# Tag Info

8

You should look at Paul Willmott's Frequently Asked Questions In Quantitative Finance. He offers 12 (I think) ways of deriving BS and I think you'll find what you look for there. The cool thing is that you really have many different approaches; one is the classic PDE, one is done using change of measure, one is done using binary trees, and so on.... Really ...

5

Specifically, we have a generic conditional claim, $C$, that is a function of the diffusion process for the underlying, $S(t)$, and time $t$ so $C = C(S(t), t)$. As you pointed out, $C$ is an Ito process becuase it is a function of a stochastic process so we use Ito's Lemma to determine how the contingent claim varies as a function of the diffusion process ...

2

I agree with vanguard2k's comment: A few more details on the notation would be helpful. But, as far as I can tell, the second equality is a simple expansion. First, $\mathbf{1}'\mathbf{1} = T$ (assuming the vectors are elements of $\mathbb{R}^T$). The expression $\mathbf{1} (\mathbf{1}'\mathbf{1})^{-1} \mathbf{1}'$ is therefore nothing else than a $T\times ... 2 Since$Y=e^{(r-\frac{\sigma^2}{2})\tau + \sigma \sqrt{\tau}Z}, then \begin{align*} xY > K \Leftrightarrow Z > -d_2, \end{align*} where \begin{align*} d_2 = \frac{\ln \frac{x}{K} + (r-\frac{\sigma^2}{2})\tau}{\sigma\sqrt{\tau}}. \end{align*} Consequently, \begin{align*} e^{-r\tau}\mathbb{E}\big(Y \mathbb{1}_{\{xY >K\}} \big) &= ... 1 let\frac{\partial C}{\partial S}=\delta_c$let$\frac{\partial^2 C}{\partial S^2}=\Gamma_c$let$\frac{\partial C_0}{\partial S}=\delta_0$let$\frac{\partial^2 C_0}{\partial S^2}=\Gamma_0$we want$\frac{\partial V}{\partial S}=\frac{\partial C}{\partial S}=\delta_c$and$\frac{\partial^2 V}{\partial S^2}=\frac{\partial^2 C}{\partial S^2}=\Gamma_c$... 1 What is the data basis that you start from? If you just have the covariance matrix, then you can only calculate portfolio variance or volatility by $$w^T \Sigma w$$ where$w$are the portfolio weights and$\Sigma$is the covariance matrix. If you have the individual asset continuously compounded returns$r^j_t$where$j$indexes assets,$j=1,\ldots,N\$, and ...

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