# Questions tagged [geometric-brownian]

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### Find a formula for the price of a derivative paying $\max(S_T^2-K,0)$ [duplicate]

Develop a formula for the price of a derivative paying $$\max(S_T^2-K,0)$$ in the Black Scholes model. What I tried: \begin{align*} e^{-rT}\mathbb{E}^\mathbb{Q^0}[\max\{S_T^2-KS_T,0\}] = e^{-rT}\left(...
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### 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 ...
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### Optimal strategy assuming Geometric Brownian motion?

Let's assume that some stock price follows a geometric Brownian motion with known parameters $\log X_t-\log X_0 \sim (\mu t, \sigma^2 t).$ I'm trying to model the size of the order book (normalized) ...
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### Bound on path length of a stock price

Consider a time series $(S_i)$ representing a stock price (say close prices of one minute candles). Let $\Delta$ be a quantization step (could be the price step in the strike prices of the ...
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### Volatility is not becoming sigma in simulated GBM [closed]

I am trying to simulate GBM using values for sigma and then after calculating the price, calculate the log of returns and the associated volatility. I was expecting the calculated volatility of log ...
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### Present value of an FX Forward contract at each simulation and time point node of a Monte Carlo simulation

Recently I started dealing with the xVA and the associated EPE and ENE concepts. In a numerical example of an FX Forward, after simulating the underlying FX spot $S_t$ (units of domestic per unit of ...
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### Distribution of Geometric Brownian with time-dependant volatility

The process $S(t) =\exp\left(\mu.t + \int_0^t\sigma(s) \text{d}W(s) - \int_0^t \frac{1}{2}\sigma^2(s)\text{d}s\right)$ where $\sigma(s) = 0.03s$ is log-normally distributed, but i'm not sure about the ...
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### Ito Process: How to calculate expected return?

On page 300 of Hull's Options, Futures and Other Derivatives It is tempting to suggest that a stock price follows a generalized Wiener process; that is, that it has a constant expected drift rate and ...
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### How can we simulate daily return based on multi-factor model?

This is an interesting question for simulation. The question is a bit lengthy but I'm trying my best to make it super clear here. Now I have some multi-factor model, say some US barra risk model from ...
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### High/Low range under GBM - Analytic solution?

Does anybody know of an analytical solution to the expected high / low range for an asset that follows a GBM process over sampling frequency dt? I have ran numerical simulations and find that the ...
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### Reinvesting the dividends of a dividend paying stock

Suppose we have a dividend paying stock that has the following dynamics: $$dS_t=S_t((\mu-q)dt+\sigma dW_t)$$ With a continuous dividend yield $q$. What is the portfolio $Y_t$ that results out of ...
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### Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$

