# Questions tagged [geometric-brownian]

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### 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 ...
• 339
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### 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 ...
• 339
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### 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 ...
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### 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, $\mu$ ...
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1 vote
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### GBM - How to make make annualized dividend reflected in one quarter

I want to simulate the price path of a stock for one quarter using geometric Brownian motion. The stock has a continuous dividend yield of 5% based on the annual dividend yield. However, historically ...
160 views

### Drift rate in Geometric Brownian Motion

I have two questions regarding the drift term in the geometric Brownian motion that I cannot find any clear answers to online. When would we use risk-free rate as drift and when would we use the ...
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1 vote
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### Equivalence of Call Option on $S_T$ and Put Option on $\frac{1}{S_T}$ in FX Markets

Part 1: I am trying to price an option in the FX world. It naturally pays in the domestic currency, but in this case the payout currency must be the foreign currency. For example, consider the payoff: ...
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### How to compute the Present Value of this path-dependent option?

I have an option whose payoff depends on its value at two times $T_1$ and $T_2$ as follows. $$V(t) = \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B} (S(T_2)-K)^+)],$$ where the stock price follows the GBM ...
• 589
1 vote
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### Deriving Law of Motion by Ito's Lemma

I've been trying to derive the law of motion for the stochastic process above using Ito's Lemma, given Geometric Brownian Motion with it's law of motion shown below: I've managed to take the partial ...
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### Do simulated values for IV need to be linked to the simulated series of underlying prices when used together in a Monte Carlo Simulation?

I've been using thousands of simulated stock price series generated with mean and standard deviation of daily returns and Geometric Brownian Motion, and then running these simulated price series ...
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### Geometric Brownian Motion simulation in Python: strange results

I am trying to simulate Geometric Brownian Motion in Python, however the results that I get seem very strange and in my opinion they can't be correct. My goal is to simulate each day of 1 year. ...
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1 vote
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### Copula analytic formula for $max(S_T^1 - K, 0) 1_{\{L<S_T^2<U\}}$

Consider the payoff function $$V_T = max(S_T^1 - K, 0) 1_{\{L<S_T^2<U\}} = (S_T^1 - K)1_{\{S_T^1 > K\}}1_{\{L<S_T^2<U\}}$$ where $S_T^1$ and $S_T^2$ are two GBM distributed stocks with ...
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### Volatility of a function of an asset

Suppose that $G$ is a function of the underlying asset $S$, which follows a geometric Brownian motion. Suppose that $\sigma_{S}$ and $\sigma_{G}$ are the volatilities of $S$ and $G$, ...
1 vote
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