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2

I would definitely recommend Volopta as a reliable source of self-contained and commented financial engineering source codes (useful for prototyping/understanding but clearly not production code). I have for instance copy-pasted, the explicit PDE solver you are looking for (centred in space, backward in time) below (+ edited for clarity + improved ...


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Monte Carlo simulation in the context of Quantitative Finance refers to a set of techniques to generate artificial time series of the stock price overtime, from which option prices can be derived. In this case, the Least-Squares Monte Carlo can be applied to any stock price stochastic process that lends itself to simulation. It is especially useful for ...


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Assuming deterministic interest rates, the price of an American call option struck at $K$ and expiring at $T$ is given by $$ V_0 = \text{sup}_{\tau \in \mathcal{T}[0,T]} \mathbb{E}_0^\mathbb{Q}\left[ e^{-r\tau} \max(S_{\tau}-K, 0) \right] $$ where $\mathcal{T}[0,T]$ denotes a family of stopping times with values in $[0,T]$ and where, under the risk-neutral ...


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I would suggest to use : $$f(t,S^\theta_t)=\max(K-S^\theta_t,0)\exp(-\theta W_t\color{red}{\mathbf{-}}\frac{1}{2}\theta^2t)$$ where $dS^\theta_t=(r+\sigma\theta)S^\theta_t dt + \sigma S^\theta_t dW_t$


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Typically when running a Monte Carlo simulation we might simulate an SDE similar to $$ \dfrac{dS}{S} = \mu\:dt + \sigma \: dW(t) $$ by some appropriate method (e.g. Euler-Maruyama, Milstein, etc). We notice by dimensional analysis that if $t$ is in units of $\textrm{years}$ then $\mu \sim \textrm{years}^{-1}$ and $\sigma \sim \textrm{years}^{-1/2}$. ...


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Overall you are not mistaken, although it is worth revisiting a few steps in your question. We assume $S$ follows the SDE $$ \dfrac{dS}{S} = \mu\:dt+ \sigma\:dW^\mathbb{P}(t) $$ under the physical measure $\mathbb{P}$. If we change to the risk neutral measure $\mathbb{Q}$ (using Girsanov's theorem) then $\mu \to r$ and we have the following SDE $$ \dfrac{dS}...


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[Short answer] IMHO there is a fundamental problem with wanting to extract a sound implied volatility figure out of a deep ITM option's price. You should use out-of-the-money forward options (OTMF) instead: put options for strikes smaller than the forward price (left wing of the volatility surface) and call options otherwise (right wing of the volatility ...


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There's nothing wrong with your formulation, in my opinion. If you model the rate z_30 with a fixed mean, then indeed the forward ZCB price is long vega. This means that the forward interest rate is short vega (i.e. the 30yr into 10yr forward rate goes down when vol goes up). This is self-consistent. In most textbooks, however, the forward interest ...


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I just made things clearer hoping it would help. Let define $\mathbb{Q}_\theta$ as $$\frac{d\mathbb{Q}_\theta}{d\mathbb{P}}|_{\mathcal{F}_t}=\exp(\theta W_t -\frac{1}{2}\theta^2 t)=Z^\theta_t$$ By girsanov, if $W$ is a brownian motion under $\mathbb{P}$, then $W^\theta_t=W_t-\theta t$ is a brownian motion under $\mathbb{Q}^\theta$ $$\begin{split} \mathbb{...


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Applying Itô's lemma to the Black-Scholes SDE and integrating from $t$ to $t+\Delta t$ gives: $$ S_{t+\Delta t} = S_t e^{(r-\frac{1}{2}\sigma^2)\Delta t + \sigma \sqrt{\Delta t}Z} $$ with $Z \sim N(0,1)$, showing that $S_{t+\Delta t}$ given $S_t$ is log-normally distributed. It is then straightforward to write, for any compact $\mathcal{A} = [a_1,a_2]$ ...


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Note that $${{f}_{W(t)\left| W(s) \right.}}\left(x\left| y \right. \right)=\frac{{{f}_{ W(s),W(t)}}\left( x,y \right)}{{{f}_{ W(s)}}\left( y \right)}=\frac{1}{\sqrt{2\pi(t-s)}}\exp \left[-\frac{{{(x-y)}^{2}}}{2(t-s)} \right]$$ By application of Ito's lemma we have $$ln\,S_{t+\Delta t}=ln\,S_t+\left((\mu-\frac{1}{2}\sigma^2)\Delta t+\sigma(W_{t+\Delta t}-...


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You are trying to price an option through Monte Carlo simulations. Here is how it should work, assuming the Black-Scholes diffusion framework. Under the Black-Scholes model's assumptions, the value of a risky asset $S$ at the time $t=T$ is a random variable which reads $$ S_T = S_0 e^{\left(\mu-\frac{\sigma^2}{2}\right)T + \sigma \sqrt{T} Z}\tag{1}$$ with ...



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