Two stocks are modelled as follows: $$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$ $$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$ with $dW_{1t}dW_{2t}=\rho dt$....
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Let's assume that we have the following stochastic differential equation: $dX_t = \mu X_t dt + \sigma X_tdW_t$ and that we have to prove that this is its solution: $X_t = X_0 \exp\left(\left(\mu -{\... 1 vote 2 answers 561 views ### Why do we adjust the drift in the geometric brownian motion I am building a monte carlo based on the GMB, and I am having a hard time understanding why we subtract 1/2 variance from the drift. If I have a drift of 12% and a volatility of 50%, that would give ... • 11 1 vote 1 answer 266 views ### Geometric Brownian motion and semi-martingality I am trying to learn by myself stochastic calculus, and I have a question regarding semi-martingale stochastic process and geometric Brownian motion (GBM). We know that a stochastic process$S_t$is ... • 127 2 votes 0 answers 45 views ### Pricing equation with two correlated states Consider the following asset pricing setting for a perpetual defaultable coupon bond with price$P(V,c)$, where$V$is the value of the underlying asset and$c$is a poisson payment that occurs with ... • 327 1 vote 0 answers 122 views ### multivariate geometric brownian motion equivalent martingale measure Suppose$W$is a$\mathbb{P}$-Brownian motion and the process$S$follows a geometric$\mathbb{P}$-Brownian motion model with respect to$W$.$S$is given by dS(t) = S(t)\big((\mu - ... • 141 1 vote 1 answer 141 views ### Confusion about the formula for gain process in a financial market In this wikipedia page, we consider the following financial market The formulas for the stocks are given here And the gain process of a portfolio$\pi$is defined such that From what I understand, ... • 141 3 votes 0 answers 59 views ### Dynamics of independent Geometric Brownian Motions under risk-neutral measure Q Suppose I have two Geometric Brownian motions and a bank account: $$dB_t=rB_tdt$$ $$dS=S(\alpha dt + \sigma dW_t)$$ $$dY = Y(\beta dt + \delta dV_t)$$ Where$dW_t$and$dV_t$are independent ... • 31 2 votes 1 answer 56 views ### Distribution of discrete Geometric average and Stock Price If we have $$S_t = S_0 e^{(r-\frac{1}{2} \sigma ^2) +\sigma W_t}$$ and a discrete geometric average of stock prices $$G_n = (\prod_{i=1}^{n} S_{t_i})^{\frac{1}{n}}$$ where the monitoring points are ... 1 vote 1 answer 347 views ### If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns? [closed] Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why? I tried to ask ... • 25 2 votes 0 answers 46 views ### Bessel Correction and Geometric Brownian Motion Does it make sense to use bessel's correction for standard deviation and variance when fitting the drift and volatility parameters of geometric brownian motion to historical return data for a security.... • 121 0 votes 1 answer 128 views ### Floating Strike Geometric Averaged Asian Option Pricing How can I use the risk neutral evaluation to price an asian option with floating strike using the continuous geometric average? I have tried searching for ways to do it but have found almost nothing. 1 vote 0 answers 49 views ### References for path-dependent GBMs or continuous time analog of discrete time filters Consider a path-dependent GBM model for a stock price: $$dS_t = \mu(t, S_.)S_tdt + \sigma(t, S_.) S_t dB_t,$$ where$\mu, \sigma : [0,\infty)\times C_{[0,\infty)}\to \mathbb{R}$are previsible path-... • 247 4 votes 1 answer 112 views ### Discounted expectation of generic$\mathbb{C}^2$function Consider a standard geometric Brownian motion$V_t$with drift$\mu<r$and standard deviation$1$. It holds that the discounted expectation is $$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right]... • 327 1 vote 0 answers 33 views ### Mean of diffusion term not zero using NORMINV? [closed] maybe this is a question considered too basic for all of you but im new so please excuse: I wanted to buid a simulation in excel using the usual suspect(STANDNORMINV(RAND()) and i tried to calculate ... 1 vote 0 answers 181 views ### how to estimate Geometric Brownian Motion parameters on long timeseries [closed] I'm working on a 50-years financial timeseries and I would like to simulate GBM paths from it. The first thing I'm supposed to do is to estimate the drift \mu and the volatility \sigma parameters. ... 1 vote 0 answers 70 views ### Quantile function for fractional Brownian motion (fBm) If anyone could help me to understand if it is possible calculate the quantile function for fBm? I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised ... 2 votes 1 answer 376 views ### Cholesky decomposition reduces volatility of simulated Wiener Process / Brownian Motions I am trying to simulate n correlated geometric brownian motions (GBM) given a specified correlation matrix \Sigma by following this procedure which uses Cholesky decomposition. However, when I ... • 548 5 votes 2 answers 1k views ### Dynamics of FX rate I've see a couple of places where a FX rate, denoted X, such as EURUSD (quoted as "the number of USD needed to buy 1 EUR") is modeled with a diffusion process / Geometric Brownian Motion ... • 548 0 votes 1 answer 818 views ### What are common ways to realistically simulate the stock market using historical market data? I am currently using the FinRL library to try to automate Trading using Reinforcement Learning. However, I wanted to understand how FinRL simulates the stock market using historical data. I read here ... • 195 0 votes 0 answers 182 views ### What is the Kurtosis of Returns in Geometric Brownian Motion? Suppose that dS_t=S_t(\mu\mathop{dt}+\sigma\mathop{dW_t}) which has solution$$S_t=S_0\exp\left(t\left(\mu+\frac{\sigma^2}{2}\right)+\sigma W_t\right),$$such that$W_t$is a Wiener process,$\mu$... • 128 1 vote 1 answer 410 views ### American option under Ornstein-Uhlenbeck stock price I came across with the following problem: For the Ornstein-Uhlenbeck process$(X_t, 0\leq t\leq T)$with initial condition$X_0 = x$, find the stopping time$\tau$that maximizes$\mathbb{E}[e^{-r\...
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A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process. The problem is to find the probability of the ...
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### Probability the stock price (following geometric Brownian motion) hits the upper boundary U before there is a retracement from the max by amount R?

I am looking for the probability that the stock price/Geometric Brownian Motion hits the upper boundary U, before there is a retracement (from the maximum price) that exceeds amount R. In other words,...
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### Modelling the instantaneous funding spread as a log-normal process

Let us consider a stochastic market model with a fixed short (risk-free) rate $r\in\mathbb{R}$. A trader can obtain unsecured funding at a rate $f_t:=r+s_t$ where $s_t$ is its stochastic funding ...
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### Estimating volatility of a geometric Brownian motion at different sample rates

I have troubles estimating volatility (= standard deviation of log returns) when the data is re-sampled at different sample frequencies. Problem I have generated a time series data using a geometric ...
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### Understanding the expected value of the average

I've been looking into Asian Options pricing. Part of the process is about looking for the expected value of the average of a time series undergoing e.g. geometric brownian motion. I came across this ...
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### Why would exchange rates follow a geometric brownian motion?

I'm reading Shreve's Stochastic Calculus for Finance. On page 382, he begins talking about exchange rates: Finally, there is an exchange rate $Q(t)$, which gives units of domestic currency per unit ...
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### How to simulate correlated stock prices (not returns)

Suppose we have two stocks following GBMs. Drift and volatility are calculated based on historical data. Furthermore the stocks are assumed to be correlated (i.e. they move together, if stock 1 goes ...
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### How to compute this current value using no arbitrage condition?

Suppose $X_t$ is a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. $X_0$ is known. You have a machine that produces something worth $X_t$ at random times $t$ generated by a Poisson ...
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### GBM drift when simulating correlation betwenn GBM with Cholesky Decomposition

I am currently trying to simulate correlated GBM paths and I found the Cholesky Composition for it. From my understanding, the Cholesky Decomposition can be used to create correlated random variables ...
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Geometric Brownian motion related to house prices can be written as $dH_t=(r-g)H_tdt+\sigma H_tdW_t$. How should I interpret $g$? The literature calls this a deferment rate. But I don't understand how ...