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The price of the April option will be more than $5.00, correct. How much more depends on the implied volatility ($\sigma$) of the option and the interest rates ($r$). The higher$\sigma$and$r$are, the higher the time value of money and the value of the April option. I highly recommend playing around with this calculator to gain an intuitive ... 2 A few tips. First note that$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. 2 First note that the price of binary call is related to the price of an ordinary call in any model by $$BinC(T,K) = e^{-rT}\mathbb{E}^{\mathbb{Q}}[1_{S_T>K}] = - \frac{\partial}{\partial K}e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)_+] = - \frac{\partial}{\partial K}C(T,K)$$ Now the volatility smile is implicitly defined by$$C(T,K) = ... 1 In the link you provided, by noting the construction of array p[], p0 and p1 are respectively the discounted$\texttt{down}$and$\texttt{up}$probabilities. Since$d=\frac{1}{u}, then \begin{align*} p0 &= e^{-r \Delta T}\, \frac{u-e^{(r-q)\Delta T}}{u-d}\\ &= \frac{\big(u\,e^{-r \Delta T} -e^{-q\Delta T}\big)u }{u^2-1}, \end{align*} and ... 1 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. 1 if we take a digital option and price under BS then you can do the whole thing by direct verification. i.e.N(d_2)\$ solves the PDE and converges to the final pay-off pointwise. So if the final pay-off has a finite number of jump discontinuities then subtract a linear combination of digitals to reduce to the continuous case.

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It is because theta is not premium / days to expiration. Theta is a "local" decay, measure of current rate of option decay, which is not assumed to stay constant. In the example you provided, theta will be closer to zero (decay rate will slow down) as you approach expiration.

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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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