know it is a cash-or-nothing option but I have no idea how to get the expected rate of return even I use put-call parity. Could anybody guide me I ...
• 1
54 views

### Orthogonalizing brownian path

I want to improve the stability of my SDE sample (statistical properties do not change much when using a different seed). I am using a sobol brownian bridge to generate the brownian path increments dw....
96 views

### Option pricing boundary condition

I am currently working on this paper "https://arxiv.org/abs/2305.02523" about travel time options and I am stuck at Theorem 14 page 20. The proof is similar to Theorem 7.5.1, "...
• 135
166 views

### Pricing PDE of Asian option by Shreve

I am currently working on "Stochastic Calculus for finance II, continuous time model" from Shreve. In chapter 7.5 Theo 7.5.1 he derives a pricing PDE with boundary conditions for an Asian ...
• 135
99 views

### Volatility of a stochastic Process given by an SDE

I am currently working on this thesis: http://arks.princeton.edu/ark:/88435/dsp01vd66w212h and i am stuck on page 199. There we have a portfolio $P=\alpha F+\beta G$ with $\alpha +\beta =1$ and ...
• 135
1 vote
137 views

### Moments of the integral of the exponential of Brownian motion/Normal random variable

I'm studying arithmetic Asian options and there is integral of the following form: $$X_T=\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$ where $W_t$ is a Brownian motion/Wiener process....
• 113
35 views

### The conditionnal law of a brownian motion

Please, I have a question about the conditionnal law of a brownian motion. Here is the statement: We have $\mathcal{B}_{h}$ the $\sigma$-field generated by the $\left(S_{t_{k}}, k=0, \ldots, N\right)$...
85 views

### Deriving probability of hitting stop loss given annual return and Sharpe

Suppose I have a strategy with a mean return and defined Sharpe. Given a preset stop loss, I want to calculate the probability of the stop being hit. In the example below I use the following ...
• 141
78 views

1 vote