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We construct a locally risk-free self-financing portfolio $X_t$, at time $t$, with $\Delta_t^1$ share of debt and $\Delta_t^2$ share of equity. That is, \begin{align*} X_t = \Delta_t^1 D_t + \Delta_t^2 E_t. \end{align*} Then, \begin{align*} dX_t &=\Delta_t^1 dD_t + \Delta_t^2 dE_t\\ &=\Delta_t^1\bigg[\Big(\frac{\partial D_t}{\partial t} + \mu ...

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Use Ito's lemma to get dD Use dE = dV - dD Form a weighted portfolio of D and E, lets call it V, where the weights sum to 1 Use the self-ﬁnancing to get the dynamics of V Find the weights that makes V risk-free Set the drift of V equal to r

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We can replicate the security with the bond and the option, and obtain the Black-Scholes PDE. We form the portfolio $A_t=D_t+\alpha E_t$ where $\alpha$ needs to be determined. Applying the self-financing assumption implies that $$dA_t=dD_t+\alpha dE_t$$ so we can write \mu A dt+\sigma A dW_t=rD\,dt+\alpha\big(\frac{\partial{E}}{\partial ... 0 You need the derivation of BS eq. https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation#Derivation where in your setup you need S = V and case 1: V = E, case 2: V = D. 2 I assume this is a plot of option value versus price of the underlying. The only case where it ought to be symmetric is if the pdf of the underlying is symmetric eg normally distributed. I'm guessing your chart assumes a lognormal underlying, which is a non symmetric pdf, so the graph is non symmetric. 3 To show whether it is self-financing, we need to show whether the equation \begin{align*} dV_t = a_t dS_t+b_t d\beta_t \end{align*} holds. Note that \begin{align*} V_t &= a_t S_t + b_t \beta_t\\ &=\frac{1}{2} S_t + \frac{1}{2} S_t e^{-rt} e^{rt}\\ &=S_t. \end{align*} Then \begin{align*} dV_t = dS_t. \end{align*} On the other hand, ... 2 Note that, \begin{align*} \frac{\partial{C}}{\partial{\sigma}} &=\frac{S_0}{\sqrt{2\pi}}{e^\frac{-d_+^2}{2}}(\frac{-1}{\sigma})(d_-)-\frac{Ke^{-rt}}{\sqrt{2\pi}}e^{\frac{-d_-^2}{2}}(\frac{-1}{\sigma})(d_+)\\ &=\frac{1}{\sqrt{2\pi}}e^{\frac{-d_+^2}{2}}\left[-\frac{S_0 d_-}{\sigma} + \frac{Ke^{-rt}d_+}{\sigma} e^{\frac{d_+^2}{2} - \frac{d_-^2}{2}} ... 2 Yes, your broker could have used one or combination of many factors: estimated volatility surface from historical returns of your target index, historical returns of similar indexes, implied volatility of similar indexes, existing inventory,etc. Check out these two approaches to deriving surfaces from returns starting slide 14 1 Generalizing, some people who write options trading software are not aware of a few small, but important details, resulting in some pricing idiosyncrasies. That is often the case with retail trading platforms, and you often read statements like "implied vol blows up in days before expiration", "greeks become unreliable before expiration", or suggestion of ... 0s\partial_{s}\Phi = S_T I_{S_T>K}.$$so no. (I am not absolutely sure whether you want to differentiate wrt S_T or S_0 however.) 2 The answer is yes. In fact, there always exist a 'Black Scholes like' formula. Easy to show too. If the risk neutral distribution of the price has cumulative density P and probability density p, then$$ E(S-K)^+=E((S-K)\ 1_{S>K})=E(S\ 1_{S>K})-K\ E(1_{S>K})  The second expectation is just $P(K)$, ie the probability that the option ends up in ...

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[Short answer] No closed-form formula in general. You need to resort to numerical methods. Monte Carlo is preferred by most practitioners but you could also use Finite Difference schemes (and sometimes even Fourier inversion techniques depending on the model used and the instruments to be priced). [Long answer] One usually distinguishes between 2 classes ...

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To my point of view, the answer is hidden in your question. You correctly stated some of the BS assumptions and empirically it is proven that they are not true (volatility is not constant and the assumption regarding the distribution of returns is unrealistic due to fat tails). The model is as good as its assumptions are. Given that volatility is the ...

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Well, hopefully your calculations are right. There are a few things to remember: The carry can be higher than what you are thinking. Very often you will get charged if you are long or short. That can cost a lot depending on the name. Implied is theoretically always higher than realized. You are selling insurance. You should collect a premium more ...

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In order to define option price we should follow Black Scholes construction to construct riskless portfolio at t then to state that instantaneous rate of return of this portfolio equal risk free rate r ( t ) where r is a random on [ t , t + dt ] interval. We actually then arrive at the problem which could not be embedded in BS pricing world.

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