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No fancy theory is needed to understand why a GBM is applied to model stock prices. To get an intuitive understand, simple Macro-economics should suffice to understand why it is being applied: it has a Brownian component it has (exponential) drift - this makes the model able to deal with stock prices growing in line with GDP (actually faster than gdp ...


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first, there is a formula for the continuously monitored case. second, if you use log coordinates the Euler discretization is exact so this should be done. third, the convergence for discretely monitored to continuously is actually very slow so you will need a lot of steps. fourth, it's actually better to draw the hitting time to the barrier rather than ...


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Suppose we have no dividends like in Black-Scholes-Merton and in your example. Expected return between time $t$ and $t+\Delta t$ is defined as $$ \mathbb{E}_t\left[R_{t+\Delta t}\right]\equiv\mathbb{E}_t\left[\frac{S_{t+\Delta t} - S_t}{S_t}\right] = \mathbb{E}_t\left[\frac{\Delta S_t}{S_t}\right] $$ You can see that, as $\Delta t \to dt$, ...


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Girsanov'Theorem let $\theta_t$ be an adapted procee such that the solution of SDE $$dL_t=-L_t\, \theta_t \,dW_t , \, L_0=1$$ is a Martingale.We set $Q{{|}_{\mathcal{F}_t}}=L_t\,P{{|}_{\mathcal{F}_t}}$,then $$W_{t}^{Q}=W_{t}^{P}+\int_{0}^{t} \theta_s\,ds$$ is a standard wiener process under Q measure. Result Now we assume $\{S_t\}_{t\geq0}$ be a ...


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$r-\frac{\sigma^2}{2}$ for the drift only applies to the log-returns. The Euler discretisation simply discretises the SDE directly. You'd use the risk-free rate for you drift under the risk-neutral measure for your question. For your reference: Please read the wikipedia for more details.


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Monte Carlo simulation in the context Financial Modeling refers to a set of techniques to generate artificial time series of the stock price,volatility and interest rate and... overtime, from which option prices can be derived. There are several choices available in this regard. The first choice is to apply a standard method such as the Euler, Milstein, or ...


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the LIBOR market model the Heston model -- Euler and Milstein are actually bad for this and much more sophisticated methods are necessary local volatility models


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Your procedure is correct. However, given that the stock follows a GBM it has a closed form solution, which will yield more accurate results. $S_{t+\Delta t}=S_te^{(\mu-0.5\sigma^2)\Delta t+\sigma \sqrt{\Delta t}X_{t+\Delta t}}$ Here's a matlab code with the method above: clear all % GBM stock price t = 250; nsim = 1000; S = NaN(nsim, t); Sminus = ...


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let $Y_t=\ln X_t$ by application of Ito lemma we have \begin{align} dY_t=\frac{1}{X_t}dX_t-\frac{1}{2X_t^2}d[X,X](t)=(v-\frac{1}{2}\sigma^2)dt+\sigma\,dW_t \end{align} by integration on $[t_1,t_2]$, we have \begin{align} Y_{t_2}=Y_{t_1}+(v-\frac{1}{2}\sigma^2)(t_2-t_1)+\sigma\,(W_{t_2}-W_{t_1}) \end{align} then \begin{align} ...



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