# 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 ...

7

I provided an answer, based on an elementary approach, to an exactly same question yesterday. However, that question has disappeared, even though I like to keep a record for what I wrote. I would suggest that people do not delete their questions as they may be helpful for others. Here, I re-post that answer. We assume that, under the risk-neutral ...

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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 ...

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See this excellent paper by @MarkJoshi which defines/discusses the use of power numeraires. Starting from a dynamics specified under the risk-neutral measure $\mathbb{Q}$ \begin{align} &\frac{dS_t}{S_t} = (r-q) dt + \sigma dW_t^{\mathbb{Q}}\\ \iff& S_T\ \vert\ \mathcal{F}_t = S_t e^{(r-q-\frac{\sigma^2}{2})(T-t) + \sigma(W_T-W_t)} \tag{EQ.0} ...

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 $T\times ... 2 Let's suppose$P$is total annual deposits made continuously, then the change in value of total deposits$dV_t$is (assuming no condition on additional deposits) $$dV_t= V_t r dt + P dt$$ where we assumed$r$is constant. Solving above differential equation, we have: $$V_T = V_0 e^{rT} + \frac{P}{r} (e^{rT} -1)$$ Assuming$t_1$is the time period at ... 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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