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American options pricing (swaption is just a kind of option) is a bit tricky due to the early exercise. Here is a page listing possible approaches, including some numeric methods, and some close form approximation formula. As I understand, lattice methods (tree, PDE discretization such as forward shooting) are fine to price American options. There're ...


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$\Pi$ is the value of a delta-hedged portfolio (option plus a short position in Δ underlying). The notation for $\Delta$ is overloaded. Here it represents the number of underlying contracts (f.ex shares) in your delta hedged portfolio, equal to the greek $\Delta$ when the portfolio is created. Therefore in the calculation of $d \Pi$, $\Delta$ (the number of ...


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Recall that the delta of an option is the sensitivity of its price to changes in the underlying's stock price: $$\Delta = \frac{\partial V}{\partial S} $$ Now, if you assume the BS framework, you find that: $$V(t,T,K,\sigma,r) = S_t \Phi(d_1) - e^{-r(T-t)} K \Phi(d_2)$$ Clearly, $\Delta = \frac{\partial V}{\partial S}= \Phi(d_1)$. Note that $d_1$ is a ...


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BS is increasing with respect to volatility, and bounded from above, i.e. the call by $F$ and put by $K$, as volatility goes to infinity. So it can not be convex with respect to volatility.


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Theory: First of all you must decide which implied volatility you want. Probably you are looking for the Black and Scholes implied vol. (but one could also caclute Heston implied vols etc) If we are in a B&S setting one desires to retrieve the implied vol by solving the B&S pricing equation for $\sigma$. Unfortunately there is no analytical solution ...


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Implied volatility cannot be calculated analytically with a closed formula. Instead, you have to approximate it numerically. There are multiple methods to compute IV on an option: Bi-section method Newton-Raphson method Secant method A quick google search came up with the following code for C++ using bi-section and newton methods: Implied ...



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