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It appears that you are plotting your analytical delta as a % of the delta of the underlying. This is why the delta converges to 100% As for the numerical delta, it could be that you are not adjusting for the DV01 of the underlying. This would explain why the numerical delta still increases as the option gets more in the money and why the distortion is ...


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The $P$ dynamics of the underlying asset are: \begin{align*} dS=S(\mu dt+\sigma dB_t) \end{align*} That has the following solution under the $\mathcal{Q}$ dynamics: \begin{align*} S_t=S_0 e^{(r-\frac{\sigma^2}{2})t+\sigma W_t} \end{align*} Where $W_t$ is the equivalent martingale with respect to the original geometric brownian motion. Define $Y_t=\int_0^t ...


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Yes it is known in closed form. See https://www.rocq.inria.fr/mathfi/Premia/free-version/doc/premia-doc/pdf_html/asian_doc/asian_doc.html section 5.1 which references an older Geman-Yor paper.


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A few tips. First note that $e^{-rt}S_t$ is a martingale. So make it appear and then integrate by part to rewrite $\int S_u du$ as a stochastic integral. Finally use the Ito isometry property.


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Great question, I love to visualize data! A visualization is really the most efficient way to display a large amount of information to be processed by the human brain IMO. Depending on what exactly you are trying to plot and visualize, I would suggest trying the javascript API for WebGL called Three.js. Examples of Three.js are here: ...


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Don't look at the structure as consisting of 3 parts (i.e. a forward plus a cap plus a floor) look at it as 2 options one bought with the Floor as Strike1 and one sold with the Cap as Strike2. That way the time value changes of bought and sold option should offset - which by the way they will already do right now even wit the forward since that does not have ...


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It is the former. Martingales are defined by filtration and probability space. The probability space for the filtration need not be the same as the probability space for the martingale. I think? That's what my prof said (iirc).


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Could you please be more specific with your question and post the text here? This will be more helpful for other people visiting the site. Now as far as to where the 1/2 went, usually people put 1/2 in front of the second order term because this will simplify to 1 after the derivation: $$ \frac{\partial x^2}{\partial x} = 2x $$ vs $$ \frac{1}{2} \cdot ...



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