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1

Agreeing to your first observation: After orthogonalization, with independent W and W’, and using self explanatory notation for the new diffusion coefficients, which obviously depend on $\rho$, we can show that, under $\mathbb{Q}_Y$, we have: $$dR = R[(\sigma_{XW} - \sigma_{YW})dW + (\sigma_{XW’} - \sigma_{YW’})dW’],$$ where $R=XY^{-1}$ (used only Ito ...

2

Might you be using the tower law in a wrong way? I have the impression you derive your second equation by conditioning by the $\sigma$-algebra generated by $(Y_t)_{t\geq0}$, however note that: $$\mathscr{F}_t\nsubseteq\sigma(Y_t)_{0\leq t\leq T}$$ Hence: E\left((X_T-Y_T)_+|\mathscr{F}_t\right)\ \not= \ E\left(E(X_T-Y_T)_+|Y_T)|\mathscr{F}_t\right) \ = \ E(...

0

Assume we are in the Black Scholes for call option settings, and let’s ignore the dividend. For the implied vol, we can treat all other variables as constant, and focus on the price of the call option as a function of implied vol. $C\left( \sigma\right)=SN\left(d_1\right)-Xe^{-rT}N\left(d_2\right)$ Where: $d_1=\frac{ln \frac{F}{X}}{\sigma \sqrt{T}}+\frac{... 0 The option price should be superior than the intrinsic value of the option. In your case: 31590-29800=1790>1768.05. if you want to test the IV given by your algorithm you can use my website [https://www.valometrics.com]. it is a web platform coded using javascript that contains an IV calculator. please let me know for more information. 0 At least two ways to price this: Use Carr-Madan Use$S^2$as a (power) numeraire, in which case you can price the payoff$(S_T - 1)_+under the power numeraire measure. EDIT: Put-call symmetry. Maybe I can get another -1 for my answer. Is the purpose of answering questions here to do homework for someone else or to stimulate further study and generate ... 0 You should review the difference between implied volatility and realised volatility. Historical volatility is the realised volatility that happen in the past but an option price will have to be determined by the view of the market about volatility in the future for a given period. Normally you do the inverse, ie the option price is given by the market but ... 0 If drift is driven by earnings retention policies then the value of an option does depend on drift! I'm 99% on the following reasoning and would welcome input from others to tighten this up. Consider what happens with Save Co., a hypothetical company that owns a pile of cash sitting in a savings account earning 1% APY. Suppose Save Co. is required to ... 1 I assume you work in the Black Scholes framework. Then, \begin{align*} P(S_0,K,T) = Ke^{-rT}\Phi(-d_2)-S_0\Phi(-d_1), \end{align*} where \begin{align*} d_1 &= \frac{\ln\left(\frac{S_0}{K}\right)+\left(r+\frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}, \\ d_2 &= \frac{\ln\left(\frac{S_0}{K}\right)+\left(r-\frac{1}{2}\sigma^2\right)T}{\sigma\sqrt{T}}= ... 2 Nevermind, i'm just confusing myself. Now I understand what I misunderstood. The implied volatility surface of a prices of calls generated by a stochastic volatility model will not be constant since the implied volatility is found using the Black-Scholes model. The Black-Scholes model and a stochastic volatility model of course disagree on prices, and hence ... 1 The first part of your question:\frac{\partial y}{\partial S} = \frac{\partial ln S}{\partial S} = \frac{1}{S} \frac{\partial^2 V}{\partial S \partial y} = \frac{\partial}{\partial y} \frac{\partial V}{\partial S} = \frac{\partial}{\partial y} (\frac{\partial y}{\partial S}\frac{\partial V}{\partial y})= \frac{\partial}{\partial y} (\frac{1}{ S}\frac{\...

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